Finite Group Databasebeta

← #29836423

Subgroups — #29836423

Order 15616.

476 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456479C211yesmaximal, characteristic, complemented
2#21456460C211yesmaximal, characteristic
3#21447973C211yesmaximal, characteristic, complemented
4#21447970C211yesmaximal, characteristic, complemented
5#21447960C211yesmaximal, characteristic, complemented
6#21447933C211yesmaximal, characteristic
7#21447926C211yesmaximal, characteristic, complemented
8#21447916C211yesmaximal, characteristic
9#21447909C211yesmaximal, characteristic, complemented
10#21447884C211yesmaximal, characteristic, complemented
11#21447884C211yesmaximal, characteristic, complemented
12#21447869C211yesmaximal, characteristic, complemented
13#21447865C211yesmaximal, characteristic, complemented
14#21447576C211yesmaximal, characteristic, complemented
15C61×#2229C211yesmaximal, characteristic, complemented
16#16840384—21no—
17#16840375—21no—
18C2×#14519723C2211yescharacteristic
19C2×#14519723C2211yescharacteristic
20C2×#14519722C2211yescharacteristic
21C2×#14519722C2211yescharacteristic
22C2×#14519716C2211yescharacteristic
23C2×C16×D61C2211yescharacteristic
24#16840166C2211yescharacteristic
25#16840147C2211yescharacteristic
26#16840142C2211yescharacteristic
27#16840132—21no—
28#16840123—21no—
29#16840021—21no—
30#16840011—21no—
31#16840004C2211yescharacteristic
32#16840004C2211yescharacteristic
33#16840000C2211yescharacteristic
34#16839999C2211yescharacteristic
35#16839868—21no—
36#16839860—21no—
37#16839856C2211yescharacteristic
38#16839856C2211yescharacteristic
39C4×D488C2211yescharacteristic
40#16839716C2211yescharacteristic
41#16839589C2211yescharacteristic
42#16839587C2211yescharacteristic
43C4⋊#11758185C2211yescharacteristic
44#16839580C2211yescharacteristic
45#16839580C2211yescharacteristic
46#16839567C2211yescharacteristic
47#16839567C2211yescharacteristic
48#16839565C2211yescharacteristic
49#16839565C2211yescharacteristic
50#16839562C2211yescharacteristic
51#16839562C2211yescharacteristic
52C61×#499C2211yescharacteristic
53C4×Q16×C61C2211yescharacteristic
54C61×#367C2211yescharacteristic
55C61×#366C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#358C2211yescharacteristic
59C2×Dic244—21no—
60C2×Dic244C2311yescharacteristic
61C2×Dic244C2311yescharacteristic
62C2×#11758201—21no—
63C2×D488C2311yescharacteristic
64C2×#11758189D411yescharacteristic
65C2×C8×D61C2311yescharacteristic
66#14519777—21no—
67#14519776—21no—
68#14519772—41no—
69#14519772—41no—
70#14519771—41no—
71#14519768—41no—
72#14519768—41no—
73#14519766—21no—
74#14519765—41no—
75#14519762—41no—
76#14519757—21no—
77#14519757—21no—
78#14519755—41no—
79#14519753—21no—
80#14519752—21no—
81#14519742—21no—
82#14519742—21no—
83C4×D244—21no—
84#14519734D411yescharacteristic
85#14519723—41no—
86#14519723—41no—
87#14519722—41no—
88#14519722—41no—
89#14519716—41no—
90C2×#11758185C2311yescharacteristic
91C2×#11758185C2311yescharacteristic
92C16×D61—41no—
93#14519711C2311yescharacteristic
94#14519710—21no—
95#14519701—21no—
96#14519700—21no—
97#14519700—21no—
98#14519700—21no—
99#14519700—21no—
100#14519699—21no—
101#14519697—21no—
102#14519688D411yescharacteristic
103#14519687D411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C2×Q16×C61C2311yescharacteristic
106#125×C61—21no—
107C4×Q8×C61—21no—
108C2×C16×C61C2311yescharacteristic
109C2×C16×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#106×C61—21no—
114#103×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C2×Q8×C61—21no—
117C2×D244—21no—
118C2×D244—41no—
119C2×C4×D61—41no—
120C2×C4×D61C2×D411yescharacteristic
