Finite Group Databasebeta

← #29568592

Subgroups — #29568592

Order 15616.

500 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 111 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#21454171C211yesmaximal, characteristic
2#21445887C211yesmaximal, characteristic
3#21443641C214yesmaximal, complemented
4C61×#1749C211yesmaximal, characteristic
5C2×#14519777C411yescharacteristic
6C2×#14519756—21no—
7C2×C22⋊#92418—21no—
8#16839668—21no—
9#16839660—21no—
10#16839660C2212yes—
11#16839632C411yescharacteristic
12#16839622—21no—
13#16839622C412yes—
14#16839517—21no—
15#16839517C2212yes—
16C22×C42×C61C2211yescharacteristic
17C2×C22⋊C8×C61—21no—
18C2×#97×C61—21no—
19C61×#336C2212yes—
20C22×Dic122—21no—
21C22×#92418—22no—
22#14519777—42no—
23#14519756—44no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61C2×C412yes—
28C2×C4⋊Dic61C2×C412yes—
29C2×#11758196—22no—
30C2×C22⋊Dic61—22no—
31C22⋊#92418—42no—
32#14519684—22no—
33#14519684—22no—
34C23×C4×C61C2×C411yescharacteristic
35C23×C4×C61D411yescharacteristic
36C23×C4×C61D411yescharacteristic
37C22×C8×C61—22no—
38C2×C42×C61—42no—
39C2×C42×C61C2×C411yescharacteristic
40C2×C42×C61D411yescharacteristic
41C2×C42×C61D411yescharacteristic
42C2×C42×C61C2311yescharacteristic
43C22⋊C8×C61—24no—
44#97×C61—42no—
45C24×C61C22⋊C411yescharacteristic
46C22×C4×C61—22no—
47C22×C4×C61—22no—
48C22×C4×C61—22no—
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—41no—
53C22×C4×C61—41no—
54C22×C4×C61—41no—
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22×C411yescharacteristic
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61C2×D411yescharacteristic
61C22×Dic61—22no—
62C22×Dic61—22no—
63C2×Dic122—42no—
64C2×Dic122—42no—
65C2×Dic122—42no—
66C2×C8×C61—24no—
67C2×C8×C61—42no—
68C2×C8×C61—42no—
69C42×C61—22no—
70C42×C61—22no—
71C42×C61—41no—
72C42×C61—41no—
73C22⋊Dic61—44no—
74C4⋊Dic61—24no—
75C4⋊Dic61—42no—
76C4⋊Dic61—42no—
77C4⋊Dic61—42no—
78C4⋊Dic61—42no—
79#11758196—44no—
80C2×#92418—42no—
81C2×#92418—42no—
82C2×#92418—42no—
83C23×C61—21no—
84C23×C61—21no—
85C23×C61—21no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61#10112yes—
90C23×C61C2×C22⋊C411yescharacteristic
91C2×C4×C61—21no—
92C2×C4×C61—21no—
93C2×C4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—44no—
105C2×C4×C61—44no—
106C2×C4×C61—44no—
107C2×C4×C61#9912yes—
108C2×C4×C61#10412yes—
109C2×C4×C61#13112yes—
110C2×C4×C61AGL(1,ℤ/8)12yes—
111C2×C4×C61#13712yes—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117C2×Dic61—42no—
118Dic122—84no—
119C8×C61—44no—
120#92418—82no—
121#7708—611nomaximal, Sylow
122C22×C61—22no—
123C22×C61—22no—
124C22×C61—22no—
125C22×C61—41no—
126C22×C61—41no—
127C22×C61—41no—
128C22×C61—41no—
129C22×C61#40811yescharacteristic
130C22×C61C2×#9911yescharacteristic
131C22×C61C2×#10111yescharacteristic
132C22×C61#41811yescharacteristic
133C22×C61#41911yescharacteristic
134C22×C61C2×#10411yescharacteristic
135C22×C61#42111yescharacteristic
136C4×C61—22no—
137C4×C61—22no—
138C4×C61—22no—
139C4×C61—22no—
140C4×C61—22no—
141C4×C61—41no—
142Dic61—82no—
143Dic61—82no—
144#2857—611no—
145#1817—611no—
146#1749D6111yescharacteristic
147#1271—614no—
148C2×C61—42no—
149C2×C61#147211yescharacteristic
