Finite Group Databasebeta

← #29733381

Subgroups — #29733381

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840450C211yesmaximal, characteristic
2C2×#16839969C211yesmaximal, characteristic, complemented
3#21450773C211yesmaximal, characteristic
4#21446828C212yesmaximal, complemented
5#21446828C212yesmaximal, complemented
6#21446777C211yesmaximal, characteristic, complemented
7#21446617C211yesmaximal, characteristic
8#21446128C212yesmaximal, complemented
9#21446123C212yesmaximal, complemented
10#21445650C211yesmaximal, characteristic, complemented
11C2×C61×#491C211yesmaximal, characteristic, complemented
12C22×#11758202C2212yes—
13#16840450—24no—
14#16840450—24no—
15C2×#14519772—21no—
16C2×#14519768—21no—
17C2×#14519762—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519757C2211yescharacteristic
20C2×#14519753C2212yes—
21C2×#14519753C2212yes—
22C2×C4×Dic122C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24C2×#14519735—21no—
25C2×#14519735C2211yescharacteristic
26C2×C4⋊C4×D61C2211yescharacteristic
27#16839969—24no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2212yes—
30C2×#14519701C2212yes—
31C2×#14519689C2212yes—
32C2×#14519689C2212yes—
33#16839697—21no—
34#16839697—21no—
35#16839663—21no—
36#16839631—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#125×C61C2211yescharacteristic
43C61×#491—24no—
44C2×#103×C61C2212yes—
45C2×#103×C61C2212yes—
46C2×#97×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C22×D244D411yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61C2311yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C2×#11758202—24no—
53C2×#11758202—42no—
54C2×#11758202—42no—
55C22×#92418C2312yes—
56#14519772—28no—
57#14519768—28no—
58#14519762—28no—
59#14519757—22no—
60#14519757—22no—
61#14519757—22no—
62#14519757—22no—
63#14519753—24no—
64#14519753—24no—
65#14519753—42no—
66#14519753—42no—
67C4×Dic122—22no—
68C4×Dic122D414yes—
69C4×D244—22no—
70C4×D244—22no—
71#14519735—28no—
72#14519735—22no—
73#14519735—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61D411yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2312yes—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—22no—
82C2×#11758196—22no—
83C4⋊C4×D61—22no—
84C4⋊C4×D61—22no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—22no—
89C2×#11758198—22no—
90C2×#11758198D411yescharacteristic
91C2×#11758198D411yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C4⋊#92418—22no—
96#14519702—24no—
97#14519701—24no—
98#14519689—42no—
99#14519689—42no—
100#14519684—21no—
101C22×Q8×C61C2312yes—
102C22×C8×C61C2312yes—
103C4⋊Q8×C61—22no—
104C4⋊Q8×C61—22no—
105#125×C61—22no—
106#125×C61D414yes—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#103×C61—24no—
115#103×C61—24no—
116#97×C61—22no—
117C23×D61C2×D411yescharacteristic
118C23×D61C2×D411yescharacteristic
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—41no—
139C2×C4×D61—24no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—21no—
153C2×Dic122C2×D412yes—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—42no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61C2×D412yes—
164C4⋊C4×C61C2×D412yes—
165C42×C61C2×D412yes—
166C42×C61C2×D412yes—
167#11758202—48no—
168#11758198—24no—
169#11758198—24no—
170#11758198—24no—
171#11758198—22no—
172#11758198—22no—
173#11758198—44no—
174#11758198—44no—
175C4⋊Dic61—24no—
176C4⋊Dic61—24no—
177C4⋊Dic61—22no—
178C4⋊Dic61—21no—
179C4⋊Dic61—22no—
