Finite Group Databasebeta

← #29834940

Subgroups — #29834940

Order 15616.

610 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456400C211yesmaximal, characteristic, complemented
2C2×D8×Dic61C211yesmaximal, characteristic
3#21448017C211yesmaximal, characteristic, complemented
4#21448014C211yesmaximal, characteristic, complemented
5#21447964C212yesmaximal, complemented
6#21447862C212yesmaximal, complemented
7#21447748C211yesmaximal, characteristic
8#21447674C212yesmaximal, complemented
9SD32×Dic61C212yesmaximal, complemented
10#21447615C211yesmaximal, characteristic, complemented
11C61×#2187C211yesmaximal, characteristic
12#16840314—21no—
13C2×D4×Dic61—21no—
14C2×#14519721C2212yes—
15#16840166C2211yescharacteristic, complemented
16#16840140C2211yescharacteristic, complemented
17#16840111—21no—
18#16840003C2211yescharacteristic
19#16840001—21no—
20Q16×Dic61C2211yescharacteristic, complemented
21#16839856C2211yescharacteristic, complemented
22#16839854—21no—
23D8×Dic61—21no—
24D8×Dic61—21no—
25D8×Dic61C2212yes—
26D8×Dic61C2212yes—
27C2×#14519711C2211yescharacteristic
28C2×#14519698—21no—
29C2×C8×Dic61C2211yescharacteristic
30#16839590C2212yescomplemented
31#16839589C2212yescomplemented
32#16839580C2212yescomplemented
33#16839567C2212yescomplemented
34#16839566C2212yescomplemented
35#16839563C2212yes—
36C22⋊#11758185C2211yescharacteristic
37C16×Dic61C2212yescomplemented
38C22×D8×C61C2211yescharacteristic
39C2×SD32×C61C2212yes—
40C61×#467C2211yescharacteristic
41C61×#357C2212yes—
42C61×C22⋊C16C2211yescharacteristic
43C23×Dic61—41no—
44C2×Dic244C2311yescharacteristic
45C2×#11758202—21no—
46C22×#92418D411yescharacteristic
47#14519776—21no—
48#14519775—21no—
49#14519767—41no—
50#14519756—21no—
51#14519754—41no—
52Q8×Dic61—21no—
53#14519742—21no—
54#14519742—41no—
55D4×Dic61—22no—
56D4×Dic61—22no—
57D4×Dic61—41no—
58D4×Dic61—41no—
59#14519733—41no—
60C2×C4⋊Dic61—21no—
61C2×C22⋊Dic61—41no—
62C2×C4×Dic61D411yescharacteristic
63#14519721—24no—
64C2×#11758185C2312yes—
65#14519711—21no—
66#14519711C2311yescharacteristic
67#14519711C2312yes—
68#14519710—21no—
69#14519702—21no—
70#14519701—21no—
71#14519700—21no—
72#14519698—22no—
73#14519698—22no—
74C8×Dic61D411yescharacteristic
75C8×Dic61D411yescharacteristic
76C8×Dic61C2311yescharacteristic
77C8×Dic61C2311yescharacteristic
78C22×D4×C61—21no—
79C2×Q16×C61C2311yescharacteristic
80C2×D8×C61—24no—
81C2×D8×C61—21no—
82C2×D8×C61C2312yes—
83C22×C8×C61C2311yescharacteristic
84C22⋊Q8×C61—21no—
85SD32×C61—24no—
86C2×C16×C61C2312yes—
87#107×C61C2311yescharacteristic
88#103×C61—21no—
89C24×C61—41no—
90C2×Q8×C61—21no—
91C2×D4×C61—24no—
92C2×D4×C61—22no—
93C2×D4×C61—42no—
94C2×D4×C61—41no—
95C22×C4×C61C2×D411yescharacteristic
96C22×Dic61—41no—
97C22×Dic61—42no—
98C22×Dic61—42no—
99C22×Dic61—84no—
100C22×Dic61D811yescharacteristic
101C22×Dic61D811yescharacteristic
102C2×Dic122—21no—
103C2×Dic122—41no—
104Q16×C61—22no—
105D8×C61—24no—
106D8×C61—22no—
107D8×C61#4114yes—
108C2×C8×C61—22no—
109C2×C8×C61C2×D411yescharacteristic
110C2×C8×C61C2411yescharacteristic
111C4⋊C4×C61—21no—
112C4⋊C4×C61—41no—
113C22⋊C4×C61—41no—
114C22⋊Dic61—41no—
115C22⋊Dic61—42no—
