Finite Group Databasebeta

← #29585421

Subgroups — #29585421

Order 15616.

1298 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840230C211yesmaximal, characteristic
2C2×#16839744C211yesmaximal, characteristic
3C2×#16839735C211yesmaximal, characteristic
4C2×#16839523C211yesmaximal, characteristic, complemented
5C2×#16839521C211yesmaximal, characteristic, complemented
6C2×#16839518C211yesmaximal, characteristic, complemented
7C2×C61×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10#16840230—22no—
11#16840230—22no—
12#16840230—22no—
13#16840230—22no—
14C2×#14519742—21no—
15C2×#14519739—21no—
16C2×#14519737—22no—
17C2×#14519736C2211yescharacteristic
18C2×#14519734C412yescomplemented
19C22×C4×Dic61C2211yescharacteristic
20C2×C42×D61C412yescomplemented
21#16839744—22no—
22#16839744C414yescomplemented
23C2×#14519695—21no—
24C2×#14519693—21no—
25#16839735—22no—
26#16839735C414yescomplemented
27C2×#14519691—22no—
28#16839523—22no—
29#16839521—22no—
30#16839518—22no—
31C22×M16×C61C2211yescharacteristic
32C2×#117×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C61×H(3,ℤ/4)—22no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222C2×C412yes—
44C2×#11758222C2×C412yes—
45C2×#11758222D412yes—
46C2×#11758222Q812yes—
47C22×Dic122D411yescharacteristic
48C22×D244D411yescharacteristic
49C22×C22⋊D61D412yes—
50C23×Dic61—21no—
51C22×C4×D61—22no—
52C22×C4×D61C2×C412yes—
53C2×#11758194—21no—
54C2×#11758194—21no—
55C2×#11758194—21no—
56C2×#11758194—21no—
57C2×#11758194—21no—
58C2×#11758194C2×C412yescomplemented
59C22×#92418—21no—
60#14519742—24no—
61#14519742—24no—
62#14519739—24no—
63#14519739—24no—
64#14519737—28no—
65#14519737—28no—
66#14519736—21no—
67#14519736—21no—
68#14519736—21no—
69#14519736C2×C412yescomplemented
70#14519734—24no—
71#14519734—24no—
72C2×C4⋊Dic61—21no—
73C2×#11758196—22no—
74C2×C22⋊Dic61—21no—
75C2×#11758198—22no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61C2×C411yescharacteristic
82C2×C4×Dic61C2×C411yescharacteristic
83C2×C4×Dic61D412yes—
84C2×C4×Dic61D412yes—
85C42×D61—24no—
86C42×D61—24no—
87#14519695—24no—
88#14519693—24no—
89#14519691—28no—
90C23×C4×C61C2311yescharacteristic
91C2×M16×C61—21no—
92C2×M16×C61—21no—
93C2×M16×C61—21no—
94C2×M16×C61—21no—
95C2×M16×C61—21no—
96C2×M16×C61C2×C412yescomplemented
97C22×C8×C61—21no—
98#117×C61—21no—
99#117×C61—21no—
100#117×C61—21no—
101#117×C61—21no—
102#117×C61—21no—
103#117×C61C2×C412yescomplemented
104#117×C61D412yes—
105#117×C61D412yes—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C42×C61—21no—
111C2×C42×C61C2×C412yes—
112C24×C61C2×D411yescharacteristic
113C23×D61C22⋊C412yes—
114C22×C4×C61—22no—
115C22×C4×C61—41no—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×Q811yescharacteristic
126C22×C4×C61#4111yescharacteristic
127C22×C4×C61#4111yescharacteristic
128C22×C4×C61#4111yescharacteristic
129C22×C4×C61#4111yescharacteristic
130C2×C22⋊D61—22no—
131C2×C22⋊D61—22no—
132C2×C22⋊D61—22no—
133C2×C22⋊D61—22no—
134C2×C22⋊D61C22⋊C414yes—
135C2×C22⋊D61C4⋊C414yes—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—44no—
141C22×Dic61—41no—
142C22×Dic61C22⋊C412yes—
143#11758222—22no—
144#11758222—22no—
145#11758222—22no—
146#11758222—22no—
147#11758222—22no—
148#11758222—22no—
149#11758222—22no—
150#11758222—22no—
151#11758222—22no—
152#11758222—22no—
153#11758222—22no—
154#11758222—22no—
155#11758222—22no—
156#11758222—22no—
157#11758222—22no—
158#11758222—22no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C22⋊C412yes—
164C2×D244C4⋊C412yes—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—28no—
170C2×C4×D61—48no—
171C2×C4×D61C4214yes—
172C2×C4×D61C4⋊C414yes—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122C22⋊C412yes—
