Finite Group Databasebeta

← #30605429

Subgroups — #30605429

Order 15616.

418 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C2×#16840657C211yesmaximal, characteristic
2#21461696C211yesmaximal, characteristic
3#21460808C212yesmaximal, complemented
4#21460807C212yesmaximal, complemented
5#467×D61C211yesmaximal, characteristic
6C22×#11758187—21no—
7C2×#14519846—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—
25C2×#11758187—22no—
26C2×#11758187—42no—
27#14519846—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—41no—
105C22×D61—41no—
106C22×D61#10211yescharacteristic
107C22×D61SD3212yes—
108C22×D61Q3212yes—
109C22×D61C2×D811yescharacteristic
110C2×#6923—82no—
111C2×#6923—82no—
112C2×#6923—82no—
113Q8×C61—41no—
114Q8×C61—41no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61#10211yescharacteristic
120C2×C4×C61C2×C22⋊C411yescharacteristic
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61#10211yescharacteristic
125C2×Dic61C2×SD1611yescharacteristic
126C4×D61—21no—
127C4×D61—21no—
128C4×D61—21no—
129C4×D61—21no—
130C4×D61—41no—
131C4×D61—41no—
132C4×D61—82no—
133C4×D61—82no—
134Dic122—41no—
135Dic122—41no—
136Dic122—82no—
137C8×C61—21no—
138C8×C61—21no—
139#92418—21no—
140#92418#9912yes—
141#11887—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61#35812yes—
145C22×C61C2×#10211yescharacteristic
146C2×D61—21no—
147C2×D61—21no—
148C2×D61—21no—
149C2×D61—21no—
150C2×D61—42no—
151C2×D61—42no—
152C2×D61#35812yes—
153C2×D61C2×SD3211yescharacteristic
154C2×D61C2×Q3211yescharacteristic
155#6923—162no—
156C4×C61—21no—
157C4×C61—81no—
158C4×C61—81no—
159C4×C61C2×#9911yescharacteristic
160C4×C61#42111yescharacteristic
161Dic61—21no—
162Dic61—21no—
163Dic61—81no—
164Dic61—81no—
165C2×#467—611no—
166C2×C22⋊C16—611no—
167#1808—611no—
168#1348—612no—
169#1348—612no—
170C2×C61—22no—
171C2×C61#151911yescharacteristic
172C2×C61C2×#35811yescharacteristic
173C2×C61#214511yescharacteristic
174D61—21no—
175D61—22no—
176D61—21no—
177C22×Q16—611no—
178C23×C8—611no—
179C2×C22⋊Q8—1221no—
180C22×C16—1221no—
181#467—611no—
182#467—611no—
183#467—611no—
184#467—1221no—
185#467—1221no—
186#467#692311yescharacteristic, complemented
187C2×#107—611no—
188C2×#107—1221no—
189C2×#103—1221no—
190#380—1221no—
191C22⋊C16—612no—
192C22⋊C16—1221no—
193#340—612no—
194C61#1188711yescharacteristic, complemented, Sylow
195C22×Q8—1221no—
196C23×C4—611no—
197C2×Q16—611no—
198C2×Q16—1221no—
199C2×Q16—2442no—
200C2×Q16C2×#692311yescharacteristic
201C22×C8—611no—
202C22×C8—611no—
203C22×C8—611no—
204C22×C8—611no—
205C22×C8—611no—
206C22×C8—2442no—
207C22×C8C2×#692311yescharacteristic
208C22⋊Q8—21no—
209C22⋊Q8—1221no—
210C22⋊Q8—1221no—
211C22⋊Q8—1221no—
212C22⋊Q8—2441no—
213C22⋊Q8—2441no—
214C2×C4⋊C4—1221no—
215C2×C4⋊C4—1222no—
216C2×C4⋊C4—2441no—
217C2×C22⋊C4—2441no—
218C2×C22⋊C4—2441no—
219C2×C16—1222no—
220C2×C16—2442no—
221#107—611no—
222#107—1221no—
223#107—2442no—
224#107C2×#692311yescharacteristic
225#103—21no—
226#103—1221no—
227#103—1221no—
228#103—1221no—
229#95—2441no—
230C24—611no—
231C2×Q8—21no—
232C2×Q8—1221no—
233C2×Q8—2441no—
234C2×Q8—2442no—
235C2×Q8—2442no—
236C22×C4—611no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—2441no—
242C22×C4—2441no—
243C22×C4—2441no—
244C22×C4—2441no—
245C22×C4—2442no—
246C22×C4C22⋊#692311yescharacteristic
247Q16—41no—
248Q16—2441no—
249Q16—2442no—
250C2×C8—41no—
251C2×C8—611no—
252C2×C8—611no—
253C2×C8—1222no—
254C2×C8—1222no—
255C2×C8—1222no—
256C2×C8—2441no—
257C2×C8—2441no—
258C2×C8C22⋊#692311yescharacteristic
259C2×C8C22×#692311yescharacteristic
260C4⋊C4—21no—
261C4⋊C4—41no—
262C4⋊C4—1221no—
263C4⋊C4—2441no—
264C4⋊C4—2441no—
265C4⋊C4—2441no—
266C4⋊C4—2442no—
267C4⋊C4—2442no—
268C4⋊C4—2441no—
269C22⋊C4—41no—
270C22⋊C4—4882no—
271C22⋊C4—2441no—
272C22⋊C4—2441no—
273C22⋊C4—2441no—
274C16—2442no—
275C23—611no—
276C23—611no—
277C23—612no—
278C23—612no—
279C23—2441no—
280C23—2441no—
281C23#1451983911yescharacteristic
282Q8—41no—
283Q8—41no—
284Q8—4882no—
285Q8—2441no—
286Q8—2441no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—611no—
292C2×C4—611no—
293C2×C4—1221no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—4882no—
298C2×C4—4882no—
299C2×C4—4882no—
300C2×C4—4882no—
301C2×C4—4882no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4#1451983911yescharacteristic
308C2×C4C2×C22⋊#692311yescharacteristic
309C8—21no—
310C8—21no—
311C8—612no—
312C8—1221no—
313C22—21no—
314C22—21no—
315C22—611no—
316C22—612no—
317C22—611no—
318C22—1221no—
319C22—1221no—
320C22—1221no—
321C22—1221no—
322C22—2442no—
323C22—2442no—
324C22#1684067512yes—
325C22C2×#1451983911yescharacteristic
326C4—21no—
327C4—81no—
328C4—81no—
329C4—1221no—
330C4—1221no—
331C4—9762no—
332C4—4881no—
333C4—4881no—
334C4D61⋊#9911yescharacteristic
335C4#1684075911yescharacteristic
336C2—22no—
337C2—1221no—
338C2—1222no—
339C2—1221no—
340C2#2146125411yescharacteristic
341C2C2×#1684067511yescharacteristic
342C2#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#11887
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
186. #467#6923semidirect
194. C61#11887semidirect

Maximal subgroups

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

subgroupindex
C2×#168406572
#214616962
#214608082
#214608072
#467×D612
#1188761