Finite Group Databasebeta

← #29654921

Subgroups — #29654921

Order 15616.

870 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 191 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#21459629C211yesmaximal, characteristic
2#21456972C212yesmaximal
3#21451031C211yesmaximal, characteristic
4#21444863C218yesmaximal
5#21444319C212yesmaximal
6C61×#1461C211yesmaximal, characteristic
7Q8×Dic122—22no—
8#16840491—22no—
9#16840488—24no—
10#16840393—21no—
11#16840393C2214yes—
12#16840319—22no—
13#16840319C2212yes—
14C2×#14519777C2211yescharacteristic
15#16840240—21no—
16#16840167—22no—
17#16840113—22no—
18#16840098C2218yes—
19#16840058—24no—
20#16839701C2214yes—
21#16839669—21no—
22#16839668—22no—
23#16839627—21no—
24C4×C4⋊Dic61C2211yescharacteristic
25#16839500C2218yes—
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#431C2212yes—
28C61×#324C2214yes—
29C22×Dic122—21no—
30#14519777—22no—
31#14519777—22no—
32#14519777—22no—
33#14519777—22no—
34#14519777D412yes—
35#14519777D412yes—
36#14519776—24no—
37#14519776—28no—
38#14519769—24no—
39#14519769—22no—
40#14519768—24no—
41#14519768—28no—
42#14519756—22no—
43#14519756—24no—
44#14519756C2314yes—
45Q8×Dic61—24no—
46Q8×Dic61—24no—
47C4×Dic122—24no—
48C4×Dic122—24no—
49C4×Dic122—22no—
50C4×Dic122D414yes—
51#14519736—24no—
52#14519736—22no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2312yes—
56C2×C4×Dic61—21no—
57#14519711—28no—
58#14519702—28no—
59#14519684—24no—
60#14519684—24no—
61#14519684C2312yes—
62C22×Q8×C61D411yescharacteristic
63C4×Q8×C61—22no—
64C4×Q8×C61—22no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61C2312yes—
67C2×C42×C61D412yes—
68C2×C42×C61C2311yescharacteristic
69C4⋊C8×C61—22no—
70C22⋊C8×C61C2314yes—
71C4×C8×C61—22no—
72C2×Q8×C61—22no—
73C2×Q8×C61—22no—
74C2×Q8×C61Q1614yes—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D412yes—
77C22×C4×C61C2×D412yes—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2411yescharacteristic
80C22×Dic61—22no—
81C22×Dic61—22no—
82C2×Dic122—22no—
83C2×Dic122—24no—
84C2×Dic122—44no—
85C2×Dic122—41no—
86C2×Dic122—41no—
87C2×Dic122C2×D414yes—
88C2×C8×C61—24no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—24no—
91C42×C61—22no—
92C42×C61—22no—
93C42×C61C2×D412yes—
94C42×C61C2×D412yes—
95C22⋊Dic61—24no—
96C4⋊Dic61—24no—
97C4⋊Dic61—24no—
98C4⋊Dic61—24no—
99C4⋊Dic61—22no—
100C4⋊Dic61—22no—
101C4⋊Dic61—22no—
102C4⋊Dic61—24no—
103C4⋊Dic61—42no—
104C4⋊Dic61—42no—
105C4⋊Dic61C2×D414yes—
106#11758196—28no—
107#11758196—24no—
108#11758196—24no—
109C4×Dic61—28no—
110C4×Dic61—24no—
111C4×Dic61—42no—
112C23×C61C22×D411yescharacteristic
113Q8×C61—28no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—22no—
118C2×C4×C61—41no—
119C2×C4×C61—42no—
120C2×C4×C61#12011yescharacteristic
121C2×C4×C61#12012yes—
122C2×C4×C61#12011yescharacteristic
123C2×C4×C61C2×Q1614yes—
124C2×C4×C61C2×Q1612yes—
125C2×C4×C61#13712yes—
126C2×C4×C61C22×D412yes—
127C2×Dic61—24no—
128C2×Dic61—24no—
129C2×Dic61—42no—
130C2×Dic61—44no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134Dic122—44no—
135Dic122—44no—
136Dic122—84no—
137Dic122—82no—
138Dic122—82no—
139C8×C61—44no—
140#18411—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61C2×#12011yescharacteristic
144C22×C61C22×Q1611yescharacteristic
145C22×C61C2×#13711yescharacteristic
146C4×C61—24no—
147C4×C61—21no—
148C4×C61—22no—
149C4×C61—42no—
150C4×C61#45114yes—
151C4×C61#45412yes—
152Dic61—44no—
153Dic61—44no—
154Dic61—82no—
155Dic61—82no—
156#3502—611no—
157#3230—612no—
158#2568—611no—
