Finite Group Databasebeta

← #30600290

Subgroups — #30600290

Order 15616.

522 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 98 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#21462717C211yesmaximal, characteristic, complemented
2#21461118C212yesmaximal
3#21461063C211yesmaximal, characteristic, complemented
4#21460643C212yesmaximal, complemented
5#21448529C211yesmaximal, characteristic
6#16840812—24no—
7#16840783—22no—
8#16840778C2212yes—
9#16840778C2212yes—
10D4⋊#92420—24no—
11#16840608—22no—
12C2×#14519764C411yescharacteristic
13C2×#14519742C2211yescharacteristic
14C2×D4×Dic61C2211yescharacteristic
15#16839668C411yescharacteristic
16#16839656—22no—
17#16839634—22no—
18#16839631C411yescharacteristic
19#16839608C2211yescharacteristic
20C61×#390C411yescharacteristic
21C4⋊#92420—24no—
22C4⋊#92420—22no—
23C4⋊#92420—22no—
24C22⋊#92420—22no—
25C22⋊#92420—22no—
26#14519832—24no—
27C4×#92420—22no—
28C4×#92420D414yes—
29C22×Dic122C2×C411yescharacteristic
30C23×Dic61—21no—
31#14519764—22no—
32#14519764—22no—
33#14519742—22no—
34#14519742—22no—
35D4×Dic61—22no—
36D4×Dic61—22no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—22no—
39C2×#11758196D412yes—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61—21no—
42C2×C4×Dic61C2×C411yescharacteristic
43C2×C4×Dic61D412yes—
44C2×C4×Dic61C2311yescharacteristic
45#14519684—22no—
46#14519684—21no—
47#14519684—21no—
48C22×D4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61—22no—
51C2×C42×C61C2×C411yescharacteristic
52C24×C61—21no—
53C2×D4×C61—22no—
54C2×D4×C61—21no—
55C2×D4×C61—22no—
56C2×D4×C61M1612yes—
57C22×C4×C61—21no—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22×C411yescharacteristic
60C22×Dic61—21no—
61C22×Dic61—21no—
62C22×Dic61—42no—
63C22×Dic61—82no—
64C22×Dic61C22⋊C412yes—
65C22×Dic61C2×D411yescharacteristic
66C22×Dic61C2×D411yescharacteristic
67C2×Dic122—22no—
68C2×Dic122—21no—
69C2×Dic122—21no—
70C2×Dic122—22no—
71C2×#92420—24no—
72C2×#92420—42no—
73C4⋊C4×C61—22no—
74C22⋊C4×C61—82no—
75C42×C61—22no—
76C22⋊Dic61—28no—
77C22⋊Dic61—41no—
78C22⋊Dic61—41no—
79C4⋊Dic61—22no—
80#11758196—24no—
81#11758196—82no—
82C4×Dic61—21no—
83C4×Dic61—22no—
84C4×Dic61—22no—
85C4×Dic61M1612yes—
86C4×Dic61C2×D412yes—
87C4×Dic61#4112yes—
88C23×C61—22no—
89C23×C61—42no—
90C23×C61—42no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61C2×C22⋊C411yescharacteristic
94D4×C61—24no—
95D4×C61—42no—
96C2×C4×C61—22no—
97C2×C4×C61—42no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61C4×D412yes—
102C2×C4×C61C2×M1611yescharacteristic
103C2×C4×C61C2×M1611yescharacteristic
104C2×C4×C61#13112yes—
105C2×Dic61—22no—
106C2×Dic61—22no—
107C2×Dic61—22no—
108C2×Dic61—22no—
109C2×Dic61—41no—
110C2×Dic61—41no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61—42no—
114C2×Dic61—41no—
115C2×Dic61—84no—
116C2×Dic61#12011yescharacteristic
117C2×Dic61#12112yes—
118C2×Dic61#12311yescharacteristic
119Dic122—42no—
120Dic122—42no—
121#92420—44no—
122#92420—82no—
123#9106—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—44no—
127C22×C61—42no—
128C22×C61—82no—
129C22×C61—81no—
130C22×C61—81no—
131C22×C61#38611yescharacteristic
132C22×C61#40711yescharacteristic
133C22×C61#40811yescharacteristic
134C4×C61—42no—
135C4×C61—81no—
136C4×C61#43512yes—
137C4×C61#43612yes—
