Finite Group Databasebeta

← #29595190

Subgroups — #29595190

Order 15616.

532 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 115 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#21448362C211yesmaximal, characteristic
2C2×#16839558C212yesmaximal
3#101×Dic61C212yesmaximal, complemented
4#21443946C211yesmaximal, characteristic, complemented
5C61×#1394C211yesmaximal, characteristic, complemented
6C2×#14519766C2211yescharacteristic
7C2×Q8×Dic61C2211yescharacteristic
8C2×#14519739C411yescharacteristic
9C2×#14519696C2212yes—
10M16×Dic61—22no—
11#16839664C411yescharacteristic
12#16839654—22no—
13#16839627C411yescharacteristic
14#16839609—22no—
15C42×Dic61C411yescharacteristic
16#16839558C418yescomplemented
17#16839530—22no—
18C2×#101×C61C2212yes—
19C61×#391C2211yescharacteristic
20C22×D244C2×C411yescharacteristic
21C22×C4×D61—21no—
22C2×#11758194—22no—
23#14519766—21no—
24#14519766—21no—
25#14519766C2×C414yes—
26Q8×Dic61—41no—
27Q8×Dic61—41no—
28#14519739—41no—
29#14519739—41no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×C4⋊Dic61D412yes—
32C2×#11758198—22no—
33C2×#11758198—21no—
34C2×#11758198—22no—
35C2×C4×Dic61—21no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61C2×C411yescharacteristic
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61D412yes—
40#14519696—22no—
41#14519696C2×C414yescomplemented
42#14519687—42no—
43C8×Dic61—42no—
44#14519684—22no—
45C22×Q8×C61C2311yescharacteristic
46C2×M16×C61—22no—
47C2×C4⋊C4×C61C2×C411yescharacteristic
48C2×C42×C61C2×C411yescharacteristic
49#101×C61—22no—
50#101×C61C2×C414yescomplemented
51#95×C61—22no—
52C23×D61—21no—
53C2×Q8×C61—41no—
54C2×Q8×C61—41no—
55C2×Q8×C61C22×C412yes—
56C2×Q8×C61#4111yescharacteristic
57C2×Q8×C61#4111yescharacteristic
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61C22×C411yescharacteristic
61C22×C4×C61C2×D412yes—
62C22×Dic61—21no—
63C22×Dic61C22⋊C411yescharacteristic
64C22×Dic61C22⋊C412yes—
65C22×Dic61C22⋊C411yescharacteristic
66C2×D244—21no—
67C2×D244—41no—
68C2×D244—41no—
69C2×D244C4212yes—
70C2×C4×D61—44no—
71C2×C4×D61—41no—
72C2×C4×D61—41no—
73M16×C61—24no—
74M16×C61—42no—
75C2×C8×C61—42no—
76C4⋊C4×C61—41no—
77C42×C61—41no—
78#11758198—24no—
79#11758198—24no—
80#11758198—42no—
81#11758198—44no—
82C4⋊Dic61—41no—
83C4⋊Dic61—42no—
84C4×Dic61—21no—
85C4×Dic61—22no—
86C4×Dic61—22no—
87C4×Dic61—21no—
88C4×Dic61—21no—
89C4×Dic61—42no—
90C4×Dic61—42no—
91C4×Dic61C4212yes—
92#11758194—24no—
93#11758194—42no—
94C2×#92418—42no—
95C23×C61C2×C22⋊C411yescharacteristic
96C22×D61—22no—
97C22×D61—42no—
98C22×D61—42no—
99C22×D61—41no—
100C22×D61—41no—
101Q8×C61—42no—
102Q8×C61—41no—
103Q8×C61—41no—
104C2×C4×C61—22no—
105C2×C4×C61—21no—
106C2×C4×C61—22no—
107C2×C4×C61—21no—
108C2×C4×C61—22no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—42no—
112C2×C4×C61—41no—
113C2×C4×C61C2×C4211yescharacteristic
114C2×C4×C61#11711yescharacteristic
115C2×C4×C61C4×D412yes—
116C2×C4×C61C4×D412yes—
117C2×Dic61—22no—
118C2×Dic61—22no—
119C2×Dic61—21no—
120C2×Dic61—22no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—42no—
125C2×Dic61#9911yescharacteristic
126C2×Dic61#9911yescharacteristic
127D244—42no—
128D244—42no—
129C4×D61—82no—
130C4×D61—82no—
131C8×C61—44no—
132#92418—44no—
133C4×#355—611nomaximal, Sylow
134C22×C61—21no—
135C22×C61—21no—
136C22×C61C4×C22⋊C411yescharacteristic
137C22×C61C2×#9911yescharacteristic
138C22×C61#41011yescharacteristic
139C2×D61—44no—
140C2×D61—42no—
141C2×D61—82no—
142C2×D61—81no—
143C2×D61—81no—
144C4×C61—22no—
145C4×C61—22no—
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—42no—
149C4×C61—81no—
150Dic61—21no—
151Dic61—42no—
152Dic61—41no—
153Dic61—42no—
154Dic61#35512yescomplemented