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122—41no—
124Q16×C61—21no—
125Q16×C61—21no—
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2411yescharacteristic
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—41no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C42×C61—41no—
135C42×C61C2×D411yescharacteristic
136#11758201—42no—
137#11758198—41no—
138#11758198—41no—
139#11758198—41no—
140#11758198—41no—
141#11758198—41no—
142#11758198—41no—
143#11758198—41no—
144#11758198D811yescharacteristic
145#11758198D811yescharacteristic
146C4⋊Dic61—21no—
147C4⋊Dic61—41no—
148C4⋊Dic61—41no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151#11758196—41no—
152#11758196—41no—
153#11758196—41no—
154#11758196—41no—
155#11758196D811yescharacteristic
156#11758196D811yescharacteristic
157C4×Dic61—41no—
158C4×Dic61—41no—
159C4×Dic61C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162Dic244—21no—
163Dic244—21no—
164Dic244—21no—
165Dic244—21no—
166Dic244—42no—
167D488—21no—
168D488—21no—
169#11758189—41no—
170C8×D61—21no—
171C8×D61—21no—
172C16×C61—21no—
173C16×C61—21no—
174#11758185—21no—
175#11758185—21no—
176C22×D61—41no—
177C22×D61C2×D811yescharacteristic
178Q8×C61—41no—
179Q8×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61C2×D811yescharacteristic
185C2×C4×C61C2×D811yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61C2×D811yescharacteristic
191C2×Dic61C2×D811yescharacteristic
192C2×Dic61C2×D811yescharacteristic
193D244—41no—
194D244—41no—
195D244—161no—
196C4×D61—21no—
197C4×D61—21no—
198C4×D61—161no—
199Dic122—41no—
200Dic122—41no—
201Dic122—41no—
202Dic122—41no—
203Dic122—161no—
204C8×C61—21no—
205C8×C61H(5,𝔽2)11yescharacteristic
206C8×C612−1+411yescharacteristic
207#92418—21no—
208#92418—21no—
209#33824—611nomaximal, Sylow
210C22×C61C22×D811yescharacteristic
211C2×D61—21no—
212C2×D61—21no—
213C2×D61—21no—
214C2×D61—81no—
215C2×D61—81no—
216C2×D61—81no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—81no—
220C4×C61—81no—
221C4×C61—81no—
222C4×C61—81no—
223C4×C61#53711yescharacteristic
224C4×C61#57511yescharacteristic
225Dic61—21no—
226Dic61—21no—
227Dic61—21no—
228Dic61—81no—
229Dic61—81no—
230Dic61—81no—
231#3177—611no—
232#3175—611no—
233#2229D6111yescharacteristic, complemented
234#2228—611no—
235#2227—611no—
236#2225—611no—
237#2220—611no—
238#2219—611no—
239#2217—611no—
240#2216—611no—
241#2211—611no—
242#2211—611no—
243#2209—611no—
244#2208—611no—
245#2151—611no—
246C2×C61#324511yescharacteristic
247C2×C61#341611yescharacteristic
248C2×C61#341711yescharacteristic
249D61—41no—
250D61—161no—
251#542—1221no—
252#541—1221no—
253C2×Q32—611no—
254C2×Q32—611no—
255C2×SD32—611no—
256C2×SD32—611no—
257C2×M32—611no—
258C22×C16—611no—
259#500—611no—
260#499C2×D6111yescharacteristic
261#495—611no—
262#494—611no—
263#491—1221no—
264#489—1221no—
265#470—1221no—
266#468—1221no—
267#467—611no—
268#467—611no—
269#467—611no—
270#466—611no—
271#441—1221no—
272#440—1221no—
273C4×Q16—611no—
274C4×Q16—611no—
275C4×Q16C2×D6111yescharacteristic
276C4×D8—611no—
277#405—611no—
278#367—611no—
279#367C2×D6111yescharacteristic
280#366—611no—
281#366C2×D6111yescharacteristic
282C4⋊C16—611no—
283C4⋊C16—611no—
284C4⋊C16—611no—
285C4⋊C16C2×D6111yescharacteristic
286#358—611no—
287#358—611no—
288#358—611no—
289#358—611no—
290#358C2×D6111yescharacteristic
291#358C2×D6111yescharacteristic
292#357—611no—
293#357—611no—
294C61#3382411yescharacteristic, complemented, Sylow
295C2×Q16—611no—
296C2×Q16—611no—
297C2×Q16—1221no—
298C2×Q16C22×D6111yescharacteristic
299C2×SD16—1221no—
300C2×D8—611no—
301C2×M16—611no—
302C22×C8—611no—
303C4⋊Q8—1221no—
304C4⋊Q8—1221no—
305#126—2441no—
306#126—2441no—
