Finite Group Databasebeta

← #30599019

Subgroups — #30599019

Order 15616.

486 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461683C212yesmaximal
2#21461376C212yesmaximal, complemented
3#21460689C212yesmaximal, complemented
4C2×C8×C22⋊D61C211yesmaximal, characteristic
5#16840765C2214yes—
6C2×#14519837—22no—
7#16840642—22no—
8#16840626—22no—
9#16840625—22no—
10C22×C8×D61C411yescharacteristic
11C2×C4×C22⋊D61C2211yescharacteristic
12C8×C22⋊D61—22no—
13C8×C22⋊D61—22no—
14C2×#14519706C2211yescharacteristic
15C2×C22⋊#92418C411yescharacteristic
16C2×#14519689C411yescharacteristic
17C2×C8×Dic61C2211yescharacteristic
18C23×C8×C61C411yescharacteristic
19C2×#11758217—24no—
20#14519844—24no—
21C4⋊#92420—24no—
22#14519837—44no—
23C22×C22⋊D61D411yescharacteristic
24C22×C4×D61C2×C411yescharacteristic
25C22×#92418C2×C411yescharacteristic
26C22×#92418C2×C411yescharacteristic
27C2×C8×D61—21no—
28C2×C8×D61—21no—
29C2×C8×D61—42no—
30C4×C22⋊D61—22no—
31C4×C22⋊D61—22no—
32C2×#11758196D411yescharacteristic
33C2×C22⋊Dic61Q811yescharacteristic
34C2×#11758198Q811yescharacteristic
35C2×C4×Dic61C2311yescharacteristic
36#14519706—22no—
37C22⋊#92418—22no—
38#14519689—22no—
39C8×Dic61D412yes—
40C8×Dic61D412yes—
41C23×C4×C61C2×C411yescharacteristic
42C22×C8×C61—21no—
43C22×C8×C61—21no—
44C22×C8×C61—42no—
45C22×C8×C61C2×C411yescharacteristic
46C22×C8×C61C2×C411yescharacteristic
47C24×C61C4⋊C411yescharacteristic
48C23×D61C4⋊C411yescharacteristic
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61—41no—
52C22×C4×C61—41no—
53C22×C4×C61C4⋊C411yescharacteristic
54C22×C4×C61C22×C411yescharacteristic
55C2×C22⋊D61—22no—
56C2×C22⋊D61—22no—
57C2×C22⋊D61D812yes—
58C2×C22⋊D61Q1612yes—
59C22×Dic61C4⋊C411yescharacteristic
60C22×Dic61C2×D411yescharacteristic
61C22×Dic61C2×Q811yescharacteristic
62C2×C4×D61—21no—
63C2×C4×D61—21no—
64C2×C4×D61—41no—
65C2×C4×D61—41no—
66#11758217—44no—
67#11758217—44no—
68C2×#92420—44no—
69C2×C8×C61—22no—
70C2×C8×C61—21no—
71C2×C8×C61—42no—
72C2×C8×C61—42no—
73C2×C8×C61—41no—
74C2×C8×C61C22⋊C411yescharacteristic
75C2×C8×C61C22⋊C411yescharacteristic
76C22⋊Dic61—22no—
77#11758198—22no—
78#11758196—22no—
79C4×Dic61C2×D412yes—
80C4×Dic61#4112yes—
81C2×#92418—21no—
82C2×#92418—41no—
83C2×#92418—41no—
84C2×#92418C22⋊C412yes—
85C8×D61—42no—
86C8×D61—42no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61—41no—
90C23×C61—41no—
91C23×C61#10712yes—
92C23×C61C2×C4⋊C411yescharacteristic
93C22×D61—21no—
94C22×D61—21no—
95C22×D61—41no—
96C22×D61—41no—
97C22×D61#10712yes—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—42no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61C2×C22⋊C411yescharacteristic
106C2×C4×C61#11711yescharacteristic
107C22⋊D61—24no—
108C22⋊D61—24no—
109C2×Dic61—21no—
110C2×Dic61—41no—
111C2×Dic61—41no—
112C2×Dic61#12112yes—
113C2×Dic61C22⋊Q812yes—
114C2×Dic61C2×D811yescharacteristic
115C2×Dic61C2×Q1611yescharacteristic
116C2×Dic61#13512yes—
117C4×D61—42no—
118C4×D61—41no—
119C4×D61—41no—
120#92420—84no—
121C8×C61—21no—
122C8×C61—41no—
123C8×C61#10012yescomplemented
124#92418—22no—
125#92418—41no—
126#8616—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—21no—
129C22×C61—22no—
130C22×C61—42no—
131C22×C61—42no—
132C22×C61—41no—
133C22×C61—41no—
134C22×C61#38011yescharacteristic
135C22×C61C2×#10711yescharacteristic
136C22×C61#42711yescharacteristic
137C2×D61—22no—
138C2×D61—42no—
139C2×D61—41no—
140C2×D61—41no—
141C2×D61—42no—
142C4×C61—21no—
143C4×C61—41no—
144C4×C61C2×#10011yescharacteristic
145C4×C61#41311yescharacteristic
146Dic61—22no—
