Finite Group Databasebeta

← #29679862

Subgroups — #29679862

Order 15616.

1260 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459020C211yesmaximal, characteristic
2C2×#16839744C211yesmaximal, characteristic
3C2×#16839735C211yesmaximal, characteristic
4#21445922C218yesmaximal, complemented
5#21445529C211yesmaximal, characteristic, complemented
6C2×#16839519C211yesmaximal, characteristic, complemented
7C2×#16839518C211yesmaximal, characteristic, complemented
8C61×#1733C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11#16840243—22no—
12#16840242—22no—
13#16840239—21no—
14#16840236—21no—
15#16840235—22no—
16C4×#11758222C2214yes—
17#16840230C414yescomplemented
18C2×#14519736C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20#117×D61C414yescomplemented
21#16839744—22no—
22#16839744C2214yes—
23C2×#14519695—21no—
24C2×#14519693—21no—
25#16839735—22no—
26#16839735C2214yes—
27C2×#14519691—22no—
28#16839522C2214yescomplemented
29#16839519C2214yescomplemented
30#16839518C2214yescomplemented
31C22×M16×C61C2211yescharacteristic
32C2×#117×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C61×#339C2214yescomplemented
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×#11758222D412yes—
45C2×#11758222D412yes—
46C2×#11758222C2312yes—
47C22×Dic122D411yescharacteristic
48C22×D244D411yescharacteristic
49C22×C22⋊D61D412yes—
50C22×C4×D61C2×C412yes—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194—21no—
54C2×#11758194—21no—
55C2×#11758194—21no—
56C2×#11758194C2312yes—
57C22×#92418—21no—
58C4×Dic122—24no—
59#14519742—24no—
60C4×D244—24no—
61#14519739—24no—
62C4×C22⋊D61—28no—
63#14519737—28no—
64#14519736—21no—
65#14519736—21no—
66#14519736—21no—
67#14519736—21no—
68#14519736—21no—
69#14519736C2×C412yes—
70#14519736C2312yes—
71#14519736C2312yes—
72#14519735—24no—
73#14519734C2×C418yescomplemented
74#14519733—24no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C4⋊C4×D61—24no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C22⋊C4×D61—24no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2×C412yes—
83C42×D61C2×C418yescomplemented
84#14519695—24no—
85#14519693—24no—
86#14519691—28no—
87C23×C4×C61C2311yescharacteristic
88C2×M16×C61—21no—
89C2×M16×C61—21no—
90C2×M16×C61—21no—
91C2×M16×C61—21no—
92C2×M16×C61—21no—
93C2×M16×C61C2312yes—
94C22×C8×C61—21no—
95#117×C61—21no—
96#117×C61—21no—
97#117×C61—21no—
98#117×C61—21no—
99#117×C61—21no—
100#117×C61C2×C412yes—
101#117×C61C2×C412yes—
102#117×C61C2312yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2312yes—
109C24×C61C2×D411yescharacteristic
110C23×D61C22⋊C412yes—
111C22×C4×C61—22no—
112C22×C4×C61—41no—
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C22×C4×C61#4111yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—22no—
128C2×C22⋊D61—22no—
129C2×C22⋊D61—22no—
130C2×C22⋊D61—22no—
131C2×C22⋊D61C22⋊C414yes—
132C2×C22⋊D61C2×D414yes—
133C22×Dic61—22no—
134C22×Dic61—41no—
135C22×Dic61C22⋊C412yes—
136#11758222—22no—
137#11758222—22no—
138#11758222—22no—
139#11758222—22no—
140#11758222—22no—
141#11758222—22no—
142#11758222—22no—
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—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244C22⋊C412yes—
157C2×D244C2×D412yes—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—28no—
163C2×C4×D61C22⋊C414yes—
164C2×C4×D61C22×C414yes—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C22⋊C412yes—
170C2×Dic122C2×D412yes—
171M16×C61—22no—
172M16×C61—22no—
173M16×C61—41no—
174M16×C61—41no—
175C2×C8×C61—22no—
176C2×C8×C61—41no—
