Finite Group Databasebeta

← #30605435

Subgroups — #30605435

Order 15616.

412 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 78 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#21462048C211yesmaximal, characteristic
2#21461695C211yesmaximal, characteristic
3#21460808C212yesmaximal, complemented
4#21460807C212yesmaximal, complemented
5#467×D61C211yesmaximal, characteristic
6C2×#14519851—21no—
7C2×#14519845—21no—
8#16840704—21no—
9#16840657—21no—
10#16840657C2212yes—
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
23C2×C4⋊#6923—22no—
24C2×C22⋊#6923—41no—
25#14519851—42no—
26C2×#11758187—22no—
27#14519845—42no—
28(C2×#6923)⋊C4—41no—
29C2×Q8×D61—21no—
30C22×C4×D61D411yescharacteristic
31C2×Dic244C2×C411yescharacteristic
32C2×#11758202—21no—
33Q16×D61—42no—
34C22×#92418D411yescharacteristic
35C2×C8×D61—42no—
36C2×C8×D61C2×C411yescharacteristic
37C2×C8×D61D411yescharacteristic
38C2×C8×D61D411yescharacteristic
39C2×C8×D61C2311yescharacteristic
40#14519757—21no—
41#14519756—21no—
42#14519755—41no—
43#14519754—41no—
44#14519753—21no—
45C4⋊C4×D61—21no—
46C4⋊C4×D61—41no—
47C22⋊C4×D61—41no—
48#14519711C2×C411yescharacteristic
49#14519710—21no—
50#14519702—21no—
51#14519701—21no—
52#14519700—21no—
53C2×Q16×C61C2×C411yescharacteristic
54C22×C8×C61C2×C411yescharacteristic
55C22⋊Q8×C61—21no—
56#107×C61C2×C411yescharacteristic
57#103×C61—21no—
58C23×D61D811yescharacteristic
59C22×#6923—42no—
60C2×Q8×C61—21no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×Dic61SD1611yescharacteristic
63Q8×D61—42no—
64Q8×D61—42no—
65C2×C4×D61—41no—
66C2×C4×D61—41no—
67C2×C4×D61—41no—
68C2×C4×D61—41no—
69C2×C4×D61C22⋊C411yescharacteristic
70C2×C4×D61D811yescharacteristic
71C2×C4×D61SD1611yescharacteristic
72C2×C4×D61C2×D411yescharacteristic
73C2×Dic122—21no—
74C2×Dic122—41no—
75C22⋊#6923—82no—
76C4⋊#6923—42no—
77C4⋊#6923—42no—
78Q16×C61—41no—
79C2×C8×C61—41no—
80C2×C8×C61C22⋊C411yescharacteristic
81C2×C8×C61C22×C411yescharacteristic
82C4⋊C4×C61—21no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—41no—
85C22⋊Dic61—41no—
86#11758202—42no—
87#11758198—41no—
88#11758198—41no—
89C4⋊Dic61—21no—
90C4⋊Dic61—41no—
91#11758196—41no—
92#11758196—41no—
93#11758196—41no—
94C2×#92418—22no—
95C2×#92418C22⋊C411yescharacteristic
96C2×#92418C2×D411yescharacteristic
97Dic244—41no—
98C8×D61—22no—
99C8×D61—22no—
100C8×D61—41no—
101C8×D61—41no—
102#11758187—42no—
103C23×C61#10211yescharacteristic
104C22×D61—22no—
105C22×D61—22no—
106C22×D61—41no—
107C22×D61#10211yescharacteristic
108C22×D61C2×D811yescharacteristic
109C2×#6923—82no—
110C2×#6923—82no—
111C2×#6923—82no—
112Q8×C61—41no—
113Q8×C61—41no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61#10211yescharacteristic
119C2×C4×C61C2×C22⋊C411yescharacteristic
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61#10211yescharacteristic
124C2×Dic61C2×SD1611yescharacteristic
125C4×D61—21no—
126C4×D61—21no—
127C4×D61—21no—
128C4×D61—21no—
129C4×D61—41no—
130C4×D61—41no—
131C4×D61—82no—
132C4×D61—82no—
133Dic122—41no—
134Dic122—41no—
135Dic122—82no—
136C8×C61—21no—
137C8×C61—21no—
138#92418—21no—
139#92418#9912yes—
140#11889—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61C2×#10211yescharacteristic
145C2×D61—21no—
146C2×D61—21no—
147C2×D61—21no—
148C2×D61—21no—
149C2×D61—44no—
150C2×D61#36012yes—
151C2×D61#51012yes—
152#6923—162no—
153C4×C61—21no—
154C4×C61—81no—
155C4×C61—81no—
156C4×C61C2×#9911yescharacteristic
157C4×C61#42111yescharacteristic
158Dic61—21no—
159Dic61—21no—
160Dic61—81no—
