Finite Group Databasebeta

← #30604859

Subgroups — #30604859

Order 15616.

470 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 70 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
1#21461566C211yesmaximal, characteristic
2#21461057C211yesmaximal, characteristic
3#21460813C212yesmaximal, complemented
4#21460799C212yesmaximal, complemented
5#21449104C211yesmaximal, characteristic
6C2×#14519835—21no—
7C2×C22⋊#92420—21no—
8C4×C22⋊#6923—21no—
9#16840640C2212yes—
10#16840608—21no—
11#16840606C2212yes—
12C2×#14519742C411yescharacteristic, complemented
13C2×#14519737C2211yescharacteristic
14#16839703—21no—
15#16839688C2211yescharacteristic
16#16839677—21no—
17#16839675C2211yescharacteristic
18#16839674C411yescharacteristic, complemented
19C22⋊C4×Dic61—21no—
20#16839609—21no—
21#16839607C411yescharacteristic, complemented
22C61×#397C411yescharacteristic, complemented
23C22×#92420—41no—
24C2×C22⋊#6923—22no—
25C2×C4×#6923—41no—
26#14519835—24no—
27C22⋊#92420—22no—
28C22⋊#92420—41no—
29(C2×#6923)⋊C4—41no—
30C22×Dic122C2×C411yescharacteristic
31C22×C22⋊D61D411yescharacteristic
32C23×Dic61D411yescharacteristic
33C22×C4×D61D411yescharacteristic
34#14519742—41no—
35#14519742—41no—
36#14519737—41no—
37#14519737—41no—
38C2×#11758196—21no—
39C2×#11758196D411yescharacteristic
40C2×C22⋊Dic61—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198C2311yescharacteristic
44C2×C4×Dic61—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61C2×C411yescharacteristic
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684C2×C411yescharacteristic
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61—21no—
54C2×C22⋊C4×C61C2×C411yescharacteristic
55#95×C61—21no—
56#95×C61C2×C411yescharacteristic
57C24×C61C22⋊C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—42no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C2×C22⋊D61—22no—
65C2×C22⋊D61—22no—
66C2×C22⋊D61—41no—
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—42no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—42no—
78C2×Dic122—41no—
79C2×Dic122—42no—
80C22⋊#6923—24no—
81C22⋊#6923—42no—
82C2×#92420—42no—
83C2×#92420—82no—
84C4×#6923—82no—
85C4⋊C4×C61—41no—
86C22⋊C4×C61—41no—
87C22⋊C4×C61—42no—
88C22⋊Dic61—42no—
89#11758198—41no—
90#11758198—42no—
91#11758198—42no—
92#11758196—41no—
93#11758196—42no—
94C4×Dic61—21no—
95C4×Dic61—21no—
96C4×Dic61—42no—
97C4×Dic61—41no—
98C4×Dic61—41no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61#10412yes—
104C23×C61C2×C22⋊C411yescharacteristic
105C22×D61—21no—
106C22×D61—21no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61#13512yes—
110C2×#6923—44no—
111C2×#6923—82no—
112C2×#6923—82no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C22⋊D61—44no—
124C2×Dic61—22no—
125C2×Dic61—22no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—42no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—81no—
138C2×Dic61#9911yescharacteristic
139C2×Dic61#9911yescharacteristic
140C4×D61—42no—
141C4×D61—81no—
142Dic122—42no—
143Dic122—42no—
144#92420—82no—
145#11728—611nomaximal, Sylow
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—22no—
149C22×C61—42no—
150C22×C61—41no—
151C22×C61—41no—
152C22×C61—81no—
153C22×C61C2×#9911yescharacteristic
154C22×C61#41611yescharacteristic
155C22×C61C2×#10411yescharacteristic
156C2×D61—22no—
157C2×D61—41no—
158C2×D61—41no—
159C2×D61—42no—
160C2×D61—81no—
161#6923—84no—
162C4×C61—41no—
163C4×C61—41no—
164C4×C61—81no—
165C4×C61—81no—
166Dic61—21no—
167Dic61—42no—
168Dic61—41no—
169Dic61—41no—
