Finite Group Databasebeta

← #30602082

Subgroups — #30602082

Order 15616.

454 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 96 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#21462004C211yesmaximal, characteristic
2#21462004C211yesmaximal, characteristic
3C4×#14519831C211yesmaximal, characteristic
4#21461375C211yesmaximal, characteristic
5#21461169C211yesmaximal, characteristic
6#21461168C211yesmaximal, characteristic
7#21456404C211yesmaximal, characteristic
8#16840766C2211yescharacteristic
9#16840766C2211yescharacteristic
10C2×#14519831—22no—
11C4×D61⋊C8C2211yescharacteristic
12C42×#6923C2211yescharacteristic
13#16840632—22no—
14#16840632—22no—
15#16840624—22no—
16#16840612—22no—
17Q8⋊#92420—22no—
18#16840317C2211yescharacteristic
19#16840317C2211yescharacteristic
20#16840165C411yescharacteristic, complemented
21#16840141C411yescharacteristic, complemented
22#16840113C411yescharacteristic, complemented
23#16840002—22no—
24#16839854—22no—
25C4×C8×D61C2211yescharacteristic
26C61×#487C411yescharacteristic, complemented
27C2×#11758213—21no—
28C2×#11758213—21no—
29C2×D61⋊C8—21no—
30C2×C4×#6923—21no—
31C2×C4×#6923—21no—
32C2×C4×#6923D412yes—
33C4⋊#92420—21no—
34C4⋊#92420—21no—
35C4⋊#92420—21no—
36C4⋊#92420—21no—
37#14519831—28no—
38C4×#92420—21no—
39C4×#92420D412yes—
40C2×#11758202—22no—
41C2×C8×D61D412yes—
42#14519777C2×C411yescharacteristic
43#14519777C2×C411yescharacteristic
44#14519768—21no—
45#14519768—21no—
46#14519757—21no—
47#14519757—21no—
48#14519753—21no—
49#14519753—21no—
50Q8×Dic61—21no—
51Q8×Dic61—21no—
52#14519742—21no—
53#14519742—21no—
54#14519735—21no—
55#14519735—21no—
56C42×D61C2311yescharacteristic
57#14519710—22no—
58#14519702—22no—
59C4×#92418C2×C411yescharacteristic
60C8×Dic61D412yes—
61C4⋊Q8×C61C2×C411yescharacteristic
62C4⋊Q8×C61C2×C411yescharacteristic
63#103×C61—22no—
64C4×C8×C61C2×C411yescharacteristic
65C22×#6923—22no—
66C2×Q8×C61—21no—
67C2×Q8×C61—21no—
68C2×C4×D61C2×D411yescharacteristic
69C2×C4×D61C2×D411yescharacteristic
70C2×C4×D61#4111yescharacteristic
71C2×Dic122—21no—
72C2×Dic122—21no—
73C2×#92420—22no—
74C2×#92420—41no—
75C2×#92420—41no—
76C4×#6923—24no—
77C4×#6923—22no—
78C4×#6923—22no—
79C4×#6923—42no—
80#11758213—42no—
81#11758213—42no—
82D61⋊C8—24no—
83C2×C8×C61C22⋊C412yes—
84C4⋊C4×C61—21no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61—21no—
88C42×C61C22×C411yescharacteristic
89#11758202—44no—
90#11758198—41no—
91#11758198—41no—
92C4⋊Dic61—21no—
93C4⋊Dic61—21no—
94C4⋊Dic61—21no—
95C4⋊Dic61—21no—
96C4⋊Dic61—41no—
97C4⋊Dic61—41no—
98#11758196—41no—
99#11758196—41no—
100C4×Dic61—41no—
101C4×Dic61—41no—
102C4×Dic61C2×D411yescharacteristic
103C4×Dic61C2×D411yescharacteristic
104C4×Dic61#4111yescharacteristic
105C2×#92418C22⋊C412yes—
106C8×D61—24no—
107C22×D61#12111yescharacteristic
108C2×#6923—24no—
109C2×#6923—42no—
110C2×#6923—42no—
111Q8×C61—42no—
112Q8×C61—42no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61C2×C22⋊C411yescharacteristic
116C2×C4×C61C4×D411yescharacteristic
117C2×C4×C61C4×D411yescharacteristic
118C2×Dic61—41no—
119C2×Dic61—41no—
120C2×Dic61#12111yescharacteristic
121C2×Dic61#12411yescharacteristic
122C2×Dic61#12711yescharacteristic
123C4×D61—22no—
124C4×D61—22no—
125C4×D61#13712yes—
126C4×D61#13712yes—
127Dic122—42no—
128Dic122—42no—
129#92420—24no—
130#92420—81no—
131#92420—81no—
132C8×C61—22no—
133#92418#10414yes—
134#9850—611nomaximal, Sylow
135C22×C61#39011yescharacteristic
136C2×D61—21no—
137C2×D61—21no—
138C2×D61#47012yes—
139#6923—44no—
140C4×C61—81no—
141C4×C61—81no—
142C4×C61C2×#10412yes—
