Finite Group Databasebeta

← #30601779

Subgroups — #30601779

Order 15616.

596 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 102 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#21461604C211yesmaximal, characteristic, complemented
2#21461536C211yesmaximal, characteristic, complemented
3C2×#16840618C214yesmaximal, complemented
4#21454177C211yesmaximal, characteristic
5C2×#14519844—21no—
6C2×#14519832C2212yes—
7C2×#14519832C2212yes—
8#16840628—21no—
9#16840628—21no—
10#16840628—21no—
11#16840618—28no—
12C2×#14519774C411yescharacteristic
13C2×#14519766C412yes—
14C2×#14519766C2212yes—
15#16839663—21no—
16#16839663—21no—
17#16839663—21no—
18#16839663—21no—
19C42×Dic61C2211yescharacteristic
20C2×C4⋊Q8×C61C411yescharacteristic
21C22×#92420—22no—
22C22×#92420—22no—
23#14519844—24no—
24#14519832—24no—
25#14519832—22no—
26#14519832D414yescomplemented
27C22×D244—21no—
28C22×D244—21no—
29C22×D244C2×C412yes—
30#14519774—42no—
31#14519766—24no—
32#14519766—24no—
33#14519766—44no—
34C2×#11758198—22no—
35C2×#11758198—22no—
36C2×#11758198—22no—
37C2×#11758198—22no—
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43C2×C4×Dic61C2311yescharacteristic
44C22×Q8×C61C2×C412yes—
45C4⋊Q8×C61—42no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C42×C61C2×C411yescharacteristic
49C23×D61—22no—
50C2×Q8×C61—24no—
51C2×Q8×C61—42no—
52C2×Q8×C61—42no—
53C22×C4×C61—22no—
54C22×C4×C61C22⋊C411yescharacteristic
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22×C411yescharacteristic
57C22×Dic61C22⋊C411yescharacteristic
58C22×Dic61C22⋊C411yescharacteristic
59C22×Dic61C2×D411yescharacteristic
60C22×Dic61C2×D411yescharacteristic
61C2×D244—24no—
62C2×D244—42no—
63C2×D244—42no—
64C2×D244—42no—
65C2×D244—42no—
66C2×D244—42no—
67C2×D244—42no—
68C2×D244—42no—
69C2×D244—42no—
70C2×#92420—24no—
71C2×#92420—24no—
72C2×#92420—42no—
73C2×#92420—42no—
74C2×#92420—42no—
75C2×#92420—42no—
76C4⋊C4×C61—42no—
77C4⋊C4×C61—42no—
78C42×C61—22no—
79#11758198—44no—
80#11758198—44no—
81#11758198—44no—
82#11758198—42no—
83#11758198—42no—
84C4×Dic61—22no—
85C4×Dic61—22no—
86C4×Dic61—41no—
87C4×Dic61—41no—
88C4×Dic61—41no—
89C4×Dic61C2×D412yes—
90C4×Dic61#4112yes—
91C23×C61C2×C22⋊C411yescharacteristic
92C22×D61—42no—
93C22×D61—42no—
94C22×D61—42no—
95C22×D61—42no—
96C22×D61—42no—
97C22×D61—42no—
98C22×D61—42no—
99Q8×C61—84no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61#10012yes—
110C2×C4×C61#10412yes—
111C2×C4×C61C4×D412yes—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61#10012yes—
122C2×Dic61#12112yes—
123C2×Dic61#12411yescharacteristic
124C2×Dic61#12711yescharacteristic
125D244—84no—
126D244—84no—
127D244—84no—
128#92420—44no—
129#92420—44no—
130#9689—611nomaximal, Sylow
131C22×C61#39011yescharacteristic
132C22×C61C2×#10011yescharacteristic
133C22×C61C2×#10011yescharacteristic
134C22×C61C2×#10411yescharacteristic
135C22×C61#42111yescharacteristic
136C22×C61#42111yescharacteristic
137C22×C61#42111yescharacteristic
138C2×D61—82no—
139C2×D61—82no—
140C2×D61—82no—
141C2×D61—82no—
142C2×D61—82no—
143C2×D61—82no—
144C2×D61—82no—
145C4×C61—22no—
146C4×C61—22no—
147C4×C61—22no—
148C4×C61—82no—
149Dic61—22no—
150Dic61—21no—
151Dic61—21no—
152Dic61—22no—
153Dic61—41no—
154#2858—611no—
155#1767—611no—
156#1745—611no—
157C2×#329—614no—
158C2×C61#152711yescharacteristic
159C2×C61#153011yescharacteristic
160C2×C61#153311yescharacteristic
