Finite Group Databasebeta

← #30609201

Subgroups — #30609201

Order 15616.

1382 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C2×#6923)⋊Q8C211yesmaximal, characteristic
2C2×(C2×#6923)⋊Q8C211yesmaximal, characteristic
3C2×#16840713C211yesmaximal, characteristic
4C2×#16840698C211yesmaximal, characteristic
5#21461006C214yesmaximal
6#21461005C214yesmaximal
7#21460997C211yesmaximal, characteristic
8#21460996C211yesmaximal, characteristic
9#391×D61C211yesmaximal, characteristic
10C22×C4⋊#6923C2211yescharacteristic
11C22×C4⋊#6923C2211yescharacteristic
12C22×C4⋊#6923C2212yes—
13C22×C4×#6923C2211yescharacteristic
14C22×C4×#6923C2211yescharacteristic
15(C2×#6923)⋊Q8—24no—
16(C2×#6923)⋊Q8—24no—
17#16840713—24no—
18#16840698—24no—
19C2×(C2×#6923)⋊C4C2212yes—
20C2×(C2×#6923)⋊C4C2212yes—
21C2×(C2×#6923)⋊C4C2214yes—
22C2×(C2×#6923)⋊C4C2212yes—
23C2×(C2×#6923)⋊C4C2212yes—
24C2×(C2×#6923)⋊C4C2212yes—
25C2×(C2×#6923)⋊C4C2214yes—
26C2×(C2×#6923)⋊C4C2214yes—
27C22×Q8×D61C2211yescharacteristic
28C2×C4⋊C4×D61C2211yescharacteristic
29C2×C42×D61C2211yescharacteristic
30#16839669C411yescharacteristic, complemented
31#16839668C411yescharacteristic, complemented
32#16839667C411yescharacteristic, complemented
33#16839666C414yescomplemented
34#95×D61C2214yes—
35C61×#391C411yescharacteristic, complemented
36C23×#6923C2312yes—
37C23×#6923C2312yes—
38C23×#6923C2311yescharacteristic
39C23×#6923C2311yescharacteristic
40C23×#6923C2312yes—
41C2×C4⋊#6923—24no—
42C2×C4⋊#6923—22no—
43C2×C4⋊#6923—22no—
44C2×C4⋊#6923—22no—
45C2×C4⋊#6923—22no—
46C2×C4⋊#6923—22no—
47C2×C4⋊#6923—22no—
48C2×C4⋊#6923—44no—
49C2×C4×#6923—22no—
50C2×C4×#6923—22no—
51C2×C4×#6923—22no—
52C2×C4×#6923—22no—
53C2×C4×#6923D414yes—
54C2×C4×#6923D414yes—
55(C2×#6923)⋊C4—24no—
56(C2×#6923)⋊C4—28no—
57(C2×#6923)⋊C4—24no—
58(C2×#6923)⋊C4—24no—
59(C2×#6923)⋊C4—24no—
60(C2×#6923)⋊C4—24no—
61(C2×#6923)⋊C4—44no—
62(C2×#6923)⋊C4—44no—
63C22×Dic122C2×C412yes—
64C22×Dic122C2×C411yescharacteristic
65C2×Q8×D61—22no—
66C2×Q8×D61—22no—
67C2×Q8×D61—22no—
68C2×Q8×D61—22no—
69C2×Q8×D61—22no—
70C2×Q8×D61—22no—
71C22×C4×D61C2312yes—
72C22×C4×D61C2312yes—
73C22×C4×D61C2312yes—
74C22×C4×D61C2311yescharacteristic
75C2×C4⋊Dic61C2×C411yescharacteristic
76C2×#11758196C2×C412yes—
77C4⋊C4×D61—22no—
78C4⋊C4×D61—22no—
79C2×C4×Dic61C2×C412yes—
80C2×C4×Dic61C2×C411yescharacteristic
81C42×D61—22no—
82C42×D61—22no—
83#14519684C2×C414yes—
84#14519684C2×C414yes—
85#14519684C2×C414yes—
86C22×Q8×C61C2×C411yescharacteristic
87C2×C4⋊C4×C61C2×C411yescharacteristic
88C2×C42×C61C2×C411yescharacteristic
89#95×C61C2×C414yes—
90C23×D61C2411yescharacteristic
91C22×#6923—21no—
92C22×#6923—21no—
93C22×#6923—21no—
94C22×#6923—21no—
95C22×#6923—22no—
96C22×#6923—22no—
97C22×#6923—22no—
98C22×#6923—22no—
99C22×#6923—22no—
100C22×#6923—22no—
101C22×#6923—22no—
102C22×#6923—22no—
103C22×#6923—22no—
