Finite Group Databasebeta

← #30601612

Subgroups — #30601612

Order 15616.

618 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 142 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#21461736C211yesmaximal, characteristic, complemented
2C42×#92420C211yesmaximal, characteristic, complemented
3C2×Q8⋊#92420C212yesmaximal, complemented
4C2×#16840610C212yesmaximal
5#21448362C211yesmaximal, characteristic
6C2×C4⋊#92420C2212yes—
7C2×C4×#92420—22no—
8C2×C4×#92420C2212yes—
9#16840628—22no—
10Q8⋊#92420—24no—
11#16840610—24no—
12C2×#14519766C411yescharacteristic
13C2×Q8×Dic61C2211yescharacteristic
14C2×#14519739C2211yescharacteristic
15#16839664C411yescharacteristic
16#16839654—22no—
17#16839627C411yescharacteristic
18#16839609—22no—
19C42×Dic61C2211yescharacteristic
20C61×#391C411yescharacteristic
21C22×#92420—22no—
22C22×#92420—22no—
23C4⋊#92420—22no—
24C4⋊#92420—22no—
25C4×#92420—28no—
26C4×#92420—22no—
27C4×#92420D414yes—
28C22×D244C2×C411yescharacteristic
29C22×C4×D61—21no—
30#14519766—22no—
31#14519766C814yescomplemented
32Q8×Dic61—22no—
33Q8×Dic61—22no—
34#14519739—22no—
35#14519739—22no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758198—22no—
39C2×#11758198—22no—
40C2×#11758198—21no—
41C2×C4×Dic61—21no—
42C2×C4×Dic61C2×C411yescharacteristic
43C2×C4×Dic61D412yes—
44C2×C4×Dic61D412yes—
45C2×C4×Dic61C2311yescharacteristic
46#14519684—22no—
47C22×Q8×C61C2×C411yescharacteristic
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C42×C61C2×C411yescharacteristic
50#95×C61—22no—
51C23×D61—21no—
52C2×Q8×C61—21no—
53C2×Q8×C61—22no—
54C2×Q8×C61—22no—
55C2×Q8×C61C2×C812yes—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22×C411yescharacteristic
60C22×Dic61—21no—
61C22×Dic61C22⋊C412yes—
62C22×Dic61C2×D411yescharacteristic
63C22×Dic61C2×D411yescharacteristic
64C2×D244—22no—
65C2×D244—22no—
66C2×D244—21no—
67C2×D244C2×C812yes—
68C2×C4×D61—41no—
69C2×C4×D61—41no—
70C2×C4×D61—44no—
71C2×#92420—28no—
72C2×#92420—24no—
73C2×#92420—42no—
74C2×#92420—44no—
75C4⋊C4×C61—22no—
76C42×C61—22no—
77#11758198—28no—
78#11758198—44no—
79#11758198—42no—
80C4⋊Dic61—22no—
81C4⋊Dic61—42no—
82C4×Dic61—24no—
83C4×Dic61—24no—
84C4×Dic61—41no—
85C4×Dic61—41no—
86C4×Dic61C2×C812yes—
87C4×Dic61M1612yes—
88C4×Dic61C2×D412yes—
89C4×Dic61#4112yes—
90C23×C61C2×C22⋊C411yescharacteristic
91C22×D61—22no—
92C22×D61—41no—
93C22×D61—41no—
94C22×D61—42no—
95C22×D61—42no—
96Q8×C61—24no—
97Q8×C61—42no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—42no—
105C2×C4×C61#10414yes—
106C2×C4×C61C4×D412yes—
107C2×C4×C61C22×C811yescharacteristic
108C2×C4×C61C2×M1611yescharacteristic
109C2×C4×C61#13112yes—
110C2×Dic61—22no—
111C2×Dic61—22no—
112C2×Dic61—22no—
113C2×Dic61—41no—
114C2×Dic61—42no—
115C2×Dic61C22⋊C814yes—
116C2×Dic61#12112yes—
117C2×Dic61#12411yescharacteristic
118C2×Dic61#12711yescharacteristic
119C2×Dic61#13514yes—
120D244—24no—
121D244—42no—
122C4×D61—82no—
123C4×D61—82no—
124#92420—28no—
125#92420—82no—
126#9639—611nomaximal, Sylow
127C22×C61#39011yescharacteristic
128C22×C61C2×C22⋊C811yescharacteristic
129C22×C61#40811yescharacteristic
130C22×C61#41612yes—
131C22×C61C2×#10412yes—
132C2×D61—42no—
133C2×D61—44no—
134C2×D61—81no—
135C2×D61—81no—
136C2×D61—82no—
137C4×C61—24no—
138C4×C61—42no—
139C4×C61—81no—
140C4×C61C8×D412yes—
141C4×C61#43612yes—
142Dic61—22no—
143Dic61—24no—
144Dic61—81no—
145Dic61#48812yes—
146Dic61#49512yes—