180C4⋊Dic61C2×D412yes—
181C4⋊Dic61#4114yes—
182#11758196—24no—
183#11758196—24no—
184#11758196—44no—
185#11758196—44no—
186C4×Dic61—24no—
187C4×Dic61—24no—
188C4×Dic61—42no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C23×C61C22×D411yescharacteristic
193C22×D61—21no—
194C22×D61—22no—
195C22×D61—21no—
196C22×D61—22no—
197C22×D61—21no—
198C22×D61—22no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—22no—
202C22×D61#13712yes—
203Q8×C61—42no—
204Q8×C61—42no—
205Q8×C61—42no—
206Q8×C61—42no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61#12012yes—
219C2×C4×C61#12012yes—
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C2×#4112yes—
223C2×C4×C61H(5,𝔽2)12yes—
224C2×Dic61—22no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—21no—
228C2×Dic61—24no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—42no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—41no—
235C2×Dic61—42no—
236D244—44no—
237C4×D61—48no—
238C4×D61—41no—
239C4×D61—41no—
240C4×D61—41no—
241C4×D61—41no—
242C4×D61—42no—
243C4×D61—42no—
244Dic122—22no—
245Dic122—22no—
246Dic122—42no—
247C8×C61—42no—
248#92418—42no—
249#25584—611nomaximal, Sylow
250C22×C61C2×#12011yescharacteristic
251C22×C61#54612yes—
252C22×C61C2×#13711yescharacteristic
253C22×C61#57511yescharacteristic
254C22×C61#57812yes—
255C2×D61—22no—
256C2×D61—24no—
257C2×D61—41no—
258C2×D61—41no—
259C2×D61—42no—
260C2×D61—41no—
261C2×D61—41no—
262C2×D61—41no—
263C2×D61—42no—
264C2×D61—42no—
265C4×C61—22no—
266C4×C61—22no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61—82no—
270C4×C61#53211yescharacteristic
271C4×C61#53511yescharacteristic
272C4×C61#55012yes—
273Dic61—42no—
274Dic61—41no—
275Dic61—41no—
276Dic61—42no—
277Dic61—44no—
278C2×#552—611no—
279C2×#491D6111yescharacteristic, complemented
280C2×#462—611no—
281#2542—611no—
282#2001—612no—
283#2001—612no—
284#1989—611no—
285#1955—611no—
286#1873—612no—
287#1872—612no—
288#1763—611no—
289C2×C61#243111yescharacteristic
290C2×C61#300711yescharacteristic
291C2×C61#301111yescharacteristic
292C2×C61#331212yes—
293C2×C61#331512yes—
294D61—42no—
295D61—81no—
296C22×Q16—612no—
297#552—1224no—
298#552—1224no—
299C2×C4⋊Q8C2×D6111yescharacteristic
300C2×#126—1221no—
301C2×#125—1221no—
302C2×#125C2×D6111yescharacteristic
303C2×#123—1221no—
304C2×C22⋊Q8—612no—
305C2×C22⋊Q8—612no—
306C2×C22⋊Q8—611no—
307C2×C22⋊Q8—611no—
308C2×C4×Q8—611no—
309C2×C4×D4—611no—
310C2×#117—611no—
311C2×#117—1221no—
312C22×C4⋊C4—611no—
313#491—24no—
314#462—1224no—
315C2×C4⋊C8—611no—
316C2×#103—612no—
317C2×#103—612no—
318C2×#103C2×D6112yes—
319C2×#103C2×D6112yes—
320C2×C22⋊C8—612no—
321C2×C22⋊C8—612no—
322C2×#97C2×D6111yescharacteristic
323#399—1221no—
324#399—1221no—
325#390—1221no—
326#384—1221no—
327C4×C4⋊C4—611no—
328#340—612no—
329#340—612no—
330#340—612no—
331C61#2558411yescharacteristic, complemented, Sylow
332C22×Q8—611no—
333C22×Q8C22×D6112yes—
334C22×D4—611no—
335C23×C4—611no—
336C23×C4—611no—
337C23×C4—1221no—
338C2×Q16—1224no—
339C2×Q16—2442no—
340C2×Q16—2442no—
341C22×C8—612no—
342C22×C8C22×D6112yes—
343C4⋊Q8—22no—
344C4⋊Q8—22no—
345#126—1228no—
346#125—22no—
347#125—1228no—
348#125C22⋊D6114yes—
349#123—1228no—
350C22⋊Q8—1222no—
351C22⋊Q8—1222no—
352C22⋊Q8—1222no—
353C22⋊Q8—1222no—
354C22⋊Q8—1224no—
355C22⋊Q8—1224no—
356C22⋊Q8—2442no—
357C22⋊Q8—2442no—
358C4×Q8—614no—
359C4×Q8—1222no—
360C4×D4—1222no—
361C4×D4—1222no—
362#117—1222no—
363#117—1228no—