150C2×C61#148411yescharacteristic
151C2×C61#149711yescharacteristic
152C2×C61#150811yescharacteristic
153C2×C61#151011yescharacteristic
154C2×C61#153111yescharacteristic
155C2×C61#155611yescharacteristic
156C2×C4⋊Q8—611no—
157C2×C22⋊Q8—1221no—
158C22×C42C2×D6111yescharacteristic
159C2×C22⋊C8—21no—
160C2×C22⋊C8—1221no—
161C2×#97—21no—
162#391—1221no—
163#389—612no—
164#389—1221no—
165#384—611no—
166#380—612no—
167#380—1221no—
168#336—612no—
169#336—1221no—
170#336C2×D6112yes—
171C61#770811yescharacteristic, complemented, Sylow
172C22×Q8—1221no—
173C23×C4C4×D6111yescharacteristic
174C23×C4D24411yescharacteristic
175C23×C4C22⋊D6111yescharacteristic
176C22×C8—22no—
177C22×C8—1222no—
178C4⋊Q8—2442no—
179C22⋊Q8—2444no—
180C2×C4⋊C4—612no—
181C2×C4⋊C4—612no—
182C2×C4⋊C4—1221no—
183C2×C4⋊C4—1221no—
184C2×C4⋊C4—1221no—
185C2×C4⋊C4—1222no—
186C2×C22⋊C4—1222no—
187C2×C42—42no—
188C2×C42C4×D6111yescharacteristic
189C2×C42D24411yescharacteristic
190C2×C42C22⋊D6111yescharacteristic
191C2×C42C22×D6111yescharacteristic
192C22⋊C8—24no—
193C22⋊C8—2442no—
194#97—42no—
195#95—1222no—
196#95—1222no—
197C24#1175819811yescharacteristic
198C2×Q8—2442no—
199C2×Q8—2442no—
200C2×Q8—2442no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—1222no—
211C22×C4—1222no—
212C22×C4#1175819811yescharacteristic
213C22×C4#1175819811yescharacteristic
214C22×C4#1175819811yescharacteristic
215C22×C4C2×C4×D6111yescharacteristic
216C22×C4C2×D24411yescharacteristic
217C22×C4C2×C22⋊D6111yescharacteristic
218C2×C8—24no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—2442no—
222C2×C8—2442no—
223C2×C8—2442no—
224C4⋊C4—1224no—
225C4⋊C4—2442no—
226C4⋊C4—2442no—
227C4⋊C4—2442no—
228C4⋊C4—2442no—
229C4⋊C4—2444no—
230C22⋊C4—2444no—
231C42—22no—
232C42—22no—
233C42—41no—
234C42—41no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23#1451969512yes—
242C23C2×#1175819811yescharacteristic
243Q8—4884no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—2442no—
261C2×C4—2442no—
262C2×C4—2442no—
263C2×C4—2442no—
264C2×C4—2442no—
265C2×C4—2442no—
266C2×C4#1451969112yes—
267C2×C4#1451970512yes—
268C2×C4#1451978512yes—
269C2×C4#1451980512yes—
270C2×C4#1451980712yes—
271C8—44no—
272C8—4882no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22#1683973011yescharacteristic
281C22C2×#1451969111yescharacteristic
282C22C2×#1451969511yescharacteristic
283C22#1683976811yescharacteristic
284C22#1683977011yescharacteristic
285C22C2×#1451970511yescharacteristic
286C22#1683977811yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—41no—
293C4—4882no—
294C4—4882no—
295C2—42no—
296C2#2144418411yescharacteristic
297C2#2144423711yescharacteristic
298C2#2144429111yescharacteristic
299C2#2144433811yescharacteristic
300C2#2144434911yescharacteristic
301C2#2144443311yescharacteristic
302C2#2144453511yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1749C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1749D61
hypocenterC61#7708
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #21443641C2semidirect
171. C61#7708semidirect

Maximal subgroups

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

subgroupindex
#214541712
#214458872
#214436412
C61×#17492
#770861