Finite Group Databasebeta

← #29679555

Subgroups — #29679555

Order 15616.

1482 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840230C211yesmaximal, characteristic
2C2×#16839785C211yesmaximal, characteristic, complemented
3C2×#16839780C211yesmaximal, characteristic
4C2×#16839778C211yesmaximal, characteristic
5#21448222C211yesmaximal, characteristic, complemented
6#21445838C218yesmaximal, complemented
7#21445528C211yesmaximal, characteristic, complemented
8C61×#1733C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11#16840230—21no—
12#16840230—21no—
13#16840230—22no—
14#16840230C412yescomplemented
15#16840230C412yescomplemented
16#16840230C2214yes—
17C2×#14519742C412yescomplemented
18C2×#14519739C412yescomplemented
19C2×#14519737—22no—
20C2×#14519736C2211yescharacteristic
21C2×#14519734—21no—
22C22×C4×Dic61C2211yescharacteristic
23C2×C42×D61—21no—
24#16839785—21no—
25#16839785—21no—
26#16839785C2214yescomplemented
27C2×#14519706—22no—
28#16839780—24no—
29#16839778—24no—
30C2×#14519705C2214yes—
31C2×#14519703C2214yes—
32#16839631—21no—
33#16839627—21no—
34#16839620—22no—
35C4×#11758196C2214yescomplemented
36#16839521C2214yescomplemented
37C22×M16×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C61×#339C2214yescomplemented
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222C2×C412yes—
46C2×#11758222D414yes—
47C2×#11758222Q814yes—
48C2×#11758222C2312yes—
49C22×Dic122C2×C411yescharacteristic
50C22×D244C2×C411yescharacteristic
51C22×C22⋊D61—21no—
52C23×Dic61D412yes—
53C22×C4×D61—21no—
54C22×C4×D61—22no—
55C2×#11758194—21no—
56C2×#11758194—22no—
57C2×#11758194—22no—
58C2×#11758194C2312yes—
59C22×#92418—21no—
60#14519742—22no—
61#14519742—22no—
62#14519742C2×C414yes—
63#14519742C2×C414yes—
64#14519739—22no—
65#14519739—22no—
66#14519739C2×C414yes—
67#14519739C2×C414yes—
68#14519737—24no—
69#14519737—24no—
70#14519737—24no—
71#14519737—24no—
72#14519736—22no—
73#14519736—22no—
74#14519734—22no—
75#14519734—22no—
76#14519734—24no—
77C2×C4⋊Dic61—21no—
78C2×#11758196—22no—
79C2×#11758196—24no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—22no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61C2×C412yes—
84C2×C4×Dic61C2×C411yescharacteristic
85C2×C4×Dic61C2×C411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61D412yes—
91C2×C4×Dic61C2312yes—
92C2×C4×Dic61C2312yes—
93C42×D61—22no—
94C42×D61—22no—
95C42×D61—24no—
96#14519706—24no—
97#14519706—24no—
98#14519705—28no—
99#14519703—28no—
100#14519684—24no—
101C23×C4×C61C2311yescharacteristic
102C2×M16×C61—21no—
103C2×M16×C61—22no—
104C2×M16×C61—22no—
105C2×M16×C61C2312yes—
106C22×C8×C61—21no—
107#117×C61—21no—
108#117×C61—21no—
109#117×C61—22no—
110#117×C61—22no—
111#117×C61C2×C412yes—
112#117×C61C2312yes—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2×C412yes—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2312yes—
119C24×C61C2×D411yescharacteristic
120C23×D61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×Q812yes—
130C22×C4×C61#4112yes—
131C22×C4×C61#4112yes—
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—22no—
134C2×C22⋊D61—24no—
135C2×C22⋊D61—24no—
136C2×C22⋊D61—22no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—44no—
140C22×Dic61C22⋊C412yes—
141C22×Dic61C22⋊C412yes—
142C22×Dic61C22⋊C412yes—
143C22×Dic61C22⋊C412yes—
