Finite Group Databasebeta

← #30605176

Subgroups — #30605176

Order 15616.

386 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 66 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#21461718C211yesmaximal, characteristic
2#21461069C211yesmaximal, characteristic
3#21460820C212yesmaximal, complemented
4#21460818C212yesmaximal, complemented
5#21451634C211yesmaximal, characteristic
6#16840732—21no—
7#16840732—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608—21no—
11#16840608C2212yes—
12C22×#11758196C2211yescharacteristic
13#16839700C411yescharacteristic
14#16839700C2211yescharacteristic
15#16839695—21no—
16#16839695—21no—
17#16839681C411yescharacteristic
18#16839681C2211yescharacteristic
19#16839634—21no—
20#16839623—21no—
21#16839622C411yescharacteristic
22C2×C22⋊Q8×C61C411yescharacteristic
23C2×#11758217—22no—
24C2×#11758217—41no—
25C22⋊#92420—22no—
26C22⋊#92420—41no—
27C22⋊#92420—41no—
28C22⋊#92420—41no—
29C23×Dic61D411yescharacteristic
30C23×Dic61D411yescharacteristic
31C2×C4⋊Dic61—21no—
32C2×#11758196—21no—
33C2×#11758196—21no—
34C2×#11758196—41no—
35C2×#11758196—41no—
36C2×#11758196C2×C411yescharacteristic
37C2×#11758196D411yescharacteristic
38C2×#11758196D411yescharacteristic
39C2×#11758196C2311yescharacteristic
40C2×C22⋊Dic61—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684C2×C411yescharacteristic
48#14519684C2×C411yescharacteristic
49C22×Q8×C61C2×C411yescharacteristic
50C23×C4×C61C2×C411yescharacteristic
51C22⋊Q8×C61—41no—
52C22⋊Q8×C61—41no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C22⋊C4×C61—21no—
56C24×C61C22⋊C411yescharacteristic
57C2×Q8×C61—41no—
58C2×Q8×C61—81no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—81no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×Dic61—21no—
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—22no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—41no—
74C22×Dic61—81no—
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79#11758217—42no—
80#11758217—42no—
81#11758217—82no—
82C2×#92420—42no—
83C2×#92420—42no—
84C4⋊C4×C61—42no—
85C4⋊C4×C61—41no—
86C22⋊C4×C61—41no—
87C22⋊C4×C61—41no—
88C22⋊Dic61—42no—
89C4⋊Dic61—81no—
90#11758196—41no—
91#11758196—41no—
92#11758196—41no—
93#11758196—41no—
94#11758196—81no—
95#11758196—81no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61#9911yescharacteristic
101C23×C61#9911yescharacteristic
102C23×C61C2×C22⋊C411yescharacteristic
103Q8×C61—82no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—42no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—81no—
114C2×C4×C61—81no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—81no—
134C2×Dic61#13712yes—
135#92420—82no—
136#92420—82no—
137#11817—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—42no—
147C22×C61—42no—
148C22×C61C2×#9911yescharacteristic
149C22×C61#41911yescharacteristic
150C22×C61#42111yescharacteristic
151C4×C61—41no—
152C4×C61—41no—
153C4×C61—81no—
154C4×C61—81no—
155Dic61—22no—
156Dic61—41no—
157Dic61—41no—
158Dic61—41no—
159Dic61—81no—
160Dic61—81no—
161#2627—611no—
162#1821—611no—
163#1463—611no—
164#1354—612no—
165#1353—612no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—41no—
169C2×C61—41no—
170C2×C61#151111yescharacteristic
171C2×C61#212111yescharacteristic
172C2×C61#213311yescharacteristic
173C2×C22⋊Q8#692311yescharacteristic
174C22×C4⋊C4—611no—
175#407—1221no—
176#407—1221no—
177#400—611no—
178#400—611no—
179#398—1221no—
180#398—1221no—
181#395—611no—
182#395—611no—
183#385—1221no—
184#381—1221no—
185#380—611no—
186#343—612no—
187#343—1221no—
188#324—612no—
189#324—1221no—
190C61#1181711yescharacteristic, complemented, Sylow
191C22×Q8C2×#692311yescharacteristic
192C23×C4—611no—
193C23×C4—611no—
194C23×C4C2×#692311yescharacteristic
195C2×M16—1222no—
196C2×M16—2441no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—611no—
201C2×C4⋊C4—611no—
202C2×C4⋊C4—611no—
203C2×C4⋊C4—611no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4—2441no—
208C2×C4⋊C4—2441no—
209C2×C4⋊C4C2×#692311yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—1221no—
212C22⋊C8—1222no—
213C22⋊C8—2441no—
214C22⋊C8—2441no—
215C22⋊C8—2441no—
216#95—611no—
217#95—611no—
218#95—1221no—
219#95—1221no—
220#95—1221no—
221#95—1221no—
222#95—1221no—
223#95—1221no—
224C24C22⋊#692311yescharacteristic
225C2×Q8—41no—
226C2×Q8—81no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—81no—
232C22×C4—611no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—1222no—
237C22×C4—1221no—
238C22×C4—1221no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—4881no—
244C22×C4—2441no—
245C22×C4C22⋊#692311yescharacteristic
246C22×C4C22×#692311yescharacteristic
247M16—4882no—
248M16—2442no—
249M16—2442no—
250C2×C8—2442no—
251C2×C8—2442no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—4881no—
255C4⋊C4—4881no—
256C4⋊C4—4881no—
257C4⋊C4—2441no—
258C4⋊C4—2441no—
259C4⋊C4—2441no—
260C4⋊C4—2441no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—2442no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23#1451983511yescharacteristic
269C23#1451983511yescharacteristic
270C23C2×C22⋊#692311yescharacteristic
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—612no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—1221no—
287C2×C4—1221no—
288C2×C4—1221no—
289C2×C4—1221no—
290C2×C4—1221no—
291C2×C4—4881no—
292C2×C4—2442no—
293C2×C4—2442no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2442no—
301C2×C4—2441no—
302C2×C4—2441no—
303C8—4882no—
304C8—4882no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#1451983511yescharacteristic
316C22#1684075511yescharacteristic
317C22#1684076111yescharacteristic
318C4—41no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—1222no—
323C4—4881no—
324C4—4881no—
325C4—2441no—
326C4—2441no—
327C4—2441no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#2146122911yescharacteristic
333C2#2146207411yescharacteristic
334C2#2146211411yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×#11758196C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#11817
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21460820C2semidirect
4. #21460818C2semidirect
190. C61#11817semidirect

Maximal subgroups

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

subgroupindex
#214617182
#214610692
#214608202
#214608182
#214516342
#1181761