Finite Group Databasebeta

← #30598234

Subgroups — #30598234

Order 15616.

826 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 132 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#21461545C211yesmaximal, characteristic
2#21461545C211yesmaximal, characteristic
3C2×#16840619C214yesmaximal, complemented
4C22×#14519768C211yesmaximal, characteristic
5C2×C22⋊#92420—21no—
6C2×C22⋊#92420—21no—
7C2×#14519832C2212yes—
8C2×#14519832C2212yes—
9#16840628—21no—
10#16840628—21no—
11#16840619—28no—
12C2×#14519768—28no—
13C2×#14519768—21no—
14C2×#14519768—21no—
15C2×#14519768C412yes—
16C2×#14519768C2212yes—
17C22×C4⋊Dic61—21no—
18C22×C4⋊Dic61C411yescharacteristic
19C22×#11758196—21no—
20C22×C4×Dic61C2211yescharacteristic
21C22×C4⋊C4×C61C411yescharacteristic
22C22×#92420—22no—
23C22×#92420—22no—
24C22⋊#92420—24no—
25C22⋊#92420—24no—
26#14519832—24no—
27#14519832—24no—
28C23×Dic61—21no—
29C23×Dic61D411yescharacteristic
30C23×Dic61D411yescharacteristic
31#14519768—28no—
32#14519768—28no—
33#14519768—24no—
34#14519768—24no—
35#14519768—24no—
36#14519768D418yes—
37C2×C4⋊Dic61—28no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—22no—
40C2×C4⋊Dic61—22no—
41C2×C4⋊Dic61—24no—
42C2×C4⋊Dic61C2×C412yes—
43C2×#11758196—28no—
44C2×#11758196—22no—
45C2×#11758196—22no—
46C2×C4×Dic61—24no—
47C2×C4×Dic61C2×C411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61C2311yescharacteristic
51C23×C4×C61—21no—
52C23×C4×C61C2×C411yescharacteristic
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61—24no—
55C2×C4⋊C4×C61C2×C412yes—
56C24×C61C22⋊C411yescharacteristic
57C22×C4×C61—22no—
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61—24no—
61C22×C4×C61—22no—
62C22×C4×C61—42no—
63C22×C4×C61—42no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×Dic61—24no—
67C22×Dic61—22no—
68C22×Dic61—22no—
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—22no—
72C22×Dic61—41no—
73C22×Dic61—41no—
74C22×Dic61—42no—
75C22×Dic61—42no—
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79C22×Dic61C2×D411yescharacteristic
80C2×#92420—24no—
81C2×#92420—24no—
82C2×#92420—42no—
83C2×#92420—42no—
84C2×#92420—42no—
85C2×#92420—42no—
86C4⋊C4×C61—24no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61C22⋊C414yes—
89C4⋊Dic61—28no—
90C4⋊Dic61—24no—
91C4⋊Dic61—22no—
92C4⋊Dic61—28no—
93C4⋊Dic61C22⋊C414yes—
94#11758196—28no—
95#11758196—28no—
96C4×Dic61—22no—
97C4×Dic61—22no—
98C4×Dic61—22no—
99C4×Dic61C2×D414yes—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61#10112yes—
105C23×C61#10412yes—
106C23×C61#10412yes—
107C23×C61C2×C22⋊C411yescharacteristic
108C2×C4×C61—28no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—24no—
114C2×C4×C61—48no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61#10112yes—
118C2×C4×C61C2×C22⋊C412yes—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—24no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—44no—
131C2×Dic61—44no—
132C2×Dic61—48no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61#12012yes—
136C2×Dic61#12012yes—
137#92420—44no—
138#92420—44no—
139#8335—611nomaximal, Sylow
140C22×C61—22no—
141C22×C61—22no—
142C22×C61—22no—
143C22×C61—22no—
144C22×C61—22no—
145C22×C61—22no—
146C22×C61#33014yescomplemented
147C22×C61#37911yescharacteristic
148C22×C61C2×#10111yescharacteristic
149C22×C61C2×#10111yescharacteristic
150C22×C61C2×#10411yescharacteristic
151C22×C61C2×#10411yescharacteristic
152C22×C61#42111yescharacteristic
153C22×C61#42111yescharacteristic
154C4×C61—22no—
155C4×C61—22no—
156C4×C61—44no—
157Dic61—22no—
158Dic61—22no—
159Dic61—41no—
160Dic61—41no—
161Dic61—44no—
162C22×#125—611no—
163#1749—611no—
164#1749—611no—
165C2×#330—614no—
