Finite Group Databasebeta

← #29663884

Subgroups — #29663884

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456399C211yesmaximal, characteristic
2#21455943C212yesmaximal, complemented
3C8×Q8×D61C211yesmaximal, characteristic, complemented
4#21445238C212yesmaximal, complemented
5#21445237C212yesmaximal, complemented
6#21445220C211yesmaximal, characteristic, complemented
7#21445180C211yesmaximal, characteristic, complemented
8#21445179C211yesmaximal, characteristic, complemented
9#21444581C211yesmaximal, characteristic, complemented
10#21444513C212yesmaximal, complemented
11C61×#1676C211yesmaximal, characteristic, complemented
12#16840313C2212yes—
13C4×Q8×D61C2211yescharacteristic
14#16840139C2211yescharacteristic
15#16840110C2212yes—
16#16840077—22no—
17#16840054C2212yes—
18#16839999—21no—
19#16839999—21no—
20#16839963C2212yes—
21#16839962C2212yes—
22#16839902—22no—
23Q8×#92418C2211yescharacteristic
24C8×Dic122—21no—
25C8×Dic122C2211yescharacteristic
26#16839886—21no—
27#16839886C2211yescharacteristic
28#16839865—22no—
29#16839861—22no—
30#16839844—21no—
31#16839844—21no—
32#16839818C2212yes—
33C4⋊C8×D61—21no—
34C4⋊C8×D61C2211yescharacteristic
35C4×C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839512C2212yes—
39#16839511C2212yes—
40#16839505C2212yes—
41#16839505C2212yes—
42C61×#500C2211yescharacteristic
43C61×#477C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#333C2212yes—
47C2×#11758226—22no—
48C2×Q8×D61D411yescharacteristic
49C2×#11758190—22no—
50C2×#11758201—22no—
51C2×D488—21no—
52C2×D488—21no—
53C2×C8×D61—21no—
54C2×C8×D61D411yescharacteristic
55C2×C8×D61D411yescharacteristic
56#14519774C2312yes—
57#14519766—22no—
58#14519752—22no—
59#14519752—22no—
60Q8×Dic61D411yescharacteristic
61C4×Dic122—21no—
62C4×Dic122C2311yescharacteristic
63#14519742—21no—
64#14519742D411yescharacteristic
65#14519739—22no—
66#14519739—22no—
67C4⋊C4×D61—21no—
68C4⋊C4×D61D411yescharacteristic
69C42×D61—21no—
70C42×D61C2311yescharacteristic
71#14519710—22no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2311yescharacteristic
75#14519706—21no—
76#14519706—21no—
77#14519706—21no—
78#14519700—22no—
79#14519699—22no—
80#14519697—22no—
81#14519697—22no—
82#14519697—22no—
83#14519689—22no—
84#14519687—22no—
85C4×#92418—21no—
86C4×#92418C2311yescharacteristic
87C8×Dic61—21no—
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C4⋊Q8×C61C2312yes—
92C4×Q8×C61C2311yescharacteristic
93#107×C61—21no—
94#107×C61—21no—
95#107×C61—22no—
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2312yes—
98C4⋊C8×C61C2311yescharacteristic
99#103×C61—22no—
100C4×C8×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61C2×D411yescharacteristic
104#11758226—84no—
105Q8×D61—22no—
106Q8×D61—22no—
107C2×D244—22no—
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—42no—
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—42no—
114C2×C4×D61C2×D411yescharacteristic
115C2×C4×D61C2×D411yescharacteristic
116C2×C4×D61#4111yescharacteristic
117C2×Dic122—21no—
118C2×Dic122C2×D411yescharacteristic
119C2×C8×C61—21no—
120C2×C8×C61—22no—
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D411yescharacteristic
123C4⋊C4×C61—21no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61C2×D411yescharacteristic
127C42×C61—21no—
128C42×C61C2411yescharacteristic
129#11758201—44no—
130#11758198—42no—
131#11758198—42no—
