Finite Group Databasebeta

← #29572280

Subgroups — #29572280

Order 15616.

656 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 111 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#21459640C211yesmaximal, characteristic
2#21444484C211yesmaximal, characteristic, complemented
3#21444251C212yesmaximal
4#21444176C212yesmaximal, complemented
5C61×#1537C211yesmaximal, characteristic, complemented
6#16840490—22no—
7#16840472—22no—
8#16840466—24no—
9#16840390C412yes—
10#16840389—22no—
11#16840318—21no—
12C2×#14519769C2211yescharacteristic
13#125×D61—21no—
14#16840242C412yes—
15C4×#11758222C2211yescharacteristic
16#16839815C2212yes—
17#16839508—22no—
18#16839492—22no—
19#16839489—22no—
20#16839487—24no—
21C61×#533C2211yescharacteristic
22C61×#432C2212yes—
23C61×#330—21no—
24C61×#330—21no—
25C2×#11758222C2×C411yescharacteristic
26#14519777—22no—
27#14519776—22no—
28#14519775—24no—
29#14519769—22no—
30#14519769—21no—
31#14519769—21no—
32#14519769—22no—
33#14519768—22no—
34#14519767—24no—
35#14519767—24no—
36#14519755—24no—
37#14519755—24no—
38#14519754—24no—
39#14519753—24no—
40C4×Dic122—22no—
41C4×Dic122D412yes—
42#14519742—24no—
43C4×D244—22no—
44C4×D244D412yes—
45C4×C22⋊D61—22no—
46#14519737—24no—
47#14519736C2×C411yescharacteristic
48#14519735—22no—
49#14519734—21no—
50C2×C4⋊Dic61C2×C412yes—
51C2×#11758196D414yes—
52C4⋊C4×D61—24no—
53C4⋊C4×D61—22no—
54C42×D61—21no—
55C4⋊#92418—22no—
56C22⋊#92418—22no—
57C4×#92418—22no—
58C4⋊Q8×C61D412yes—
59#125×C61D411yescharacteristic
60#125×C61D411yescharacteristic
61C22⋊Q8×C61—22no—
62C4×Q8×C61—22no—
63#117×C61C2×C412yes—
64C2×C42×C61C2311yescharacteristic
65C4⋊C8×C61—22no—
66C22⋊C8×C61—22no—
67#97×C61—22no—
68C2×Q8×C61—22no—
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D412yes—
71C2×C22⋊D61—21no—
72C22×Dic61C22⋊C414yes—
73#11758222—22no—
74#11758222—22no—
75C2×D244C22⋊C411yescharacteristic
76C2×C4×D61—21no—
77C2×C4×D61—24no—
78C2×C4×D61—22no—
79C2×Dic122—24no—
80C2×Dic122C22⋊C411yescharacteristic
81C2×C8×C61—42no—
82C4⋊C4×C61—22no—
83C4⋊C4×C61—22no—
84C4⋊C4×C61—22no—
85C4⋊C4×C61C22⋊C412yes—
86C4⋊C4×C61C22⋊C412yes—
87C22⋊C4×C61—22no—
88C42×C61—22no—
89C42×C61C2×D411yescharacteristic
90C42×C61C2×D412yes—
91C42×C61C2×D411yescharacteristic
92C22⋊Dic61—21no—
93#11758198—24no—
94#11758198—22no—
95C4⋊Dic61—24no—
96C4⋊Dic61—22no—
97C4⋊Dic61—22no—
98C4⋊Dic61C22⋊C412yes—
99#11758196—24no—
100#11758196—24no—
101#11758196—24no—
102#11758196—24no—
103#11758196—24no—
104#11758196—22no—
105C4×Dic61—24no—
106C4×Dic61—21no—
107C4×Dic61—22no—
108C2×#92418—42no—
109C23×C61C2×C22⋊C411yescharacteristic
110C22×D61—21no—
111Q8×C61—82no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—42no—
117C2×C4×C61C2×C22⋊C411yescharacteristic
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C2×C4×C61#12012yes—
120C2×C4×C61#12012yes—
121C22⋊D61—24no—
122C2×Dic61—21no—
123C2×Dic61—22no—
124C2×Dic61—24no—
125C2×Dic61—44no—
126D244—21no—
127D244#10112yes—
128C4×D61—24no—
129C4×D61—82no—
130C4×D61—84no—
131Dic122—21no—
132Dic122—84no—
133Dic122#10112yes—
134C8×C61—82no—
135#92418—82no—
136#8478—611nomaximal, Sylow
137C22×C61—22no—
138C22×C61—21no—
139C22×C61#37911yescharacteristic
140C2×D61—22no—
141C2×D61—42no—
142C4×C61—22no—
143C4×C61—22no—
144C4×C61—21no—
145C4×C61—42no—
146C4×C61—42no—
147C4×C61C2×#10111yescharacteristic
148C4×C61C2×#10111yescharacteristic
149Dic61—22no—
150Dic61—42no—
151Dic61—44no—
