Finite Group Databasebeta

← #29689867

Subgroups — #29689867

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840385C211yesmaximal, characteristic
2#21454454C211yesmaximal, characteristic, complemented
3#21454382C211yesmaximal, characteristic, complemented
4#21446806C212yesmaximal, complemented
5#21446667C212yesmaximal, complemented
6#21446288C211yesmaximal, characteristic, complemented
7#21446282C211yesmaximal, characteristic
8#21446280C211yesmaximal, characteristic
9#21445813C212yesmaximal, complemented
10#21445810C212yesmaximal, complemented
11C61×#1901C211yesmaximal, characteristic, complemented
12C22×#11758194C2211yescharacteristic
13#16840385—22no—
14#16840385—22no—
15#16840385—22no—
16#16840385—22no—
17C2×#14519776C2211yescharacteristic
18C2×#14519772C412yes—
19C2×#14519766C2211yescharacteristic
20C2×#14519763C412yes—
21C2×#14519759C412yes—
22C2×#14519757C412yes—
23C2×#14519751C2211yescharacteristic
24C2×Q8×Dic61C2211yescharacteristic
25C2×#14519739C2211yescharacteristic
26C2×#14519736C2211yescharacteristic
27#16839820—22no—
28#16839820—22no—
29#16839790—22no—
30#16839788—22no—
31#16839785C2212yes—
32#16839785C2212yes—
33#16839785C2212yes—
34C2×#14519696C2212yes—
35C2×#14519693C2212yes—
36C2×#14519693C2212yes—
37#16839726C2211yescharacteristic
38M16×Dic61C2212yes—
39#16839531C2212yes—
40#16839530C2212yes—
41#16839530C2212yes—
42C22×M16×C61C2211yescharacteristic
43C2×C22⋊Q8×C61C2211yescharacteristic
44C2×#101×C61C2212yes—
45C61×#407C2211yescharacteristic
46C61×#343C2212yes—
47C22×Dic122C2×C412yes—
48C22×D244C2311yescharacteristic
49C22×C22⋊D61C2×C412yes—
50C22×C4×D61C2×C412yes—
51C2×#11758194—21no—
52C2×#11758194—42no—
53C2×#11758194C2312yes—
54C2×#11758194C2312yes—
55C22×#92418—21no—
56#14519776—22no—
57#14519776—22no—
58#14519772—24no—
59#14519772—24no—
60#14519766—22no—
61#14519766—22no—
62#14519763—24no—
63#14519763—24no—
64#14519759—24no—
65#14519759—24no—
66#14519757—24no—
67#14519757—24no—
68#14519751—21no—
69#14519751—21no—
70#14519751—21no—
71#14519751—21no—
72Q8×Dic61—22no—
73Q8×Dic61—22no—
74#14519739—22no—
75#14519739—22no—
76#14519736D412yes—
77#14519736D412yes—
78#14519736D412yes—
79#14519736D412yes—
80C2×C4⋊Dic61C2×C412yes—
81C2×C4⋊Dic61C2×C411yescharacteristic
82C2×C4⋊Dic61C2×C411yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196C2×C412yes—
85C2×C22⋊Dic61C2×C411yescharacteristic
86C2×C22⋊Dic61C2×C411yescharacteristic
87C2×#11758198C2×C412yes—
88C2×#11758198C2×C412yes—
89C2×#11758198C2×C412yes—
90C2×#11758198C2×C412yes—
91C2×C4×Dic61C2×C412yes—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519706—24no—
95#14519706—24no—
96#14519706—24no—
97#14519696—42no—
98#14519693—42no—
99#14519693—42no—
100C22⋊#92418—21no—
101C22⋊#92418—21no—
102#14519687—22no—
103C8×Dic61—22no—
104C22×Q8×C61C2311yescharacteristic
105C23×C4×C61C2311yescharacteristic
106C2×M16×C61—21no—
107C2×M16×C61—42no—
108C2×M16×C61C2312yes—
109C2×M16×C61C2312yes—
110C22×C8×C61—21no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61—21no—
114C22⋊Q8×C61—21no—
115C2×C4⋊C4×C61C2×C412yes—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61C2×C412yes—
118#101×C61—42no—
119C22⋊C8×C61—21no—
120C22⋊C8×C61—21no—
121C24×C61C22×C411yescharacteristic
122C23×D61C22×C412yes—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C22×C411yescharacteristic
131C22×C4×C61C22×C412yes—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—22no—
138C2×C22⋊D61—22no—
139C2×C22⋊D61—82no—
140C22×Dic61C22×C412yes—
141C22×Dic61C22×C411yescharacteristic
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—22no—
148C2×D244—22no—
149C2×C4×D61—24no—
150C2×C4×D61—82no—
151C2×Dic122—24no—
152C2×Dic122—82no—
153M16×C61—22no—
154M16×C61—22no—
155M16×C61—22no—
156M16×C61—22no—