161Dic61—81no—
162C2×#467—611no—
163#2112—611no—
164#1807—611no—
165#1348—612no—
166#1348—612no—
167C2×C61—41no—
168C2×C61#151911yescharacteristic
169C2×C61#214111yescharacteristic
170C2×C61C2×#36011yescharacteristic
171D61—22no—
172D61—22no—
173C22×Q16—611no—
174C23×C8—611no—
175C2×C22⋊Q8—1221no—
176C2×M32—1221no—
177#467—611no—
178#467—611no—
179#467—611no—
180#467—1221no—
181#467—1221no—
182#467#692311yescharacteristic, complemented
183C2×#107—611no—
184C2×#106—1221no—
185C2×#103—1221no—
186#380—1221no—
187C22⋊C16—612no—
188C22⋊C16—1221no—
189#340—612no—
190C61#1188911yescharacteristic, complemented, Sylow
191C22×Q8—1221no—
192C23×C4—611no—
193C2×Q16—611no—
194C2×Q16—1221no—
195C2×Q16—2442no—
196C2×Q16C2×#692311yescharacteristic
197C22×C8—611no—
198C22×C8—611no—
199C22×C8—611no—
200C22×C8—611no—
201C22×C8—611no—
202C22×C8—2442no—
203C22×C8C2×#692311yescharacteristic
204C22⋊Q8—21no—
205C22⋊Q8—1221no—
206C22⋊Q8—1221no—
207C22⋊Q8—1221no—
208C22⋊Q8—2441no—
209C22⋊Q8—2441no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1222no—
212C2×C4⋊C4—2441no—
213C2×C22⋊C4—2441no—
214C2×C22⋊C4—2441no—
215M32—2442no—
216C2×C16—1222no—
217#107—611no—
218#107—1221no—
219#107C2×#692311yescharacteristic
220#106—2442no—
221#103—21no—
222#103—1221no—
223#103—1221no—
224#103—1221no—
225#95—2441no—
226C24—611no—
227C2×Q8—21no—
228C2×Q8—1221no—
229C2×Q8—2441no—
230C2×Q8—2442no—
231C2×Q8—2442no—
232C22×C4—611no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—2441no—
238C22×C4—2441no—
239C22×C4—2441no—
240C22×C4—2441no—
241C22×C4—2442no—
242C22×C4C22⋊#692311yescharacteristic
243Q16—41no—
244Q16—2441no—
245Q16—2442no—
246C2×C8—41no—
247C2×C8—611no—
248C2×C8—611no—
249C2×C8—1222no—
250C2×C8—1222no—
251C2×C8—1222no—
252C2×C8—2441no—
253C2×C8—2441no—
254C2×C8C22⋊#692311yescharacteristic
255C2×C8C22×#692311yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—1221no—
259C4⋊C4—2441no—
260C4⋊C4—2441no—
261C4⋊C4—2441no—
262C4⋊C4—2442no—
263C4⋊C4—2442no—
264C4⋊C4—2441no—
265C22⋊C4—41no—
266C22⋊C4—4882no—
267C22⋊C4—2441no—
268C22⋊C4—2441no—
269C22⋊C4—2441no—
270C16—2442no—
271C23—611no—
272C23—611no—
273C23—1222no—
274C23—1222no—
275C23—2441no—
276C23#1451983911yescharacteristic
277Q8—41no—
278Q8—41no—
279Q8—4882no—
280Q8—2441no—
281Q8—2441no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—611no—
287C2×C4—611no—
288C2×C4—1221no—
289C2×C4—1221no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—4882no—
293C2×C4—4882no—
294C2×C4—4882no—
295C2×C4—4882no—
296C2×C4—4882no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4#1451983911yescharacteristic
303C2×C4C2×C22⋊#692311yescharacteristic
304C8—21no—
305C8—21no—
306C8—612no—
307C8—1221no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—612no—
312C22—612no—
313C22—1221no—
314C22—1221no—
315C22—1221no—
316C22—1221no—
317C22—2444no—
318C22C2×#1451983911yescharacteristic
319C4—21no—
320C4—81no—
321C4—81no—
322C4—1221no—
323C4—1221no—
324C4—9762no—
325C4—4881no—
326C4—4881no—
327C4D61⋊#9911yescharacteristic
328C4#1684075911yescharacteristic
329C2—41no—
330C2—1222no—
331C2—1222no—
332C2#2146125411yescharacteristic
333C2#2146212911yescharacteristic
334C2D61⋊#36011yescharacteristic
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#11889
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
182. #467#6923semidirect
190. C61#11889semidirect

Maximal subgroups

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

subgroupindex
#214620482
#214616952
#214608082
#214608072
#467×D612
#1188961