152#3504—611no—
153#1544—611no—
154#1537D6111yescharacteristic, complemented
155#1488—612no—
156#1470—612no—
157C2×C61—41no—
158C2×C61#178411yescharacteristic
159C2×C61#179412yes—
160D61—42no—
161#557—1222no—
162#554—1224no—
163#554—1222no—
164#544—612no—
165#544—1222no—
166#533—1221no—
167#533C2×D6111yescharacteristic
168C2×#125—611no—
169C2×#125—1221no—
170#520—612no—
171C4×#41—611no—
172#432C2×D6112yes—
173#431—612no—
174#332—1222no—
175#330—21no—
176#330—21no—
177Q8⋊C8—1222no—
178D4⋊C8—1222no—
179#324—1224no—
180C61#847811yescharacteristic, complemented, Sylow
181C2×#41—611no—
182C4⋊Q8—1222no—
183C4⋊Q8—1224no—
184C4⋊Q8—1222no—
185C4⋊Q8D24412yes—
186#125—1222no—
187#125—1224no—
188#125—1224no—
189#125—1222no—
190#125—1221no—
191#125—1222no—
192#125—1221no—
193#125C22⋊D6111yescharacteristic
194#125C22⋊D6111yescharacteristic
195C22⋊Q8—22no—
196C22⋊Q8—1224no—
197C22⋊Q8—1224no—
198C22⋊Q8—1224no—
199C22⋊Q8—1224no—
200C4×Q8—22no—
201C4×Q8—612no—
202C4×Q8—1222no—
203C4×Q8—1224no—
204C4×D4—612no—
205C4×D4—1222no—
206C4×D4—1224no—
207C4×D4—1222no—
208#117—611no—
209#117—1222no—
210#117—1221no—
211#117C4×D6112yes—
212C2×C4⋊C4—612no—
213C2×C4⋊C4—614no—
214C2×C4⋊C4—1224no—
215C2×C4⋊C4—1222no—
216C2×C42—1221no—
217C2×C42C22×D6111yescharacteristic
218C4⋊C8—22no—
219C4⋊C8—1222no—
220C22⋊C8—22no—
221C22⋊C8—1222no—
222#97—22no—
223C4×C8—1222no—
224#41—1222no—
225#41—1222no—
226C2×Q8—22no—
227C2×Q8—611no—
228C2×Q8—1224no—
229C2×D4—611no—
230C2×D4—1221no—
231C22×C4—614no—
232C22×C4—1224no—
233C22×C4—1221no—
234C22×C4—1222no—
235C22×C4C2×C4×D6111yescharacteristic
236C22×C4D4×D6112yes—
237C2×C8—42no—
238C2×C8—2442no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—612no—
243C4⋊C4—1224no—
244C4⋊C4—1224no—
245C4⋊C4—1224no—
246C4⋊C4—1224no—
247C4⋊C4—1222no—
248C4⋊C4—1222no—
249C4⋊C4—1224no—
250C4⋊C4—1224no—
251C4⋊C4—1222no—
252C4⋊C4#1175819812yes—
253C4⋊C4#1175819812yes—
254C22⋊C4—22no—
255C22⋊C4—1224no—
256C22⋊C4—1221no—
257C22⋊C4—1222no—
258C42—22no—
259C42—1224no—
260C42—1221no—
261C42—1222no—
262C42C2×D24411yescharacteristic
263C42D4×D6112yes—
264C42C2×C22⋊D6111yescharacteristic
265C23—1221no—
266C23C22⋊C4×D6111yescharacteristic
267Q8—82no—
268Q8—612no—
269Q8—1221no—
270Q8—4884no—
271D4—612no—
272D4—1224no—
273D4—1221no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—1224no—
280C2×C4—1221no—
281C2×C4—1222no—
282C2×C4—1224no—
283C2×C4—4882no—
284C2×C4—4884no—
285C2×C4—2444no—
286C2×C4C22⋊C4×D6111yescharacteristic
287C2×C4C2×#1175819811yescharacteristic
288C2×C4#1451974512yes—
289C2×C4#1451974712yes—
290C8—82no—
291C8—4882no—
292C22—21no—
293C22—22no—
294C22—1222no—
295C22—2442no—
296C22#1683961711yescharacteristic
297C4—21no—
298C4—22no—
299C4—22no—
300C4—42no—
301C4—42no—
302C4—1222no—
303C4—2444no—
304C4—2442no—
305C4#101×D6111yescharacteristic
306C4#101×D6111yescharacteristic
307C2—41no—
308C2—2442no—
309C2#2144574111yescharacteristic
310C2#2144578112yes—
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1537C2
socleC22×C61#379
prime radicalC2×#11758222C2×C4
cosocleC2×C42×C61C23
hypercenter#1537D61
hypocenterC61#8478
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
2. #21444484C2semidirect
4. #21444176C2semidirect
5. C61×#1537C2semidirect
154. #1537D61semidirect
180. C61#8478semidirect

Maximal subgroups

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

subgroupindex
#214596402
#214444842
#214442512
#214441762
C61×#15372
#847861