Finite Group Databasebeta

← #29835167

Subgroups — #29835167

Order 15616.

532 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456459C211yesmaximal, characteristic
2#21454687C211yesmaximal, characteristic, complemented
3#21448024C211yesmaximal, characteristic, complemented
4#21448011C211yesmaximal, characteristic
5#21447969C211yesmaximal, characteristic, complemented
6#21447959C211yesmaximal, characteristic, complemented
7#21447883C211yesmaximal, characteristic, complemented
8#21447883C211yesmaximal, characteristic, complemented
9#21447736C211yesmaximal, characteristic
10#21447690C211yesmaximal, characteristic, complemented
11#21447686C211yesmaximal, characteristic, complemented
12Q32×Dic61C211yesmaximal, characteristic, complemented
13#21447668C211yesmaximal, characteristic, complemented
14#21447623C211yesmaximal, characteristic, complemented
15C61×#2188C211yesmaximal, characteristic, complemented
16#16840385—21no—
17#16840240—21no—
18C2×#14519720C2211yescharacteristic
19C2×#14519720C2211yescharacteristic
20#16840166C2211yescharacteristic
21#16840147C2211yescharacteristic
22#16840123—21no—
23#16840010—21no—
24#16839998C2211yescharacteristic
25#16839869—21no—
26#16839869—21no—
27#16839865—21no—
28Q16×Dic61C2211yescharacteristic
29Q16×Dic61C2211yescharacteristic
30Q16×Dic61C2211yescharacteristic
31Q16×Dic61C2211yescharacteristic
32Q16×Dic61C2211yescharacteristic
33#16839844C2211yescharacteristic
34#16839800C2211yescharacteristic
35#16839771—21no—
36#16839717C2211yescharacteristic
37#16839590C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839588C2211yescharacteristic
40#16839587C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839564C2211yescharacteristic
48#16839564C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C22×Q16×C61C2211yescharacteristic
53C2×Q32×C61C2211yescharacteristic
54C2×Q32×C61C2211yescharacteristic
55C61×#467C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic
59C2×#11758201—21no—
60C2×D488C2311yescharacteristic
61C2×#11758194D411yescharacteristic
62#14519776—21no—
63#14519772—41no—
64#14519766—21no—
65#14519763—41no—
66#14519759—41no—
67#14519757—41no—
68#14519751—21no—
69Q8×Dic61—21no—
70Q8×Dic61—21no—
71Q8×Dic61—21no—
72Q8×Dic61—21no—
73Q8×Dic61—21no—
74Q8×Dic61—41no—
75Q8×Dic61—41no—
76#14519739—21no—
77#14519736—21no—
78#14519736—41no—
79#14519736—41no—
80#14519736D411yescharacteristic
81#14519720—22no—
82#14519720—22no—
83C2×#11758185C2311yescharacteristic
84C2×#11758185C2311yescharacteristic
85#14519711C2311yescharacteristic
86#14519711C2311yescharacteristic
87#14519711C2311yescharacteristic
88#14519710—21no—
89#14519709—21no—
90#14519701—21no—
91#14519701—21no—
92#14519701—21no—
93#14519701—21no—
94#14519701—21no—
95#14519699—21no—
96#14519697—21no—
97#14519687D411yescharacteristic
98#14519687D411yescharacteristic
99C8×Dic61C2311yescharacteristic
100C8×Dic61C2311yescharacteristic
101C22×Q8×C61—21no—
102C2×Q16×C61—22no—
103C2×Q16×C61—22no—
104C2×Q16×C61—21no—
105C2×Q16×C61C2311yescharacteristic
106C2×Q16×C61C2311yescharacteristic
107C2×Q16×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22⋊Q8×C61—21no—
110Q32×C61—22no—
111Q32×C61—22no—
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114#107×C61C2311yescharacteristic
115#103×C61—21no—
116C2×Q8×C61—24no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61—42no—
121C2×Q8×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61C2×D411yescharacteristic
124C2×C22⋊D61—41no—
125C2×D244—21no—
126C2×C4×D61—41no—
127C2×Dic122—41no—
128Q16×C61—24no—
129Q16×C61—21no—
130Q16×C61—21no—
131Q16×C61—21no—
132Q16×C61—21no—
133Q16×C61#4112yes—