177C2×C8×C61—41no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—41no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—41no—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61—41no—
189C42×C61C22×C414yes—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—41no—
194#11758198—28no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—41no—
199#11758196—28no—
200C4×Dic61—22no—
201C4×Dic61—28no—
202C4×Dic61—22no—
203C4×Dic61—41no—
204C4×Dic61C22×C414yes—
205#11758194—22no—
206#11758194—22no—
207#11758194—41no—
208#11758194—41no—
209C2×#92418—22no—
210C2×#92418—41no—
211C2×#92418—41no—
212C23×C61—81no—
213C23×C61C2×C22⋊C411yescharacteristic
214C23×C61C2×C22⋊C411yescharacteristic
215C23×C61#12311yescharacteristic
216C23×C61#12311yescharacteristic
217C23×C61#12311yescharacteristic
218C23×C61#12311yescharacteristic
219C23×C61C22×D411yescharacteristic
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—82no—
223C22×D61C2×C22⋊C414yes—
224C2×C4×C61—24no—
225C2×C4×C61—42no—
226C2×C4×C61—82no—
227C2×C4×C61—82no—
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61#11711yescharacteristic
231C2×C4×C61#11711yescharacteristic
232C2×C4×C61#11711yescharacteristic
233C2×C4×C61#11711yescharacteristic
234C2×C4×C61#11711yescharacteristic
235C2×C4×C61#11711yescharacteristic
236C2×C4×C61#11711yescharacteristic
237C2×C4×C61#11711yescharacteristic
238C2×C4×C61#12311yescharacteristic
239C2×C4×C61#12311yescharacteristic
240C2×C4×C61#12311yescharacteristic
241C2×C4×C61#12311yescharacteristic
242C2×C4×C61#12311yescharacteristic
243C2×C4×C61#12311yescharacteristic
244C2×C4×C61#12311yescharacteristic
245C2×C4×C61#12311yescharacteristic
246C2×C4×C61#12311yescharacteristic
247C2×C4×C61#12311yescharacteristic
248C2×C4×C61#12311yescharacteristic
249C2×C4×C61#12311yescharacteristic
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C22⋊D61—24no—
257C22⋊D61—24no—
258C22⋊D61—24no—
259C22⋊D61—24no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—24no—
263C2×Dic61—42no—
264C2×Dic61—82no—
265C2×Dic61—82no—
266C2×Dic61C2×C22⋊C414yes—
267D244—22no—
268D244—22no—
269D244—22no—
270D244—22no—
271C4×D61—24no—
272C4×D61—24no—
273C4×D61—24no—
274C4×D61—24no—
275C4×D61—88no—
276Dic122—22no—
277Dic122—22no—
278Dic122—22no—
279Dic122—22no—
280C8×C61—42no—
281#92418—42no—
282#20708—611nomaximal, Sylow
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61#38111yescharacteristic
291C22×C61#38111yescharacteristic
292C22×C61#38111yescharacteristic
293C22×C61#38111yescharacteristic
294C22×C61C22×C22⋊C411yescharacteristic
295C22×C61C2×#12311yescharacteristic
296C22×C61C2×#12311yescharacteristic
297C2×D61—24no—
298C2×D61—44no—
299C2×D61—82no—
300C2×D61—82no—
301C4×C61—44no—
302C4×C61—82no—
303C4×C61#38111yescharacteristic
304C4×C61#38111yescharacteristic
305C4×C61#38111yescharacteristic
306C4×C61#38111yescharacteristic
307C4×C61C2×#11711yescharacteristic
308C4×C61C2×#11711yescharacteristic
309C4×C61C2×#12311yescharacteristic
310C4×C61C2×#12311yescharacteristic
311Dic61—24no—
312Dic61—44no—
313Dic61—82no—
314#3428—611no—
315C2×#413—611no—
316C2×#410—611no—
317#1827—618no—
318#1733—611no—
319#1733D6111yescharacteristic, complemented
320C2×H(3,ℤ/4)—611no—
321C2×H(3,ℤ/4)—611no—
322C2×C61—21no—
323C2×C61—21no—
324C2×C61—21no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—21no—
328C2×C61—21no—
329C2×C61C2×#38111yescharacteristic
330D61—44no—
331C22×#41—611no—
332C22×M16—611no—
333C22×M16C2×D6111yescharacteristic
334#520—1222no—
335#520—1222no—
336#519—1221no—
