Finite Group Databasebeta

← #30601635

Subgroups — #30601635

Order 15616.

610 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 130 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#21461751C211yesmaximal, characteristic, complemented
2C4×#14519832C211yesmaximal, characteristic, complemented
3C2×Q8⋊#92420C212yesmaximal, complemented
4C2×#16840610C212yesmaximal
5#21448362C211yesmaximal, characteristic
6C2×C4⋊#92420—22no—
7C2×C4⋊#92420C2212yes—
8C2×#14519832—22no—
9C2×C4×#92420C2212yes—
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—44no—
26#14519832—28no—
27C4×#92420—22no—
28C4×#92420D414yes—
29C22×D244C2×C411yescharacteristic
30C22×C4×D61—21no—
31#14519766—22no—
32#14519766—22no—
33Q8×Dic61—22no—
34Q8×Dic61—22no—
35#14519739—22no—
36#14519739—22no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×#11758198—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61C2×C411yescharacteristic
44C2×C4×Dic61D412yes—
45C2×C4×Dic61D412yes—
46C2×C4×Dic61C2311yescharacteristic
47#14519684—22no—
48C22×Q8×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C42×C61C2×C411yescharacteristic
51#95×C61—22no—
52C23×D61—21no—
53C2×Q8×C61—21no—
54C2×Q8×C61—22no—
55C2×Q8×C61—22no—
56C2×Q8×C61M1612yes—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61C22⋊C412yes—
60C22×C4×C61C22×C411yescharacteristic
61C22×Dic61—21no—
62C22×Dic61C22⋊C412yes—
63C22×Dic61C2×D411yescharacteristic
64C22×Dic61C2×D411yescharacteristic
65C2×D244—22no—
66C2×D244—22no—
67C2×D244—21no—
68C2×D244M1612yes—
69C2×C4×D61—41no—
70C2×C4×D61—41no—
71C2×C4×D61—42no—
72C2×C4×D61—42no—
73C2×#92420—28no—
74C2×#92420—24no—
75C2×#92420—42no—
76C2×#92420—44no—
77C4⋊C4×C61—22no—
78C42×C61—22no—
79#11758198—28no—
80#11758198—44no—
81#11758198—42no—
82C4⋊Dic61—22no—
83C4⋊Dic61—42no—
84C4×Dic61—24no—
85C4×Dic61—21no—
86C4×Dic61—21no—
87C4×Dic61—22no—
88C4×Dic61—22no—
89C4×Dic61—41no—
90C4×Dic61—41no—
91C4×Dic61C2×D412yes—
92C4×Dic61#4112yes—
93C23×C61C2×C22⋊C411yescharacteristic
94C22×D61—22no—
95C22×D61—41no—
96C22×D61—41no—
97C22×D61—42no—
98C22×D61—42no—
99Q8×C61—24no—
100Q8×C61—42no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—42no—
108C2×C4×C61#10414yes—
109C2×C4×C61C4×D412yes—
110C2×C4×C61C2×M1611yescharacteristic
111C2×C4×C61C2×M1611yescharacteristic
112C2×C4×C61#13112yes—
113C2×Dic61—22no—
114C2×Dic61—22no—
115C2×Dic61—22no—
116C2×Dic61—22no—
117C2×Dic61—41no—
118C2×Dic61—42no—
119C2×Dic61#12112yes—
120C2×Dic61#12411yescharacteristic
121C2×Dic61#12711yescharacteristic
122C2×Dic61AGL(1,ℤ/8)12yes—
123C2×Dic61#13712yes—
124D244—24no—
125D244—42no—
126C4×D61—82no—
127C4×D61—81no—
128C4×D61—81no—
129#92420—28no—
130#92420—82no—
131#9646—611nomaximal, Sylow
132C22×C61#39011yescharacteristic
133C22×C61#40711yescharacteristic
134C22×C61#40811yescharacteristic
135C22×C61#41712yes—
136C22×C61C2×#10412yes—
137C2×D61—42no—
138C2×D61—44no—
139C2×D61—81no—
140C2×D61—81no—
141C2×D61—82no—
142C4×C61—24no—
143C4×C61—42no—
144C4×C61—81no—
145C4×C61#43512yes—
146C4×C61#43512yes—
147Dic61—22no—
148Dic61—42no—
149Dic61—81no—
150Dic61#48912yes—
151Dic61#49611yescharacteristic
152Dic61#49711yescharacteristic
153C4×#124—611no—
154#1842—611no—
155C4×#97—611no—
156C2×Q8⋊C8—612no—
157C2×D4⋊C8—612no—
158C2×C61#148312yes—
159C2×C61#148912yes—
160C2×C61#195511yescharacteristic
