Finite Group Databasebeta

← #29568652

Subgroups — #29568652

Order 15616.

444 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 103 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#21454158C211yesmaximal, characteristic
2#21445887C211yesmaximal, characteristic
3#21443644C212yesmaximal, complemented
4#21443643C212yesmaximal, complemented
5C61×#1817C211yesmaximal, characteristic
6C2×#14519756C411yescharacteristic
7C2×C22⋊#92418—21no—
8#16839668C411yescharacteristic
9#16839668C2211yescharacteristic
10#16839660C411yescharacteristic
11#16839660C2211yescharacteristic
12#16839646—21no—
13#16839646—21no—
14#16839635—21no—
15#16839625—21no—
16#16839622C411yescharacteristic
17#16839517—21no—
18#16839517C2212yes—
19C22×C42×C61C2211yescharacteristic
20C2×C22⋊C8×C61—21no—
21C61×#336—21no—
22C61×#336C2212yes—
23C22×Dic122C2×C411yescharacteristic
24C22×#92418—22no—
25#14519756—42no—
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C22⋊#92418—42no—
35#14519684—21no—
36#14519684—21no—
37#14519684—21no—
38#14519684—21no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42C23×C4×C61C2×C411yescharacteristic
43C23×C4×C61D411yescharacteristic
44C23×C4×C61D411yescharacteristic
45C22×C8×C61—22no—
46C2×C42×C61—42no—
47C2×C42×C61C2×C411yescharacteristic
48C2×C42×C61D411yescharacteristic
49C2×C42×C61D411yescharacteristic
50C2×C42×C61C2311yescharacteristic
51C22⋊C8×C61—42no—
52C24×C61C22⋊C411yescharacteristic
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—22no—
58C22×C4×C61—22no—
59C22×C4×C61—41no—
60C22×C4×C61—41no—
61C22×C4×C61—41no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C2×Dic122—22no—
73C2×Dic122—41no—
74C2×Dic122—41no—
75C2×C8×C61—42no—
76C2×C8×C61—42no—
77C2×C8×C61—42no—
78C42×C61—41no—
79C42×C61—41no—
80C42×C61—41no—
81C42×C61—41no—
82C22⋊Dic61—42no—
83C22⋊Dic61—42no—
84C4⋊Dic61—22no—
85C4⋊Dic61—41no—
86C4⋊Dic61—41no—
87#11758196—42no—
88#11758196—42no—
89#11758196—42no—
90C2×#92418—42no—
91C2×#92418—42no—
92C2×#92418—42no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61#9912yes—
100C23×C61C2×C22⋊C411yescharacteristic
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—44no—
117C2×C4×C61—44no—
118C2×C4×C61—44no—
119C2×C4×C61#13112yes—
120C2×C4×C61#13512yes—
121C2×C4×C61#13712yes—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134Dic122—82no—
135C8×C61—82no—
136#92418—82no—
137#7721—611nomaximal, Sylow
138C22×C61—22no—
139C22×C61—22no—
140C22×C61—22no—
141C22×C61—41no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61#40811yescharacteristic
146C22×C61C2×#9911yescharacteristic
147C22×C61#41311yescharacteristic
148C22×C61#41611yescharacteristic
149C22×C61#41911yescharacteristic
150C22×C61#42111yescharacteristic
151C22×C61#42111yescharacteristic
152C4×C61—22no—
153C4×C61—22no—
154C4×C61—22no—
155C4×C61—41no—
156C4×C61—41no—
157C4×C61—41no—
158Dic61—81no—
159Dic61—81no—
160Dic61—81no—
161Dic61—81no—
162#2855—611no—
163#1817—611no—
164#1817D6111yescharacteristic
165#1272—612no—
166#1271—612no—
167C2×C61—42no—
168C2×C61#146811yescharacteristic
169C2×C61#148211yescharacteristic
170C2×C61#149711yescharacteristic
171C2×C61#151111yescharacteristic
172C2×C61#151611yescharacteristic
173C2×C61#153511yescharacteristic
174C2×C61#155411yescharacteristic
175C2×C22⋊Q8—611no—
176C22×C42C2×D6111yescharacteristic
177C2×C22⋊C8—21no—
178C2×C22⋊C8—1221no—
179#391—611no—
180#391—611no—
181#389—611no—
182#389—611no—
183#387—1221no—
184#387—1221no—
185#385—1221no—
186#381—1221no—
187#380—611no—
188#336—21no—
189#336—612no—
190#336—1221no—
191#336C2×D6112yes—
192C61#772111yescharacteristic, complemented, Sylow
193C22×Q8—611no—
194C23×C4C4×D6111yescharacteristic
195C23×C4D24411yescharacteristic
196C23×C4C22⋊D6111yescharacteristic
197C22×C8—22no—
198C22×C8—1222no—
199C22⋊Q8—2442no—
200C2×C4⋊C4—611no—
201C2×C4⋊C4—611no—
202C2×C4⋊C4—611no—
203C2×C4⋊C4—1221no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4—1221no—
206C2×C22⋊C4—1221no—
207C2×C22⋊C4—1221no—
208C2×C42—42no—
209C2×C42C4×D6111yescharacteristic
210C2×C42D24411yescharacteristic
211C2×C42C22⋊D6111yescharacteristic
212C2×C42C22×D6111yescharacteristic
213C22⋊C8—42no—
214C22⋊C8—2442no—
215#95—1221no—
216#95—1221no—
217#95—1221no—
218#95—1221no—
219#95—1221no—
220#95—1221no—
221#95—1221no—
222C24#1175819811yescharacteristic
223C2×Q8—1222no—
224C2×Q8—2441no—
225C2×Q8—2441no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—1221no—
236C22×C4—1221no—
237C22×C4—1221no—
238C22×C4—1221no—
239C22×C4#1175819811yescharacteristic
240C22×C4#1175819811yescharacteristic
241C22×C4#1175819811yescharacteristic
242C22×C4C2×C4×D6111yescharacteristic
243C22×C4C2×D24411yescharacteristic
244C22×C4C2×C22⋊D6111yescharacteristic
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—2442no—
249C2×C8—2442no—
250C2×C8—2442no—
251C4⋊C4—1222no—
252C4⋊C4—2441no—
253C4⋊C4—2441no—
254C4⋊C4—2442no—
255C4⋊C4—2442no—
256C4⋊C4—2442no—
257C22⋊C4—2442no—
258C22⋊C4—2442no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42—41no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23#1451969112yes—
270C23C2×#1175819811yescharacteristic
271Q8—4882no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—2441no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4#1451978312yes—
303C2×C4#1451979712yes—
304C2×C4#1451981012yes—
305C8—82no—
306C8—4882no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22#1683972811yescharacteristic
315C22C2×#1451969111yescharacteristic
316C22#1683974411yescharacteristic
317C22#1683975611yescharacteristic
318C22#1683977211yescharacteristic
319C22#1683977811yescharacteristic
320C22#1683978011yescharacteristic
321C4—22no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—4881no—
328C4—4881no—
329C4—4881no—
330C4—4881no—
331C2—42no—
332C2#2144416511yescharacteristic
333C2#2144422511yescharacteristic
334C2#2144429311yescharacteristic
335C2#2144435311yescharacteristic
336C2#2144437311yescharacteristic
337C2#2144444811yescharacteristic
338C2#2144453011yescharacteristic
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#7721
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. #21443644C2semidirect
4. #21443643C2semidirect
192. C61#7721semidirect

Maximal subgroups

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

subgroupindex
#214541582
#214458872
#214436442
#214436432
C61×#18172
#772161