170Dic61—81no—
171Dic61#35412yes—
172#2398—611no—
173C4×#99—611no—
174#1459—611no—
175#1351—612no—
176#1344—612no—
177C2×C61—21no—
178C2×C61—21no—
179C2×C61—42no—
180C2×C61#150111yescharacteristic
181C2×C61#212011yescharacteristic
182C2×C61C2×#35411yescharacteristic
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#692311yescharacteristic, complemented
191#394—1221no—
192#393—611no—
193#393—611no—
194C4×C22⋊C4—1221no—
195C4×C22⋊C4—1221no—
196C4×C22⋊C4—1221no—
197#376—611no—
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—1221no—
213C2×C4⋊C4C2×#692311yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—611no—
216C2×C22⋊C4—1222no—
217C2×C22⋊C4—1221no—
218C2×C22⋊C4—1221no—
219C2×C22⋊C4—1221no—
220C2×C22⋊C4C2×#692311yescharacteristic
221C2×C42—611no—
222C2×C42—1221no—
223C2×C42—1221no—
224C2×C42—2441no—
225#99—1224no—
226C22⋊C8—1222no—
227C22⋊C8—2441no—
228#95—21no—
229#95—611no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234#95—2441no—
235#95C2×#692311yescharacteristic
236C24—611no—
237C24C22⋊#692311yescharacteristic
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—611no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4—1221no—
252C22×C4—1221no—
253C22×C4—1221no—
254C22×C4—2442no—
255C22×C4—2442no—
256C22×C4—2442no—
257C22×C4C22⋊#692311yescharacteristic
258C22×C4C22×#692311yescharacteristic
259C2×C8—4882no—
260C2×C8—2442no—
261C4⋊C4—41no—
262C4⋊C4—2441no—
263C4⋊C4—2442no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—1224no—
267C22⋊C4—2441no—
268C22⋊C4—2442no—
269C22⋊C4—2442no—
270C22⋊C4—2442no—
271C22⋊C4—2442no—
272C42—1221no—
273C42—1221no—
274C42—4882no—
275C42—2441no—
276C42—2441no—
277C42—2442no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—612no—
283C23—1221no—
284C23—1221no—
285C23—2441no—
286C23—2441no—
287C23#1451984112yes—
288C23C2×C22⋊#692311yescharacteristic
289Q8—2442no—
290Q8—2442no—
291D4—2444no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—611no—
303C2×C4—611no—
304C2×C4—1222no—
305C2×C4—1222no—
306C2×C4—1221no—
307C2×C4—1221no—
308C2×C4—1221no—
309C2×C4—1221no—
310C2×C4—4881no—
311C2×C4—4881no—
312C2×C4—4882no—
313C2×C4—4882no—
314C2×C4—2442no—
315C2×C4—2442no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2444no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4—2441no—
323C8—4882no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22—1222no—
332C22—4881no—
333C22—2442no—
334C22—2441no—
335C22—2441no—
336C22D61⋊#9911yescharacteristic
337C22#1684075211yescharacteristic
338C22C2×#1451984111yescharacteristic
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—612no—
344C4—1221no—
345C4—4881no—
346C4—4884no—
347C4—2442no—
348C4—2441no—
349C4—2441no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—2442no—
354C2#2146119511yescharacteristic
355C2#2146207311yescharacteristic
356C2D61⋊#35411yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#397C4
socleC22×C61C2×#99
prime radicalC22×C22⋊D61D4
cosocleC2×#11758198C23
hypercenter#397#6923
hypocenterC61#11728
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #397

Complements

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

NH ≅ G/Ntype
3. #21460813C2semidirect
4. #21460799C2semidirect
12. C2×#14519742C4semidirect
18. #16839674C4semidirect
21. #16839607C4semidirect
22. C61×#397C4semidirect
190. #397#6923semidirect
201. C61#11728semidirect

Maximal subgroups

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

subgroupindex
#214615662
#214610572
#214608132
#214607992
#214491042
#1172861