161C2×C61#195911yescharacteristic
162C2×C61#196211yescharacteristic
163D61—82no—
164C2×#124—611no—
165C2×C4×Q8—611no—
166C2×C4×D4—611no—
167C2×C4⋊C8—612no—
168C2×C4⋊C8—1222no—
169C2×#97—1222no—
170C2×C4×C8—612no—
171#391#692311yescharacteristic
172#390—611no—
173#388—1222no—
174#382—611no—
175C4×C22⋊C4—1222no—
176C43—611no—
177Q8⋊C8—1224no—
178D4⋊C8—1224no—
179C61#964611yescharacteristic, complemented, Sylow
180C22×Q8C2×#692311yescharacteristic
181C22×D4—611no—
182C23×C4—1221no—
183C22×C8—1222no—
184C22×C8—1222no—
185#124—1222no—
186#124—1222no—
187C4×Q8—1222no—
188C4×Q8—1222no—
189C4×D4—1222no—
190C4×D4—1222no—
191C2×C4⋊C4—611no—
192C2×C4⋊C4—1221no—
193C2×C4⋊C4C2×#692311yescharacteristic
194C2×C22⋊C4—1222no—
195C2×C22⋊C4—1221no—
196C2×C22⋊C4—1222no—
197C2×C42—611no—
198C2×C42—612no—
199C2×C42—612no—
200C2×C42—611no—
201C2×C42—1221no—
202C2×C42C2×#692311yescharacteristic
203C4⋊C8—1222no—
204C4⋊C8—1222no—
205C4⋊C8—2444no—
206#97—1228no—
207C4×C8—614no—
208C4×C8—1222no—
209#95—22no—
210#95—1222no—
211C24—1221no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8#1175821312yes—
216C2×D4—612no—
217C2×D4—1222no—
218C2×D4—1221no—
219C2×D4—1222no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—611no—
223C22×C4—611no—
224C22×C4—612no—
225C22×C4—1221no—
226C22×C4—2441no—
227C22×C4—2441no—
228C22×C4—2442no—
229C22×C4—2442no—
230C22×C4C22⋊#692312yes—
231C22×C4C22×#692311yescharacteristic
232C2×C8—1224no—
233C2×C8—1228no—
234C2×C8—2442no—
235C2×C8—2444no—
236C4⋊C4—22no—
237C4⋊C4—1222no—
238C4⋊C4—2442no—
239C22⋊C4—1228no—
240C22⋊C4—2442no—
241C22⋊C4—2444no—
242C42—22no—
243C42—612no—
244C42—612no—
245C42—1222no—
246C42—1222no—
247C42—1224no—
248C42—1221no—
249C42—1221no—
250C42—2441no—
251C42—2441no—
252C23—1222no—
253C23—2441no—
254C23—2441no—
255C23—2442no—
256C23—2442no—
257C23C2×C22⋊#692311yescharacteristic
258Q8—24no—
259Q8—42no—
260D4—1224no—
261D4—2442no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—611no—
270C2×C4—611no—
271C2×C4—612no—
272C2×C4—612no—
273C2×C4—612no—
274C2×C4—1222no—
275C2×C4—1222no—
276C2×C4—1222no—
277C2×C4—1222no—
278C2×C4—4882no—
279C2×C4—4881no—
280C2×C4—4881no—
281C2×C4—2441no—
282C2×C4—2442no—
283C2×C4#1451984214yes—
284C2×C4D4×#692312yes—
285C2×C4D61⋊M1611yescharacteristic
286C2×C4C2×#1175821311yescharacteristic
287C2×C4#1451986412yes—
288C8—1228no—
289C8—4882no—
290C22—4882no—
291C22—4881no—
292C22—4881no—
293C22—2444no—
294C22—2442no—
295C22#1684071411yescharacteristic
296C22#1684073311yescharacteristic
297C22#1684073511yescharacteristic
298C22#1684075312yes—
299C22C2×#1451984212yes—
300C4—24no—
301C4—42no—
302C4—81no—
303C4—612no—
304C4—611no—
305C4—611no—
306C4—1222no—
307C4—4881no—
308C4—2442no—
309C4#1684079112yes—
310C4#1684079412yes—
311C2—4882no—
312C2#2146113612yes—
313C2#2146115712yes—
314C2#2146194411yescharacteristic
315C2#2146194911yescharacteristic
316C2#2146195211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC2×C4×Dic61C23
hypercenter#391#6923
hypocenterC61#9646
LCSG ▷ C22×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. #21461751C2semidirect
2. C4×#14519832C2semidirect
3. C2×Q8⋊#92420C2semidirect
179. C61#9646semidirect

Maximal subgroups

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

subgroupindex
#214617512
C4×#145198322
C2×Q8⋊#924202
C2×#168406102
#214483622
#964661