Finite Group Databasebeta

← #30604268

Subgroups — #30604268

Order 15616.

366 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 68 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#21461070C211yesmaximal, characteristic
2#21461050C211yesmaximal, characteristic
3#21460762C212yesmaximal, complemented
4#21460749C212yesmaximal, complemented
5#21450273C211yesmaximal, characteristic
6#16840782—21no—
7#16840778—21no—
8#16840607C2212yes—
9#16840607C2212yes—
10#16840605—21no—
11#16840605—21no—
12C2×#14519736C2211yescharacteristic
13#16839690—21no—
14#16839690—21no—
15#16839669C411yescharacteristic
16#16839669C2211yescharacteristic
17#16839643—21no—
18#16839643—21no—
19#16839631C2211yescharacteristic
20#16839627C411yescharacteristic
21C4×C22⋊Dic61C411yescharacteristic
22C2×C22⋊Q8×C61C411yescharacteristic
23#14519844—41no—
24C4⋊#92420—41no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27#14519832—41no—
28C4×#92420—41no—
29#14519736—41no—
30#14519736—41no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×C4⋊Dic61C2311yescharacteristic
33C2×#11758196—21no—
34C2×#11758196—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61D411yescharacteristic
37C2×C22⋊Dic61D411yescharacteristic
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61C2×C411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
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×Dic61C22⋊C411yescharacteristic
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C2×#92420—42no—
75C2×#92420—42no—
76C4⋊C4×C61—41no—
77C4⋊C4×C61—41no—
78C4⋊C4×C61—41no—
79C22⋊C4×C61—42no—
80C22⋊Dic61—41no—
81C22⋊Dic61—41no—
82C22⋊Dic61—42no—
83C4⋊Dic61—41no—
84C4⋊Dic61—41no—
85#11758196—42no—
86#11758196—42no—
87C4×Dic61—21no—
88C4×Dic61—21no—
89C4×Dic61—21no—
90C4×Dic61—21no—
91C4×Dic61—22no—
92C4×Dic61—41no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61C2×C22⋊C411yescharacteristic
99Q8×C61—82no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61#9911yescharacteristic
113C2×C4×C61#9911yescharacteristic
114C2×C4×C61#13112yes—
115C2×Dic61—22no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129#92420—82no—
130#92420—82no—
131#11552—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—41no—
135C22×C61—41no—
136C22×C61—41no—
137C22×C61—81no—
138C22×C61—81no—
139C22×C61#40811yescharacteristic
140C22×C61C2×#9911yescharacteristic
141C22×C61#41311yescharacteristic
142C4×C61—21no—
143C4×C61—21no—
144C4×C61—22no—
145C4×C61—81no—
146C4×C61—81no—
147Dic61—42no—
148Dic61—41no—
149Dic61—41no—
150Dic61—41no—
151Dic61—41no—
152Dic61—81no—
153#2499—611no—
154#1464—611no—
155#1458—611no—
156#1325—612no—
157#1319—612no—
158C2×C61—21no—
159C2×C61—21no—
160C2×C61—81no—
161C2×C61#146811yescharacteristic
162C2×C61#212611yescharacteristic
163C2×C61#213511yescharacteristic
164C2×C22⋊Q8#692311yescharacteristic
165C2×#117—611no—
166#432—1221no—
167#431—1221no—
168#397—1221no—
169#397—1221no—
170#391—611no—
171#391—611no—
172#387—1221no—
173#387—1221no—
174#384—611no—
175#382—611no—
176C4×C22⋊C4—611no—
177#324—612no—
178#324—612no—
179#323—1221no—
180#323—1221no—
181C61#1155211yescharacteristic, complemented, Sylow
182C22×Q8C2×#692311yescharacteristic
183C23×C4C2×#692311yescharacteristic
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186#117—2441no—
187#117—2441no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—611no—
190C2×C4⋊C4—611no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4—1221no—
193C2×C4⋊C4C2×#692311yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—611no—
196C2×C22⋊C4—611no—
197C2×C22⋊C4—1221no—
198C2×C42—611no—
199C2×C42—611no—
200C2×C42—611no—
201C2×C42—611no—
202C4⋊C8—2441no—
203C4⋊C8—2441no—
204C22⋊C8—1222no—
205C22⋊C8—1222no—
206#97—2441no—
207C4×C8—2441no—
208#95—1221no—
209#95—1221no—
210#95—1221no—
211#95—1221no—
212#95—1221no—
213#95—1221no—
214#95—1221no—
215C24C22⋊#692311yescharacteristic
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—611no—
226C22×C4—611no—
227C22×C4—611no—
228C22×C4—611no—
229C22×C4—1221no—
230C22×C4—1221no—
231C22×C4C22⋊#692311yescharacteristic
232C22×C4C22×#692311yescharacteristic
233C2×C8—2442no—
234C2×C8—2442no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—2441no—
239C4⋊C4—2441no—
240C4⋊C4—2442no—
241C4⋊C4—2442no—
242C22⋊C4—42no—
243C22⋊C4—2441no—
244C22⋊C4—2441no—
245C22⋊C4—2442no—
246C42—1221no—
247C42—1221no—
248C42—1221no—
249C42—1221no—
250C42—1222no—
251C42—2441no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊#692311yescharacteristic
258Q8—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—1222no—
272C2×C4—1221no—
273C2×C4—1221no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—2441no—
277C2×C4—2441no—
278C2×C4—2441no—
279C2×C4—2441no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4—2441no—
284C2×C4—2441no—
285C2×C4#1451983511yescharacteristic
286C2×C4#1451983511yescharacteristic
287C2×C4#1451986412yes—
288C8—4882no—
289C8—4882no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1684073511yescharacteristic
298C22C2×#1451983511yescharacteristic
299C22#1684074911yescharacteristic
300C4—21no—
301C4—21no—
302C4—22no—
303C4—81no—
304C4—81no—
305C4—4881no—
306C4—2442no—
307C4—2441no—
308C4—2441no—
309C4—2441no—
310C4—2441no—
311C2—21no—
312C2—21no—
313C2—81no—
314C2#2146108411yescharacteristic
315C2#2146209011yescharacteristic
316C2#2146212111yescharacteristic
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#11552
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21460762C2semidirect
4. #21460749C2semidirect
181. C61#11552semidirect

Maximal subgroups

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

subgroupindex
#214610702
#214610502
#214607622
#214607492
#214502732
#1155261