166C2×C61—24no—
167C2×C61C2×#33011yescharacteristic
168C2×C61#152811yescharacteristic
169C2×C61#152811yescharacteristic
170C2×C61#153411yescharacteristic
171C2×C61#178411yescharacteristic
172C2×C61#178911yescharacteristic
173C2×C61#178911yescharacteristic
174C2×#125—612no—
175C2×#125—612no—
176C2×#125—1221no—
177C2×#125—1221no—
178C2×#125—1228no—
179C22×C4⋊C4—611no—
180C22×C4⋊C4—1221no—
181C22×C4⋊C4—1221no—
182C22×C4⋊C4#692311yescharacteristic
183C22×C42—611no—
184C2×C22⋊C8—1221no—
185C2×C22⋊C8—1221no—
186C2×#97—612no—
187C2×#97—612no—
188#336—1221no—
189#336—1221no—
190#330—1228no—
191C61#833511yescharacteristic, complemented, Sylow
192C23×C4—21no—
193C23×C4—611no—
194C23×C4—611no—
195C23×C4—1221no—
196C23×C4C2×#692311yescharacteristic
197C22×C8—1222no—
198C22×C8—1222no—
199#125—618no—
200#125—1224no—
201#125—1224no—
202#125—1228no—
203#125—1228no—
204#125—1224no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—612no—
208C2×C4⋊C4—1224no—
209C2×C4⋊C4—1222no—
210C2×C4⋊C4—1222no—
211C2×C4⋊C4—1222no—
212C2×C4⋊C4—1222no—
213C2×C4⋊C4—1228no—
214C2×C4⋊C4—1228no—
215C2×C4⋊C4—1221no—
216C2×C4⋊C4C2×#692312yes—
217C2×C42—611no—
218C2×C42—611no—
219C2×C42—611no—
220C2×C42—611no—
221C2×C42—1224no—
222C22⋊C8—1224no—
223C22⋊C8—1224no—
224#97—1224no—
225#97—1224no—
226C24C22⋊#692311yescharacteristic
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—611no—
238C22×C4—1222no—
239C22×C4—1222no—
240C22×C4—1224no—
241C22×C4—1222no—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4—2441no—
245C22×C4—2441no—
246C22×C4—2442no—
247C22×C4—2442no—
248C22×C4C22⋊#692311yescharacteristic
249C22×C4C22×#692311yescharacteristic
250C2×C8—1224no—
251C2×C8—1224no—
252C2×C8—2442no—
253C2×C8—2442no—
254C2×C8—2442no—
255C2×C8—2442no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—614no—
259C4⋊C4—1228no—
260C4⋊C4—1228no—
261C4⋊C4—1228no—
262C4⋊C4—1224no—
263C4⋊C4—1228no—
264C4⋊C4—1222no—
265C4⋊C4C22⋊#692314yes—
266C42—614no—
267C42—1222no—
268C42—1222no—
269C42—1222no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23#1451983812yes—
275C23#1451984112yes—
276C23#1451984212yes—
277C23C2×C22⋊#692311yescharacteristic
278C2×C4—28no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—24no—
284C2×C4—48no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—612no—
288C2×C4—612no—
289C2×C4—1221no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1221no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1224no—
300C2×C4—2444no—
301C2×C4—2444no—
302C2×C4—2448no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4#1451983812yes—
306C2×C4C2×C22⋊#692312yes—
307C8—2444no—
308C8—2444no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22#1684061914yescomplemented
316C22#1684070311yescharacteristic
317C22C2×#1451983811yescharacteristic
318C22C2×#1451983811yescharacteristic
319C22C2×#1451984111yescharacteristic
320C22C2×#1451984211yescharacteristic
321C22#1684076011yescharacteristic
322C22#1684076111yescharacteristic
323C4—22no—
324C4—22no—
325C4—44no—
326C4—1222no—
327C4—1222no—
328C4—2441no—
329C4—2441no—
330C4—2444no—
331C2—24no—
332C2C2×#1684061911yescharacteristic
333C2#2146128111yescharacteristic
334C2#2146128311yescharacteristic
335C2#2146130111yescharacteristic
336C2#2146165311yescharacteristic
337C2#2146166011yescharacteristic
338C2#2146166211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#8335
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#16840619C2semidirect
146. C22×C61#330semidirect
191. C61#8335semidirect
315. C22#16840619semidirect

Maximal subgroups

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

subgroupindex
#214615452
C2×#168406192
C22×#145197682
#833561