337#518—1222no—
338#518—1221no—
339C4×#41—614no—
340C4×#41—614no—
341C2×#117—611no—
342C2×#117—614no—
343C2×#117—611no—
344C2×#117C2×D6111yescharacteristic
345C2×#117C2×D6111yescharacteristic
346#413—614no—
347#413—1222no—
348C2×#101—1221no—
349C2×#100—1221no—
350#410—614no—
351#410—1222no—
352C2×#99—1222no—
353#339—614no—
354#339C2×D6114yescomplemented
355H(3,ℤ/4)—614no—
356H(3,ℤ/4)—614no—
357C61#2070811yescharacteristic, complemented, Sylow
358C2×#41—612no—
359C2×#41—612no—
360C2×#41—612no—
361C2×#41—612no—
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1221no—
365C2×#41—1221no—
366C2×#41—1221no—
367C2×#41—1221no—
368C2×#41—1221no—
369C2×#41—1221no—
370C22×Q8—611no—
371C22×D4—612no—
372C22×D4—611no—
373C23×C4—612no—
374C23×C4C22×D6111yescharacteristic
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—21no—
379C2×M16—21no—
380C2×M16—612no—
381C2×M16—1221no—
382C2×M16—1221no—
383C2×M16—1221no—
384C2×M16—1221no—
385C2×M16—1221no—
386C2×M16C22×D6112yes—
387C22×C8—21no—
388C22×C8—1221no—
389C4×Q8—1224no—
390C4×Q8—1224no—
391C4×D4—1224no—
392C4×D4—1228no—
393C4×D4—1224no—
394C4×D4—1228no—
395#117—21no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—612no—
401#117—612no—
402#117—618no—
403#117—612no—
404#117—1224no—
405#117—1224no—
406#117—1221no—
407#117—1221no—
408#117—1221no—
409#117—1221no—
410#117—1221no—
411#117C4×D6112yes—
412#117C4×D6112yes—
413#117C22×D6112yes—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—1224no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1224no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—21no—
425C2×C42—612no—
426C2×C42—618no—
427C2×C42—1221no—
428C2×C42C22×D6112yes—
429#101—1224no—
430#100—1224no—
431#99—1228no—
432C24—612no—
433C24D4×D6111yescharacteristic
434#41—1222no—
435#41—1222no—
436#41—1222no—
437#41—1222no—
438#41—1222no—
439#41—1222no—
440#41—1222no—
441#41—1222no—
442#41—1222no—
443#41—1222no—
444#41—1222no—
445#41—1222no—
446#41—1222no—
447#41—1222no—
448#41—1222no—
449#41—1222no—
450C2×Q8—612no—
451C2×Q8—612no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×D4—614no—
457C2×D4—614no—
458C2×D4—612no—
459C2×D4—612no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4—612no—
471C22×C4—614no—
472C22×C4—614no—
473C22×C4—1228no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—2441no—
480C22×C4C2×C4×D6111yescharacteristic
481C22×C4C2×C4×D6111yescharacteristic
482C22×C4#1175822211yescharacteristic
483C22×C4#1175822211yescharacteristic
484C22×C4#1175822211yescharacteristic
485C22×C4#1175822211yescharacteristic
486C22×C4D4×D6111yescharacteristic
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4#1175822411yescharacteristic
490C22×C4#1175822411yescharacteristic
491C22×C4#1175822611yescharacteristic
492C22×C4#1175822611yescharacteristic
493C22×C4C23×D6111yescharacteristic
494M16—22no—
495M16—22no—
496M16—41no—
497M16—41no—
498M16—1222no—
499M16—1222no—
500M16—2441no—
501M16—2441no—
502C2×C8—22no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—1222no—
506C2×C8—2441no—
507C2×C8—2441no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—41no—
512C4⋊C4—1228no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—2441no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—41no—
521C22⋊C4—1228no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—2441no—
526C42—22no—
527C42—22no—
528C42—41no—
529C42—614no—
530C42—1228no—
531C42—1222no—
532C42—1222no—
533C42—2441no—
534C42C2×C4×D6114yes—
535C23—81no—
536C23—614no—
537C23—1222no—
538C23—1222no—
539C23—4882no—
