Finite Group Databasebeta

← #29835231

Subgroups — #29835231

Order 15616.

520 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456471C211yesmaximal, characteristic
2#21454687C211yesmaximal, characteristic
3#21448028C211yesmaximal, characteristic
4#21448024C211yesmaximal, characteristic
5#21447985C211yesmaximal, characteristic, complemented
6#21447981C211yesmaximal, characteristic, complemented
7#21447867C211yesmaximal, characteristic, complemented
8#21447867C211yesmaximal, characteristic, complemented
9#21447753C211yesmaximal, characteristic
10#21447690C211yesmaximal, characteristic, complemented
11#21447687C211yesmaximal, characteristic, complemented
12Q32×Dic61C211yesmaximal, characteristic, complemented
13#21447675C211yesmaximal, characteristic, complemented
14#21447623C211yesmaximal, characteristic
15C61×#2188C211yesmaximal, characteristic
16#16840372—21no—
17#16840240—21no—
18C2×#14519723C2211yescharacteristic
19C2×#14519723C2211yescharacteristic
20#16840163C2211yescharacteristic
21#16840140C2211yescharacteristic
22#16840129—21no—
23#16840018—21no—
24#16840003C2211yescharacteristic
25#16839869—21no—
26#16839869—21no—
27#16839866—21no—
28Q16×Dic61C2211yescharacteristic
29Q16×Dic61C2211yescharacteristic
30Q16×Dic61C2211yescharacteristic
31Q16×Dic61C2211yescharacteristic
32Q16×Dic61C2211yescharacteristic
33#16839856C2211yescharacteristic
34#16839800C2211yescharacteristic
35#16839771—21no—
36#16839717C2211yescharacteristic
37#16839590C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839588C2211yescharacteristic
40#16839587C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839566C2211yescharacteristic
48#16839566C2211yescharacteristic
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×Dic244C2311yescharacteristic
60C2×#11758202—21no—
61C2×#11758194D411yescharacteristic
62#14519775—21no—
63#14519770—41no—
64#14519770—41no—
65#14519768—21no—
66#14519768—41no—
67#14519767—41no—
68#14519756—21no—
69Q8×Dic61—21no—
70Q8×Dic61—21no—
71Q8×Dic61—21no—
72Q8×Dic61—21no—
73Q8×Dic61—21no—
74Q8×Dic61—41no—
75Q8×Dic61—41no—
76#14519742—21no—
77#14519736—21no—
78#14519736—41no—
79#14519736—41no—
80#14519736D411yescharacteristic
81#14519723—22no—
82#14519723—22no—
83C2×#11758185C2311yescharacteristic
84C2×#11758185C2311yescharacteristic
85#14519711C2311yescharacteristic
86#14519711C2311yescharacteristic
87#14519711C2311yescharacteristic
88#14519709—21no—
89#14519708—21no—
90#14519702—21no—
91#14519701—21no—
92#14519701—21no—
93#14519701—21no—
94#14519701—21no—
95#14519701—21no—
96#14519700—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×Dic122—21no—
125Q16×C61—24no—
126Q16×C61—21no—
127Q16×C61—21no—
128Q16×C61—21no—
129Q16×C61—21no—
130Q16×C61#4112yes—
131Q16×C61#4112yes—
132C2×C8×C61—21no—
133C2×C8×C61—21no—
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2411yescharacteristic
136C4⋊C4×C61—21no—
137C4⋊C4×C61—41no—
138C22⋊C4×C61—41no—
139C22⋊Dic61—41no—
140C22⋊Dic61—41no—
141C22⋊Dic61D811yescharacteristic
142C22⋊Dic61D811yescharacteristic
143#11758202—42no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—41no—