104C22×#6923—22no—
105C22×#6923—22no—
106C22×#6923—22no—
107C22×#6923—22no—
108C22×#6923—22no—
109C22×#6923—22no—
110C22×#6923—22no—
111C22×#6923—24no—
112C22×#6923—24no—
113C22×#6923—24no—
114C22×#6923—24no—
115C22×#6923C2×D412yes—
116C22×#6923C2×D414yes—
117C22×#6923C2×D412yes—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C22×C411yescharacteristic
125C22×Dic61C22×C412yes—
126C22×Dic61C22×C412yes—
127C22×Dic61C22×C412yes—
128C22×Dic61C22×C411yescharacteristic
129Q8×D61—44no—
130Q8×D61—44no—
131Q8×D61—44no—
132Q8×D61—44no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—24no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—24no—
148C2×C4×D61—22no—
149C2×C4×D61C2×D412yes—
150C2×C4×D61C2×D412yes—
151C2×C4×D61#4112yes—
152C2×C4×D61#4112yes—
153C2×C4×D61#4112yes—
154C2×C4×D61#4112yes—
155C2×Dic122—22no—
156C2×Dic122—22no—
157C2×Dic122—22no—
158C2×Dic122—24no—
159C2×Dic122—24no—
160C2×Dic122—24no—
161C4⋊#6923—48no—
162C4⋊#6923—44no—
163C4⋊#6923—44no—
164C4×#6923—28no—
165C4×#6923—28no—
166C4⋊C4×C61—22no—
167C42×C61—22no—
168C4⋊Dic61—22no—
169#11758196—24no—
170C4×Dic61—22no—
171C4×Dic61—24no—
172C23×C61C23×C411yescharacteristic
173C22×D61C22×D411yescharacteristic
174C22×D61C22×D411yescharacteristic
175C22×D61C2×#4111yescharacteristic
176C22×D61C2×#4111yescharacteristic
177C22×D61C2×#4111yescharacteristic
178C22×D61C2×#4111yescharacteristic
179C22×D61H(5,𝔽2)12yes—
180C22×D612−1+412yes—
181C22×D612−1+412yes—
182C22×D612−1+412yes—
183C2×#6923—22no—
184C2×#6923—22no—
185C2×#6923—24no—
186C2×#6923—22no—
187C2×#6923—22no—
188C2×#6923—22no—
189C2×#6923—22no—
190C2×#6923—24no—
191C2×#6923—24no—
192C2×#6923—42no—
193C2×#6923—42no—
194C2×#6923—42no—
195C2×#6923—42no—
196C2×#6923—42no—
197C2×#6923—42no—
198C2×#6923—42no—
199C2×#6923—42no—
200C2×#6923—42no—
201C2×#6923—42no—
202C2×#6923—42no—
203C2×#6923—42no—
204C2×#6923—42no—
205C2×#6923—42no—
206C2×#6923—44no—
207C2×#6923—44no—
208C2×#6923—44no—
209C2×#6923—44no—
210Q8×C61—44no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61#11712yes—
220C2×C4×C61C4×D412yes—
221C2×C4×C61C4×D412yes—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—24no—
230C2×Dic61#11712yes—
231C2×Dic61C4×D412yes—
232C2×Dic61C4×D412yes—
233C4×D61—42no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61—42no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—42no—
241C4×D61—42no—
242C4×D61—42no—
243C4×D61—44no—
244C4×D61#12114yes—
245C4×D61#12314yes—
246C4×D61#12414yes—
247C4×D61#12614yes—
248Dic122—44no—
249Dic122—48no—
250#14261—611nomaximal, Sylow
251C22×C61C2×#11711yescharacteristic
252C22×C61C2×C4×D411yescharacteristic
253C22×C61C2×C4×D411yescharacteristic
254C22×C61#51912yes—
255C22×C61#52012yes—
256C2×D61C2×#12111yescharacteristic
257C2×D61C2×#12311yescharacteristic
258C2×D61C2×#12411yescharacteristic
259C2×D61C2×#12611yescharacteristic