134Q16×C61#4112yes—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2411yescharacteristic
139C4⋊C4×C61—21no—
140C4⋊C4×C61—41no—
141C22⋊C4×C61—41no—
142C22⋊Dic61—41no—
143C22⋊Dic61D811yescharacteristic
144C22⋊Dic61D811yescharacteristic
145#11758201—42no—
146#11758198—41no—
147#11758198—41no—
148#11758198—41no—
149#11758198—41no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—41no—
154C4⋊Dic61—41no—
155C4⋊Dic61—41no—
156C4⋊Dic61—41no—
157C4⋊Dic61—41no—
158C4⋊Dic61D811yescharacteristic
159C4⋊Dic61D811yescharacteristic
160#11758196—41no—
161C4×Dic61—41no—
162C4×Dic61—41no—
163C4×Dic61—41no—
164C4×Dic61—41no—
165C4×Dic61—41no—
166C4×Dic61—41no—
167C4×Dic61C2×D411yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169#11758194—41no—
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172D488—21no—
173D488—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61C2×D811yescharacteristic
179C22×D61—41no—
180Q8×C61—22no—
181Q8×C61—22no—
182Q8×C61—44no—
183Q8×C61—41no—
184Q8×C61—41no—
185Q8×C61—41no—
186Q8×C61—41no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—44no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61C2×D811yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C22⋊D61—161no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61C2×D811yescharacteristic
201C2×Dic61C2×D811yescharacteristic
202C2×Dic61C2×D811yescharacteristic
203C2×Dic61C2×D811yescharacteristic
204D244—41no—
205D244—41no—
206C4×D61—161no—
207Dic122—161no—
208C8×C61—21no—
209C8×C61C2×#4111yescharacteristic
210C8×C61C2×#4111yescharacteristic
211#92418—21no—
212#92418—21no—
213#33681—611nomaximal, Sylow
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61C22×D811yescharacteristic
218C2×D61—81no—
219C2×D61—81no—
220C2×D61—81no—
221C4×C61—21no—
222C4×C61—42no—
223C4×C61—42no—
224C4×C61—81no—
225C4×C61—81no—
226C4×C61#52511yescharacteristic
227C4×C61#57511yescharacteristic
228Dic61—21no—
229Dic61—21no—
230Dic61—21no—
231Dic61—21no—
232Dic61—81no—
233Dic61—81no—
234Dic61—81no—
235#3175—611no—
236#2950—611no—
237#2237—611no—
238#2234—611no—
239#2227—611no—
240#2225—611no—
241#2211—611no—
242#2211—611no—
243#2188D6111yescharacteristic, complemented
244#2185—611no—
245#2176—611no—
246#2175—611no—
247C4×Q32—611no—
248C4×SD32—611no—
249#2164—611no—
250C2×C61—41no—
251C2×C61#310011yescharacteristic
252C2×C61#341611yescharacteristic
253C2×C61#341711yescharacteristic
254D61—161no—
255C22×Q16C2×D6111yescharacteristic
256#542—1221no—
257#519—1221no—
258C2×Q32C2×D6111yescharacteristic
259C2×Q32C2×D6111yescharacteristic
260C2×SD32—611no—
261C2×SD32—611no—
262#500—611no—
263#495—611no—
264#489—1221no—
265#468—1221no—
266#467C2×D6111yescharacteristic
267#466—611no—
268#441—1221no—
269#441—1221no—
270#440—1221no—
271C4×Q16—611no—
272C4×Q16—611no—
273C4×Q16—611no—
274C4×Q16—611no—
275C4×Q16—611no—
276C4×D8—611no—
277#427—611no—
278#419—1221no—
279#405—611no—
280#367—611no—
281#367—611no—
282#366—611no—
283#366—611no—
284C4⋊C16—611no—
285C4⋊C16—611no—
286#358—611no—
287#358—611no—
288#358—611no—
289#358—611no—
290#358C2×D6111yescharacteristic
291#358C2×D6111yescharacteristic
292#357—611no—
293#357—611no—
294C22⋊C16—611no—
295C22⋊C16C2×D6111yescharacteristic
296#346—611no—
297C4×C16—611no—
298C61#3368111yescharacteristic, complemented, Sylow
299C22×Q8—21no—
300C2×Q16—22no—
301C2×Q16—22no—
302C2×Q16—21no—
303C2×Q16C22×D6111yescharacteristic
304C2×Q16C22×D6111yescharacteristic
305C2×Q16C22×D6111yescharacteristic
306C2×SD16—1221no—
307C2×D8—611no—
308C2×M16—611no—
309C22×C8C22×D6111yescharacteristic
310C4⋊Q8—1221no—
311#126—2441no—
312#124—1221no—
313#124—2441no—
314#123—2441no—
315C22⋊Q8—21no—
316C22⋊Q8—2441no—
317#121—1221no—
