Finite Group Databasebeta

← #29568684

Subgroups — #29568684

Order 15616.

494 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 123 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#21445799C211yesmaximal, characteristic
3#21443641C214yesmaximal, complemented
4C61×#1817C211yesmaximal, characteristic
5C2×#14519777C411yescharacteristic
6C2×#14519756—21no—
7C2×C4⋊#92418—21no—
8C2×C22⋊#92418—21no—
9#16839668—21no—
10#16839660—21no—
11#16839660C2212yes—
12#16839632C411yescharacteristic
13#16839622—21no—
14#16839622C412yes—
15#16839517C2212yes—
16C22×C42×C61C2211yescharacteristic
17C2×C22⋊C8×C61—21no—
18C61×#336—21no—
19C61×#336C2212yes—
20C22×Dic122—21no—
21C22×#92418—22no—
22#14519777—42no—
23#14519756—42no—
24#14519756—42no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61C2×C412yes—
29C2×C4⋊Dic61C2×C412yes—
30C2×#11758196—22no—
31C2×C22⋊Dic61—22no—
32C4⋊#92418—42no—
33C22⋊#92418—42no—
34#14519684—22no—
35#14519684—22no—
36C23×C4×C61C2×C411yescharacteristic
37C23×C4×C61D411yescharacteristic
38C23×C4×C61D411yescharacteristic
39C22×C8×C61—22no—
40C2×C42×C61—42no—
41C2×C42×C61C2×C411yescharacteristic
42C2×C42×C61D411yescharacteristic
43C2×C42×C61D411yescharacteristic
44C2×C42×C61C2311yescharacteristic
45C22⋊C8×C61—42no—
46C24×C61C22⋊C411yescharacteristic
47C22×C4×C61—22no—
48C22×C4×C61—22no—
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—41no—
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×C61D812yes—
59C22×C4×C61SD1612yes—
60C22×C4×C61C22×C411yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61C2×D411yescharacteristic
63C22×Dic61—22no—
64C22×Dic61—22no—
65C2×Dic122—42no—
66C2×Dic122—42no—
67C2×Dic122—42no—
68C2×C8×C61—42no—
69C2×C8×C61—42no—
70C2×C8×C61—42no—
71C42×C61—22no—
72C42×C61—22no—
73C42×C61—41no—
74C42×C61—41no—
75C22⋊Dic61—44no—
76C4⋊Dic61—24no—
77C4⋊Dic61—42no—
78C4⋊Dic61—42no—
79C4⋊Dic61—42no—
80C4⋊Dic61—42no—
81#11758196—42no—
82#11758196—42no—
83C2×#92418—42no—
84C2×#92418—42no—
85C2×#92418—42no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61#10212yes—
93C23×C61C2×C22⋊C411yescharacteristic
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—21no—
105C2×C4×C61—21no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—44no—
109C2×C4×C61—44no—
110C2×C4×C61#9912yes—
111C2×C4×C61#10112yes—
112C2×C4×C61#10212yes—
113C2×C4×C61#13112yes—
114C2×C4×C61C2×D811yescharacteristic
115C2×C4×C61C2×SD1611yescharacteristic
116C2×C4×C61#13712yes—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123Dic122—84no—
124C8×C61—82no—
125#92418—82no—
126#7728—611nomaximal, Sylow
127C22×C61—22no—
128C22×C61—22no—
129C22×C61—22no—
130C22×C61—41no—
131C22×C61—41no—
132C22×C61—41no—
133C22×C61—41no—
134C22×C61#40811yescharacteristic
135C22×C61C2×#9911yescharacteristic
136C22×C61C2×#10111yescharacteristic
137C22×C61C2×#10211yescharacteristic
138C22×C61#41911yescharacteristic
139C22×C61#42111yescharacteristic
140C22×C61#42111yescharacteristic
141C4×C61—22no—
142C4×C61—22no—
143C4×C61—22no—
144C4×C61—22no—
145C4×C61—21no—
146C4×C61—22no—
147C4×C61—21no—
148Dic61—82no—
149Dic61—82no—
150#2857—611no—
151#1817D6111yescharacteristic
152#1797—611no—
153#1271—614no—
154C2×C61—42no—
155C2×C61#147211yescharacteristic
156C2×C61#148011yescharacteristic
157C2×C61#149711yescharacteristic
158C2×C61#151111yescharacteristic