161C2×C61#153611yescharacteristic
162C2×C61#196311yescharacteristic
163C2×C61#196511yescharacteristic
164C2×C61#198011yescharacteristic
165D61—162no—
166C2×C4⋊Q8#692311yescharacteristic
167C2×#127—611no—
168C2×#124—612no—
169C2×#124—612no—
170C2×C4⋊C8—1221no—
171C2×#97—612no—
172C2×#97—612no—
173#390—1221no—
174#390—1221no—
175#390—1221no—
176#390—1221no—
177C43—611no—
178#336—1221no—
179#336—1221no—
180#336—1221no—
181#329—1228no—
182C61#968911yescharacteristic, complemented, Sylow
183C22×Q8C2×#692312yes—
184C22×D4—612no—
185C22×D4—1221no—
186C22×D4—1221no—
187C22×C8—1222no—
188C22×C8—1222no—
189C4⋊Q8—42no—
190#127—2442no—
191#124—1224no—
192#124—1224no—
193#124—2444no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C22⋊C4—1222no—
197C2×C22⋊C4—1222no—
198C2×C22⋊C4—1222no—
199C2×C22⋊C4—1222no—
200C2×C42—611no—
201C2×C42—611no—
202C2×C42—611no—
203C2×C42—611no—
204C2×C42—611no—
205C2×C42—611no—
206C2×C42C2×#692311yescharacteristic
207C4⋊C8—1224no—
208#97—614no—
209#97—1222no—
210#97—1224no—
211C24—1222no—
212C2×Q8—24no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×D4—1224no—
216C2×D4—2442no—
217C2×D4—2442no—
218C2×D4—2442no—
219C2×D4—2442no—
220C2×D4—2442no—
221C2×D4—2442no—
222C2×D4—2442no—
223C2×D4—2442no—
224C22×C4—22no—
225C22×C4—611no—
226C22×C4—611no—
227C22×C4—611no—
228C22×C4—611no—
229C22×C4C22⋊#692311yescharacteristic
230C22×C4C22⋊#692311yescharacteristic
231C22×C4C22×#692311yescharacteristic
232C2×C8—1224no—
233C2×C8—1224no—
234C2×C8—2442no—
235C2×C8—2442no—
236C2×C8—2442no—
237C2×C8—2442no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C22⋊C4—2444no—
241C22⋊C4—2442no—
242C22⋊C4—2442no—
243C22⋊C4—2444no—
244C22⋊C4—2444no—
245C42—22no—
246C42—612no—
247C42—612no—
248C42—1222no—
249C42—1222no—
250C42—2441no—
251C42—2441no—
252C42—2441no—
253C23—2442no—
254C23—2442no—
255C23—2442no—
256C23—2442no—
257C23—2442no—
258C23—2442no—
259C23—2442no—
260C23C2×C22⋊#692311yescharacteristic
261Q8—84no—
262D4—4884no—
263D4—4884no—
264D4—4884no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—612no—
275C2×C4—611no—
276C2×C4—611no—
277C2×C4—612no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—1221no—
287C2×C4#1451983712yes—
288C2×C4#1451984212yes—
289C2×C4D4×#692312yes—
290C8—2444no—
291C8—2444no—
292C22—4882no—
293C22—4882no—
294C22—4882no—
295C22—4882no—
296C22—4882no—
297C22—4882no—
298C22—4882no—
299C22#1684071411yescharacteristic
300C22C2×#1451983711yescharacteristic
301C22D61⋊#10011yescharacteristic
302C22C2×#1451984211yescharacteristic
303C22#1684076111yescharacteristic
304C22#1684076111yescharacteristic
305C22#1684076111yescharacteristic
306C4—22no—
307C4—22no—
308C4—22no—
309C4—82no—
310C4—1222no—
311C4—1222no—
312C4—1221no—
313C4—1221no—
314C4—2441no—
315C2—9762no—
316C2#2146128011yescharacteristic
317C2#2146129011yescharacteristic
318C2#2146130011yescharacteristic
319C2#2146130911yescharacteristic
320C2#2146195511yescharacteristic
321C2#2146196111yescharacteristic
322C2#2146198211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519774C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#9689
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21461604C2semidirect
2. #21461536C2semidirect
3. C2×#16840618C2semidirect
26. #14519832D4semidirect
182. C61#9689semidirect

Maximal subgroups

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

subgroupindex
#214616042
#214615362
C2×#168406182
#214541772
#968961