144C22×Dic61C2×D412yes—
145C22×Dic61C2×D412yes—
146#11758222—24no—
147#11758222—24no—
148#11758222—24no—
149#11758222—24no—
150#11758222—24no—
151#11758222—22no—
152#11758222—22no—
153#11758222C4⋊C418yescomplemented
154#11758222C2×D414yes—
155#11758222C2×Q814yes—
156C2×D244—21no—
157C2×D244—22no—
158C2×D244C4⋊C414yescomplemented
159C2×D244C22×C412yes—
160C2×C4×D61—22no—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—22no—
164C2×C4×D61—24no—
165C2×C4×D61—24no—
166C2×C4×D61—48no—
167C2×Dic122—21no—
168C2×Dic122—22no—
169C2×Dic122C4⋊C414yescomplemented
170C2×Dic122C22×C412yes—
171M16×C61—22no—
172M16×C61—22no—
173M16×C61—42no—
174C2×C8×C61—22no—
175C2×C8×C61—42no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61C22×C414yes—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—42no—
182C42×C61—22no—
183C42×C61—22no—
184C42×C61—22no—
185C42×C61—22no—
186C22⋊Dic61—42no—
187#11758198—24no—
188#11758198—24no—
189C4⋊Dic61—42no—
190#11758196—24no—
191#11758196—24no—
192#11758196—48no—
193C4×Dic61—24no—
194C4×Dic61—24no—
195C4×Dic61—22no—
196C4×Dic61—22no—
197C4×Dic61C22⋊C411yescharacteristic
198C4×Dic61C22⋊C411yescharacteristic
199C4×Dic61C22⋊C411yescharacteristic
200C4×Dic61C22⋊C411yescharacteristic
201C4×Dic61C22⋊C412yes—
202C4×Dic61C22⋊C412yes—
203C4×Dic61C22×C412yes—
204C4×Dic61C22×C412yes—
205C4×Dic61C2×D412yes—
206C4×Dic61C2×D412yes—
207#11758194—22no—
208#11758194—22no—
209#11758194—42no—
210C2×#92418—22no—
211C2×#92418—42no—
212C23×C61—42no—
213C23×C61C2×C22⋊C411yescharacteristic
214C23×C61C2×C22⋊C411yescharacteristic
215C23×C61#12112yes—
216C23×C61C22⋊Q812yes—
217C23×C61C22×D411yescharacteristic
218C22×D61—22no—
219C22×D61—24no—
220C22×D61—44no—
221C2×C4×C61—24no—
222C2×C4×C61—44no—
223C2×C4×C61—44no—
224C2×C4×C61—42no—
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C4⋊C412yes—
228C2×C4×C61C2×C4⋊C412yes—
229C2×C4×C61#11712yes—
230C2×C4×C61#11712yes—
231C2×C4×C61#12112yes—
232C2×C4×C61#12112yes—
233C2×C4×C61#12112yes—
234C2×C4×C61C22⋊Q812yes—
235C2×C4×C61C22⋊Q812yes—
236C2×C4×C61C22⋊Q812yes—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×Q811yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C22⋊D61—24no—
244C22⋊D61—24no—
245C22⋊D61—28no—
246C2×Dic61—22no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—44no—
255C2×Dic61—42no—
256C2×Dic61C2×C22⋊C412yes—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61C2×C22⋊C412yes—
260D244—22no—
261D244—24no—
262D244C2×C4⋊C414yes—
263C4×D61—24no—
264C4×D61—24no—
265C4×D61—28no—
266C4×D61—48no—
267C4×D61—48no—
268Dic122—22no—
269Dic122—24no—
270Dic122C2×C4⋊C414yes—
271C8×C61—42no—
272#92418—42no—
273#20654—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61#38012yes—
280C22×C61#38012yes—
281C22×C61C22×C22⋊C411yescharacteristic
282C22×C61C2×#12111yescharacteristic
283C22×C61C2×C22⋊Q811yescharacteristic
284C2×D61—24no—
285C2×D61—48no—
286C2×D61—44no—
287C4×C61—44no—
288C4×C61—82no—
289C4×C61#38012yes—
290C4×C61#38012yes—
291C4×C61C22×C4⋊C411yescharacteristic
292C4×C61C2×#11711yescharacteristic
293C4×C61C2×#12111yescharacteristic
294C4×C61C2×C22⋊Q811yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—22no—
298Dic61—24no—
299Dic61—82no—
300Dic61#42114yescomplemented
301C2×C4×#41—611no—
302C2×#423—611no—
303C2×#421—611no—
304C2×#421—611no—
305#2285—611no—
