Finite Group Databasebeta

← #29596284

Subgroups — #29596284

Order 15616.

566 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 87 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#21448402C211yesmaximal, characteristic
2#21445741C212yesmaximal
3#21443946C211yesmaximal, characteristic, complemented
4#21443945C212yesmaximal, complemented
5C61×#1394C211yesmaximal, characteristic, complemented
6C22×Q8×D61C2211yescharacteristic
7C2×#14519735C411yescharacteristic
8C2×#14519734C411yescharacteristic
9C2×#14519696C2212yes—
10C2×#14519695—22no—
11#16839725—22no—
12#16839669C2211yescharacteristic
13#16839668C411yescharacteristic
14#16839667C411yescharacteristic
15#16839666—22no—
16#16839624—22no—
17C2×#101×C61C2212yes—
18C61×#391C2211yescharacteristic
19C22×Dic122C2×C411yescharacteristic
20C22×Dic122D412yes—
21C2×Q8×D61—22no—
22C2×Q8×D61—21no—
23C2×Q8×D61—21no—
24C2×Q8×D61—42no—
25C2×Q8×D61—42no—
26C22×C4×D61C2×C411yescharacteristic
27C22×C4×D61D412yes—
28C2×#11758194—22no—
29#14519735—42no—
30#14519734—42no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×#11758196—21no—
33C2×#11758198—21no—
34C2×#11758198—21no—
35C2×C4×Dic61—21no—
36C2×C4×Dic61C2×C411yescharacteristic
37#14519696—22no—
38#14519696—22no—
39#14519695—44no—
40#14519689—44no—
41#14519684—22no—
42#14519684—22no—
43#14519684—22no—
44C22×Q8×C61C2311yescharacteristic
45C2×M16×C61—22no—
46C2×C4⋊C4×C61C2×C411yescharacteristic
47C2×C42×C61C2×C411yescharacteristic
48#101×C61—22no—
49#101×C61—22no—
50#95×C61—22no—
51C23×D61C22⋊C411yescharacteristic
52C2×Q8×C61—41no—
53C2×Q8×C61—41no—
54C2×Q8×C61#4112yes—
55C2×Q8×C61#4111yescharacteristic
56C2×Q8×C61#4111yescharacteristic
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61C22×C411yescharacteristic
60C22×C4×C61C2×D412yes—
61C22×Dic61—21no—
62C22×Dic61—21no—
63C22×Dic61C22⋊C412yes—
64C22×Dic61C22⋊C411yescharacteristic
65Q8×D61—44no—
66Q8×D61—48no—
67Q8×D61—44no—
68C2×C4×D61—22no—
69C2×C4×D61—21no—
70C2×C4×D61—21no—
71C2×C4×D61—22no—
72C2×C4×D61—24no—
73C2×C4×D61—22no—
74C2×C4×D61—22no—
75C2×C4×D61—42no—
76C2×Dic122—21no—
77C2×Dic122—21no—
78C2×Dic122—22no—
79C2×Dic122—22no—
80C2×Dic122—24no—
81C2×Dic122—41no—
82C2×Dic122—41no—
83C2×Dic122—42no—
84M16×C61—42no—
85M16×C61—42no—
86C2×C8×C61—42no—
87C4⋊C4×C61—41no—
88C42×C61—41no—
89#11758198—42no—
90#11758198—42no—
91C4⋊Dic61—41no—
92#11758196—42no—
93C4×Dic61—42no—
94C4×Dic61—41no—
95#11758194—42no—
96#11758194—42no—
97C2×#92418—42no—
98C23×C61C2×C22⋊C411yescharacteristic
99C22×D61—21no—
100C22×D61—21no—
101C22×D61—41no—
102C22×D61—41no—
103C22×D61#9912yes—
104Q8×C61—41no—
105Q8×C61—42no—
106Q8×C61—41no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—42no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61#11711yescharacteristic
117C2×C4×C61#11711yescharacteristic
118C2×C4×C61#12312yes—
119C2×C4×C61#12312yes—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—22no—
123C2×Dic61—22no—
124C2×Dic61—22no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—42no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61#9912yes—
131C4×D61—24no—
132C4×D61—24no—
133C4×D61—44no—
134C4×D61—44no—
135Dic122—44no—
136Dic122—44no—
137Dic122—42no—
138Dic122—42no—
139C8×C61—82no—
140#92418—82no—
141#12862—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61#38111yescharacteristic
145C22×C61C2×#9911yescharacteristic
146C22×C61C2×#9911yescharacteristic
147C2×D61—22no—
148C2×D61—22no—
149C2×D61—22no—
150C2×D61—42no—
151C2×D61—42no—
152C4×C61—22no—
153C4×C61—22no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—81no—
157C4×C61—81no—
158Dic61—22no—
159Dic61—42no—
160Dic61—81no—
161Dic61—81no—
162#2310—611no—
163#1784—612no—