138Dic61—22no—
139Dic61—21no—
140Dic61—21no—
141Dic61—42no—
142Dic61—42no—
143Dic61—81no—
144#2983—611no—
145#2326—611no—
146#1478—612no—
147#1461—611no—
148#1274—612no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61—82no—
152C2×C61#187011yescharacteristic
153C2×C61#191011yescharacteristic
154C2×C61#191211yescharacteristic
155C2×#124—611no—
156C2×C4×Q8—611no—
157C2×C4×D4—611no—
158#446—1224no—
159#432—1222no—
160#431—612no—
161#431—612no—
162#391—611no—
163#390#692311yescharacteristic
164#388—1222no—
165#385—1222no—
166#384—611no—
167#376—611no—
168D4⋊C8—1224no—
169#324—1222no—
170C61#910611yescharacteristic, complemented, Sylow
171C22×Q8—611no—
172C22×D4C2×#692311yescharacteristic
173C23×C4—1221no—
174#124—1222no—
175#124—1222no—
176C4×Q8—1222no—
177C4×Q8—1222no—
178C4×D4—1222no—
179C4×D4—1222no—
180C2×C4⋊C4—612no—
181C2×C4⋊C4—611no—
182C2×C4⋊C4—1222no—
183C2×C4⋊C4C2×#692311yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—1222no—
186C2×C22⋊C4—1221no—
187C2×C42—611no—
188C2×C42—611no—
189C2×C42—612no—
190C2×C42C2×#692311yescharacteristic
191C4⋊C8—1222no—
192C4⋊C8—1222no—
193C4⋊C8—1224no—
194C22⋊C8—1222no—
195C22⋊C8—1222no—
196#97—1224no—
197C4×C8—614no—
198C4×C8—1222no—
199#95—1221no—
200#95—1221no—
201#95—1222no—
202C24—21no—
203C2×Q8—1222no—
204C2×Q8—1221no—
205C2×Q8—1221no—
206C2×Q8—1222no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—22no—
210C2×D4#1175821312yes—
211C22×C4—21no—
212C22×C4—611no—
213C22×C4—611no—
214C22×C4—612no—
215C22×C4—1221no—
216C22×C4—1221no—
217C22×C4—4882no—
218C22×C4—2442no—
219C22×C4C22⋊#692312yes—
220C22×C4C22×#692311yescharacteristic
221C2×C8—1224no—
222C2×C8—2442no—
223C4⋊C4—22no—
224C4⋊C4—1224no—
225C4⋊C4—1222no—
226C4⋊C4—4882no—
227C22⋊C4—82no—
228C22⋊C4—1228no—
229C22⋊C4—2441no—
230C22⋊C4—2441no—
231C42—22no—
232C42—612no—
233C42—612no—
234C42—612no—
235C42—1222no—
236C42—1222no—
237C42—1221no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23C2×C22⋊#692311yescharacteristic
244Q8—2442no—
245Q8—2442no—
246D4—24no—
247D4—42no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—611no—
254C2×C4—612no—
255C2×C4—611no—
256C2×C4—1222no—
257C2×C4—1222no—
258C2×C4—1222no—
259C2×C4—1222no—
260C2×C4—4884no—
261C2×C4—2441no—
262C2×C4—2441no—
263C2×C4—2442no—
264C2×C4—2441no—
265C2×C4—2441no—
266C2×C4—2441no—
267C2×C4D4×#692312yes—
268C2×C4D61⋊M1611yescharacteristic
269C2×C4C2×#1175821311yescharacteristic
270C2×C4#1451986412yes—
271C8—4882no—
272C8—2444no—
273C22—21no—
274C22—21no—
275C22—44no—
276C22—42no—
277C22—82no—
278C22—81no—
279C22—81no—
280C22#1684071011yescharacteristic
281C22#1684073311yescharacteristic
282C22#1684073611yescharacteristic
283C4—42no—
284C4—81no—
285C4—1222no—
286C4—1221no—
287C4—1221no—
288C4—4881no—
289C4—2442no—
290C4—2442no—
291C4#1684079112yes—
292C4#1684079512yes—
293C2—21no—
294C2—21no—
295C2—82no—
296C2#2146179711yescharacteristic
297C2#2146186211yescharacteristic
298C2#2146186711yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#9106
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21462717C2semidirect
3. #21461063C2semidirect
4. #21460643C2semidirect
170. C61#9106semidirect

Maximal subgroups

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

subgroupindex
#214627172
#214611182
#214610632
#214606432
#214485292
#910661