260C2×D61#53511yescharacteristic
261C2×D61#53611yescharacteristic
262C2×D61#53612yes—
263C2×D61#53611yescharacteristic
264C2×D61#53611yescharacteristic
265C2×D61#53712yes—
266C2×D61#54112yes—
267C2×D61#54112yes—
268C2×D61#54112yes—
269C2×D61#54112yes—
270C2×D61#54212yes—
271C2×D61#54212yes—
272C2×D61#54212yes—
273C2×D61#54212yes—
274#6923—42no—
275#6923—42no—
276#6923—44no—
277#6923—82no—
278#6923—82no—
279C4×C61—42no—
280C4×C61—42no—
281C4×C61—42no—
282C4×C61#38814yes—
283Dic61—42no—
284Dic61—42no—
285Dic61—42no—
286Dic61#38814yes—
287C2×#391—611no—
288C2×#391—611no—
289C2×#391—611no—
290C2×#389—611no—
291C2×#376—611no—
292#1443—614no—
293#1442—614no—
294C4⋊#95—611no—
295C4⋊#95—611no—
296C2×C61C2×#38811yescharacteristic
297C2×C61#232012yes—
298C2×C61#232312yes—
299C2×C61#232611yescharacteristic
300C2×C61#232711yescharacteristic
301D61#262011yescharacteristic
302D61#262111yescharacteristic
303D61#262512yes—
304D61#264011yescharacteristic
305D61#264211yescharacteristic
306D61#264512yes—
307C23×Q8—611no—
308C22×C4⋊C4—611no—
309C22×C4⋊C4—611no—
310C22×C4⋊C4—611no—
311C22×C4⋊C4—612no—
312C22×C42—611no—
313C22×C42—611no—
314C22×C42—611no—
315#391—611no—
316#391—611no—
317#391—611no—
318#391—614no—
319#391—1224no—
320#391—1224no—
321#391#692311yescharacteristic, complemented
322#389—1224no—
323#376—1224no—
324C2×#95—614no—
325C2×#95—612no—
326C2×#95—612no—
327C2×#95—614no—
328C2×#95—612no—
329C2×#95—612no—
330C2×#95—612no—
331C2×#95—614no—
332C2×#95—614no—
333C61#1426111yescharacteristic, complemented, Sylow
334C22×Q8—612no—
335C22×Q8—611no—
336C22×Q8—1222no—
337C22×Q8—1222no—
338C22×Q8—1222no—
339C22×Q8—1222no—
340C22×Q8—1222no—
341C22×Q8—1222no—
342C22×Q8C2×#692311yescharacteristic
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—612no—
347C23×C4—612no—
348C23×C4—612no—
349C23×C4—612no—
350C23×C4—612no—
351C23×C4—612no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1224no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4—2444no—
364C2×C4⋊C4C2×#692311yescharacteristic
365C2×C42—614no—
366C2×C42—614no—
367C2×C42—612no—
368C2×C42—611no—
369C2×C42—1222no—
370C2×C42—1222no—
371C2×C42—1222no—
372C2×C42—1222no—
373C2×C42—1222no—
374C2×C42—1222no—
375C2×C42C2×#692311yescharacteristic
376#95—614no—
377#95—614no—
378#95—614no—
379#95—1224no—
380#95—1224no—
381#95—1224no—
382#95—1228no—
383#95—1224no—
384#95—1224no—
385#95—2444no—
386#95—2444no—
387#95C2×#692314yes—
388C24—611no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—1224no—
393C2×Q8—1224no—
394C2×Q8—1224no—
395C2×Q8—1222no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×Q8—2444no—
399C2×Q8—2444no—
400C2×Q8—2444no—
401C2×Q8—2444no—
402C22×C4—611no—
403C22×C4—612no—
404C22×C4—612no—
405C22×C4—612no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—612no—
409C22×C4—614no—