155C4×#124—611no—
156C2×#355—612no—
157C4×#101—612no—
158#1394—611no—
159#1394D6111yescharacteristic, complemented
160C2×C61—21no—
161C2×C61—21no—
162C2×C61C4×#9911yescharacteristic
163C2×C61C2×#35511yescharacteristic
164C2×C61#213211yescharacteristic
165D61—82no—
166C2×#124—611no—
167C2×C4×Q8—611no—
168C2×C4×D4—611no—
169C2×#101—612no—
170C2×#101C2×D6112yes—
171C4×M16—1222no—
172#391C2×D6111yescharacteristic
173#390—611no—
174#388—1222no—
175#382—611no—
176C4×C22⋊C4—1222no—
177C43—611no—
178#355—618no—
179#343—1222no—
180C61C4×#35511yescharacteristic, complemented, Sylow
181C22×Q8C22×D6111yescharacteristic
182C22×D4—611no—
183C23×C4—1221no—
184C2×M16—22no—
185C2×M16—1222no—
186#124—614no—
187#124—1221no—
188#124—1221no—
189C4×Q8—2441no—
190C4×Q8—2441no—
191C4×D4—2441no—
192C4×D4—2441no—
193C2×C4⋊C4—611no—
194C2×C4⋊C4—612no—
195C2×C4⋊C4C4×D6111yescharacteristic
196C2×C22⋊C4—1222no—
197C2×C22⋊C4—1221no—
198C2×C22⋊C4—1222no—
199C2×C42—611no—
200C2×C42—611no—
201C2×C42—612no—
202C2×C42—1221no—
203C2×C42—1221no—
204C2×C42C4×D6111yescharacteristic
205#101—22no—
206#101—614no—
207#101—1222no—
208#101C4×D6114yescomplemented
209#97—2442no—
210C4×C8—2442no—
211#95—22no—
212#95—1222no—
213C24—1221no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8C2×C4×D6112yes—
217C2×Q8#1175822411yescharacteristic
218C2×Q8#1175822611yescharacteristic
219C2×D4—612no—
220C2×D4—1221no—
221C2×D4—2441no—
222C2×D4—2441no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—611no—
226C22×C4—612no—
227C22×C4—611no—
228C22×C4—1221no—
229C22×C4—2441no—
230C22×C4—2441no—
231C22×C4—2444no—
232C22×C4C2×C4×D6111yescharacteristic
233C22×C4D4×D6112yes—
234M16—24no—
235M16—42no—
236M16—1224no—
237M16—2442no—
238C2×C8—42no—
239C2×C8—2442no—
240C4⋊C4—41no—
241C4⋊C4—2441no—
242C4⋊C4—2442no—
243C22⋊C4—1224no—
244C22⋊C4—1224no—
245C22⋊C4—2444no—
246C22⋊C4—2442no—
247C42—41no—
248C42—612no—
249C42—1222no—
250C42—1222no—
251C42—1221no—
252C42—1221no—
253C42—1221no—
254C42—2442no—
255C42—2442no—
256C23—1222no—
257C23—2442no—
258C23—2442no—
259C23—2441no—
260C23—2441no—
261C23C22⋊C4×D6111yescharacteristic
262Q8—42no—
263Q8—41no—
264Q8—41no—
265D4—2442no—
266D4—2442no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—611no—
277C2×C4—611no—
278C2×C4—1222no—
279C2×C4—1221no—
280C2×C4—1222no—
281C2×C4—1222no—
282C2×C4—4882no—
283C2×C4—4882no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4—2442no—
288C2×C4C42×D6111yescharacteristic
289C2×C4#1451973511yescharacteristic
290C2×C4#1451973712yes—
291C2×C4#1451973912yes—
292C8—44no—
293C8—2444no—
294C22—21no—
295C22—21no—
296C22—4882no—
297C22—4881no—
298C22—4881no—
299C22—2444no—
300C22—2442no—
301C22#1683960911yescharacteristic
302C22#99×D6111yescharacteristic
303C22#1683973411yescharacteristic
304C4—22no—
305C4—22no—
306C4—21no—
307C4—21no—
308C4—42no—
309C4—81no—
310C4—612no—
311C4—1221no—
312C4—2442no—
313C4—2441no—
314C4—2442no—
315C2—21no—
316C2—21no—
317C2—4882no—
318C2#2144560611yescharacteristic
319C2#355×D6111yescharacteristic
320C2#2144750311yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC2×C4×C61C2×C42
FrattiniC22×Q8C22×D61
FittingC61×#1394C2
socleC22×C61C2×#99
prime radicalC22×D244C2×C4
cosocleC22×Q8×C61C23
hypercenter#1394D61
hypocenterC61C4×#355
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #101×Dic61C2semidirect
4. #21443946C2semidirect
5. C61×#1394C2semidirect
16. #16839558C4semidirect
41. #14519696C2×C4semidirect
50. #101×C61C2×C4semidirect
154. Dic61#355semidirect
159. #1394D61semidirect
180. C61C4×#355semidirect
208. #101C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214483622
C2×#168395582
#101×Dic612
#214439462
C61×#13942
C4×#35561