157M16×C61—42no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—42no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊Dic61C22⋊C412yes—
168C22⋊Dic61C22⋊C412yes—
169C22⋊Dic61C22⋊C412yes—
170C22⋊Dic61C22⋊C412yes—
171#11758198—22no—
172#11758198—22no—
173#11758198—24no—
174#11758198—24no—
175#11758198—24no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—24no—
179C4⋊Dic61C22⋊C412yes—
180C4⋊Dic61C22⋊C412yes—
181C4⋊Dic61C22⋊C412yes—
182C4⋊Dic61C22⋊C412yes—
183#11758196—24no—
184C4×Dic61—24no—
185C4×Dic61C2×D412yes—
186C4×Dic61C2×D412yes—
187C4×Dic61C2×D412yes—
188C4×Dic61C2×D412yes—
189#11758194—22no—
190#11758194—22no—
191#11758194—22no—
192#11758194—22no—
193#11758194—42no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—42no—
197C23×C61—22no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61C2×C22⋊C411yescharacteristic
201C23×C61C2×C22⋊C411yescharacteristic
202C23×C61C23×C411yescharacteristic
203C22×D61—22no—
204C22×D61—24no—
205C22×D61—42no—
206C22×D61—42no—
207Q8×C61—41no—
208Q8×C61—41no—
209Q8×C61—42no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—24no—
220C2×C4×C61—42no—
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61H(5,𝔽2)12yes—
226C2×C4×C612−1+412yes—
227C22⋊D61—82no—
228C22⋊D61—82no—
229C2×Dic61—22no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61C2×C22⋊C412yes—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61C2×C22⋊C412yes—
238C2×Dic61C2×C22⋊C412yes—
239D244—41no—
240D244—41no—
241D244—42no—
242C4×D61—84no—
243Dic122—84no—
244C8×C61—42no—
245C8×C61—42no—
246#92418—42no—
247#92418—42no—
248#21881—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—82no—
258C22×C61C22×C22⋊C411yescharacteristic
259C22×C61#52012yes—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—84no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61#53511yescharacteristic
269C4×C61#53611yescharacteristic
270C4×C61#53712yes—
271Dic61—22no—
272Dic61—22no—
273Dic61—22no—
274Dic61—22no—
275Dic61—42no—
276C2×#542—611no—
277#2921—611no—
278#2905—611no—
279#1997—612no—
280#1967—612no—
281#1901—611no—
282#1901D6111yescharacteristic, complemented
283#1900—611no—
284#1900—611no—
285#1801—612no—
286#1800—612no—
287C2×C61—22no—
288C2×C61—41no—
289C2×C61—41no—
290C2×C61#231111yescharacteristic
291C2×C61#288811yescharacteristic
292C2×C61#288911yescharacteristic
293D61—82no—
294C22×M16—611no—
295C22×M16C2×D6111yescharacteristic
296#542—1222no—
297#542—1222no—
298#542—1222no—
299#542—1222no—
300C2×C4⋊Q8—611no—
301C2×#126—612no—
302C2×#124—611no—
303C2×#124—612no—
304C2×#123—612no—
305C2×C22⋊Q8—612no—
306C2×C22⋊Q8C2×D6111yescharacteristic
307C2×#121—611no—
308C2×C4×Q8—611no—
309C2×C4×D4—611no—
310C2×#117—611no—
311#433—1222no—
312#433—1222no—
313#424—1222no—
314#424—1222no—
315#423—612no—
316#423—612no—
317#423—612no—
318C2×#101—612no—
319C2×#101C2×D6112yes—
320C2×#100—612no—
321C2×#100—612no—
322#407—611no—
323#407C2×D6111yescharacteristic
324C4×M16—612no—
325#343—612no—
326#343—612no—
327#343—612no—
328#343C2×D6112yes—
329C61#2188111yescharacteristic, complemented, Sylow
330C22×Q8—612no—
331C22×Q8C22×D6111yescharacteristic
332C22×D4—612no—
333C22×D4—611no—
334C23×C4—612no—
335C23×C4C22×D6111yescharacteristic
336C2×M16—21no—
337C2×M16—42no—
338C2×M16—612no—
339C2×M16—612no—
340C2×M16—1221no—
341C2×M16—2442no—
342C2×M16C22×D6112yes—
343C2×M16C22×D6112yes—
344C22×C8—21no—
345C22×C8—1221no—
346C4⋊Q8—1222no—
347C4⋊Q8—1222no—
348#126—1224no—
349#126—1224no—
350#124—1222no—
351#124—1222no—
352#124—1224no—
353#124—1224no—
354#123—1224no—
355#123—1224no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—1224no—
361C22⋊Q8—1224no—
362#121—1221no—
363#121—1221no—
364#121—1221no—
365#121—1221no—
366C4×Q8—1222no—
367C4×Q8—1222no—
368C4×D4—1222no—
369C4×D4—1222no—
370#117—612no—
371#117—612no—
372#117—612no—