132C4⋊Dic61—21no—
133C4⋊Dic61C2×D411yescharacteristic
134#11758196—21no—
135#11758196—21no—
136#11758196—21no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139C4×Dic61—21no—
140C4×Dic61C2×D411yescharacteristic
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61#4111yescharacteristic
143C2×#92418—21no—
144C2×#92418—22no—
145C2×#92418C2×D411yescharacteristic
146C2×#92418C2×D411yescharacteristic
147D488—42no—
148D488—42no—
149#11758190—44no—
150C8×D61—22no—
151C8×D61—41no—
152C8×D61—42no—
153C22×D61—42no—
154C22×D61#12111yescharacteristic
155Q8×C61—21no—
156Q8×C61—82no—
157Q8×C61#13512yes—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—42no—
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C2×#4111yescharacteristic
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61#12111yescharacteristic
167C2×Dic61#12111yescharacteristic
168C2×Dic61#12111yescharacteristic
169D244—42no—
170D244—42no—
171D244—42no—
172D244—42no—
173D244—82no—
174D244—84no—
175C4×D61—24no—
176C4×D61—22no—
177C4×D61—21no—
178C4×D61—21no—
179C4×D61—21no—
180C4×D61—21no—
181C4×D61—42no—
182C4×D61—84no—
183Dic122—24no—
184Dic122—21no—
185Dic122#13512yes—
186C8×C61—21no—
187C8×C61—42no—
188C8×C61—41no—
189C8×C61#13712yes—
190#92418—21no—
191#92418—21no—
192#92418—42no—
193#92418—41no—
194#19264—611nomaximal, Sylow
195C22×C61C2×#12111yescharacteristic
196C2×D61—21no—
197C2×D61—21no—
198C2×D61—82no—
199C2×D61—82no—
200C2×D61—82no—
201C2×D61#47212yes—
202C4×C61—22no—
203C4×C61—21no—
204C4×C61—41no—
205C4×C61—82no—
206C4×C61C2×#13511yescharacteristic
207C4×C61C2×#13511yescharacteristic
208C4×C61C2×#13711yescharacteristic
209C4×C61#57511yescharacteristic
210Dic61—21no—
211Dic61—21no—
212Dic61—22no—
213Dic61—21no—
214Dic61—41no—
215#3169—611no—
216#3112—612no—
217C2×C8×Q8—611no—
218#1678—612no—
219#1678—612no—
220#1676—611no—
221#1676D6111yescharacteristic, complemented
222#1668—611no—
223#1668—611no—
224#1568—611no—
225#1551—612no—
226C2×C61#303711yescharacteristic
227C2×C61#305011yescharacteristic
228C2×C61C2×#47211yescharacteristic
229D61—22no—
230D61—162no—
231#532—612no—
232C2×C4×Q8—611no—
233#500C2×D6111yescharacteristic
234#493—611no—
235#486—612no—
236#480—1222no—
237#477—612no—
238#477C2×D6112yes—
239#466—1221no—
240#466—1221no—
241#460—612no—
242#460—612no—
243#448—1222no—
244C8×Q8—611no—
245C8×Q8—611no—
246C8×Q8—611no—
247C8×Q8—1221no—
248C8×Q8—1221no—
249C8×Q8C2×D6111yescharacteristic
250#440—1222no—
251#440—1222no—
252C4×D8—1221no—
253C4×D8—1221no—
254#432—612no—
255C2×C4⋊C8—611no—
256C2×C4⋊C8—1221no—
257C2×C4×C8—611no—
258C2×C4×C8—1221no—
259#335—611no—
260#335C2×D6111yescharacteristic
261#334—612no—
262#333—612no—
263#333C2×D6112yes—
264#331—612no—
265#331—612no—
266C61#1926411yescharacteristic, complemented, Sylow
267C2×#41—1222no—
268C22×Q8—611no—
269C2×SD16—1222no—
270C2×SD16—1222no—
271C2×D8—1221no—
272C2×D8—1221no—
273C22×C8—611no—
274C22×C8—611no—
275C22×C8—1221no—
276C4⋊Q8C22×D6112yes—
277#127—612no—
278#124—1222no—
279#121—1222no—
280#121—1222no—
281C4×Q8—611no—
282C4×Q8—611no—
283C4×Q8—611no—
284C4×Q8—1221no—
285C4×Q8—1221no—
286C4×Q8C22×D6111yescharacteristic
287C4×D4—1222no—
288C4×D4—1222no—
289C2×C4⋊C4—611no—
290C2×C4⋊C4—1221no—
291C2×C42—611no—
292C2×C42—1221no—
293#107—21no—
294#107—21no—
295#107—22no—
296#107—1222no—
297C4⋊C8—21no—
298C4⋊C8—611no—
299C4⋊C8—612no—