540C23C22⋊C4×D6111yescharacteristic
541C23C22⋊C4×D6111yescharacteristic
542C23#1451975911yescharacteristic
543C23#1451975911yescharacteristic
544C23#1451976211yescharacteristic
545C23#1451976211yescharacteristic
546C23C2×D4×D6111yescharacteristic
547Q8—1222no—
548Q8—1222no—
549Q8—1222no—
550Q8—1222no—
551D4—1224no—
552D4—1224no—
553D4—1224no—
554D4—1224no—
555D4—1222no—
556D4—1222no—
557D4—1222no—
558D4—1222no—
559C2×C4—24no—
560C2×C4—42no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—614no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1224no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—1224no—
570C2×C4—1224no—
571C2×C4—4888no—
572C2×C4—4882no—
573C2×C4—4882no—
574C2×C4—2442no—
575C2×C4C22⋊C4×D6111yescharacteristic
576C2×C4C22⋊C4×D6111yescharacteristic
577C2×C4#1451973411yescharacteristic
578C2×C4#1451973411yescharacteristic
579C2×C4#1451973411yescharacteristic
580C2×C4#1451973411yescharacteristic
581C2×C4#1451973511yescharacteristic
582C2×C4#1451973511yescharacteristic
583C2×C4#1451973511yescharacteristic
584C2×C4#1451973511yescharacteristic
585C2×C4#1451975911yescharacteristic
586C2×C4#1451975911yescharacteristic
587C2×C4#1451975911yescharacteristic
588C2×C4#1451975911yescharacteristic
589C2×C4#1451975911yescharacteristic
590C2×C4#1451975911yescharacteristic
591C2×C4#1451976211yescharacteristic
592C2×C4#1451976211yescharacteristic
593C2×C4#1451976211yescharacteristic
594C2×C4#1451976211yescharacteristic
595C2×C4#1451976211yescharacteristic
596C2×C4#1451976211yescharacteristic
597C2×C4C22×C4×D6111yescharacteristic
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4C2×#1175822411yescharacteristic
600C2×C4C2×#1175822211yescharacteristic
601C2×C4C2×#1175822211yescharacteristic
602C2×C4C2×#1175822611yescharacteristic
603C8—42no—
604C8—2442no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—1224no—
613C22—4882no—
614C22—4882no—
615C22—2444no—
616C22#1683962411yescharacteristic
617C22#1683962411yescharacteristic
618C22#1683962411yescharacteristic
619C22#1683962411yescharacteristic
620C22C2×C22⋊C4×D6111yescharacteristic
621C22C2×#1451975911yescharacteristic
622C22C2×#1451976211yescharacteristic
623C4—44no—
624C4—82no—
625C4—1224no—
626C4—4882no—
627C4—2444no—
628C4#1683962411yescharacteristic
629C4#1683962411yescharacteristic
630C4#1683962411yescharacteristic
631C4#1683962411yescharacteristic
632C4C2×#1451973411yescharacteristic
633C4C2×#1451973511yescharacteristic
634C4C2×#1451975911yescharacteristic
635C4C2×#1451976211yescharacteristic
636C2—21no—
637C2—21no—
638C2—21no—
639C2—21no—
640C2—21no—
641C2—21no—
642C2—21no—
643C2—2444no—
644C2C2×#1683962411yescharacteristic
subgroupG/H
centerC4C2×#14519735
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1733C2
socleC2×C61C2×#381
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1733D61
hypocenterC61#20708
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
4. #21445922C2semidirect
5. #21445529C2semidirect
6. C2×#16839519C2semidirect
7. C2×#16839518C2semidirect
8. C61×#1733C2semidirect
17. #16840230C4semidirect
20. #117×D61C4semidirect
28. #16839522C22semidirect
29. #16839519C22semidirect
30. #16839518C22semidirect
34. C61×#339C22semidirect
73. #14519734C2×C4semidirect
83. C42×D61C2×C4semidirect
319. #1733D61semidirect
354. #339C2×D61semidirect
357. C61#20708semidirect

Maximal subgroups

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

subgroupindex
#214590202
C2×#168397442
C2×#168397352
#214459222
#214455292
C2×#168395192
C2×#168395182
C61×#17332
#2070861