164#1394—612no—
165#1394—611no—
166#1394D6111yescharacteristic, complemented
167C2×C61—21no—
168C2×C61—21no—
169C2×C61#181911yescharacteristic
170C2×C61#213311yescharacteristic
171C2×C61#213311yescharacteristic
172D61—42no—
173C23×Q8—611no—
174C2×#117—611no—
175C2×#117—611no—
176C2×#101—612no—
177C2×#101—1222no—
178C2×#101C2×D6112yes—
179#407—1222no—
180#391—611no—
181#391—611no—
182#391—611no—
183#391—1222no—
184#391C2×D6111yescharacteristic
185#381—1222no—
186C61#1286211yescharacteristic, complemented, Sylow
187C22×Q8—611no—
188C22×Q8—612no—
189C22×Q8—1222no—
190C22×Q8—1221no—
191C22×Q8—1221no—
192C22×Q8—2442no—
193C22×Q8—2442no—
194C22×Q8C22×D6111yescharacteristic
195C23×C4—611no—
196C23×C4—612no—
197C2×M16—22no—
198C2×M16—1222no—
199#117—2442no—
200#117—2442no—
201C2×C4⋊C4—611no—
202C2×C4⋊C4—1221no—
203C2×C4⋊C4C4×D6111yescharacteristic
204C2×C22⋊C4—1221no—
205C2×C22⋊C4—1221no—
206C2×C42—611no—
207C2×C42—1221no—
208C2×C42C4×D6111yescharacteristic
209#101—22no—
210#101—22no—
211#101—1222no—
212#101—1222no—
213#101—2444no—
214C22⋊C8—2444no—
215#95—22no—
216#95—1222no—
217#95—1222no—
218#95—1222no—
219C24—611no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—1224no—
223C2×Q8—1221no—
224C2×Q8—1221no—
225C2×Q8—1222no—
226C2×Q8—1222no—
227C2×Q8—2441no—
228C2×Q8—2441no—
229C2×Q8—2442no—
230C2×Q8—2448no—
231C2×Q8—2444no—
232C2×Q8—2444no—
233C2×Q8#1175822212yes—
234C2×Q8#1175822411yescharacteristic
235C2×Q8#1175822611yescharacteristic
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—611no—
239C22×C4—612no—
240C22×C4—1222no—
241C22×C4—1222no—
242C22×C4—1224no—
243C22×C4—1221no—
244C22×C4—1221no—
245C22×C4—1222no—
246C22×C4—1222no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—2442no—
250C22×C4C2×C4×D6111yescharacteristic
251C22×C4D4×D6112yes—
252M16—42no—
253M16—42no—
254M16—2442no—
255M16—2442no—
256C2×C8—42no—
257C2×C8—2442no—
258C4⋊C4—41no—
259C4⋊C4—2441no—
260C4⋊C4—2442no—
261C22⋊C4—2442no—
262C22⋊C4—2442no—
263C42—41no—
264C42—2441no—
265C42—2442no—
266C23—612no—
267C23—1221no—
268C23—1221no—
269C23—2441no—
270C23—2441no—
271C23C22⋊C4×D6111yescharacteristic
272Q8—42no—
273Q8—41no—
274Q8—41no—
275Q8—2444no—
276Q8—2444no—
277Q8—2442no—
278Q8—2442no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—612no—
289C2×C4—1224no—
290C2×C4—1224no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1222no—
294C2×C4—1222no—
295C2×C4—1222no—
296C2×C4—2444no—
297C2×C4—2444no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2442no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4#1451973411yescharacteristic
304C2×C4#1451973511yescharacteristic
305C2×C4#1451975912yes—
306C2×C4#1451976212yes—
307C8—82no—
308C8—4882no—
309C22—21no—
310C22—21no—
311C22—1222no—
312C22—1222no—
313C22—1222no—
314C22—2442no—
315C22—2442no—
316C22#1683962411yescharacteristic
317C22#99×D6111yescharacteristic
318C22#99×D6111yescharacteristic
319C4—22no—
320C4—22no—
321C4—21no—
322C4—21no—
323C4—81no—
324C4—81no—
325C4—1222no—
326C4—4881no—
327C4—4881no—
328C4—2442no—
329C2—21no—
330C2—21no—
331C2—2442no—
332C2#2144589211yescharacteristic
333C2#2144750511yescharacteristic
334C2#2144750711yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1394C2
socleC22×C61C2×#99
prime radicalC23×D61C22⋊C4
cosocleC22×Q8×C61C23
hypercenter#1394D61
hypocenterC61#12862
LCSG ▷ C22×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. #21443946C2semidirect
4. #21443945C2semidirect
5. C61×#1394C2semidirect
166. #1394D61semidirect
186. C61#12862semidirect

Maximal subgroups

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

subgroupindex
#214484022
#214457412
#214439462
#214439452
C61×#13942
#1286261