Finite Group Databasebeta

← #30599167

Subgroups — #30599167

Order 15616.

414 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 92 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#21461744C211yesmaximal, characteristic, complemented
2D61⋊#334C211yesmaximal, characteristic, complemented
3#21461330C212yesmaximal
4#21460688C212yesmaximal, complemented
5#21454526C211yesmaximal, characteristic
6#16840767C2212yes—
7D61⋊(C4⋊C8)C2212yes—
8#16840638—21no—
9#16840638—21no—
10#16840624—21no—
11#16840623—21no—
12#16840621—22no—
13#16840620—22no—
14C2×#14519783—21no—
15#16840309C2211yescharacteristic
16#16839840C411yescharacteristic
17#16839839C411yescharacteristic
18#16839837—21no—
19C8×D244C411yescharacteristic
20C8×C22⋊D61—21no—
21#16839784C2211yescharacteristic
22#16839727—21no—
23C4×C8×D61C2211yescharacteristic
24C8×D4×C61C411yescharacteristic
25C2×#11758213—22no—
26C2×D61⋊C8—22no—
27#14519844—22no—
28#14519844—22no—
29#14519844—22no—
30C4⋊#92420—22no—
31C2×#11758222—21no—
32#14519783—44no—
33C2×#11758194—21no—
34C2×#11758189—21no—
35C2×#11758189—21no—
36C2×C8×D61—21no—
37C2×C8×D61D411yescharacteristic
38C2×C8×D61D411yescharacteristic
39#14519776D411yescharacteristic
40#14519773D411yescharacteristic
41#14519763—21no—
42#14519752Q811yescharacteristic
43#14519750—21no—
44#14519748Q811yescharacteristic
45C4×D244C2×C411yescharacteristic
46C4×C22⋊D61—21no—
47C42×D61C2311yescharacteristic
48C4⋊#92418C2×C411yescharacteristic
49#14519706—21no—
50C22⋊#92418—21no—
51#14519689—21no—
52#14519689—21no—
53C4×#92418C2×C411yescharacteristic
54C8×Dic61D411yescharacteristic
55C8×Dic61D411yescharacteristic
56C22×C8×C61—21no—
57C4×D4×C61C2×C411yescharacteristic
58C4⋊C8×C61C2×C411yescharacteristic
59C22⋊C8×C61—21no—
60C4×C8×C61C2×C411yescharacteristic
61C2×D4×C61C4⋊C411yescharacteristic
62C22×C4×C61—21no—
63C2×C22⋊D61—21no—
64C2×C22⋊D61—21no—
65C2×C22⋊D61—21no—
66#11758222—42no—
67#11758222—42no—
68C2×D244—21no—
69C2×D244C4⋊C411yescharacteristic
70C2×C4×D61—21no—
71C2×C4×D61C2×D411yescharacteristic
72C2×C4×D61C2×Q811yescharacteristic
73C2×C4×D61#4111yescharacteristic
74C2×Dic122—21no—
75C2×#92420—42no—
76C2×#92420—42no—
77#11758213—42no—
78#11758213—42no—
79D61⋊C8—82no—
80C2×C8×C61—21no—
81C2×C8×C61—42no—
82C2×C8×C61C22⋊C411yescharacteristic
83C2×C8×C61C22⋊C411yescharacteristic
84C4⋊C4×C61C4⋊C411yescharacteristic
85C22⋊C4×C61—21no—
86C42×C61C22×C411yescharacteristic
87C22⋊Dic61—21no—
88#11758198—21no—
89#11758198—21no—
90C4⋊Dic61C4⋊C411yescharacteristic
91#11758196—21no—
92C4×Dic61C2×D411yescharacteristic
93C4×Dic61C2×D411yescharacteristic
94C4×Dic61#4111yescharacteristic
95#11758194—42no—
96C2×#92418—21no—
97C2×#92418C22⋊C411yescharacteristic
98C2×#92418C22⋊C411yescharacteristic
99#11758189—42no—
100#11758189—42no—
101C8×D61—22no—
102C8×D61—22no—
103C8×D61—42no—
104C23×C61—21no—
105C22×D61—21no—
106C22×D61C22⋊Q811yescharacteristic
107D4×C61—41no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61C2×C22⋊C411yescharacteristic
112C2×C4×C61C2×C4⋊C411yescharacteristic
113C2×C4×C61#11711yescharacteristic
114C22⋊D61—42no—
115C22⋊D61—42no—
116C22⋊D61—42no—
117C2×Dic61—21no—
118C2×Dic61#12111yescharacteristic
119C2×Dic61#12111yescharacteristic
120C2×Dic61C22⋊Q811yescharacteristic
121D244—41no—
122D244—42no—
123C4×D61—22no—
124C4×D61—41no—
125C4×D61—41no—
126C4×D61—41no—
127C4×D61#13512yes—
128C4×D61AGL(1,ℤ/8)12yes—
129Dic122—42no—
130#92420—82no—
131#92420—82no—
132C8×C61—21no—
133C8×C61—21no—
134C8×C61—41no—
135#92418—21no—
136#92418—21no—
137#92418—41no—
138#8686—611nomaximal, Sylow
139C22×C61—41no—
140C22×C61—41no—
141C22×C61—41no—
142C22×C61#38011yescharacteristic
143C2×D61—21no—
144C2×D61—21no—
145C2×D61—41no—
146C2×D61—41no—
147C2×D61—41no—
148C2×D61#47812yes—
149C4×C61—21no—