159C2×C61#151911yescharacteristic
160C2×C61#153711yescharacteristic
161C2×C61#154611yescharacteristic
162C2×C4⋊Q8—611no—
163C2×C22⋊Q8—1221no—
164C22×C42C2×D6111yescharacteristic
165C2×C4⋊C8—1221no—
166C2×C22⋊C8—21no—
167C2×C22⋊C8—1221no—
168#391—1221no—
169#389—612no—
170#389—1221no—
171#384—611no—
172#380—612no—
173#380—1221no—
174#336—21no—
175#336—612no—
176#336C2×D6112yes—
177C61#772811yescharacteristic, complemented, Sylow
178C22×Q8—1221no—
179C23×C4C4×D6111yescharacteristic
180C23×C4D24411yescharacteristic
181C23×C4C22⋊D6111yescharacteristic
182C22×C8—22no—
183C22×C8—1222no—
184C4⋊Q8—2442no—
185C22⋊Q8—2442no—
186C22⋊Q8—2442no—
187C2×C4⋊C4—612no—
188C2×C4⋊C4—612no—
189C2×C4⋊C4—1221no—
190C2×C4⋊C4—1221no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4—1222no—
193C2×C22⋊C4—1222no—
194C2×C42—42no—
195C2×C42C4×D6111yescharacteristic
196C2×C42D24411yescharacteristic
197C2×C42C22⋊D6111yescharacteristic
198C2×C42C22×D6111yescharacteristic
199C4⋊C8—2442no—
200C22⋊C8—42no—
201C22⋊C8—2442no—
202#95—1222no—
203#95—1222no—
204C24#1175819811yescharacteristic
205C2×Q8—2442no—
206C2×Q8—2442no—
207C2×Q8—2442no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—1222no—
217C22×C4—1222no—
218C22×C4#1175819012yes—
219C22×C4D48812yes—
220C22×C4#1175819811yescharacteristic
221C22×C4#1175819811yescharacteristic
222C22×C4#1175819811yescharacteristic
223C22×C4C2×C4×D6111yescharacteristic
224C22×C4C2×D24411yescharacteristic
225C22×C4C2×C22⋊D6111yescharacteristic
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—2442no—
230C2×C8—2442no—
231C2×C8—2442no—
232C4⋊C4—1224no—
233C4⋊C4—2442no—
234C4⋊C4—2442no—
235C4⋊C4—2442no—
236C4⋊C4—2442no—
237C4⋊C4—2442no—
238C4⋊C4—2442no—
239C22⋊C4—2444no—
240C42—22no—
241C42—22no—
242C42—41no—
243C42—41no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#1451969912yes—
251C23C2×#1175819811yescharacteristic
252Q8—4884no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—2442no—
270C2×C4—2442no—
271C2×C4—2442no—
272C2×C4—2442no—
273C2×C4—2442no—
274C2×C4—2442no—
275C2×C4#1451969112yes—
276C2×C4#1451969512yes—
277C2×C4#1451969912yes—
278C2×C4#1451978512yes—
279C2×C4C2×D48811yescharacteristic
280C2×C4C2×#1175819011yescharacteristic
281C2×C4#1451980712yes—
282C8—82no—
283C8—4882no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22#1683973011yescharacteristic
292C22C2×#1451969111yescharacteristic
293C22C2×#1451969511yescharacteristic
294C22C2×#1451969911yescharacteristic
295C22#1683977011yescharacteristic
296C22#1683977811yescharacteristic
297C22#1683978011yescharacteristic
298C4—21no—
299C4—22no—
300C4—21no—
301C4—22no—
302C4—22no—
303C4—22no—
304C4—22no—
305C4—4882no—
306C4—4882no—
307C2—42no—
308C2#2144418411yescharacteristic
309C2#2144421911yescharacteristic
310C2#2144429111yescharacteristic
311C2#2144435111yescharacteristic
312C2#2144438811yescharacteristic
313C2#2144445611yescharacteristic
314C2#2144449011yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1817C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1817D61
hypocenterC61#7728
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #21443641C2semidirect
177. C61#7728semidirect

Maximal subgroups

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

subgroupindex
#214541712
#214457992
#214436412
C61×#18172
#772861