116C22⋊Dic61—42no—
117#11758202—42no—
118C4⋊Dic61—22no—
119C4⋊Dic61—21no—
120C4⋊Dic61—41no—
121C4⋊Dic61—41no—
122#11758196—41no—
123#11758196—41no—
124#11758196—41no—
125#11758196—41no—
126#11758196—41no—
127C4×Dic61—41no—
128C4×Dic61—41no—
129C4×Dic61D811yescharacteristic
130C4×Dic61D811yescharacteristic
131C4×Dic61C2×D411yescharacteristic
132C4×Dic61C2×D411yescharacteristic
133C2×#92418—41no—
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136Dic244—22no—
137C16×C61—22no—
138#11758185—22no—
139C23×C61—44no—
140C23×C61—42no—
141C23×C61—82no—
142C23×C61—82no—
143C23×C61C2×D811yescharacteristic
144Q8×C61—41no—
145Q8×C61—41no—
146D4×C61—24no—
147D4×C61—44no—
148D4×C61—42no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61C2×D811yescharacteristic
154C2×C4×C61C22×D411yescharacteristic
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—42no—
159C2×Dic61—82no—
160C2×Dic61—84no—
161C2×Dic61—84no—
162C2×Dic61SD3214yescomplemented
163C2×Dic61C2×D811yescharacteristic
164C2×Dic61C2×D811yescharacteristic
165C2×Dic61C2×D811yescharacteristic
166C2×Dic61C2×D811yescharacteristic
167Dic122—41no—
168Dic122—41no—
169Dic122—82no—
170C8×C61—21no—
171C8×C61C2×#4112yes—
172#92418—21no—
173#92418—21no—
174#33661—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—44no—
178C22×C61—88no—
179C22×C61—82no—
180C22×C61—82no—
181C22×C61C2×SD3212yes—
182C22×C61C22×D811yescharacteristic
183C4×C61—21no—
184C4×C61—81no—
185C4×C61—81no—
186C4×C61#52511yescharacteristic
187C4×C61#57511yescharacteristic
188Dic61—21no—
189Dic61—21no—
190Dic61—81no—
191Dic61—82no—
192Dic61—81no—
193Dic61C2×SD3212yes—
194Dic61C2×SD3212yes—
195#3170—611no—
196C2×C4×D8—611no—
197#2235—611no—
198#2235—611no—
199#2226—612no—
200#2208—612no—
201#2187—611no—
202#2187D6111yescharacteristic
203C4×SD32—612no—
204C4×SD32—612no—
205#2162—611no—
206C2×C61—22no—
207C2×C61—84no—
208C2×C61#310011yescharacteristic
209C2×C61C22×SD3211yescharacteristic
210C2×C61#341411yescharacteristic
211C22×D8C2×D6111yescharacteristic
212#533—1221no—
213C2×C4×D4—1221no—
214C2×SD32—612no—
215C2×SD32C2×D6112yes—
216#500—611no—
217#494—611no—
218#487—1221no—
219#467—611no—
220#467—1221no—
221#467C2×D6111yescharacteristic
222C4×Q16—611no—
223C4×Q16—611no—
224C4×Q16—1221no—
225C4×D8—612no—
226C4×D8—612no—
227C4×D8—1221no—
228C4×D8—1221no—
229C2×#107—611no—
230C2×#102—1221no—
231C2×C4×C8—611no—
232#367—612no—
233#367—612no—
234C4⋊C16—612no—
235#358—612no—
236#358—612no—
237#357—612no—
238#357C2×D6112yes—
239C22⋊C16—611no—
240C22⋊C16C2×D6111yescharacteristic
241C4×C16—612no—
242C61#3366111yescharacteristic, complemented, Sylow
243C22×D4—21no—
244C23×C4—2441no—
245C2×Q16—611no—
246C2×Q16—1221no—
247C2×Q16C22×D6111yescharacteristic
248C2×D8—24no—
249C2×D8—21no—
250C2×D8C22×D6112yes—
251C22×C8—611no—
252C22×C8C22×D6111yescharacteristic
253C4⋊Q8—1221no—
254C4⋊Q8—1221no—
255#125—2441no—
256C22⋊Q8—21no—
257C22⋊Q8—1221no—
258C22⋊Q8—2441no—
259C4×Q8—1221no—
260C4×Q8—1221no—
261C4×Q8—2441no—
262C4×D4—1222no—
263C4×D4—1222no—
264C4×D4—2441no—
265C4×D4—2441no—
266#117—2441no—
267C2×C4⋊C4—1221no—
268C2×C22⋊C4—2441no—
269C2×C42—611no—
270SD32—24no—
271SD32—1224no—