178C2×Dic122C4⋊C412yes—
179M16×C61—22no—
180M16×C61—22no—
181M16×C61—41no—
182M16×C61—41no—
183C2×C8×C61—22no—
184C2×C8×C61—41no—
185C2×C8×C61—41no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—41no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—41no—
194C42×C61—22no—
195C42×C61—22no—
196C42×C61—41no—
197C42×C61C22⋊C414yes—
198C22⋊Dic61—22no—
199C22⋊Dic61—41no—
200#11758198—28no—
201C4⋊Dic61—22no—
202C4⋊Dic61—41no—
203#11758196—28no—
204C4×Dic61—28no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—41no—
211C4×Dic61C22⋊C412yes—
212C4×Dic61C22⋊C412yes—
213#11758194—22no—
214#11758194—22no—
215#11758194—41no—
216#11758194—41no—
217C2×#92418—22no—
218C2×#92418—41no—
219C2×#92418—41no—
220C23×C61—42no—
221C23×C61C2×C22⋊C411yescharacteristic
222C23×C61C2×C22⋊C411yescharacteristic
223C23×C61C2×C4⋊C411yescharacteristic
224C23×C61#12011yescharacteristic
225C23×C61#12011yescharacteristic
226C23×C61#12311yescharacteristic
227C23×C61#12311yescharacteristic
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—44no—
231C22×D61#9514yes—
232C2×C4×C61—24no—
233C2×C4×C61—44no—
234C2×C4×C61—42no—
235C2×C4×C61—82no—
236C2×C4×C61C2×C4211yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C4⋊C411yescharacteristic
241C2×C4×C61C2×C4⋊C411yescharacteristic
242C2×C4×C61#11711yescharacteristic
243C2×C4×C61#11711yescharacteristic
244C2×C4×C61C4×D411yescharacteristic
245C2×C4×C61C4×D411yescharacteristic
246C2×C4×C61C4×D411yescharacteristic
247C2×C4×C61C4×D411yescharacteristic
248C2×C4×C61C4×D411yescharacteristic
249C2×C4×C61C4×D411yescharacteristic
250C2×C4×C61C4×D411yescharacteristic
251C2×C4×C61C4×D411yescharacteristic
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12111yescharacteristic
255C2×C4×C61#12111yescharacteristic
256C2×C4×C61#12111yescharacteristic
257C2×C4×C61#12111yescharacteristic
258C2×C4×C61C22⋊Q811yescharacteristic
259C2×C4×C61C22⋊Q811yescharacteristic
260C2×C4×C61C22⋊Q811yescharacteristic
261C2×C4×C61C22⋊Q811yescharacteristic
262C2×C4×C61#12311yescharacteristic
263C2×C4×C61#12311yescharacteristic
264C22⋊D61—24no—
265C22⋊D61—24no—
266C22⋊D61—24no—
267C22⋊D61—24no—
268C2×Dic61—24no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—44no—
274C2×Dic61—44no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—82no—
278C2×Dic61#9514yes—
279D244—22no—
280D244—22no—
281D244—22no—
282D244—22no—
283C4×D61—24no—
284C4×D61—24no—
285C4×D61—24no—
286C4×D61—24no—
287C4×D61—416no—
288Dic122—22no—
289Dic122—22no—
290Dic122—22no—
291Dic122—22no—
292C8×C61—42no—
293#92418—42no—
294#10758—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61C2×#9511yescharacteristic
303C22×C61#37911yescharacteristic
304C22×C61#38111yescharacteristic
305C22×C61#38511yescharacteristic
306C22×C61#38511yescharacteristic
307C22×C61#38611yescharacteristic
308C22×C61#38611yescharacteristic
309C2×D61—24no—
310C2×D61—44no—
311C2×D61—44no—
312C2×D61—44no—
313C4×C61—44no—
314C4×C61—82no—
315C4×C61C4×C22⋊C411yescharacteristic
316C4×C61C4×C22⋊C411yescharacteristic
317C4×C61#38011yescharacteristic
318C4×C61#38011yescharacteristic
319C4×C61#38511yescharacteristic
320C4×C61#38511yescharacteristic
321C4×C61#38611yescharacteristic
322C4×C61#38611yescharacteristic
323Dic61—24no—
324Dic61—24no—
325Dic61—42no—
326Dic61—82no—
327C2×C4×#41—611no—
328C2×#413—611no—
329C2×#410—611no—
330C2×#339—611no—
331C2×#339—611no—
332C2×H(3,ℤ/4)—611no—
333C2×H(3,ℤ/4)D6111yescharacteristic, complemented
334C2×C61—21no—
335C2×C61—21no—
336C2×C61—21no—
337C2×C61—21no—
338C2×C61—21no—
339C2×C61—21no—
340C2×C61C22⋊#9511yescharacteristic
341C2×C61#182712yes—
342D61—44no—
343C22×#41—611no—
344C22×M16—611no—