300C4⋊C8—1221no—
301C4⋊C8—1221no—
302C4⋊C8—1221no—
303C4⋊C8—1221no—
304C4⋊C8C22×D6111yescharacteristic
305C4⋊C8C22×D6112yes—
306#103—22no—
307#103—1222no—
308#102—1222no—
309#102—1222no—
310#102—1222no—
311#102—1222no—
312C22⋊C8—1222no—
313#97—1222no—
314C4×C8—21no—
315C4×C8—611no—
316C4×C8—611no—
317C4×C8—611no—
318C4×C8—1221no—
319C4×C8—1221no—
320C4×C8—1221no—
321C4×C8C22×D6111yescharacteristic
322#41—4884no—
323C2×Q8—22no—
324C2×Q8—611no—
325C2×Q8—1221no—
326C2×Q8—1222no—
327C2×Q8—1222no—
328C2×Q8D4×D6111yescharacteristic
329C2×D4—1222no—
330C2×D4—1222no—
331C2×D4—1222no—
332C2×D4—2442no—
333C22×C4—611no—
334C22×C4—611no—
335C22×C4—611no—
336C22×C4—1221no—
337C22×C4—1221no—
338C22×C4—2442no—
339SD16—2444no—
340SD16—2444no—
341D8—2442no—
342D8—2442no—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—611no—
346C2×C8—611no—
347C2×C8—1222no—
348C2×C8—1221no—
349C2×C8—1222no—
350C2×C8—2441no—
351C2×C8—2442no—
352C2×C8D4×D6111yescharacteristic
353C2×C8D4×D6111yescharacteristic
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—611no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C4⋊C4—1221no—
361C4⋊C4—1221no—
362C4⋊C4D4×D6111yescharacteristic
363C22⋊C4—2442no—
364C22⋊C4—2442no—
365C42—21no—
366C42—611no—
367C42—611no—
368C42—611no—
369C42—1221no—
370C42—1221no—
371C42—1221no—
372C42C23×D6111yescharacteristic
373C23—611no—
374C23—2442no—
375Q8—21no—
376Q8—82no—
377Q8—612no—
378Q8—1224no—
379Q8—1221no—
380Q8#1451980012yes—
381D4—4882no—
382D4—4884no—
383D4—2442no—
384D4—2442no—
385D4—2442no—
386D4—2442no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—42no—
390C2×C4—611no—
391C2×C4—611no—
392C2×C4—611no—
393C2×C4—1222no—
394C2×C4—1224no—
395C2×C4—1221no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—1221no—
399C2×C4—1221no—
400C2×C4—1221no—
401C2×C4—4884no—
402C2×C4—2442no—
403C2×C4C2×D4×D6111yescharacteristic
404C2×C4C2×D4×D6111yescharacteristic
405C2×C4#41×D6111yescharacteristic
406C8—21no—
407C8—42no—
408C8—41no—
409C8—1221no—
410C8—1221no—
411C8—2442no—
412C8—2441no—
413C8#1451980912yes—
414C22—612no—
415C22—1221no—
416C22—1221no—
417C22—4882no—
418C22—4882no—
419C22—4882no—
420C22#121×D6111yescharacteristic
421C4—22no—
422C4—21no—
423C4—41no—
424C4—82no—
425C4—1221no—
426C4—1222no—
427C4—1221no—
428C4—1221no—
429C4—2441no—
430C4#135×D6111yescharacteristic
431C4C2×#1451980011yescharacteristic
432C4C2×#1451980911yescharacteristic
433C4#1684056111yescharacteristic
434C2—1222no—
435C2—9762no—
436C2#2145529211yescharacteristic
437C2#2145540211yescharacteristic
438C2#472×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1676C2
socleC22×C61C2×#121
prime radical#21456399C2
hypercenter#1676D61
hypocenterC61#19264
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1676

Complements

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

NH ≅ G/Ntype
2. #21455943C2semidirect
3. C8×Q8×D61C2semidirect
4. #21445238C2semidirect
5. #21445237C2semidirect
6. #21445220C2semidirect
7. #21445180C2semidirect
8. #21445179C2semidirect
9. #21444581C2semidirect
10. #21444513C2semidirect
11. C61×#1676C2semidirect
221. #1676D61semidirect
266. C61#19264semidirect

Maximal subgroups

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

subgroupindex
#214563992
#214559432
C8×Q8×D612
#214452382
#214452372
#214452202
#214451802
#214451792
#214445812
#214445132
C61×#16762
#1926461