Finite Group Databasebeta

← #29647882

Subgroups — #29647882

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459629C211yesmaximal, characteristic
2#21454904C211yesmaximal, characteristic
3#21444382C212yesmaximal
4#21444369C214yesmaximal
5#21444330C214yesmaximal
6#21444317C212yesmaximal
7C61×#2356C211yesmaximal, characteristic
8Q8×Dic122C414yescomplemented
9#16840491C414yescomplemented
10#16840488—24no—
11#16840393—21no—
12#16840393—22no—
13#16840319C2214yes—
14#16840319C2212yes—
15#16840240—21no—
16#16839893—24no—
17#16839819C2214yes—
18#16839815C2212yes—
19#16839498C2214yes—
20#16839498C2218yes—
21#16839498C2214yes—
22C2×C4×Q8×C61C2211yescharacteristic
23C61×#391—21no—
24C61×#387—22no—
25C61×#383C2212yes—
26C61×#382C2211yescharacteristic
27C4×C4⋊C4×C61—21no—
28C4×C4⋊C4×C61C2211yescharacteristic
29C61×#376C2212yes—
30#14519777C2×C414yes—
31#14519777C2×C412yes—
32#14519777C2×C414yes—
33#14519777C2×C412yes—
34#14519776—28no—
35#14519776—24no—
36#14519769—22no—
37#14519769—24no—
38#14519768—28no—
39#14519768—24no—
40#14519756—24no—
41#14519756—22no—
42#14519756C2314yes—
43Q8×Dic61—24no—
44Q8×Dic61—24no—
45C4×Dic122—22no—
46C4×Dic122—24no—
47C4×Dic122C2×C418yes—
48C4×Dic122C2×C414yes—
49#14519736—24no—
50#14519736—22no—
51C4⋊#92418—22no—
52C4⋊#92418—24no—
53C22⋊#92418C2314yes—
54#14519688—24no—
55C4×#92418—22no—
56C22×Q8×C61D411yescharacteristic
57C4×Q8×C61C2×C414yes—
58C4×Q8×C61D414yes—
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61D411yescharacteristic
62C2×C4⋊C4×C61D412yes—
63C2×C42×C61—22no—
64C2×C42×C61—21no—
65C2×C42×C61C2312yes—
66C2×C42×C61C2311yescharacteristic
67#95×C61—22no—
68#95×C61—22no—
69#95×C61—21no—
70#95×C61—22no—
71#95×C61C2311yescharacteristic
72#95×C61C2312yes—
73#95×C61C2311yescharacteristic
74C2×Q8×C61—22no—
75C2×Q8×C61—22no—
76C2×Q8×C61C22⋊C414yes—
77C22×C4×C61—21no—
78C22×C4×C61—22no—
79C22×C4×C61—21no—
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D412yes—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2411yescharacteristic
85C2×Dic122—22no—
86C2×Dic122—24no—
87C2×Dic122C22×C414yes—
88C4⋊C4×C61—42no—
89C4⋊C4×C61—42no—
90C4⋊C4×C61C22⋊C418yes—
91C4⋊C4×C61C22⋊C414yes—
92C42×C61—44no—
93C42×C61—42no—
94C42×C61C22×C414yes—
95C42×C61C22×C412yes—
96C42×C61C2×D412yes—
97C42×C61C2×D414yes—
98C22⋊Dic61—24no—
99C4⋊Dic61—24no—
100C4⋊Dic61—22no—
101C4⋊Dic61—24no—
102C4⋊Dic61—22no—
103C4⋊Dic61—22no—
104C4⋊Dic61—24no—
105C4⋊Dic61C22×C414yes—
106#11758196—24no—
107#11758196—28no—
108#11758196—24no—
109C4×Dic61—28no—
110C4×Dic61—24no—
111C2×#92418—24no—
112C23×C61C22×D411yescharacteristic
113Q8×C61—28no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—42no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—42no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61—42no—
128C2×C4×C61C2×C22⋊C414yes—
129C2×C4×C61C2×C22⋊C412yes—
130C2×C4×C61C2×C22⋊C414yes—
131C2×C4×C61C2×C22⋊C412yes—
132C2×C4×C61#13712yes—
133C2×C4×C61#13712yes—
134C2×C4×C61C23×C411yescharacteristic
135C2×C4×C61C22×D411yescharacteristic
136C2×Dic61—24no—
137C2×Dic61—24no—
138Dic122—44no—
139Dic122—84no—
140Dic122—82no—
141#92418—44no—
142#17632—611nomaximal, Sylow
143C22×C61—21no—
144C22×C61—21no—
145C22×C61C22×C22⋊C411yescharacteristic
146C22×C61C2×#13711yescharacteristic
147C22×C61C2×#13711yescharacteristic
148C4×C61—21no—
149C4×C61—21no—
150C4×C61—24no—
151C4×C61—24no—
152C4×C61—22no—
153C4×C61—42no—
154C4×C61—81no—
155C4×C61—82no—
156C4×C61#41914yes—
157Dic61—44no—
158Dic61—44no—
159#3502—611no—
160#2983—611no—
161#2356D6111yescharacteristic
162#1518—612no—
163#1515—614no—
164#1506—614no—