345C22×M16C2×D6111yescharacteristic
346C4×#41—1222no—
347C4×#41—1222no—
348C4×#41—1222no—
349C4×#41—1222no—
350C2×C4×Q8—1221no—
351C2×C4×D4—1222no—
352C2×C4×D4—1221no—
353C2×#117—612no—
354C2×#117—611no—
355C2×#117C2×D6111yescharacteristic
356C2×#117C2×D6111yescharacteristic
357C22×C42—611no—
358C22×C42—612no—
359#413—614no—
360#413—1222no—
361C2×#101—1221no—
362C2×#100—1221no—
363#410—614no—
364#410—1222no—
365C2×#99—1222no—
366#339—1222no—
367#339—1222no—
368H(3,ℤ/4)—22no—
369H(3,ℤ/4)—1222no—
370C61#1075811yescharacteristic, complemented, Sylow
371C2×#41—612no—
372C2×#41—612no—
373C2×#41—612no—
374C2×#41—612no—
375C2×#41—1221no—
376C2×#41—1221no—
377C2×#41—1221no—
378C2×#41—1221no—
379C2×#41—1221no—
380C2×#41—1221no—
381C2×#41—1221no—
382C2×#41—1221no—
383C22×Q8—611no—
384C22×D4—612no—
385C22×D4—611no—
386C23×C4—612no—
387C23×C4—1222no—
388C23×C4—1221no—
389C23×C4C22×D6111yescharacteristic
390C2×M16—21no—
391C2×M16—21no—
392C2×M16—21no—
393C2×M16—21no—
394C2×M16—21no—
395C2×M16—612no—
396C2×M16—1221no—
397C2×M16—1221no—
398C2×M16—1221no—
399C2×M16—1221no—
400C2×M16—1221no—
401C2×M16C4×D6112yescomplemented
402C22×C8—21no—
403C22×C8—1221no—
404C4×Q8—1224no—
405C4×Q8—1224no—
406C4×D4—1228no—
407C4×D4—1228no—
408C4×D4—1224no—
409C4×D4—1224no—
410#117—21no—
411#117—21no—
412#117—21no—
413#117—21no—
414#117—21no—
415#117—612no—
416#117—1224no—
417#117—1221no—
418#117—1224no—
419#117—1221no—
420#117—1221no—
421#117C4×D6112yescomplemented
422#117D24412yes—
423#117C22⋊D6112yes—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—1222no—
427C2×C4⋊C4—1221no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—1222no—
431C2×C22⋊C4—1221no—
432C2×C42—21no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—612no—
436C2×C42—612no—
437C2×C42—1224no—
438C2×C42—1222no—
439C2×C42—1221no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42—1224no—
443C2×C42—1221no—
444C2×C42C4×D6112yes—
445#101—1224no—
446#100—1224no—
447#99—1228no—
448C24—612no—
449C24D4×D6111yescharacteristic
450#41—1222no—
451#41—1222no—
452#41—1222no—
453#41—1222no—
454#41—1222no—
455#41—1222no—
456#41—1222no—
457#41—1222no—
458#41—1222no—
459#41—1222no—
460#41—1222no—
461#41—1222no—
462#41—1222no—
463#41—1222no—
464#41—1222no—
465#41—1222no—
466C2×Q8—612no—
467C2×Q8—612no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×D4—614no—
473C2×D4—614no—
474C2×D4—612no—
475C2×D4—612no—
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—1221no—
484C22×C4—22no—
485C22×C4—41no—
486C22×C4—612no—
487C22×C4—614no—
488C22×C4—614no—
489C22×C4—1228no—
490C22×C4—1221no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1221no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—2448no—
499C22×C4—2444no—
500C22×C4—2441no—
501C22×C4C2×C4×D6111yescharacteristic
502C22×C4C2×C4×D6111yescharacteristic
503C22×C4C2×C4×D6111yescharacteristic
504C22×C4C2×D24411yescharacteristic
505C22×C4#1175822211yescharacteristic
506C22×C4#1175822211yescharacteristic
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4#1175822411yescharacteristic
512C22×C4Q8×D6111yescharacteristic
513C22×C4#1175822611yescharacteristic
514C22×C4C2×C22⋊D6111yescharacteristic
515M16—22no—
516M16—22no—
517M16—41no—
518M16—41no—
519M16—1222no—
520M16—1222no—
521M16—2441no—
522M16—2441no—
523C2×C8—22no—
524C2×C8—41no—
525C2×C8—41no—
526C2×C8—1222no—
527C2×C8—2441no—
528C2×C8—2441no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—41no—
533C4⋊C4—1228no—
534C4⋊C4—1222no—
535C4⋊C4—2441no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—41no—
540C22⋊C4—1228no—
541C22⋊C4—1222no—
542C22⋊C4—2441no—
543C42—22no—
544C42—22no—
545C42—41no—
546C42—612no—
547C42—612no—
548C42—1228no—