150C4×C61—21no—
151C4×C61—41no—
152C4×C61#41611yescharacteristic
153C4×C61#41811yescharacteristic
154Dic61—21no—
155Dic61—41no—
156Dic61#46312yes—
157Dic61#47112yes—
158#2931—611no—
159#1837—611no—
160C2×#334—611no—
161#1547—612no—
162#1292—612no—
163C2×C61—81no—
164C2×C61#180411yescharacteristic
165C2×C61#181311yescharacteristic
166C2×C61#181511yescharacteristic
167D61—22no—
168D61—81no—
169C2×#131—1221no—
170#532—611no—
171#436—611no—
172#436—611no—
173#436—1221no—
174C8×D4—611no—
175C8×D4—1221no—
176C8×D4#692311yescharacteristic
177#423—611no—
178#423—612no—
179C2×C4⋊C8—612no—
180#408—1221no—
181C2×C4×C8—611no—
182#341—1221no—
183#341—1221no—
184#334—1221no—
185#334—1221no—
186#332—1222no—
187#331—1222no—
188C61#868611yescharacteristic, complemented, Sylow
189C2×#41—1221no—
190#131—2444no—
191C2×M16—1221no—
192C2×M16—1221no—
193C2×M16—1221no—
194C2×M16—1222no—
195C22×C8—21no—
196C22×C8—611no—
197C22×C8—611no—
198C22×C8—1221no—
199C22×C8—1222no—
200C4⋊Q8—611no—
201#127—611no—
202#124—1221no—
203#121—611no—
204#121—611no—
205#121—1221no—
206C4×D4—611no—
207C4×D4—1221no—
208C4×D4C2×#692311yescharacteristic
209C2×C42—611no—
210C4⋊C8—611no—
211C4⋊C8—1221no—
212C4⋊C8—1222no—
213C4⋊C8—1222no—
214C4⋊C8—1222no—
215C4⋊C8—1222no—
216C4⋊C8C2×#692311yescharacteristic
217C22⋊C8—21no—
218C22⋊C8—1221no—
219C22⋊C8—1221no—
220C22⋊C8—1221no—
221C4×C8—611no—
222C4×C8—611no—
223C4×C8—611no—
224C4×C8C2×#692311yescharacteristic
225#41—2442no—
226#41—2442no—
227C2×Q8—1221no—
228C2×D4—611no—
229C2×D4—1221no—
230C2×D4—1221no—
231C2×D4—1221no—
232C2×D4—1221no—
233C2×D4C4⋊#692311yescharacteristic
234C22×C4—21no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—611no—
238C22×C4—1221no—
239M16—2442no—
240M16—2442no—
241M16—2442no—
242M16—2442no—
243M16—2442no—
244C2×C8—21no—
245C2×C8—42no—
246C2×C8—611no—
247C2×C8—611no—
248C2×C8—1222no—
249C2×C8—1222no—
250C2×C8—1221no—
251C2×C8—4882no—
252C2×C8—2442no—
253C2×C8—2442no—
254C2×C8—2442no—
255C2×C8C22⋊#692311yescharacteristic
256C2×C8C22⋊#692311yescharacteristic
257C4⋊C4—611no—
258C4⋊C4—1221no—
259C4⋊C4C4⋊#692311yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—1221no—
262C22⋊C4—1221no—
263C22⋊C4—1221no—
264C42—611no—
265C42—611no—
266C42—611no—
267C42C22×#692311yescharacteristic
268C23—21no—
269C23—611no—
270C23—1221no—
271Q8—2442no—
272D4—41no—
273D4—2441no—
274D4—2442no—
275D4—2442no—
276D4—2442no—
277D4—2442no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—611no—
282C2×C4—611no—
283C2×C4—612no—
284C2×C4—612no—
285C2×C4—611no—
286C2×C4—1222no—
287C2×C4—1221no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4—2441no—
291C2×C4C2×C22⋊#692311yescharacteristic
292C2×C4C2×C4⋊#692311yescharacteristic
293C2×C4#1451985611yescharacteristic
294C8—21no—
295C8—21no—
296C8—41no—
297C8—1221no—
298C8—1221no—
299C8—4882no—
300C8—4882no—
301C8—2441no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—612no—
306C22—1221no—
307C22—1221no—
308C22—2441no—
309C22—2441no—
310C22—2441no—
311C22#1684070411yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—612no—
316C4—612no—
317C4—1221no—
318C4—2441no—
319C4#1684075211yescharacteristic
320C4#1684075411yescharacteristic
321C2—81no—
322C2—1222no—
323C2—4881no—
324C2#2146168911yescharacteristic
325C2#2146170311yescharacteristic
326C2#2146170711yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC8×D4×C61C4
socleC22×C61#380
prime radical#16840309C22
cosocleC42×D61C23
hypercenterC8×D4#6923
hypocenterC61#8686
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #21461744C2semidirect
2. D61⋊#334C2semidirect
4. #21460688C2semidirect
188. C61#8686semidirect

Maximal subgroups

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

subgroupindex
#214617442
D61⋊#3342
#214613302
#214606882
#214545262
#868661