165#1503—612no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61#288411yescharacteristic
169C2×C61C2×#41911yescharacteristic
170C2×C61#290111yescharacteristic
171Q82—614no—
172#557—614no—
173#557—1224no—
174#544—1221no—
175#544—1222no—
176#533—612no—
177#533—614no—
178#519—1221no—
179C2×C4×Q8C2×D6111yescharacteristic
180#446—1224no—
181#432—614no—
182#431—612no—
183#391—21no—
184#387—22no—
185#383C2×D6112yes—
186#382C2×D6111yescharacteristic
187C4×C4⋊C4—21no—
188C4×C4⋊C4C2×D6111yescharacteristic
189#376C2×D6112yes—
190#328—614no—
191#328—614no—
192#328—618no—
193C61#1763211yescharacteristic, complemented, Sylow
194C22×Q8C22⋊D6111yescharacteristic
195C4⋊Q8—612no—
196C4⋊Q8—614no—
197C4⋊Q8—612no—
198C4⋊Q8—614no—
199C4⋊Q8—1228no—
200C4⋊Q8—1224no—
201#125—1228no—
202#125—1224no—
203#125—1222no—
204#125—1224no—
205C22⋊Q8—614no—
206C22⋊Q8—1224no—
207C22⋊Q8—1222no—
208C4×Q8—614no—
209C4×Q8—618no—
210C4×Q8—1224no—
211C4×Q8—1224no—
212C4×Q8—1222no—
213C4×Q8—1224no—
214C4×Q8C4×D6114yes—
215C4×Q8D24414yes—
216#117—1224no—
217#117—1222no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C22⋊D6111yescharacteristic
221C2×C4⋊C4C22⋊D6112yes—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42C22×D6112yes—
225C2×C42C22×D6111yescharacteristic
226C4⋊C8—1222no—
227C4⋊C8—1224no—
228C22⋊C8—614no—
229#97—1224no—
230C4×C8—1222no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95C22×D6111yescharacteristic
236#95C22×D6112yes—
237#95C22×D6111yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—614no—
241C2×Q8—1222no—
242C2×Q8—1224no—
243C2×Q8#1175819814yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4C2×C22⋊D6112yes—
248C22×C4C2×C22⋊D6112yes—
249C22×C4C2×C22⋊D6111yescharacteristic
250C22×C4C2×C22⋊D6111yescharacteristic
251C22×C4C23×D6111yescharacteristic
252C2×C8—1224no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—614no—
256C4⋊C4—1224no—
257C4⋊C4—1224no—
258C4⋊C4—1228no—
259C4⋊C4—1224no—
260C4⋊C4—1222no—
261C4⋊C4—1224no—
262C4⋊C4—1222no—
263C4⋊C4—1222no—
264C4⋊C4—1224no—
265C4⋊C4#1175819818yes—
266C4⋊C4#1175819814yes—
267C22⋊C4—1224no—
268C42—44no—
269C42—42no—
270C42—1228no—
271C42—1224no—
272C42C2×C4×D6114yes—
273C42C2×C4×D6112yes—
274C42C2×D24412yes—
275C42C2×D24414yes—
276C23C22×C22⋊D6111yescharacteristic
277Q8—28no—
278Q8—4884no—
279Q8—4882no—
280Q8—2444no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—1224no—
296C2×C4—1224no—
297C2×C4C2×#1175819814yes—
298C2×C4C2×#1175819812yes—
299C2×C4C2×#1175819814yes—
300C2×C4C2×#1175819812yes—
301C2×C4#1451980712yes—
302C2×C4#1451980712yes—
303C2×C4C22×C4×D6111yescharacteristic
304C2×C4C22×D24411yescharacteristic
305C8—2444no—
306C22—21no—
307C22—21no—
308C22C22×#1175819811yescharacteristic
309C22C2×#1451980711yescharacteristic
310C22C2×#1451980711yescharacteristic
311C4—24no—
312C4—22no—
313C4—21no—
314C4—21no—
315C4—24no—
316C4—42no—
317C4—81no—
318C4—82no—
319C4—2444no—
320C4—2444no—
321C4#1683977014yes—
322C2—21no—
323C2—21no—
324C2#2145428211yescharacteristic
325C2C2×#1683977011yescharacteristic
326C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2356C2
socleC22×C61C22×C22⋊C4
prime radicalC23×C61C22×D4
cosocleC22×C4×C61C24
hypercenter#2356D61
hypocenterC61#17632
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2356

Complements

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

NH ≅ G/Ntype
8. Q8×Dic122C4semidirect
9. #16840491C4semidirect
193. C61#17632semidirect

Maximal subgroups

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

subgroupindex
#214596292
#214549042
#214443822
#214443692
#214443302
#214443172
C61×#23562
#1763261