272C2×C16—612no—
273C2×C16C22×D6112yes—
274#107—611no—
275#107—612no—
276#107—1221no—
277#107—1221no—
278#107C22×D6111yescharacteristic
279#103—21no—
280#103—1221no—
281#103—1221no—
282#103—1221no—
283#102—1222no—
284#102—1222no—
285C4×C8—611no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—611no—
289C24—41no—
290C2×Q8—21no—
291C2×Q8—1221no—
292C2×Q8—2441no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—42no—
296C2×D4—41no—
297C22×C4—611no—
298C22×C4—611no—
299C22×C4—4884no—
300C22×C4—2442no—
301C22×C4—2442no—
302C22×C4—2441no—
303C22×C4D4×D6111yescharacteristic
304Q16—22no—
305Q16—1222no—
306Q16—2442no—
307D8—24no—
308D8—22no—
309D8#1175822414yes—
310C2×C8—22no—
311C2×C8—611no—
312C2×C8—611no—
313C2×C8—2441no—
314C2×C8D4×D6111yescharacteristic
315C2×C8C23×D6111yescharacteristic
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—1222no—
319C4⋊C4—1221no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2441no—
323C4⋊C4—2441no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C22⋊C4—41no—
328C22⋊C4—2442no—
329C22⋊C4—2442no—
330C22⋊C4—2441no—
331C42—611no—
332C42—611no—
333C42—611no—
334C42—611no—
335C42—2441no—
336C42—2441no—
337C16—22no—
338C16—1222no—
339C23—44no—
340C23—42no—
341C23—82no—
342C23—82no—
343C23D8×D6111yescharacteristic
344Q8—41no—
345Q8—41no—
346Q8—4882no—
347Q8—2441no—
348Q8—2441no—
349D4—24no—
350D4—44no—
351D4—42no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—611no—
357C2×C4—611no—
358C2×C4—614no—
359C2×C4—611no—
360C2×C4—611no—
361C2×C4—4884no—
362C2×C4—4884no—
363C2×C4—4882no—
364C2×C4—2441no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4—2442no—
368C2×C4D8×D6111yescharacteristic
369C2×C4C2×D4×D6111yescharacteristic
370C8—21no—
371C8—1221no—
372C8—1221no—
373C8C2×#1175822412yes—
374C22—21no—
375C22—21no—
376C22—44no—
377C22—88no—
378C22—82no—
379C22—82no—
380C22SD32×D6112yes—
381C22C2×D8×D6111yescharacteristic
382C4—21no—
383C4—81no—
384C4—81no—
385C4—612no—
386C4—612no—
387C4—1221no—
388C4—1221no—
389C4—4881no—
390C4—4881no—
391C4—4882no—
392C4#1684026811yescharacteristic
393C4#1684055811yescharacteristic
394C2—22no—
395C2—84no—
396C2#2145581411yescharacteristic
397C2C2×SD32×D6111yescharacteristic
398C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2187C2
socleC22×C61C22×D8
prime radicalC22×D8×C61C22
hypercenter#2187D61
hypocenterC61#33661
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. #21456400C2semidirect
3. #21448017C2semidirect
4. #21448014C2semidirect
5. #21447964C2semidirect
6. #21447862C2semidirect
8. #21447674C2semidirect
9. SD32×Dic61C2semidirect
10. #21447615C2semidirect
15. #16840166C22semidirect
16. #16840140C22semidirect
20. Q16×Dic61C22semidirect
21. #16839856C22semidirect
30. #16839590C22semidirect
31. #16839589C22semidirect
32. #16839580C22semidirect
33. #16839567C22semidirect
34. #16839566C22semidirect
37. C16×Dic61C22semidirect
162. C2×Dic61SD32semidirect
242. C61#33661semidirect

Maximal subgroups

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

subgroupindex
#214564002
C2×D8×Dic612
#214480172
#214480142
#214479642
#214478622
#214477482
#214476742
SD32×Dic612
#214476152
C61×#21872
#3366161