Finite Group Databasebeta

← #30604784

Subgroups — #30604784

Order 15616.

456 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462047C211yesmaximal, characteristic
2#21461505C211yesmaximal, characteristic
3#21460808C212yesmaximal, complemented
4#21460807C212yesmaximal, complemented
5#467×D61C211yesmaximal, characteristic
6C2×#14519856—21no—
7C2×#14519852—21no—
8#16840657C2212yes—
9C22⋊#11758187—21no—
10#16840636—21no—
11#16840636C2212yes—
12C2×Q16×D61C411yescharacteristic, complemented
13C22×C8×D61C2211yescharacteristic
14C22⋊Q8×D61—21no—
15#16840004C2211yescharacteristic
16#16840003C411yescharacteristic, complemented
17#16840002—21no—
18#16840001—21no—
19#16840000C2211yescharacteristic
20#107×D61C411yescharacteristic, complemented
21#103×D61—21no—
22C61×#467C411yescharacteristic, complemented
23#14519856—24no—
24#14519856—42no—
25C2×C4⋊#6923—22no—
26C2×C22⋊#6923—41no—
27C2×C4×#6923—41no—
28#14519852—42no—
29C2×#11758187—22no—
30C2×Q8×D61—21no—
31C22×C4×D61D411yescharacteristic
32C2×Dic244C2×C411yescharacteristic
33C2×#11758202—21no—
34Q16×D61—42no—
35C22×#92418D411yescharacteristic
36C2×C8×D61—42no—
37C2×C8×D61C2×C411yescharacteristic
38C2×C8×D61D411yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61C2311yescharacteristic
41#14519757—21no—
42#14519756—21no—
43#14519755—41no—
44#14519754—41no—
45#14519753—21no—
46C4⋊C4×D61—21no—
47C4⋊C4×D61—41no—
48C22⋊C4×D61—41no—
49#14519711C2×C411yescharacteristic
50#14519710—21no—
51#14519702—21no—
52#14519701—21no—
53#14519700—21no—
54C2×Q16×C61C2×C411yescharacteristic
55C22×C8×C61C2×C411yescharacteristic
56C22⋊Q8×C61—21no—
57#107×C61C2×C411yescharacteristic
58#103×C61—21no—
59C23×D61D811yescharacteristic
60C22×#6923—42no—
61C2×Q8×C61—21no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×Dic61SD1611yescharacteristic
64Q8×D61—42no—
65Q8×D61—42no—
66C2×C4×D61—22no—
67C2×C4×D61—22no—
68C2×C4×D61—41no—
69C2×C4×D61—41no—
70C2×C4×D61C22⋊C411yescharacteristic
71C2×C4×D61D811yescharacteristic
72C2×C4×D61SD1611yescharacteristic
73C2×C4×D61C2×D411yescharacteristic
74C2×Dic122—21no—
75C2×Dic122—41no—
76C22⋊#6923—44no—
77C4⋊#6923—24no—
78C4⋊#6923—42no—
79C4×#6923—44no—
80Q16×C61—41no—
81C2×C8×C61—41no—
82C2×C8×C61C22⋊C411yescharacteristic
83C2×C8×C61C22×C411yescharacteristic
84C4⋊C4×C61—21no—
85C4⋊C4×C61—41no—
86C22⋊C4×C61—41no—
87C22⋊Dic61—41no—
88#11758202—42no—
89#11758198—41no—
90#11758198—41no—
91C4⋊Dic61—21no—
92C4⋊Dic61—41no—
93#11758196—41no—
94#11758196—41no—
95#11758196—41no—
96C2×#92418—41no—
97C2×#92418C22⋊C411yescharacteristic
98C2×#92418C2×D411yescharacteristic
99Dic244—41no—
100C8×D61—22no—
101C8×D61—22no—
102C8×D61—41no—
103C8×D61—41no—
104#11758187—42no—
105C23×C61#10211yescharacteristic
106C22×D61—22no—
107C22×D61—22no—
108C22×D61—41no—
109C22×D61#10211yescharacteristic
110C22×D61C2×D811yescharacteristic
111C2×#6923—44no—
112C2×#6923—82no—
113C2×#6923—82no—
114Q8×C61—41no—
115Q8×C61—41no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61#10211yescharacteristic
121C2×C4×C61C2×C22⋊C411yescharacteristic
122C2×Dic61—22no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61#10211yescharacteristic
126C2×Dic61C2×SD1611yescharacteristic
127C4×D61—22no—
128C4×D61—21no—
129C4×D61—21no—
130C4×D61—22no—
131C4×D61—82no—
132C4×D61—82no—
133C4×D61#9912yes—
134C4×D61#10412yes—
135Dic122—41no—
136Dic122—41no—
137Dic122—82no—
138C8×C61—21no—
139C8×C61—21no—
140#92418—21no—
141#92418—21no—
142#11706—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—21no—
146C22×C61C2×#10211yescharacteristic
147C2×D61—21no—
148C2×D61—21no—
149C2×D61—21no—
150C2×D61—21no—
151C2×D61—44no—
152C2×D61#32712yes—
153C2×D61#51012yes—
154#6923—84no—
155C4×C61—21no—
156C4×C61—81no—
157C4×C61—81no—
158C4×C61C2×#9911yescharacteristic
159C4×C61C2×#10411yescharacteristic
160Dic61—21no—
161Dic61—81no—
162Dic61—81no—
163Dic61#32712yes—
164C2×#467—611no—