318C4×Q8—1221no—
319C4×Q8—1221no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×Q8—1221no—
323C4×Q8—2441no—
324C4×Q8—2441no—
325C4×D4—1221no—
326#117—611no—
327#117—1221no—
328#117—2441no—
329#117—2441no—
330Q32—22no—
331Q32—22no—
332SD32—1222no—
333SD32—1222no—
334C2×C16—611no—
335C2×C16—611no—
336C2×C16C22×D6111yescharacteristic
337C2×C16C22×D6111yescharacteristic
338#107—611no—
339#107—611no—
340#107—611no—
341#107—1221no—
342#107C22×D6111yescharacteristic
343#106—1221no—
344#103—21no—
345#103—1221no—
346#103—1221no—
347#103—1221no—
348#103—1221no—
349#103—1221no—
350#102—1221no—
351#102—1221no—
352#97—611no—
353#97—611no—
354C4×C8—611no—
355C4×C8—611no—
356C2×Q8—24no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—42no—
361C2×Q8—41no—
362C2×Q8—2441no—
363C2×D4—1221no—
364C2×D4—2441no—
365C22×C4—41no—
366C22×C4—2441no—
367C22×C4D4×D6111yescharacteristic
368Q16—24no—
369Q16—21no—
370Q16—21no—
371Q16—21no—
372Q16—21no—
373Q16#1175822412yes—
374Q16#1175822412yes—
375SD16—2442no—
376D8—1221no—
377D8—1221no—
378M16—2441no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—611no—
382C2×C8—611no—
383C2×C8D4×D6111yescharacteristic
384C2×C8C23×D6111yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—41no—
387C4⋊C4—611no—
388C4⋊C4—611no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—2441no—
393C4⋊C4—2441no—
394C4⋊C4—2441no—
395C4⋊C4—2441no—
396C4⋊C4—2441no—
397C4⋊C4—2441no—
398C22⋊C4—41no—
399C22⋊C4—611no—
400C22⋊C4—611no—
401C22⋊C4—2441no—
402C22⋊C4—2441no—
403C22⋊C4—2441no—
404C22⋊C4—2441no—
405C22⋊C4—2441no—
406C42—611no—
407C42—611no—
408C42—2441no—
409C42—2441no—
410C42—2441no—
411C42—2441no—
412C42—2441no—
413C42—2441no—
414C16—21no—
415C16—21no—
416C16—1221no—
417C16—1221no—
418C23—2441no—
419C23D8×D6111yescharacteristic
420Q8—22no—
421Q8—22no—
422Q8—44no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426Q8—41no—
427Q8—9761no—
428D4—9761no—
429D4—2441no—
430D4—2441no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—611no—
439C2×C4—611no—
440C2×C4—611no—
441C2×C4—611no—
442C2×C4—9761no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4D8×D6111yescharacteristic
447C2×C4C2×D4×D6111yescharacteristic
448C8—21no—
449C8—1221no—
450C8—1221no—
451C8C2×#1175822411yescharacteristic
452C8C2×#1175822411yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—4881no—
457C22—4881no—
458C22—4881no—
459C22C2×D8×D6111yescharacteristic
460C4—21no—
461C4—42no—
462C4—42no—
463C4—81no—
464C4—81no—
465C4—1221no—
466C4—1221no—
467C4—1221no—
468C4—1221no—
469C4—4881no—
470C4—4881no—
471C4—4881no—
472C4#1684026811yescharacteristic
473C4#1684055811yescharacteristic
474C2—41no—
475C2—9761no—
476C2#2145581411yescharacteristic
477C2#2145896411yescharacteristic
478C2#2145896811yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2188C2
socleC22×C61C22×D8
prime radical#16839998C22
hypercenter#2188D61
hypocenterC61#33681
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2188

Complements

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

NH ≅ G/Ntype
2. #21454687C2semidirect
3. #21448024C2semidirect
5. #21447969C2semidirect
6. #21447959C2semidirect
7. #21447883C2semidirect
8. #21447883C2semidirect
10. #21447690C2semidirect
11. #21447686C2semidirect
12. Q32×Dic61C2semidirect
13. #21447668C2semidirect
14. #21447623C2semidirect
15. C61×#2188C2semidirect
243. #2188D61semidirect
298. C61#33681semidirect

Maximal subgroups

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

subgroupindex
#214564592
#214546872
#214480242
#214480112
#214479692
#214479592
#214478832
#214477362
#214476902
#214476862
Q32×Dic612
#214476682
#214476232
C61×#21882
#3368161