Finite Group Databasebeta

← #30604264

Subgroups — #30604264

Order 15616.

398 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 72 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#21461067C211yesmaximal, characteristic
2C2×#16840607C211yesmaximal, characteristic
3#21460760C212yesmaximal, complemented
4#21460749C212yesmaximal, complemented
5#21450305C211yesmaximal, characteristic
6#16840732—21no—
7C2×C22⋊#92420—21no—
8#16840607—21no—
9#16840607—21no—
10#16840607C2212yes—
11#16840607C2212yes—
12C22×C4⋊Dic61C2211yescharacteristic
13#16839701—21no—
14#16839701—21no—
15#16839669C411yescharacteristic
16#16839669C2211yescharacteristic
17#16839627C411yescharacteristic
18#16839627C2211yescharacteristic
19#16839623—21no—
20#16839623—21no—
21C4×C22⋊Dic61C411yescharacteristic
22C2×C22⋊Q8×C61C411yescharacteristic
23C2×#11758217—41no—
24C22×#92420—41no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27C22⋊#92420—41no—
28C22⋊#92420—41no—
29C23×Dic61D411yescharacteristic
30C23×Dic61D411yescharacteristic
31C2×C4⋊Dic61—41no—
32C2×C4⋊Dic61—41no—
33C2×C4⋊Dic61C2×C411yescharacteristic
34C2×C4⋊Dic61D411yescharacteristic
35C2×C4⋊Dic61D411yescharacteristic
36C2×C4⋊Dic61C2311yescharacteristic
37C2×C22⋊Dic61—21no—
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61C2×C411yescharacteristic
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
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—22no—
58C2×Q8×C61—41no—
59C2×Q8×C61—41no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—22no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—41no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—42no—
75C22×Dic61—42no—
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79C22×Dic61C2×D411yescharacteristic
80#11758217—82no—
81C2×#92420—42no—
82C2×#92420—42no—
83C2×#92420—82no—
84C4⋊C4×C61—42no—
85C4⋊C4×C61—41no—
86C22⋊C4×C61—41no—
87C22⋊C4×C61—41no—
88C22⋊Dic61—42no—
89C4⋊Dic61—41no—
90C4⋊Dic61—41no—
91C4⋊Dic61—41no—
92C4⋊Dic61—41no—
93C4×Dic61—22no—
94C4×Dic61—41no—
95C23×C61—22no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—41no—
99C23×C61#9911yescharacteristic
100C23×C61#9911yescharacteristic
101C23×C61C2×C22⋊C411yescharacteristic
102Q8×C61—82no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61#13112yes—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—81no—
133C2×Dic61—81no—
134#92420—82no—
135#92420—82no—
136#11551—611nomaximal, Sylow
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—42no—
145C22×C61—42no—
146C22×C61#35412yes—
147C22×C61#40811yescharacteristic
148C22×C61C2×#9911yescharacteristic
149C22×C61#41311yescharacteristic
150C4×C61—22no—
151C4×C61—41no—
152C4×C61—81no—
153C4×C61—81no—
154Dic61—41no—
155Dic61—41no—
156Dic61—41no—
157Dic61—41no—
158Dic61—81no—
159Dic61—81no—
160#2503—611no—
161#1463—611no—
162C2×#324—611no—
163#1324—612no—
164#1319—612no—
165C2×C61—21no—
166C2×C61—21no—
167C2×C61—22no—
168C2×C61—41no—
169C2×C61#146811yescharacteristic
170C2×C61C2×#35411yescharacteristic
171C2×C61#213311yescharacteristic
172C2×C22⋊Q8#692311yescharacteristic
173C22×C4⋊C4—611no—
174#407—1221no—
175C2×C22⋊C8—1221no—
176#400—1221no—
177#400—1221no—
178#391—611no—
179#391—611no—
180#382—611no—
181#382—611no—
182#381—1221no—
183#381—1221no—
184C4×C22⋊C4—611no—
185#324—612no—
186#324—612no—
187#324—1221no—
188#324—1221no—
189C61#1155111yescharacteristic, complemented, Sylow
190C22×Q8C2×#692311yescharacteristic
191C23×C4—611no—
192C23×C4—611no—
193C23×C4C2×#692311yescharacteristic
194C2×M16—2441no—
195C22×C8—2441no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—611no—
200C2×C4⋊C4—611no—
201C2×C4⋊C4—611no—
202C2×C4⋊C4—611no—
203C2×C4⋊C4—2441no—
204C2×C4⋊C4—2441no—
205C2×C4⋊C4C2×#692311yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—1221no—
208C2×C42—611no—
209C2×C42—611no—
210C22⋊C8—1222no—
211C22⋊C8—1222no—
212C22⋊C8—2441no—
213C22⋊C8—2441no—
214#95—1221no—
215#95—1221no—
216#95—1221no—
217#95—1221no—
218#95—1221no—
219#95—1221no—
220#95—1221no—
221#95—1221no—
222#95—1221no—
223C24C22⋊#692311yescharacteristic
224C2×Q8—22no—
225C2×Q8—41no—
226C2×Q8—41no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—1221no—
238C22×C4—1221no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—2442no—
244C22×C4—2442no—
245C22×C4C22⋊#692311yescharacteristic
246C22×C4C22×#692311yescharacteristic
247M16—4882no—
248C2×C8—4882no—
249C2×C8—2442no—
250C2×C8—2442no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—2441no—
254C4⋊C4—2441no—
255C4⋊C4—2441no—
256C4⋊C4—2441no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—2442no—
260C42—1222no—
261C42—2441no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—41no—
266C23#1451983511yescharacteristic
267C23#1451983511yescharacteristic
268C23C2×C22⋊#692311yescharacteristic
269Q8—82no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—4881no—
287C2×C4—4881no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4—2441no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4#1451986412yes—
301C8—4882no—
302C8—4882no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22#1684066812yes—
313C22#1684073511yescharacteristic
314C22C2×#1451983511yescharacteristic
315C22#1684074911yescharacteristic
316C4—22no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—4881no—
321C4—4881no—
322C4—2441no—
323C4—2441no—
324C4—2441no—
325C4—2441no—
326C2—21no—
327C2—21no—
328C2—22no—
329C2—41no—
330C2#2146108411yescharacteristic
331C2C2×#1684066811yescharacteristic
332C2#2146211411yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×C4⋊Dic61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#11551
LCSG ▷ C22×C4×C61 ▷ C22×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. #21460760C2semidirect
4. #21460749C2semidirect
189. C61#11551semidirect

Maximal subgroups

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

subgroupindex
#214610672
C2×#168406072
#214607602
#214607492
#214503052
#1155161