Finite Group Databasebeta

← #30604860

Subgroups — #30604860

Order 15616.

470 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519836C211yesmaximal, characteristic
2#21461056C211yesmaximal, characteristic
3#21460813C212yesmaximal, complemented
4#21460800C212yesmaximal, complemented
5#21449105C211yesmaximal, characteristic
6C2×#14519836—21no—
7D61⋊(C22⋊C8)—21no—
8C4×C22⋊#6923—21no—
9#16840640C2212yes—
10#16840607—21no—
11#16840606C2212yes—
12C2×C4×Dic122C411yescharacteristic, complemented
13C2×C4×D244C2211yescharacteristic
14#16839702—21no—
15#16839689C411yescharacteristic, complemented
16#16839689C2211yescharacteristic
17#16839680—21no—
18#16839675C411yescharacteristic, complemented
19#16839675C2211yescharacteristic
20C4×#11758198—21no—
21C4×#11758198—21no—
22C61×#376C411yescharacteristic, complemented
23C2×D61⋊C8—41no—
24C2×C22⋊#6923—22no—
25C2×C4×#6923—41no—
26#14519836—24no—
27#14519834—41no—
28C22⋊#92420—22no—
29(C2×#6923)⋊C4—41no—
30C22×Dic122C2×C411yescharacteristic
31C22×D244D411yescharacteristic
32C22×C4×D61D411yescharacteristic
33C22×C4×D61D411yescharacteristic
34C4×Dic122—41no—
35C4×Dic122—41no—
36C4×D244—41no—
37C4×D244—41no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×C4⋊Dic61D411yescharacteristic
40C2×#11758196—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198C2×C411yescharacteristic
46C2×#11758198C2311yescharacteristic
47C2×C4×Dic61—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52C2×C42×C61—21no—
53C2×C42×C61C2×C411yescharacteristic
54#95×C61—21no—
55#95×C61C2×C411yescharacteristic
56#95×C61C2×C411yescharacteristic
57C23×D61C22⋊C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—42no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×Dic61—21no—
66C22×Dic61—21no—
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C2×D411yescharacteristic
69C2×D244—22no—
70C2×D244—22no—
71C2×D244—41no—
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61—42no—
77C2×C4×D61—42no—
78C2×Dic122—41no—
79C2×Dic122—42no—
80C22⋊#6923—24no—
81C22⋊#6923—42no—
82C2×#92420—42no—
83C4×#6923—82no—
84D61⋊C8—82no—
85C42×C61—21no—
86C42×C61—21no—
87C42×C61—42no—
88#11758198—41no—
89#11758198—41no—
90#11758198—42no—
91#11758198—42no—
92#11758198—42no—
93#11758198—42no—
94C4⋊Dic61—41no—
95C4⋊Dic61—41no—
96#11758196—42no—
97C4×Dic61—41no—
98C4×Dic61—41no—
99C23×C61C2×C22⋊C411yescharacteristic
100C22×D61—21no—
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—21no—
104C22×D61—41no—
105C22×D61—41no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61#10412yes—
109C22×D61#13512yes—
110C2×#6923—44no—
111C2×#6923—82no—
112C2×#6923—82no—
113C2×C4×C61—22no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61#9911yescharacteristic
126C2×C4×C61#9911yescharacteristic
127C2×Dic61—22no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137D244—44no—
138C4×D61—42no—
139C4×D61—42no—
140C4×D61—81no—
141C4×D61—81no—
142Dic122—42no—
143Dic122—42no—
144#92420—82no—
145#11728—611nomaximal, Sylow
146C22×C61—21no—
147C22×C61—21no—
148C22×C61C2×#9911yescharacteristic
149C22×C61#41611yescharacteristic
150C22×C61C2×#10411yescharacteristic
151C2×D61—22no—
152C2×D61—22no—
153C2×D61—41no—
154C2×D61—41no—
155C2×D61—41no—
156C2×D61—41no—
157C2×D61—42no—
158C2×D61—42no—
159C2×D61—81no—
160C2×D61—81no—
161#6923—84no—
162C4×C61—21no—
163C4×C61—41no—
164C4×C61—41no—
165C4×C61—81no—
166C4×C61—81no—
167C4×C61#35412yes—
168Dic61—42no—
169Dic61—41no—
170Dic61—41no—
171Dic61—81no—
172#2398—611no—
173C4×#99—611no—
174#1459—611no—
175#1351—612no—
176#1344—612no—
177C2×C61—21no—
178C2×C61—21no—
179C2×C61#150111yescharacteristic
180C2×C61#212011yescharacteristic
181C2×C61C2×#35411yescharacteristic
