Finite Group Databasebeta

← #29590329

Subgroups — #29590329

Order 15616.

432 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 107 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#21451640C211yesmaximal, characteristic
2#21445558C211yesmaximal, characteristic
3#21443847C214yesmaximal, complemented
4C61×#1458C211yesmaximal, characteristic
5C22×#11758194—21no—
6C2×#14519777C411yescharacteristic
7#16839701C2212yes—
8#16839698C412yes—
9#16839682—21no—
10#16839674—21no—
11#16839660—21no—
12#16839628C411yescharacteristic
13#16839622—21no—
14#16839531—21no—
15#16839531C2212yes—
16C2×#117×C61C2211yescharacteristic
17C61×#431—21no—
18C61×#324C2212yes—
19C61×#323—21no—
20C22×Dic122—21no—
21C2×#11758194—24no—
22C2×#11758194—22no—
23C2×#11758194—41no—
24C2×#11758194—42no—
25C22×#92418—41no—
26#14519777—41no—
27#14519777—41no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61C2×C412yes—
31C2×#11758196—22no—
32C2×C22⋊Dic61—22no—
33#14519684—21no—
34#14519684—22no—
35#14519684—22no—
36#14519684C2×C412yes—
37C23×C4×C61D411yescharacteristic
38#117×C61—41no—
39#117×C61—41no—
40C2×C4⋊C4×C61D411yescharacteristic
41C2×C4⋊C4×C61C2311yescharacteristic
42C2×C22⋊C4×C61C2×C411yescharacteristic
43C2×C22⋊C4×C61D411yescharacteristic
44C2×C42×C61C2×C411yescharacteristic
45C2×C42×C61D411yescharacteristic
46C4⋊C8×C61—41no—
47C22⋊C8×C61—22no—
48C4×C8×C61—41no—
49C24×C61C22⋊C411yescharacteristic
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—22no—
53C22×C4×C61C22⋊C411yescharacteristic
54C22×C4×C61C22⋊C411yescharacteristic
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61SD1612yes—
57C22×C4×C61Q1612yes—
58C22×C4×C61C22×C411yescharacteristic
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61C2×D411yescharacteristic
61C22×Dic61—22no—
62C22×Dic61—22no—
63C2×Dic122—42no—
64C2×Dic122—82no—
65C2×C8×C61—42no—
66C4⋊C4×C61—41no—
67C4⋊C4×C61—41no—
68C22⋊C4×C61—41no—
69C22⋊C4×C61—41no—
70C42×C61—21no—
71C42×C61—21no—
72C42×C61—21no—
73C42×C61—21no—
74C22⋊Dic61—82no—
75C4⋊Dic61—41no—
76C4⋊Dic61—41no—
77C4⋊Dic61—42no—
78C4⋊Dic61—42no—
79#11758196—82no—
80#11758194—24no—
81#11758194—44no—
82#11758194—42no—
83C2×#92418—42no—
84C2×#92418—44no—
85C23×C61—21no—
86C23×C61—21no—
87C23×C61—41no—
88C23×C61—41no—
89C23×C61#10312yes—
90C23×C61C2×C22⋊C411yescharacteristic
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—22no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61#9911yescharacteristic
106C2×C4×C61#9911yescharacteristic
107C2×C4×C61#10312yes—
108C2×C4×C61#10412yes—
109C2×C4×C61C2×SD1611yescharacteristic
110C2×C4×C61C2×Q1611yescharacteristic
111C2×Dic61—42no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117Dic122—84no—
118C8×C61—82no—
119#92418—44no—
120#11723—611nomaximal, Sylow
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—22no—
124C22×C61—42no—
125C22×C61—41no—
126C22×C61—41no—
127C22×C61—81no—
128C22×C61C2×#9911yescharacteristic
129C22×C61C2×#10311yescharacteristic
130C22×C61C2×#10411yescharacteristic
131C4×C61—21no—
132C4×C61—21no—
133C4×C61—41no—
134C4×C61—41no—
135C4×C61—41no—
136C4×C61—41no—
137C4×C61#32812yes—
138C4×C61#32812yes—
139Dic61—82no—
140Dic61—82no—
141#2628—611no—