549C42—1221no—
550C42—1221no—
551C42—1222no—
552C42—1222no—
553C42—1222no—
554C42—2441no—
555C42#1175819814yes—
556C23—42no—
557C23—614no—
558C23—1222no—
559C23—1222no—
560C23—2444no—
561C23C22⋊C4×D6111yescharacteristic
562C23C22⋊C4×D6111yescharacteristic
563C23C4⋊C4×D6111yescharacteristic
564C23#1451974511yescharacteristic
565C23#1451974711yescharacteristic
566C23#1451975911yescharacteristic
567C23#1451976211yescharacteristic
568Q8—1222no—
569Q8—1222no—
570Q8—1222no—
571Q8—1222no—
572D4—1224no—
573D4—1224no—
574D4—1224no—
575D4—1224no—
576D4—1222no—
577D4—1222no—
578D4—1222no—
579D4—1222no—
580C2×C4—24no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—82no—
584C2×C4—614no—
585C2×C4—1224no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1222no—
594C2×C4—4882no—
595C2×C4—24416no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4C42×D6111yescharacteristic
601C2×C4C22⋊C4×D6111yescharacteristic
602C2×C4C22⋊C4×D6111yescharacteristic
603C2×C4C2×#1175819811yescharacteristic
604C2×C4C4⋊C4×D6111yescharacteristic
605C2×C4C4⋊C4×D6111yescharacteristic
606C2×C4#1451973411yescharacteristic
607C2×C4#1451973511yescharacteristic
608C2×C4#1451973711yescharacteristic
609C2×C4#1451973711yescharacteristic
610C2×C4C4×C22⋊D6111yescharacteristic
611C2×C4C4×C22⋊D6111yescharacteristic
612C2×C4#1451973911yescharacteristic
613C2×C4#1451973911yescharacteristic
614C2×C4C4×D24411yescharacteristic
615C2×C4C4×D24411yescharacteristic
616C2×C4#1451974511yescharacteristic
617C2×C4#1451974711yescharacteristic
618C2×C4#1451974811yescharacteristic
619C2×C4#1451975011yescharacteristic
620C2×C4#1451975011yescharacteristic
621C2×C4#1451975211yescharacteristic
622C2×C4#1451975311yescharacteristic
623C2×C4#1451975511yescharacteristic
624C2×C4#1451975511yescharacteristic
625C2×C4#1451975711yescharacteristic
626C2×C4#1451975911yescharacteristic
627C2×C4#1451976211yescharacteristic
628C8—42no—
629C8—2442no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—1224no—
638C22—2444no—
639C22—2444no—
640C22—2444no—
641C22#95×D6111yescharacteristic
642C22#1683961711yescharacteristic
643C22#1683962411yescharacteristic
644C22#1683963311yescharacteristic
645C22#1683963611yescharacteristic
646C22#1683963811yescharacteristic
647C22#1683964011yescharacteristic
648C4—44no—
649C4—82no—
650C4—1224no—
651C4—1224no—
652C4—4882no—
653C4—2442no—
654C4#1683960911yescharacteristic
655C4C4×#1175819811yescharacteristic
656C4#1683961911yescharacteristic
657C4#1683962111yescharacteristic
658C4#1683963311yescharacteristic
659C4#1683963611yescharacteristic
660C4#1683963811yescharacteristic
661C4#1683964011yescharacteristic
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—2444no—
669C2#2144404011yescharacteristic
670C2#2144592112yes—
subgroupG/H
centerC2×C4#14519734
commutatorC2×C4×C61C2×C42
FrattiniC23×C4C22×D61
FittingC2×C61×H(3,ℤ/4)C2
socleC22×C61#381
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×H(3,ℤ/4)D61
hypocenterC61#10758
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×#117 ◁ C2×H(3,ℤ/4)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×#16839523C2semidirect
5. C2×#16839521C2semidirect
6. C2×#16839518C2semidirect
7. C2×C61×H(3,ℤ/4)C2semidirect
18. C2×#14519734C4semidirect
20. C2×C42×D61C4semidirect
22. #16839744C4semidirect
26. #16839735C4semidirect
58. C2×#11758194C2×C4semidirect
69. #14519736C2×C4semidirect
96. C2×M16×C61C2×C4semidirect
103. #117×C61C2×C4semidirect
333. C2×H(3,ℤ/4)D61semidirect
370. C61#10758semidirect
401. C2×M16C4×D61semidirect
421. #117C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168402302
C2×#168397442
C2×#168397352
C2×#168395232
C2×#168395212
C2×#168395182
C2×C61×H(3,ℤ/4)2
#1075861