159#1617—618no—
160#1503—612no—
161#1461D6111yescharacteristic
162C2×C61—21no—
163C2×C61—21no—
164C2×C61C2×#45111yescharacteristic
165C2×C61C2×#45411yescharacteristic
166C2×C61#302811yescharacteristic
167Q82—1222no—
168#557—1224no—
169#557—1222no—
170#544—614no—
171#544—1221no—
172#533—612no—
173#533—1222no—
174C2×C4⋊Q8—611no—
175#519—1221no—
176C2×C4×Q8C2×D6111yescharacteristic
177#500—1222no—
178#487—1222no—
179#484—618no—
180#477—1224no—
181#431C2×D6112yes—
182#400—614no—
183#391—1221no—
184#391—1222no—
185#382—1221no—
186C4×C4⋊C4—611no—
187#328—618no—
188#324C2×D6114yes—
189C61#1841111yescharacteristic, complemented, Sylow
190C22×Q8—1221no—
191C22×Q8D24411yescharacteristic
192C4⋊Q8—612no—
193C4⋊Q8—612no—
194C4⋊Q8—1222no—
195C4⋊Q8—1222no—
196C4⋊Q8—1228no—
197C4⋊Q8—1224no—
198C4⋊Q8—1222no—
199C4⋊Q8—1222no—
200#125—1224no—
201#125—1228no—
202#125—1224no—
203#125—1222no—
204C22⋊Q8—614no—
205C22⋊Q8—1222no—
206C22⋊Q8—1224no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—614no—
210C4×Q8—1224no—
211C4×Q8—1224no—
212C4×Q8—1224no—
213C4×Q8—1224no—
214C4×Q8—1222no—
215#117—1222no—
216#117—1224no—
217C2×C4⋊C4—612no—
218C2×C4⋊C4—1221no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4D24411yescharacteristic
221C2×C4⋊C4C22×D6112yes—
222C2×C42—1221no—
223C2×C42D24412yes—
224C2×C42C22×D6111yescharacteristic
225#107—1228no—
226C4⋊C8—22no—
227#103—1228no—
228C22⋊C8C22×D6114yes—
229C4×C8—22no—
230#95—612no—
231#95—1224no—
232#95—1224no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—614no—
236C2×Q8—1224no—
237C2×Q8—1222no—
238C2×Q8—2444no—
239C2×Q8—2441no—
240C2×Q8—2441no—
241C2×Q8Dic24414yes—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4C2×D24411yescharacteristic
245C22×C4C2×D24412yes—
246C22×C4C2×D24412yes—
247C22×C4C2×D24411yescharacteristic
248C22×C4C23×D6111yescharacteristic
249C2×C8—24no—
250C4⋊C4—24no—
251C4⋊C4—22no—
252C4⋊C4—614no—
253C4⋊C4—1228no—
254C4⋊C4—1224no—
255C4⋊C4—1224no—
256C4⋊C4—1224no—
257C4⋊C4—1224no—
258C4⋊C4—1224no—
259C4⋊C4—1222no—
260C4⋊C4—1222no—
261C4⋊C4—1222no—
262C4⋊C4—1224no—
263C4⋊C4—2442no—
264C4⋊C4—2442no—
265C22⋊C4—1224no—
266C42—22no—
267C42—22no—
268C42—1228no—
269C42—1224no—
270C42—2442no—
271C42D4×D6112yes—
272C42D4×D6112yes—
273C23C22×D24411yescharacteristic
274Q8—28no—
275Q8—4882no—
276Q8—4884no—
277Q8—4882no—
278Q8—2444no—
279Q8—2444no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—1224no—
287C2×C4—1224no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4—2444no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4#1451974711yescharacteristic
294C2×C4#1451974711yescharacteristic
295C2×C4#1451974712yes—
296C2×C4C2×Dic24414yes—
297C2×C4C2×Dic24412yes—
298C2×C4#1451981012yes—
299C2×C4C2×D4×D6112yes—
300C8—44no—
301C22—21no—
302C22—21no—
303C22C2×#1451974711yescharacteristic
304C22C22×Dic24411yescharacteristic
305C22C2×#1451981011yescharacteristic
306C4—21no—
307C4—22no—
308C4—24no—
309C4—42no—
310C4—4882no—
311C4—4882no—
312C4—2444no—
313C4—2444no—
314C4#1683992014yes—
315C4#1683993712yes—
316C2—21no—
317C2—21no—
318C2C2×#1683992011yescharacteristic
319C2C2×#1683993711yescharacteristic
320C2#2145522611yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1461C2
socleC22×C61C2×#120
prime radicalC23×C61C22×D4
hypercenter#1461D61
hypocenterC61#18411
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
189. C61#18411semidirect

Maximal subgroups

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

subgroupindex
#214596292
#214569722
#214510312
#214448632
#214443192
C61×#14612
#1841161