147C4×#124—611no—
148#1831—611no—
149C42×C8—611no—
150C2×Q8⋊C8—612no—
151C2×D4⋊C8—612no—
152C2×C61#147612yes—
153C2×C61#149412yes—
154C2×C61#195411yescharacteristic
155C2×C61#195811yescharacteristic
156C2×C61#196211yescharacteristic
157D61—82no—
158C2×#124—611no—
159C2×C4×Q8—611no—
160C2×C4×D4—611no—
161C2×C4⋊C8—612no—
162C2×C4×C8—612no—
163C2×C4×C8—1222no—
164#391#692311yescharacteristic
165#390—611no—
166#388—1222no—
167#382—611no—
168C4×C22⋊C4—1222no—
169C43—611no—
170#336—1222no—
171Q8⋊C8—1224no—
172D4⋊C8—1224no—
173C61#963911yescharacteristic, complemented, Sylow
174C22×Q8C2×#692311yescharacteristic
175C22×D4—611no—
176C23×C4—1221no—
177C22×C8—1222no—
178C22×C8—1222no—
179#124—614no—
180#124—1222no—
181C4×Q8—1222no—
182C4×Q8—1222no—
183C4×D4—1222no—
184C4×D4—1222no—
185C2×C4⋊C4—611no—
186C2×C4⋊C4—1221no—
187C2×C4⋊C4C2×#692311yescharacteristic
188C2×C22⋊C4—1222no—
189C2×C22⋊C4—1221no—
190C2×C22⋊C4—1222no—
191C2×C42—611no—
192C2×C42—611no—
193C2×C42—612no—
194C2×C42—612no—
195C2×C42—1221no—
196C2×C42C2×#692311yescharacteristic
197C4⋊C8—1222no—
198C4⋊C8—1222no—
199C4×C8—614no—
200C4×C8—1222no—
201C4×C8—1228no—
202#95—22no—
203#95—1222no—
204C24—1221no—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8D61⋊C812yes—
209C2×D4—612no—
210C2×D4—1222no—
211C2×D4—1221no—
212C2×D4—1222no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—611no—
216C22×C4—612no—
217C22×C4—611no—
218C22×C4—1221no—
219C22×C4—2441no—
220C22×C4—2441no—
221C22×C4—2444no—
222C22×C4C22⋊#692312yes—
223C22×C4C22×#692311yescharacteristic
224C2×C8—1224no—
225C2×C8—1228no—
226C2×C8—2442no—
227C2×C8—2444no—
228C4⋊C4—22no—
229C4⋊C4—1222no—
230C4⋊C4—2442no—
231C22⋊C4—1228no—
232C22⋊C4—2442no—
233C22⋊C4—2444no—
234C42—22no—
235C42—612no—
236C42—612no—
237C42—612no—
238C42—612no—
239C42—1224no—
240C42—1224no—
241C42—2441no—
242C42—2441no—
243C23—1222no—
244C23—2441no—
245C23—2441no—
246C23—2442no—
247C23—2442no—
248C23C2×C22⋊#692311yescharacteristic
249Q8—24no—
250Q8—42no—
251D4—1224no—
252D4—2442no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—611no—
261C2×C4—611no—
262C2×C4—614no—
263C2×C4—612no—
264C2×C4—614no—
265C2×C4—1222no—
266C2×C4—1222no—
267C2×C4—1222no—
268C2×C4—4882no—
269C2×C4—4882no—
270C2×C4—2441no—
271C2×C4—2442no—
272C2×C4#1451984214yes—
273C2×C4D4×#692312yes—
274C2×C4C2×D61⋊C811yescharacteristic
275C2×C4D61⋊M1611yescharacteristic
276C2×C4#1451986412yes—
277C8—1228no—
278C8—4882no—
279C22—4882no—
280C22—4881no—
281C22—4881no—
282C22—2444no—
283C22—2442no—
284C22#1684071411yescharacteristic
285C22D61⋊(C22⋊C8)11yescharacteristic
286C22#1684073511yescharacteristic
287C22#1684075212yes—
288C22C2×#1451984212yes—
289C4—24no—
290C4—42no—
291C4—81no—
292C4—612no—
293C4—612no—
294C4—1224no—
295C4—1222no—
296C4—4881no—
297C4#1684078912yes—
298C4#1684079512yes—
299C2—4882no—
300C2#2146111312yes—
301C2#2146117212yes—
302C2#2146194211yescharacteristic
303C2#2146194811yescharacteristic
304C2#2146195211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC2×C4×Dic61C23
hypercenter#391#6923
hypocenterC61#9639
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #21461736C2semidirect
2. C42×#92420C2semidirect
3. C2×Q8⋊#92420C2semidirect
31. #14519766C8semidirect
173. C61#9639semidirect

Maximal subgroups

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

subgroupindex
#214617362
C42×#924202
C2×Q8⋊#924202
C2×#168406102
#214483622
#963961