148C4⋊Dic61—41no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152C4⋊Dic61—41no—
153C4⋊Dic61D811yescharacteristic
154C4⋊Dic61D811yescharacteristic
155#11758196—41no—
156#11758196—41no—
157#11758196—41no—
158#11758196—41no—
159#11758196—41no—
160C4×Dic61—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
172Dic244—21no—
173Dic244—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61C2×D811yescharacteristic
179Q8×C61—22no—
180Q8×C61—22no—
181Q8×C61—44no—
182Q8×C61—41no—
183Q8×C61—41no—
184Q8×C61—41no—
185Q8×C61—41no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—44no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61C2×D811yescharacteristic
200C2×Dic61C2×D811yescharacteristic
201C2×Dic61C2×D811yescharacteristic
202C2×Dic61C2×D811yescharacteristic
203Dic122—41no—
204Dic122—41no—
205C8×C61—21no—
206C8×C61C2×#4111yescharacteristic
207C8×C61C2×#4111yescharacteristic
208#92418—21no—
209#92418—21no—
210#33687—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61C22×D811yescharacteristic
215C4×C61—21no—
216C4×C61—42no—
217C4×C61—42no—
218C4×C61—81no—
219C4×C61—81no—
220C4×C61#52511yescharacteristic
221C4×C61#57511yescharacteristic
222Dic61—21no—
223Dic61—21no—
224Dic61—21no—
225Dic61—21no—
226Dic61—81no—
227Dic61—81no—
228Dic61—81no—
229Dic61—81no—
230#3177—611no—
231#2950—611no—
232#2237—611no—
233#2237—611no—
234#2229—611no—
235#2229—611no—
236#2209—611no—
237#2209—611no—
238#2188—611no—
239#2188D6111yescharacteristic
240#2176—611no—
241#2176—611no—
242C4×Q32—611no—
243C4×Q32—611no—
244#2164—611no—
245C2×C61—41no—
246C2×C61#310011yescharacteristic
247C2×C61#341711yescharacteristic
248C2×C61#341711yescharacteristic
249C22×Q16C2×D6111yescharacteristic
250#541—1221no—
251#519—1221no—
252C2×Q32—611no—
253C2×Q32—611no—
254C2×Q32C2×D6111yescharacteristic
255C2×Q32C2×D6111yescharacteristic
256#499—611no—
257#494—611no—
258#491—1221no—
259#470—1221no—
260#467—611no—
261#467C2×D6111yescharacteristic
262#441—1221no—
263#441—1221no—
264#441—1221no—
265C4×Q16—611no—
266C4×Q16—611no—
267C4×Q16—611no—
268C4×Q16—611no—
269C4×Q16—611no—
270C4×Q16—611no—
271#427—611no—
272#419—1221no—
273#405—611no—
274#367—611no—
275#367—611no—
276#366—611no—
277#366—611no—
278C4⋊C16—611no—
279C4⋊C16—611no—
280#358—611no—
281#358—611no—
282#358—611no—
283#358—611no—
284#358—611no—
285#358—611no—
286#358C2×D6111yescharacteristic
287#358C2×D6111yescharacteristic
288C22⋊C16—611no—
289C22⋊C16C2×D6111yescharacteristic
290#346—611no—
291C4×C16—611no—
292C61#3368711yescharacteristic, complemented, Sylow
293C22×Q8—21no—
294C2×Q16—22no—
295C2×Q16—22no—
296C2×Q16—21no—
297C2×Q16—611no—
298C2×Q16—1221no—
299C2×Q16C22×D6111yescharacteristic
300C2×Q16C22×D6111yescharacteristic
301C2×Q16C22×D6111yescharacteristic
302C2×M16—611no—
303C22×C8C22×D6111yescharacteristic
304C4⋊Q8—1221no—
305#126—2441no—
306#126—2441no—
307#125—1221no—
308#125—2441no—
309#125—2441no—
310C22⋊Q8—21no—
311C22⋊Q8—1221no—
312C4×Q8—1221no—
313C4×Q8—1221no—
314C4×Q8—1221no—
315C4×Q8—1221no—
316C4×Q8—1221no—
317C4×Q8—1221no—
318C4×Q8—2441no—
319C4×Q8—2441no—
320#117—611no—
321#117—1221no—
322#117—2441no—
323#117—2441no—
324Q32—22no—
