Finite Group Databasebeta

← #29589282

Subgroups — #29589282

Order 15616.

616 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 89 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#21448381C211yesmaximal, characteristic
2#21444136C212yesmaximal, complemented
3#21443765C211yesmaximal, characteristic, complemented
4#21443750C212yesmaximal, complemented
5C61×#1325C211yesmaximal, characteristic, complemented
6C2×#14519774C411yescharacteristic
7C2×#14519739—22no—
8C2×#14519739C411yescharacteristic
9C2×#14519739C2211yescharacteristic
10C22⋊C8×D61—22no—
11#16839663—22no—
12#16839663C411yescharacteristic
13#16839663C2211yescharacteristic
14C42×Dic61C411yescharacteristic
15#16839488C2212yes—
16#16839485—22no—
17C61×#384C2211yescharacteristic
18C61×#324C2212yes—
19C22×D244—22no—
20C22×D244C2×C411yescharacteristic
21C22×D244C2×C411yescharacteristic
22C22×C4×D61—21no—
23C22×C4×D61D412yes—
24C2×C8×D61—42no—
25#14519774—22no—
26#14519774C814yescomplemented
27#14519739—41no—
28#14519739—41no—
29#14519739—48no—
30#14519739—41no—
31#14519739—41no—
32C2×#11758198—22no—
33C2×#11758198—22no—
34C2×#11758198—21no—
35C2×#11758198D412yes—
36C2×C4×Dic61—21no—
37C2×C4×Dic61—21no—
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61C2×C411yescharacteristic
40C22⋊#92418—22no—
41#14519689—42no—
42C2×C4⋊C4×C61—22no—
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C4⋊C4×C61C2311yescharacteristic
45C2×C42×C61C2×C411yescharacteristic
46C22⋊C8×C61—22no—
47C23×D61—21no—
48C23×D61C22⋊C412yes—
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61C22×C411yescharacteristic
52C22×C4×C61C2×D412yes—
53C22×Dic61—21no—
54C22×Dic61C22⋊C411yescharacteristic
55C22×Dic61C22⋊C411yescharacteristic
56C2×D244—28no—
57C2×D244—21no—
58C2×D244—41no—
59C2×D244—41no—
60C2×D244—44no—
61C2×D244—44no—
62C2×D244—41no—
63C2×D244—41no—
64C2×D244C2×C812yes—
65C2×D244C2×C812yescomplemented
66C2×D244M1612yes—
67C2×C4×D61—22no—
68C2×C4×D61—22no—
69C2×C4×D61—44no—
70C2×C4×D61—44no—
71C2×C4×D61—41no—
72C2×C4×D61—41no—
73C2×C8×C61—42no—
74C4⋊C4×C61—41no—
75C4⋊C4×C61—44no—
76C4⋊C4×C61—41no—
77C42×C61—21no—
78C42×C61C2×C812yescomplemented
79#11758198—42no—
80#11758198—44no—
81#11758198—44no—
82#11758198—42no—
83C4×Dic61—24no—
84C4×Dic61—21no—
85C4×Dic61—21no—
86C4×Dic61—42no—
87C4×Dic61—41no—
88C2×#92418—42no—
89C8×D61—84no—
90C23×C61C2×C22⋊C411yescharacteristic
91C22×D61—22no—
92C22×D61—22no—
93C22×D61—22no—
94C22×D61—42no—
95C22×D61—42no—
96C22×D61—42no—
97C22×D61—42no—
98C22×D61—41no—
99C22×D61—41no—
100C22×D61C22⋊C814yes—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—22no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—41no—
110C2×C4×C61C22×C811yescharacteristic
111C2×C4×C61C2×M1611yescharacteristic
112C2×Dic61—22no—
113C2×Dic61—22no—
114C2×Dic61—21no—
115C2×Dic61—41no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61#9911yescharacteristic
119C2×Dic61#9911yescharacteristic
120D244—42no—
121D244—42no—
122D244—42no—
123D244—416no—
124D244—42no—
125C4×D61—44no—
126C4×D61—82no—
127C4×D61—82no—
128C4×D61—82no—
129C8×C61—82no—
130#92418—82no—
131#11475—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—21no—
134C22×C61C2×C22⋊C811yescharacteristic
135C22×C61C2×#9911yescharacteristic
136C22×C61#41311yescharacteristic
137C2×D61—24no—
138C2×D61—42no—
139C2×D61—42no—
140C2×D61—44no—
141C2×D61—44no—
142C2×D61—42no—
143C2×D61—82no—
144C2×D61—82no—
145C2×D61—81no—
146C2×D61—81no—
147C4×C61—24no—
148C4×C61—21no—
149C4×C61—21no—
150C4×C61—42no—
151C4×C61—81no—
152Dic61—21no—
153Dic61—42no—
154Dic61—41no—
155Dic61#35312yes—
156C4×#127—611no—
157#1459—612no—
158#1325—611no—
159#1325D6111yescharacteristic, complemented
160#1318—612no—