147Dic61—41no—
148Dic61#46612yes—
149Dic61#46712yes—
150C2×C8×D4—611no—
151#1800—612no—
152#1567—612no—
153#1292—612no—
154C2×C61—21no—
155C2×C61—21no—
156C2×C61—42no—
157C2×C61#180011yescharacteristic
158C2×C61#180811yescharacteristic
159C2×C61#181311yescharacteristic
160D61—42no—
161C23×C8—611no—
162C23×C8#692311yescharacteristic
163C2×C4×D4—611no—
164C8×D4—1222no—
165C8×D4—1222no—
166#423—614no—
167C2×C4⋊C8—611no—
168C2×#100—1222no—
169C2×C22⋊C8—611no—
170C2×C22⋊C8—611no—
171C2×C4×C8—611no—
172#343—1222no—
173#335—1222no—
174#335—1222no—
175C61#861611yescharacteristic, complemented, Sylow
176C22×D4—611no—
177C23×C4—611no—
178C23×C4C2×#692311yescharacteristic
179C2×M16—1224no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—42no—
183C22×C8—611no—
184C22×C8—611no—
185C22×C8—1221no—
186C22×C8—1221no—
187C22×C8—2442no—
188C22×C8C2×#692311yescharacteristic
189C22×C8C2×#692311yescharacteristic
190C4×D4—1222no—
191C4×D4—1222no—
192C2×C4⋊C4—611no—
193C2×C22⋊C4—611no—
194C2×C22⋊C4—611no—
195C2×C42—611no—
196C4⋊C8—1222no—
197C4⋊C8—1224no—
198C4⋊C8—1224no—
199#100—2444no—
200C22⋊C8—1222no—
201C22⋊C8—1222no—
202C4×C8—612no—
203C4×C8—612no—
204C24—611no—
205C24C4⋊#692311yescharacteristic
206C2×D4—612no—
207C2×D4—612no—
208C2×D4—1222no—
209C2×D4—1222no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—611no—
215C22×C4—611no—
216C22×C4—611no—
217C22×C4—1221no—
218C22×C4—1221no—
219C22×C4—2441no—
220C22×C4—2441no—
221C22×C4C4⋊#692311yescharacteristic
222C22×C4C22×#692311yescharacteristic
223M16—2444no—
224M16—2444no—
225C2×C8—22no—
226C2×C8—21no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—41no—
230C2×C8—612no—
231C2×C8—1221no—
232C2×C8—2442no—
233C2×C8—2441no—
234C2×C8—2444no—
235C2×C8—2442no—
236C2×C8—2441no—
237C2×C8C22⋊#692311yescharacteristic
238C2×C8C22⋊#692311yescharacteristic
239C4⋊C4—1222no—
240C22⋊C4—1222no—
241C22⋊C4—1222no—
242C42—612no—
243C42—612no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23—612no—
249C23—1221no—
250C23—1221no—
251C23—2441no—
252C23—2441no—
253C23#1451984612yes—
254C23C2×C4⋊#692311yescharacteristic
255D4—1224no—
256D4—1224no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—612no—
265C2×C4—611no—
266C2×C4—611no—
267C2×C4—612no—
268C2×C4—612no—
269C2×C4—1221no—
270C2×C4—2442no—
271C2×C4—2441no—
272C2×C4—2441no—
273C2×C4—2441no—
274C2×C4—2441no—
275C2×C4C2×C22⋊#692311yescharacteristic
276C2×C4#1451985611yescharacteristic
277C8—21no—
278C8—41no—
279C8—1222no—
280C8—4884no—
281C8—2441no—
282C8#1451983712yescomplemented
283C22—21no—
284C22—21no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—1222no—
291C22—2442no—
292C22—2442no—
293C22—2441no—
294C22—2441no—
295C22#1684070411yescharacteristic
296C22C2×#1451984611yescharacteristic
297C22#1684077211yescharacteristic
298C4—21no—
299C4—41no—
300C4—612no—
301C4—612no—
302C4—1222no—
303C4—2441no—
304C4C2×#1451983711yescharacteristic
305C4#1684074811yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—2442no—
310C2#2146168311yescharacteristic
311C2#2146169611yescharacteristic
312C2#2146170311yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×C8×C61C4
socleC22×C61#380
prime radicalC22×C22⋊D61D4
cosocleC2×C4×Dic61C23
hypercenterC23×C8#6923
hypocenterC61#8616
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
2. #21461376C2semidirect
3. #21460689C2semidirect
123. C8×C61#100semidirect
175. C61#8616semidirect
282. C8#14519837semidirect

Maximal subgroups

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

subgroupindex
#214616832
#214613762
#214606892
C2×C8×C22⋊D612
#861661