373#117—612no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4C4×D6112yes—
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C22⋊C4—611no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—612no—
386C2×C22⋊C4—612no—
387C2×C22⋊C4C4×D6112yes—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—612no—
391C4⋊C8—1224no—
392C4⋊C8—1224no—
393C4⋊C8—1224no—
394#101—42no—
395#101—2442no—
396#100—2442no—
397#100—2442no—
398C22⋊C8—21no—
399C22⋊C8—21no—
400C22⋊C8—1221no—
401C22⋊C8—1221no—
402#97—1222no—
403C4×C8—1222no—
404C24—612no—
405C24C2×C4×D6111yescharacteristic
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—1224no—
411C2×Q8—4882no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1221no—
415C2×D4—1221no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—4882no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—612no—
426C22×C4—1224no—
427C22×C4—4882no—
428C22×C4C2×C4×D6112yes—
429C22×C4C2×C4×D6111yescharacteristic
430C22×C4C2×C4×D6112yes—
431C22×C4D4×D6111yescharacteristic
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4D4×D6111yescharacteristic
435C22×C4C23×D6111yescharacteristic
436M16—22no—
437M16—22no—
438M16—22no—
439M16—22no—
440M16—42no—
441M16—1222no—
442M16—1222no—
443M16—1222no—
444M16—1222no—
445M16—2442no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—42no—
449C2×C8—1222no—
450C2×C8—1222no—
451C2×C8—2442no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—612no—
457C4⋊C4—612no—
458C4⋊C4—612no—
459C4⋊C4—612no—
460C4⋊C4—1224no—
461C4⋊C4—1224no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—612no—
467C22⋊C4—612no—
468C22⋊C4—612no—
469C22⋊C4—612no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—1222no—
474C22⋊C4—1222no—
475C42—612no—
476C42—612no—
477C42—612no—
478C42—612no—
479C42—1224no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—1224no—
484C23—1222no—
485C23—2442no—
486C23—2442no—
487C23C22⋊C4×D6111yescharacteristic
488C23C22⋊C4×D6111yescharacteristic
489C23C22×C4×D6111yescharacteristic
490Q8—41no—
491Q8—41no—
492Q8—42no—
493Q8—4884no—
494D4—4882no—
495D4—4882no—
496D4—2441no—
497D4—2441no—
498D4—2442no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—1222no—
515C2×C4—4884no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2442no—
521C2×C4C22⋊C4×D6111yescharacteristic
522C2×C4C22⋊C4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4C2×D4×D6111yescharacteristic
525C2×C4#1451982212yes—
526C2×C4#1451982412yes—
527C8—42no—
528C8—42no—
529C8—2442no—
530C8—2442no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—82no—
540C22—4884no—
541C22—2442no—
542C22—2442no—
543C22—2442no—
544C22C2×C22⋊C4×D6111yescharacteristic
545C22#1684024112yes—
546C4—21no—
547C4—21no—
548C4—42no—
549C4—42no—
550C4—1222no—
551C4—1222no—
552C4—1222no—
553C4—1222no—
554C4—2442no—
555C4#1684032711yescharacteristic
556C4#1684033411yescharacteristic
557C4#1684033912yes—
558C2—22no—
559C2—41no—
560C2—41no—
561C2—4882no—
562C2#2144840611yescharacteristic
563C2#2145429611yescharacteristic
564C2#2145429911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1901C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#1901D61
hypocenterC61#21881
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1901

Complements

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

NH ≅ G/Ntype
2. #21454454C2semidirect
3. #21454382C2semidirect
4. #21446806C2semidirect
5. #21446667C2semidirect
6. #21446288C2semidirect
9. #21445813C2semidirect
10. #21445810C2semidirect
11. C61×#1901C2semidirect
282. #1901D61semidirect
329. C61#21881semidirect

Maximal subgroups

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

subgroupindex
C2×#168403852
#214544542
#214543822
#214468062
#214466672
#214462882
#214462822
#214462802
#214458132
#214458102
C61×#19012
#2188161