410C22×C4—612no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—612no—
414C22×C4—612no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1224no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1224no—
430C22×C4—1222no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1224no—
449C22×C4—1224no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1224no—
454C22×C4—1224no—
455C22×C4C22×#692312yes—
456C22×C4C22×#692312yes—
457C22×C4C22×#692312yes—
458C22×C4C22×#692311yescharacteristic
459C4⋊C4—22no—
460C4⋊C4—1224no—
461C4⋊C4—1222no—
462C4⋊C4—2444no—
463C4⋊C4—2444no—
464C4⋊C4—2448no—
465C42—22no—
466C42—1224no—
467C42—1228no—
468C42—1228no—
469C42—1222no—
470C23—611no—
471C23—611no—
472C23—611no—
473C23—611no—
474C23—611no—
475C23—611no—
476C23—612no—
477C23—612no—
478C23—612no—
479C23—612no—
480C23C23×#692311yescharacteristic
481Q8—44no—
482Q8—2448no—
483Q8—2444no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—612no—
493C2×C4—612no—
494C2×C4—612no—
495C2×C4—614no—
496C2×C4—614no—
497C2×C4—614no—
498C2×C4—614no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1224no—
502C2×C4—1222no—
503C2×C4—1224no—
504C2×C4—1224no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1224no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2444no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2444no—
531C2×C4—2444no—
532C2×C4—2444no—
533C2×C4—2444no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4#1451985612yes—
546C2×C4D4×#692312yes—
547C2×C4D4×#692312yes—
548C22—611no—
549C22—611no—
550C22—611no—
551C22—611no—
552C22—612no—
553C22—612no—
554C22—612no—
555C22—612no—
556C22—612no—
557C22—612no—
558C22—612no—
559C22—612no—
560C22—611no—
561C22—611no—
562C22—612no—
563C22—611no—
564C22—611no—
565C22—612no—
566C22C2×#1451985611yescharacteristic
567C22C2×D4×#692311yescharacteristic
568C22C2×D4×#692311yescharacteristic
569C22#1684083012yes—
570C22#1684083112yes—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—614no—
575C4—4882no—
576C4—4882no—
577C4—2442no—
578C4—2442no—
579C4—2444no—
580C4—2442no—
581C4—2442no—
582C4—2442no—
583C4#1684071214yes—
584C2—611no—
585C2—611no—
586C2—611no—
587C2—612no—
588C2—611no—
589C2—612no—
590C2C2×#1684071211yescharacteristic
591C2#2146230312yes—
592C2#2146230612yes—
593C2#2146230911yescharacteristic
594C2#2146231011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#391C4
prime radical, cosocleC23×D61C24
hypercenter#391#6923
hypocenterC61#14261
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
30. #16839669C4semidirect
31. #16839668C4semidirect
32. #16839667C4semidirect
33. #16839666C4semidirect
35. C61×#391C4semidirect
321. #391#6923semidirect
333. C61#14261semidirect

Maximal subgroups

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

subgroupindex
C2×(C2×#6923)⋊Q82
C2×#168407132
C2×#168406982
#214610062
#214610052
#214609972
#214609962
#391×D612
#1426161