182D61—42no—
183D61—42no—
184C2×C4×Q8—611no—
185C2×C4×D4—611no—
186C2×#99—1221no—
187C2×C22⋊C8—1221no—
188#400—1221no—
189#397—611no—
190#397—611no—
191#394—1221no—
192#393—611no—
193#393—611no—
194C4×C22⋊C4—1221no—
195C4×C22⋊C4—1221no—
196C4×C22⋊C4—1221no—
197#376#692311yescharacteristic, complemented
198#342—612no—
199#324—1221no—
200#323—612no—
201C61#1172811yescharacteristic, complemented, Sylow
202C22×Q8—611no—
203C22×D4—611no—
204C23×C4—611no—
205C23×C4—611no—
206C22×C8—2441no—
207C4×Q8—2441no—
208C4×Q8—2441no—
209C4×D4—2441no—
210C4×D4—2441no—
211C2×C4⋊C4—611no—
212C2×C4⋊C4—611no—
213C2×C4⋊C4—1221no—
214C2×C22⋊C4—611no—
215C2×C22⋊C4—611no—
216C2×C22⋊C4—1222no—
217C2×C22⋊C4—1221no—
218C2×C22⋊C4—1221no—
219C2×C22⋊C4—1221no—
220C2×C22⋊C4—1221no—
221C2×C42—21no—
222C2×C42—1221no—
223C2×C42—2441no—
224C2×C42C2×#692311yescharacteristic
225#99—1224no—
226C22⋊C8—1222no—
227C22⋊C8—2441no—
228#95—21no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—2441no—
234#95C2×#692311yescharacteristic
235#95C2×#692311yescharacteristic
236C24—611no—
237C24—611no—
238C2×Q8—2441no—
239C2×Q8—2442no—
240C2×D4—1222no—
241C2×D4—1222no—
242C2×D4—2441no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—611no—
246C22×C4—611no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4—1221no—
252C22×C4—1221no—
253C22×C4—2442no—
254C22×C4—2442no—
255C22×C4—2442no—
256C22×C4C22⋊#692311yescharacteristic
257C22×C4C22⋊#692311yescharacteristic
258C22×C4C22×#692311yescharacteristic
259C2×C8—4882no—
260C2×C8—2442no—
261C4⋊C4—2441no—
262C4⋊C4—2441no—
263C4⋊C4—2442no—
264C22⋊C4—1224no—
265C22⋊C4—2441no—
266C22⋊C4—2441no—
267C22⋊C4—2442no—
268C22⋊C4—2442no—
269C22⋊C4—2442no—
270C22⋊C4—2442no—
271C22⋊C4—2442no—
272C42—21no—
273C42—21no—
274C42—42no—
275C42—4882no—
276C42—2441no—
277C42—2441no—
278C23—612no—
279C23—612no—
280C23—1221no—
281C23—1221no—
282C23—1221no—
283C23—1221no—
284C23—2441no—
285C23—2441no—
286C23—2441no—
287C23—2441no—
288C23C2×C22⋊#692311yescharacteristic
289Q8—2442no—
290Q8—2442no—
291D4—2444no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—1222no—
305C2×C4—1221no—
306C2×C4—1221no—
307C2×C4—4881no—
308C2×C4—4881no—
309C2×C4—4882no—
310C2×C4—4882no—
311C2×C4—2444no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4—2442no—
318C2×C4—2442no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4#1451983611yescharacteristic
322C2×C4#1451983611yescharacteristic
323C8—4882no—
324C22—21no—
325C22—21no—
326C22—1222no—
327C22—1222no—
328C22—4881no—
329C22—4881no—
330C22—2442no—
331C22—2442no—
332C22—2441no—
333C22—2441no—
334C22—2441no—
335C22—2441no—
336C22C2×#1451983611yescharacteristic
337C22#1684075211yescharacteristic
338C22D61⋊#10411yescharacteristic
339C4—21no—
340C4—41no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—4884no—
345C4—4881no—
346C4—2442no—
347C4—2441no—
348C4—2441no—
349C4#1684066912yes—
350C2—21no—
351C2—21no—
352C2—2442no—
353C2—2442no—
354C2#2146119611yescharacteristic
355C2#2146207211yescharacteristic
356C2C2×#1684066911yescharacteristic
subgroupG/H
centerC2×C4#14519836
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#376C4
socleC22×C61C2×#99
prime radicalC22×D244D4
cosocleC2×#11758198C23
hypercenter#376#6923
hypocenterC61#11728
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
3. #21460813C2semidirect
4. #21460800C2semidirect
12. C2×C4×Dic122C4semidirect
15. #16839689C4semidirect
18. #16839675C4semidirect
22. C61×#376C4semidirect
197. #376#6923semidirect
201. C61#11728semidirect

Maximal subgroups

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

subgroupindex
C4×#145198362
#214610562
#214608132
#214608002
#214491052
#1172861