143C4×C61#44112yes—
144C4×C61#44412yes—
145Dic61—21no—
146Dic61—21no—
147Dic61—81no—
148Dic61—81no—
149Dic61#49711yescharacteristic
150Dic61#49711yescharacteristic
151#3170—611no—
152#1994—611no—
153#1994—611no—
154C4×#97—611no—
155#1566—611no—
156#1493—611no—
157#1493—611no—
158C2×C61#196211yescharacteristic
159C2×C61#197111yescharacteristic
160C2×C61#197511yescharacteristic
161D61—22no—
162#533—611no—
163#533—611no—
164#500—611no—
165#494—611no—
166#487—611no—
167#487#692311yescharacteristic, complemented
168#467—1222no—
169C4×Q16—1222no—
170#423—611no—
171#423—611no—
172C2×#97—1222no—
173C2×C4×C8—611no—
174C2×C4×C8—611no—
175C43—611no—
176#339—1222no—
177#339—1222no—
178#334—1222no—
179Q8⋊C8—1222no—
180Q8⋊C8—1222no—
181C61#985011yescharacteristic, complemented, Sylow
182C2×Q16—1222no—
183C2×M16—1221no—
184C2×M16—1221no—
185C22×C8—612no—
186C22×C8—1221no—
187C4⋊Q8—611no—
188C4⋊Q8—611no—
189C4⋊Q8C2×#692311yescharacteristic
190C4⋊Q8C2×#692311yescharacteristic
191#125—1221no—
192#125—1221no—
193C22⋊Q8—1221no—
194C22⋊Q8—1221no—
195C22⋊Q8—1221no—
196C22⋊Q8—1221no—
197C4×Q8—1221no—
198C4×Q8—1221no—
199C4×Q8—1221no—
200C4×Q8—1221no—
201#117—1221no—
202#117—1221no—
203C2×C42—611no—
204C2×C42—612no—
205C2×C42—1221no—
206C2×C42—1221no—
207#107—1222no—
208C4⋊C8—1221no—
209C4⋊C8—1221no—
210C4⋊C8—1221no—
211C4⋊C8—1221no—
212#103—22no—
213#103—1222no—
214#97—1228no—
215C4×C8—612no—
216C4×C8—612no—
217C4×C8—611no—
218C4×C8—1221no—
219C4×C8C2×#692311yescharacteristic
220C2×Q8—21no—
221C2×Q8—21no—
222C2×Q8—1221no—
223C2×Q8—1221no—
224C22×C4—611no—
225C22×C4—611no—
226C22×C4—611no—
227C22×C4—1222no—
228Q16—2444no—
229M16—2442no—
230M16—2442no—
231C2×C8—612no—
232C2×C8—1224no—
233C2×C8—1222no—
234C2×C8—1224no—
235C2×C8—2441no—
236C2×C8—2441no—
237C2×C8C22⋊#692312yes—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—1221no—
243C4⋊C4—1221no—
244C4⋊C4—1221no—
245C4⋊C4—1221no—
246C4⋊C4—2441no—
247C4⋊C4—2441no—
248C4⋊C4—2441no—
249C4⋊C4—2441no—
250C22⋊C4—2441no—
251C22⋊C4—2441no—
252C42—611no—
253C42—611no—
254C42—611no—
255C42—1222no—
256C42—1222no—
257C42—1224no—
258C42—2441no—
259C42—2441no—
260C42—2442no—
261C42C22×#692311yescharacteristic
262C23—611no—
263Q8—42no—
264Q8—42no—
265Q8—2442no—
266Q8—2442no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—611no—
270C2×C4—612no—
271C2×C4—612no—
272C2×C4—611no—
273C2×C4—611no—
274C2×C4—1222no—
275C2×C4—1222no—
276C2×C4—1224no—
277C2×C4—2441no—
278C2×C4—2441no—
279C2×C4—2442no—
280C2×C4—2442no—
281C2×C4C2×C22⋊#692311yescharacteristic
282C2×C4D4×#692311yescharacteristic
283C2×C4D4×#692311yescharacteristic
284C8—22no—
285C8—614no—
286C8—1224no—
287C8—4881no—
288C8—4881no—
289C22—612no—
290C22—1221no—
291C22—1221no—
292C22#1684071411yescharacteristic
293C4—81no—
294C4—81no—
295C4—611no—
296C4—611no—
297C4—1221no—
298C4—1221no—
299C4—4881no—
300C4—4881no—
301C4—2444no—
302C4D61⋊#10412yes—
303C4#1684080212yes—
304C4#1684080712yes—
305C2—1222no—
306C2#2146195411yescharacteristic
307C2#2146196711yescharacteristic
308C2#2146197511yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#487C4
socleC22×C61#390
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#487#6923
hypocenterC61#9850
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
20. #16840165C4semidirect
21. #16840141C4semidirect
22. #16840113C4semidirect
26. C61×#487C4semidirect
167. #487#6923semidirect
181. C61#9850semidirect

Maximal subgroups

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

subgroupindex
#214620042
C4×#145198312
#214613752
#214611692
#214611682
#214564042
#985061