Finite Group Databasebeta

← #30600072

Subgroups — #30600072

Order 15616.

482 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 84 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#21462499C211yesmaximal, characteristic
2#21461149C212yesmaximal, complemented
3#21461051C211yesmaximal, characteristic
4#21460717C212yesmaximal, complemented
5#21449088C211yesmaximal, characteristic
6#16840786—24no—
7#16840779C2212yes—
8#16840765C2212yes—
9#16840732—21no—
10C4×#11758217—21no—
11#16840612—24no—
12#16840608—21no—
13#16840605—21no—
14C2×#14519742C2211yescharacteristic
15C2×#14519733C2211yescharacteristic
16#16839700C411yescharacteristic
17#16839693—21no—
18#16839685—21no—
19#16839683C411yescharacteristic
20#16839674C411yescharacteristic
21C4⋊C4×Dic61—21no—
22#16839613C2211yescharacteristic
23C22⋊C4×Dic61—21no—
24C61×#399C411yescharacteristic
25C2×#11758217—22no—
26#14519844—22no—
27#14519844—22no—
28C4⋊#92420D414yes—
29C22⋊#92420—24no—
30C22⋊#92420—22no—
31#14519832—22no—
32C4×#92420—22no—
33C4×#92420—22no—
34C22×Dic122C2×C411yescharacteristic
35C23×Dic61D411yescharacteristic
36#14519742—22no—
37#14519742—22no—
38#14519733—22no—
39#14519733—22no—
40C2×C4⋊Dic61—21no—
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×#11758196C2×C411yescharacteristic
44C2×#11758196D411yescharacteristic
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61D411yescharacteristic
47C2×C4×Dic61—21no—
48C2×C4×Dic61—21no—
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61C2311yescharacteristic
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684C2×C411yescharacteristic
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61C2×C411yescharacteristic
60#95×C61C2×C411yescharacteristic
61C24×C61C22⋊C411yescharacteristic
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×Dic61—21no—
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—21no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×Dic122—22no—
77C2×Dic122—42no—
78#11758217—42no—
79#11758217—42no—
80C2×#92420—24no—
81C2×#92420—42no—
82C4⋊C4×C61—22no—
83C4⋊C4×C61—42no—
84C22⋊C4×C61—22no—
85C22⋊C4×C61—42no—
86C22⋊Dic61—22no—
87C22⋊Dic61—42no—
88C4⋊Dic61—42no—
89#11758196—22no—
90#11758196—22no—
91#11758196—42no—
92#11758196—42no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C4×Dic61—41no—
96C4×Dic61—41no—
97C4×Dic61—42no—
98C4×Dic61C2×D412yes—
99C4×Dic61#4112yes—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61C2×C22⋊C411yescharacteristic
106C2×C4×C61—22no—
107C2×C4×C61—21no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61C4×D412yes—
116C2×Dic61—22no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61#12011yescharacteristic
131C2×Dic61#12112yes—
132C2×Dic61#12311yescharacteristic
133C2×Dic61#13512yes—
134C2×Dic61#13712yes—
135Dic122—24no—
136Dic122—42no—
137#92420—44no—
138#92420—82no—
139#9034—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—42no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61—81no—
147C22×C61#38611yescharacteristic
148C22×C61#41611yescharacteristic
149C22×C61#41911yescharacteristic
150C4×C61—22no—
151C4×C61—42no—
152C4×C61—41no—
153C4×C61—81no—
154Dic61—22no—
155Dic61—41no—
156Dic61—42no—
157Dic61—41no—
158Dic61—81no—
159Dic61#45212yes—
160Dic61#48512yes—
161#2921—611no—
162#2396—611no—
163#1487—612no—
164#1458—611no—
165#1303—612no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—81no—
169C2×C61#187311yescharacteristic
170C2×C61#188411yescharacteristic
171C2×C61#189111yescharacteristic
172C2×C4×Q8—611no—
173C2×#117—611no—
174#433—1224no—
175#431—612no—
176#423—612no—
177#407—1221no—
178C4×M16—1221no—
179#400—611no—
180#399#692311yescharacteristic
181#398—1221no—
182#396—1221no—
183#395—611no—
184#393—611no—
185C4×C4⋊C4—611no—
186C4×C4⋊C4—1221no—
187C4×C22⋊C4—1221no—
188Q8⋊C8—1224no—
189#324—1221no—
190#323—1221no—
191C61#903411yescharacteristic, complemented, Sylow
192C22×Q8—611no—
193C23×C4—611no—
194C2×M16—1222no—
195C4×Q8—1222no—
196C4×Q8—1222no—
197#117—1222no—
198#117—1222no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—611no—
201C2×C4⋊C4—611no—
202C2×C4⋊C4—1221no—
203C2×C4⋊C4—1221no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4C2×#692311yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—611no—
208C2×C22⋊C4—1221no—
209C2×C22⋊C4C2×#692311yescharacteristic
210C2×C42—611no—
211C2×C42—611no—
212C2×C42—1221no—
213C2×C42—1221no—
214C4⋊C8—614no—
215C4⋊C8—1222no—
216C4⋊C8—1222no—
217C22⋊C8—1224no—
218C22⋊C8—1222no—
219#97—1222no—
220C4×C8—1222no—
221C4×C8—1222no—
222#95—611no—
223#95—1221no—
224#95—1221no—
225#95—1221no—
226#95—1221no—
227#95C2×#692311yescharacteristic
228C24C22⋊#692311yescharacteristic
229C2×Q8—1222no—
230C2×Q8—2442no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—1221no—
238C22×C4—1221no—
239C22×C4—1222no—
240C22×C4—1222no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4C22⋊#692311yescharacteristic
244C22×C4C22×#692311yescharacteristic
245M16—2442no—
246M16—2442no—
247C2×C8—1224no—
248C2×C8—2442no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—1222no—
252C4⋊C4—1222no—
253C4⋊C4—2442no—
254C4⋊C4—2442no—
255C4⋊C4—2442no—
256C22⋊C4—22no—
257C22⋊C4—42no—
258C22⋊C4—1222no—
259C22⋊C4—2442no—
260C42—612no—
261C42—612no—
262C42—1221no—
263C42—1221no—
264C42—2442no—
265C42—2441no—
266C42—2441no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23C2×C22⋊#692311yescharacteristic
273Q8—1224no—
274Q8—2442no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—611no—
285C2×C4—611no—
286C2×C4—612no—
287C2×C4—612no—
288C2×C4—612no—
289C2×C4—1222no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1221no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4D4×#692312yes—
304C8—4882no—
305C8—2444no—
306C22—21no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—81no—
313C22#1684071011yescharacteristic
314C22#1684075211yescharacteristic
315C22#1684075511yescharacteristic
316C4—22no—
317C4—42no—
318C4—41no—
319C4—81no—
320C4—612no—
321C4—612no—
322C4—1222no—
323C4—4881no—
324C4—2441no—
325C4—2441no—
326C4—2442no—
327C2—21no—
328C2—21no—
329C2—81no—
330C2#2146180511yescharacteristic
331C2#2146182211yescharacteristic
332C2#2146183111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#399C4
socleC22×C61#386
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#399#6923
hypocenterC61#9034
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #21461149C2semidirect
4. #21460717C2semidirect
191. C61#9034semidirect

Maximal subgroups

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

subgroupindex
#214624992
#214611492
#214610512
#214607172
#214490882
#903461