364#117—1222no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—611no—
383C2×C22⋊C4—611no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1222no—
389C2×C42—1221no—
390C2×C42—1221no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393C4⋊C8—1222no—
394#103—24no—
395#103—24no—
396#103—1224no—
397#103—1224no—
398C22⋊C8—2442no—
399C22⋊C8—2442no—
400#97—22no—
401#95—1221no—
402C24—611no—
403C24—611no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—612no—
411C2×Q8—1221no—
412C2×Q8—1222no—
413C2×Q8—1222no—
414C2×D4—1222no—
415C2×D4—1222no—
416C2×D4—2441no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—1224no—
423C22×C4—1222no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4D4×D6111yescharacteristic
437C22×C4D4×D6111yescharacteristic
438C22×C4C23×D6111yescharacteristic
439Q16—2448no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—612no—
453C4⋊C4—614no—
454C4⋊C4—1224no—
455C4⋊C4—1224no—
456C4⋊C4—1224no—
457C4⋊C4—1221no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1224no—
461C4⋊C4—2444no—
462C4⋊C4—2444no—
463C4⋊C4C2×C22⋊D6112yes—
464C4⋊C4C2×C22⋊D6112yes—
465C22⋊C4—1224no—
466C22⋊C4—1224no—
467C22⋊C4—1224no—
468C22⋊C4—1222no—
469C22⋊C4—1222no—
470C22⋊C4—2444no—
471C22⋊C4—2444no—
472C42—1224no—
473C42—1224no—
474C42—2442no—
475C42D4×D6112yes—
476C42C2×C22⋊D6112yes—
477C23—612no—
478C23—1221no—
479C23—1221no—
480C23—1222no—
481C23—1222no—
482C23—1221no—
483C23—1221no—
484C23—1221no—
485C23—1222no—
486C23—1222no—
487C23C2×D4×D6111yescharacteristic
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—1222no—
493Q8—1222no—
494Q8—2442no—
495D4—2444no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—1222no—
508C2×C4—1221no—
509C2×C4—1222no—
510C2×C4—1221no—
511C2×C4—1224no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—2448no—
515C2×C4—2441no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451974512yes—
527C2×C4#1451974512yes—
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4C22×C22⋊D6111yescharacteristic
530C2×C4#41×D6112yes—
531C2×C4#1451982312yes—
532C8—42no—
533C8—2442no—
534C22—1224no—
535C22—1222no—
536C22—2441no—
537C22—2442no—
538C22—2441no—
539C22—2441no—
540C22—2442no—
541C22—2441no—
542C22—2441no—
543C22—2442no—
544C22C2×#1451974511yescharacteristic
545C22#1684040512yes—
546C22#137×D6111yescharacteristic
547C22#1684056111yescharacteristic
548C22#1684057812yes—
549C4—22no—
550C4—22no—
551C4—42no—
552C4—42no—
553C4—82no—
554C4—2441no—
555C4—2441no—
556C4—2442no—
557C4—2442no—
558C4—2444no—
559C4#1684031111yescharacteristic
560C4#1684033011yescharacteristic
561C4#1684043112yes—
562C2—4881no—
563C2—2442no—
564C2#2144944111yescharacteristic
565C2#2145507511yescharacteristic
566C2#2145511311yescharacteristic
567C2#2145786912yes—
568C2#2145791412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#491C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenterC2×#491D61
hypocenterC61#25584
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
2. C2×#16839969C2semidirect
4. #21446828C2semidirect
5. #21446828C2semidirect
6. #21446777C2semidirect
8. #21446128C2semidirect
9. #21446123C2semidirect
10. #21445650C2semidirect
11. C2×C61×#491C2semidirect
279. C2×#491D61semidirect
331. C61#25584semidirect

Maximal subgroups

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

subgroupindex
C2×#168404502
C2×#168399692
#214507732
#214468282
#214467772
#214466172
#214461282
#214461232
#214456502
C2×C61×#4912
#2558461