307#126—2441no—
308#125—21no—
309#125—2441no—
310#125—2441no—
311#124—1221no—
312#124—2441no—
313#123—2441no—
314C22⋊Q8—1221no—
315C22⋊Q8—1221no—
316C22⋊Q8—1221no—
317C22⋊Q8—2441no—
318#121—1221no—
319C4×Q8—21no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×D4—1221no—
323#117—611no—
324Q32—2441no—
325Q32—2441no—
326SD32—2441no—
327SD32—2441no—
328M32—2441no—
329C2×C16—611no—
330C2×C16—611no—
331C2×C16—2441no—
332C2×C16C22×D6111yescharacteristic
333C2×C16C22×D6111yescharacteristic
334#107—611no—
335#107—1221no—
336#107C22×D6111yescharacteristic
337#107C22×D6111yescharacteristic
338#107C22×D6111yescharacteristic
339#106—21no—
340#103—21no—
341#103—1221no—
342#103—1221no—
343#103—1221no—
344#103—1221no—
345#103—1221no—
346#102—1221no—
347#102—1221no—
348#97—611no—
349#97—611no—
350C4×C8—611no—
351C4×C8C22×D6111yescharacteristic
352C2×Q8—21no—
353C2×Q8—1221no—
354C2×Q8—1221no—
355C2×Q8—2441no—
356C2×D4—1221no—
357C2×D4—2441no—
358C22×C4—611no—
359C22×C4—2441no—
360Q16—21no—
361Q16—21no—
362Q16—1221no—
363Q16—1221no—
364Q16—1221no—
365Q16—1221no—
366Q16—2442no—
367SD16—2442no—
368D8—1221no—
369D8—1221no—
370M16—2441no—
371C2×C8—611no—
372C2×C8—611no—
373C2×C8—1221no—
374C2×C8—1221no—
375C2×C8D4×D6111yescharacteristic
376C2×C8C23×D6111yescharacteristic
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—611no—
384C4⋊C4—611no—
385C4⋊C4—1221no—
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—611no—
395C22⋊C4—611no—
396C22⋊C4—2441no—
397C22⋊C4—2441no—
398C22⋊C4—2441no—
399C22⋊C4—2441no—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C22⋊C4—2441no—
403C42—41no—
404C42—611no—
405C42—2441no—
406C42—2441no—
407C42D4×D6111yescharacteristic
408C16—21no—
409C16—21no—
410C16—1221no—
411C16—1221no—
412C23—611no—
413C23—2441no—
414Q8—41no—
415Q8—41no—
416Q8—9761no—
417Q8—2441no—
418Q8—2441no—
419Q8—2441no—
420Q8—2441no—
421D4—9761no—
422D4—2441no—
423D4—2441no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—611no—
429C2×C4—611no—
430C2×C4—611no—
431C2×C4—1221no—
432C2×C4—1221no—
433C2×C4—9761no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4D8×D6111yescharacteristic
438C2×C4D8×D6111yescharacteristic
439C2×C4C2×D4×D6111yescharacteristic
440C8—21no—
441C8—1221no—
442C8—1221no—
443C8#1451982311yescharacteristic
444C8#1451982511yescharacteristic
445C22—1221no—
446C22—1221no—
447C22—1221no—
448C22—4881no—
449C22—4881no—
450C22—4881no—
451C22C2×D8×D6111yescharacteristic
452C4—21no—
453C4—21no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—1221no—
459C4—1221no—
460C4—1221no—
461C4—4881no—
462C4—4881no—
463C4—4881no—
464C4#1684034211yescharacteristic
465C4#1684056111yescharacteristic
466C2—9761no—
467C2—2441no—
468C2#2145712811yescharacteristic
469C2#2145896211yescharacteristic
470C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2229C2
socleC22×C61C22×D8
prime radical#16839999C22
hypercenter#2229D61
hypocenterC61#33824
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2229

Complements

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

NH ≅ G/Ntype
1. #21456479C2semidirect
3. #21447973C2semidirect
4. #21447970C2semidirect
5. #21447960C2semidirect
7. #21447926C2semidirect
9. #21447909C2semidirect
10. #21447884C2semidirect
11. #21447884C2semidirect
12. #21447869C2semidirect
13. #21447865C2semidirect
14. #21447576C2semidirect
15. C61×#2229C2semidirect
233. #2229D61semidirect
294. C61#33824semidirect

Maximal subgroups

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

subgroupindex
#214564792
#214564602
#214479732
#214479702
#214479602
#214479332
#214479262
#214479162
#214479092
#214478842
#214478692
#214478652
#214475762
C61×#22292
#3382461