306#1806—618no—
307#1733—611no—
308#1733D6111yescharacteristic, complemented
309C2×C61—21no—
310C2×C61—22no—
311C2×C61—21no—
312C2×C61—22no—
313C2×C61C2×#38011yescharacteristic
314C2×C61C2×#42112yes—
315D61—44no—
316C22×#41—611no—
317C22×M16—611no—
318C22×M16C2×D6111yescharacteristic
319C4×#41—612no—
320C4×#41—612no—
321C4×#41—614no—
322C4×#41—1221no—
323C4×#41—1221no—
324C4×#41—1222no—
325C2×C4×Q8—612no—
326C2×C4×D4—612no—
327C2×C4×D4—1222no—
328C2×#117—611no—
329C2×#117—1221no—
330C2×#117C2×D6111yescharacteristic
331C2×#117C2×D6111yescharacteristic
332C22×C42—611no—
333C22×C42—1221no—
334#423—614no—
335#423—1221no—
336#423—1221no—
337C2×C4⋊C8—1222no—
338#421—1224no—
339#421—1224no—
340C2×#104—614no—
341C2×#104—614no—
342#384—1221no—
343#382—1221no—
344#380—1222no—
345C4×C4⋊C4—614no—
346#339—614no—
347#339C2×D6114yescomplemented
348C61#2065411yescharacteristic, complemented, Sylow
349C2×#41—612no—
350C2×#41—614no—
351C2×#41—614no—
352C2×#41—612no—
353C2×#41—1222no—
354C2×#41—1222no—
355C2×#41—1221no—
356C2×#41—1221no—
357C22×Q8—611no—
358C22×D4—611no—
359C22×D4—1221no—
360C23×C4—612no—
361C23×C4—1222no—
362C23×C4—1221no—
363C23×C4C22×D6111yescharacteristic
364C2×M16—21no—
365C2×M16—22no—
366C2×M16—22no—
367C2×M16—612no—
368C2×M16—1221no—
369C2×M16—1222no—
370C2×M16—1222no—
371C2×M16C22×D6112yes—
372C22×C8—21no—
373C22×C8—1221no—
374C4×Q8—614no—
375C4×Q8—614no—
376C4×Q8—1222no—
377C4×Q8—1222no—
378C4×D4—614no—
379C4×D4—614no—
380C4×D4—1222no—
381C4×D4—1222no—
382C4×D4—1224no—
383C4×D4—1224no—
384C4×D4—1224no—
385C4×D4—1224no—
386#117—21no—
387#117—21no—
388#117—22no—
389#117—22no—
390#117—1224no—
391#117—1222no—
392#117—1222no—
393#117—1222no—
394#117—1222no—
395#117C4×D6112yes—
396#117C22×D6112yes—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—1222no—
399C2×C4⋊C4—1224no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4C4×D6112yes—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1222no—
405C2×C22⋊C4—1221no—
406C2×C42—21no—
407C2×C42—612no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—612no—
414C2×C42—611no—
415C2×C42—612no—
416C2×C42—612no—
417C2×C42—1224no—
418C2×C42—1222no—
419C2×C42—1222no—
420C2×C42—1222no—
421C2×C42C22×D6112yes—
422C4⋊C8—1224no—
423C4⋊C8—1224no—
424#104—1228no—
425#104—1228no—
426#95—1224no—
427C24—1221no—
428C24D4×D6111yescharacteristic
429#41—614no—
430#41—614no—
431#41—618no—
432#41—1222no—
433#41—1222no—
434#41—1224no—
435#41—1224no—
436#41—1224no—
437#41—1224no—
438#41—1224no—
439C2×Q8—612no—
440C2×Q8—614no—
441C2×Q8—1221no—
442C2×Q8—1222no—
443C2×D4—612no—
444C2×D4—614no—
445C2×D4—1222no—
446C2×D4—1222no—
447C2×D4—1224no—
448C2×D4—1224no—
449C2×D4—1221no—
450C2×D4—1222no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—1224no—
460C22×C4—1224no—
461C22×C4—1221no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1224no—
465C22×C4—1224no—
466C22×C4—1222no—
467C22×C4—2448no—
468C22×C4—2444no—
469C22×C4C2×C4×D6111yescharacteristic
470C22×C4C2×C4×D6111yescharacteristic
471C22×C4#1175822212yes—
472C22×C4#1175822212yes—
473C22×C4D4×D6112yes—
474C22×C4D4×D6111yescharacteristic
475C22×C4D4×D6111yescharacteristic
476C22×C4D4×D6111yescharacteristic
477C22×C4Q8×D6112yes—
478C22×C4C23×D6111yescharacteristic
479M16—22no—
480M16—22no—
481M16—42no—
482M16—1222no—
483M16—1222no—
484M16—2442no—