161C2×C61—21no—
162C2×C61—21no—
163C2×C61#145911yescharacteristic
164C2×C61C2×#35311yescharacteristic
165C2×C61#213211yescharacteristic
166D61—44no—
167D61—82no—
168C2×#127—611no—
169C2×C4×D4—611no—
170C2×C4×D4—611no—
171C2×C4×D4—1222no—
172C2×C22⋊C8—1222no—
173#390—611no—
174#390—611no—
175#390—1222no—
176#384C2×D6111yescharacteristic
177C43—611no—
178#324—612no—
179#324C2×D6112yes—
180#323—1222no—
181C61#1147511yescharacteristic, complemented, Sylow
182C22×D4—611no—
183C22×D4—611no—
184C22×D4—1222no—
185C23×C4—612no—
186C23×C4—1221no—
187C22×C8—2442no—
188#127—614no—
189#127—1222no—
190C4×D4—2441no—
191C4×D4—2441no—
192C4×D4—2448no—
193C4×D4—2441no—
194C4×D4—2441no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C4×D6111yescharacteristic
197C2×C4⋊C4C22×D6111yescharacteristic
198C2×C22⋊C4—612no—
199C2×C22⋊C4—1221no—
200C2×C22⋊C4—1222no—
201C2×C22⋊C4—1222no—
202C2×C42—611no—
203C2×C42—611no—
204C2×C42—1221no—
205C2×C42—1221no—
206C2×C42C4×D6111yescharacteristic
207C22⋊C8—22no—
208C22⋊C8—1222no—
209C22⋊C8—2442no—
210C24—612no—
211C24—1221no—
212C2×D4—612no—
213C2×D4—612no—
214C2×D4—612no—
215C2×D4—1228no—
216C2×D4—1221no—
217C2×D4—2441no—
218C2×D4—2441no—
219C2×D4—2441no—
220C2×D4—2441no—
221C2×D4—2444no—
222C2×D4—2444no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—611no—
226C22×C4—611no—
227C22×C4—1222no—
228C22×C4—1222no—
229C22×C4—1221no—
230C22×C4—2441no—
231C22×C4—2441no—
232C22×C4—2444no—
233C22×C4—2444no—
234C22×C4C2×C4×D6111yescharacteristic
235C22×C4D4×D6112yes—
236C2×C8—42no—
237C2×C8—4884no—
238C2×C8—2442no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—44no—
242C22⋊C4—2442no—
243C22⋊C4—2444no—
244C22⋊C4—2444no—
245C22⋊C4—2442no—
246C42—21no—
247C42—1221no—
248C42—1221no—
249C42—1224no—
250C42—2441no—
251C42—2442no—
252C42C8×D6112yescomplemented
253C23—614no—
254C23—1222no—
255C23—1222no—
256C23—1222no—
257C23—2442no—
258C23—2442no—
259C23—2442no—
260C23—2442no—
261C23—2441no—
262C23—2441no—
263C23C22⋊C4×D6111yescharacteristic
264D4—2442no—
265D4—2442no—
266D4—2442no—
267D4—24416no—
268D4—2442no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—611no—
279C2×C4—611no—
280C2×C4—1222no—
281C2×C4—1222no—
282C2×C4—1221no—
283C2×C4—4882no—
284C2×C4—4882no—
285C2×C4—4882no—
286C2×C4—2444no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4C2×C8×D6111yescharacteristic
291C2×C4M16×D6111yescharacteristic
292C8—82no—
293C8—4882no—
294C22—21no—
295C22—21no—
296C22—1224no—
297C22—4882no—
298C22—4882no—
299C22—4881no—
300C22—4881no—
301C22—2444no—
302C22—2444no—
303C22—2442no—
304C22—2442no—
305C22—2442no—
306C22C22⋊C8×D6111yescharacteristic
307C22#99×D6111yescharacteristic
308C22#1683974511yescharacteristic
309C4—24no—
310C4—21no—
311C4—21no—
312C4—42no—
313C4—81no—
314C4—612no—
315C4—1221no—
316C4—2442no—
317C4—2441no—
318C2—21no—
319C2—21no—
320C2—4882no—
321C2—2444no—
322C2#2144413611yescharacteristic
323C2#353×D6111yescharacteristic
324C2#2144750311yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC2×C4×C61C22×C8
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1325C2
socleC22×C61C2×#99
prime radicalC2×#14519774C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1325D61
hypocenterC61#11475
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
2. #21444136C2semidirect
3. #21443765C2semidirect
4. #21443750C2semidirect
5. C61×#1325C2semidirect
26. #14519774C8semidirect
65. C2×D244C2×C8semidirect
78. C42×C61C2×C8semidirect
159. #1325D61semidirect
181. C61#11475semidirect
252. C42C8×D61semidirect

Maximal subgroups

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

subgroupindex
#214483812
#214441362
#214437652
#214437502
C61×#13252
#1147561