142#1742—611no—
143#1458D6111yescharacteristic
144#1353—614no—
145C2×C61—21no—
146C2×C61—21no—
147C2×C61—42no—
148C2×C61C2×#32811yescharacteristic
149C2×C61#212311yescharacteristic
150C2×C61#213411yescharacteristic
151C22×M16—1221no—
152C2×C4⋊Q8—611no—
153C2×#117C2×D6111yescharacteristic
154#431—21no—
155#400—612no—
156#399—612no—
157#395—1221no—
158#393—1221no—
159#389—1221no—
160#382—611no—
161#380—1221no—
162#343—612no—
163#343—1221no—
164#324C2×D6112yes—
165#323—21no—
166C61#1172311yescharacteristic, complemented, Sylow
167C22×Q8—1221no—
168C23×C4D24411yescharacteristic
169C2×M16—1224no—
170C2×M16—1222no—
171C2×M16—2441no—
172C2×M16—2442no—
173C22×C8—2441no—
174C4⋊Q8—2441no—
175C4⋊Q8—2441no—
176#117—41no—
177#117—41no—
178C2×C4⋊C4—612no—
179C2×C4⋊C4—1222no—
180C2×C4⋊C4—1221no—
181C2×C4⋊C4—1221no—
182C2×C4⋊C4D24411yescharacteristic
183C2×C4⋊C4C22×D6111yescharacteristic
184C2×C22⋊C4—1222no—
185C2×C22⋊C4C4×D6111yescharacteristic
186C2×C22⋊C4C22⋊D6111yescharacteristic
187C2×C42C4×D6111yescharacteristic
188C2×C42C22⋊D6111yescharacteristic
189C4⋊C8—41no—
190C22⋊C8—22no—
191C4×C8—41no—
192#95—612no—
193#95—1221no—
194#95—1222no—
195#95—1222no—
196C24#1175819811yescharacteristic
197C2×Q8—4882no—
198C2×Q8—2442no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—1222no—
203C22×C4—1222no—
204C22×C4#1175819012yes—
205C22×C4Dic24412yes—
206C22×C4#1175819811yescharacteristic
207C22×C4#1175819811yescharacteristic
208C22×C4#1175819811yescharacteristic
209C22×C4C2×C4×D6111yescharacteristic
210C22×C4C2×D24411yescharacteristic
211C22×C4C2×C22⋊D6111yescharacteristic
212M16—1224no—
213M16—2444no—
214M16—2442no—
215C2×C8—42no—
216C2×C8—2442no—
217C2×C8—2444no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—4882no—
221C4⋊C4—2442no—
222C4⋊C4—2441no—
223C4⋊C4—2441no—
224C4⋊C4—2442no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—4882no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23#1451970212yes—
237C23C2×#1175819811yescharacteristic
238Q8—4884no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—2442no—
254C2×C4—2442no—
255C2×C4—2442no—
256C2×C4—2442no—
257C2×C4—2442no—
258C2×C4—2442no—
259C2×C4#1451969111yescharacteristic
260C2×C4#1451969111yescharacteristic
261C2×C4#1451970212yes—
262C2×C4#1451970312yes—
263C2×C4C2×#1175819011yescharacteristic
264C2×C4C2×Dic24411yescharacteristic
265C8—82no—
266C8—2444no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22C2×#1451969111yescharacteristic
275C22C2×#1451970211yescharacteristic
276C22C2×#1451970311yescharacteristic
277C4—21no—
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—4882no—
284C4—4882no—
285C4#1683950012yes—
286C4#1683950012yes—
287C2—21no—
288C2—21no—
289C2—42no—
290C2C2×#1683950011yescharacteristic
291C2#2144747211yescharacteristic
292C2#2144751011yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1458C2
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1458D61
hypocenterC61#11723
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #21443847C2semidirect
166. C61#11723semidirect

Maximal subgroups

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

subgroupindex
#214516402
#214455582
#214438472
C61×#14582
#1172361