485C2×C8—22no—
486C2×C8—42no—
487C2×C8—1222no—
488C2×C8—2442no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—1224no—
492C4⋊C4—1224no—
493C4⋊C4—2448no—
494C4⋊C4—2442no—
495C4⋊C4C2×C4×D6114yes—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—42no—
499C22⋊C4—1224no—
500C22⋊C4—1224no—
501C22⋊C4—2442no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—612no—
507C42—612no—
508C42—612no—
509C42—612no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—612no—
515C42—612no—
516C42—1224no—
517C42—1224no—
518C42—1222no—
519C42—1222no—
520C23—42no—
521C23—1222no—
522C23—1224no—
523C23—2444no—
524C23C22⋊C4×D6111yescharacteristic
525C23C22⋊C4×D6111yescharacteristic
526C23#1451975012yes—
527C23#1451975512yes—
528C23C2×D4×D6111yescharacteristic
529Q8—614no—
530Q8—1222no—
531Q8—1224no—
532D4—614no—
533D4—1224no—
534D4—1224no—
535D4—1228no—
536D4—1222no—
537D4—1224no—
538C2×C4—24no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—1222no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1228no—
554C2×C4—2448no—
555C2×C4—2448no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2444no—
560C2×C4—2442no—
561C2×C4C22⋊C4×D6111yescharacteristic
562C2×C4C22⋊C4×D6111yescharacteristic
563C2×C4C4⋊C4×D6112yes—
564C2×C4C4⋊C4×D6112yes—
565C2×C4#1451973412yes—
566C2×C4#1451973412yes—
567C2×C4#1451975012yes—
568C2×C4#1451975012yes—
569C2×C4#1451975012yes—
570C2×C4#1451975512yes—
571C2×C4#1451975512yes—
572C2×C4#1451975512yes—
573C2×C4C22×C4×D6111yescharacteristic
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C2×Q8×D6111yescharacteristic
577C2×C4C2×#1175822211yescharacteristic
578C2×C4C2×#1175822211yescharacteristic
579C8—42no—
580C8—2442no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—42no—
585C22—42no—
586C22—1224no—
587C22—2448no—
588C22—2444no—
589C22#1683962112yes—
590C22#1683962112yes—
591C22C2×C22⋊C4×D6111yescharacteristic
592C22C2×#1451975011yescharacteristic
593C22C2×#1451975511yescharacteristic
594C4—44no—
595C4—82no—
596C4—614no—
597C4—1222no—
598C4—1222no—
599C4—1222no—
600C4—1224no—
601C4—4882no—
602C4#1683962112yes—
603C4#1683962112yes—
604C4C2×C4⋊C4×D6111yescharacteristic
605C4C2×#1451973411yescharacteristic
606C4C2×#1451975011yescharacteristic
607C4C2×#1451975511yescharacteristic
608C2—21no—
609C2—22no—
610C2—21no—
611C2—22no—
612C2—2444no—
613C2C2×#1683962111yescharacteristic
614C2#421×D6112yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1733C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1733D61
hypocenterC61#20654
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
2. C2×#16839785C2semidirect
5. #21448222C2semidirect
6. #21445838C2semidirect
7. #21445528C2semidirect
8. C61×#1733C2semidirect
14. #16840230C4semidirect
15. #16840230C4semidirect
17. C2×#14519742C4semidirect
18. C2×#14519739C4semidirect
26. #16839785C22semidirect
35. C4×#11758196C22semidirect
36. #16839521C22semidirect
40. C61×#339C22semidirect
153. #11758222C4⋊C4semidirect
158. C2×D244C4⋊C4semidirect
169. C2×Dic122C4⋊C4semidirect
300. Dic61#421semidirect
308. #1733D61semidirect
347. #339C2×D61semidirect
348. C61#20654semidirect

Maximal subgroups

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

subgroupindex
C2×#168402302
C2×#168397852
C2×#168397802
C2×#168397782
#214482222
#214458382
#214455282
C61×#17332
#2065461