165#2112—611no—
166#1735—611no—
167#1348—612no—
168#1348—612no—
169C2×C61—41no—
170C2×C61C2×#32711yescharacteristic
171C2×C61#214111yescharacteristic
172C2×C61#214511yescharacteristic
173D61—22no—
174D61—22no—
175C22×Q16—611no—
176C23×C8—611no—
177C2×C22⋊Q8—1221no—
178C2×#117—1221no—
179C2×M32—1221no—
180#467—611no—
181#467—611no—
182#467—611no—
183#467—1221no—
184#467—1221no—
185#467#692311yescharacteristic, complemented
186C2×#107—611no—
187C2×#103—1221no—
188C22⋊C16—612no—
189C22⋊C16—1221no—
190#340—612no—
191#340—1221no—
192C61#1170611yescharacteristic, complemented, Sylow
193C22×Q8—1221no—
194C23×C4—611no—
195C2×Q16—611no—
196C2×Q16—1221no—
197C2×Q16—2442no—
198C2×Q16C2×#692311yescharacteristic
199C22×C8—611no—
200C22×C8—611no—
201C22×C8—611no—
202C22×C8—611no—
203C22×C8—611no—
204C22×C8—2442no—
205C22×C8C2×#692311yescharacteristic
206C22⋊Q8—21no—
207C22⋊Q8—1221no—
208C22⋊Q8—1221no—
209C22⋊Q8—1221no—
210C22⋊Q8—2441no—
211C22⋊Q8—2441no—
212#117—1224no—
213#117—2442no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4—1222no—
216C2×C4⋊C4—2441no—
217C2×C22⋊C4—2441no—
218C2×C22⋊C4—2441no—
219C2×C42—2441no—
220M32—2442no—
221C2×C16—1222no—
222#107—611no—
223#107—1221no—
224#107C2×#692311yescharacteristic
225#103—21no—
226#103—1221no—
227#103—1221no—
228#103—1221no—
229C24—611no—
230C2×Q8—21no—
231C2×Q8—1221no—
232C2×Q8—2441no—
233C2×Q8—2442no—
234C2×Q8—2442no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—1222no—
241C22×C4—1222no—
242C22×C4—2441no—
243C22×C4—2441no—
244C22×C4—2442no—
245C22×C4C22⋊#692311yescharacteristic
246Q16—41no—
247Q16—2441no—
248Q16—2442no—
249C2×C8—41no—
250C2×C8—611no—
251C2×C8—611no—
252C2×C8—1222no—
253C2×C8—1222no—
254C2×C8—2441no—
255C2×C8—2441no—
256C2×C8—2441no—
257C2×C8C22⋊#692311yescharacteristic
258C2×C8C22×#692311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—1224no—
262C4⋊C4—1221no—
263C4⋊C4—2441no—
264C4⋊C4—2441no—
265C4⋊C4—2441no—
266C4⋊C4—2442no—
267C4⋊C4—2441no—
268C22⋊C4—41no—
269C22⋊C4—2441no—
270C22⋊C4—2441no—
271C22⋊C4—2444no—
272C22⋊C4—2441no—
273C42—2444no—
274C16—2442no—
275C23—611no—
276C23—611no—
277C23—1222no—
278C23—1222no—
279C23—2441no—
280C23#1451983911yescharacteristic
281Q8—41no—
282Q8—41no—
283Q8—4882no—
284Q8—2441no—
285Q8—2441no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—611no—
291C2×C4—612no—
292C2×C4—612no—
293C2×C4—611no—
294C2×C4—1222no—
295C2×C4—1222no—
296C2×C4—1222no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—4882no—
300C2×C4—4882no—
301C2×C4—4882no—
302C2×C4—4882no—
303C2×C4—2444no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4#1451983911yescharacteristic
307C2×C4C2×C22⋊#692311yescharacteristic
308C8—21no—
309C8—21no—
310C8—1221no—
311C8—1221no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—612no—
316C22—612no—
317C22—1221no—
318C22—1221no—
319C22—1221no—
320C22—1221no—
321C22—2444no—
322C22C2×#1451983911yescharacteristic
323C4—21no—
324C4—81no—
325C4—81no—
326C4—612no—
327C4—1221no—
328C4—4884no—
329C4—4881no—
330C4—4881no—
331C4D61⋊#9911yescharacteristic
332C4D61⋊#10411yescharacteristic
333C2—41no—
334C2—1222no—
335C2—1222no—
336C2D61⋊#32711yescharacteristic
337C2#2146212911yescharacteristic
338C2#2146213911yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#467C4
socleC22×C61C2×#102
prime radicalC23×D61D8
cosocleC2×C8×D61C23
hypercenter#467#6923
hypocenterC61#11706
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #467

Complements

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

NH ≅ G/Ntype
3. #21460808C2semidirect
4. #21460807C2semidirect
12. C2×Q16×D61C4semidirect
16. #16840003C4semidirect
20. #107×D61C4semidirect
22. C61×#467C4semidirect
185. #467#6923semidirect
192. C61#11706semidirect

Maximal subgroups

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

subgroupindex
#214620472
#214615052
#214608082
#214608072
#467×D612
#1170661