325Q32—22no—
326Q32—1222no—
327Q32—1222no—
328C2×C16—611no—
329C2×C16—611no—
330C2×C16C22×D6111yescharacteristic
331C2×C16C22×D6111yescharacteristic
332#107—611no—
333#107—611no—
334#107—611no—
335#107C22×D6111yescharacteristic
336#106—1221no—
337#106—1221no—
338#103—21no—
339#103—1221no—
340#103—1221no—
341#103—1221no—
342#103—1221no—
343#103—1221no—
344#103—1221no—
345#103—1221no—
346#97—611no—
347#97—611no—
348C4×C8—611no—
349C4×C8—611no—
350C2×Q8—24no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—42no—
355C2×Q8—41no—
356C2×Q8—1221no—
357C22×C4—41no—
358C22×C4D4×D6111yescharacteristic
359Q16—24no—
360Q16—21no—
361Q16—21no—
362Q16—21no—
363Q16—21no—
364Q16—1221no—
365Q16—1221no—
366Q16—2442no—
367Q16#1175822412yes—
368Q16#1175822412yes—
369M16—2441no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—611no—
373C2×C8—611no—
374C2×C8D4×D6111yescharacteristic
375C2×C8C23×D6111yescharacteristic
376C4⋊C4—21no—
377C4⋊C4—41no—
378C4⋊C4—611no—
379C4⋊C4—611no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—2441no—
384C4⋊C4—2441no—
385C4⋊C4—2441no—
386C4⋊C4—2441no—
387C4⋊C4—2441no—
388C4⋊C4—2441no—
389C4⋊C4—2441no—
390C4⋊C4—2441no—
391C4⋊C4—2441no—
392C4⋊C4—2441no—
393C4⋊C4—2441no—
394C22⋊C4—41no—
395C22⋊C4—611no—
396C22⋊C4—611no—
397C22⋊C4—2441no—
398C22⋊C4—2441no—
399C42—611no—
400C42—611no—
401C42—2441no—
402C42—2441no—
403C42—2441no—
404C42—2441no—
405C42—2441no—
406C42—2441no—
407C42—2441no—
408C16—21no—
409C16—21no—
410C16—1221no—
411C16—1221no—
412C23D8×D6111yescharacteristic
413Q8—22no—
414Q8—22no—
415Q8—44no—
416Q8—41no—
417Q8—41no—
418Q8—41no—
419Q8—41no—
420Q8—2441no—
421Q8—2441no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—44no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—611no—
430C2×C4—611no—
431C2×C4—611no—
432C2×C4—611no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4D8×D6111yescharacteristic
438C2×C4C2×D4×D6111yescharacteristic
439C8—21no—
440C8—1221no—
441C8—1221no—
442C8C2×#1175822411yescharacteristic
443C8C2×#1175822411yescharacteristic
444C22—21no—
445C22—21no—
446C22—21no—
447C22C2×D8×D6111yescharacteristic
448C4—21no—
449C4—42no—
450C4—42no—
451C4—81no—
452C4—81no—
453C4—1221no—
454C4—1221no—
455C4—1221no—
456C4—1221no—
457C4—4881no—
458C4—4881no—
459C4—4881no—
460C4—4881no—
461C4#1684026811yescharacteristic
462C4#1684055811yescharacteristic
463C2—41no—
464C2#2145581411yescharacteristic
465C2#2145896811yescharacteristic
466C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2188C2
socleC22×C61C22×D8
prime radicalC23×C61C2×D8
hypercenter#2188D61
hypocenterC61#33687
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
5. #21447985C2semidirect
6. #21447981C2semidirect
7. #21447867C2semidirect
8. #21447867C2semidirect
10. #21447690C2semidirect
11. #21447687C2semidirect
12. Q32×Dic61C2semidirect
13. #21447675C2semidirect
292. C61#33687semidirect

Maximal subgroups

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

subgroupindex
#214564712
#214546872
#214480282
#214480242
#214479852
#214479812
#214478672
#214477532
#214476902
#214476872
Q32×Dic612
#214476752
#214476232
C61×#21882
#3368761