Finite Group Databasebeta

← #30604744

Subgroups — #30604744

Order 15616.

468 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461519C211yesmaximal, characteristic
2#21461053C211yesmaximal, characteristic
3#21460800C212yesmaximal
4#21460799C212yesmaximal
5#21451587C211yesmaximal, characteristic
6C2×#14519856—21no—
7#16840779—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840607—21no—
11#16840606C2212yes—
12C2×#14519776C411yescharacteristic, complemented
13C2×#14519755—21no—
14C2×#14519735C2211yescharacteristic
15#16839695—21no—
16#16839692C411yescharacteristic, complemented
17#16839691—21no—
18#16839689C2211yescharacteristic
19#16839688C2211yescharacteristic
20#16839666—21no—
21#16839626C411yescharacteristic, complemented
22C61×#398C411yescharacteristic, complemented
23#14519856—24no—
24#14519856—42no—
25C2×C4⋊#6923—41no—
26C2×C22⋊#6923—22no—
27C2×C4×#6923—41no—
28#14519844—41no—
29C22⋊#92420—22no—
30C4×#92420—41no—
31C22×Dic122—21no—
32C22×C4×D61D411yescharacteristic
33#14519776—41no—
34#14519776—41no—
35#14519755—42no—
36#14519755—42no—
37#14519735—41no—
38#14519735—41no—
39C2×C4⋊Dic61C2×C411yescharacteristic
40C2×C4⋊Dic61D411yescharacteristic
41C2×#11758196—21no—
42C2×#11758196—21no—
43C2×#11758196—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198D411yescharacteristic
47C2×#11758198C2311yescharacteristic
48C2×C4×Dic61C2×C411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684C2×C411yescharacteristic
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C4⋊C4×C61C2×C411yescharacteristic
58#95×C61—21no—
59#95×C61C2×C411yescharacteristic
60C23×D61C2×D411yescharacteristic
61C22×#6923—42no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—22no—
75C2×C4×D61D812yes—
76C2×C4×D61SD1612yes—
77C2×Dic122—41no—
78C2×Dic122—41no—
79C2×Dic122—42no—
80C2×Dic122—42no—
81C22⋊#6923—24no—
82C22⋊#6923—42no—
83C2×#92420—42no—
84C4⋊#6923—44no—
85C4×#6923—44no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—42no—
89#11758198—41no—
90#11758198—41no—
91#11758198—41no—
92#11758198—41no—
93#11758198—41no—
94#11758198—41no—
95C4⋊Dic61—41no—
96C4⋊Dic61—41no—
97#11758196—42no—
98#11758196—42no—
99#11758196—42no—
100C4×Dic61—21no—
101C4×Dic61—21no—
102C4×Dic61—21no—
103C4×Dic61—21no—
104C23×C61C2×C22⋊C411yescharacteristic
105C22×D61—21no—
106C22×D61—21no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61C2×D811yescharacteristic
110C22×D61C2×SD1611yescharacteristic
111C2×#6923—44no—
112C2×#6923—82no—
113C2×#6923—82no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61#10212yes—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61#9911yescharacteristic
138C2×Dic61#9911yescharacteristic
139C2×Dic61#10212yes—
140C2×Dic61#10412yes—
141C4×D61—22no—
142C4×D61—22no—
143C4×D61—22no—
144C4×D61—22no—
145Dic122—82no—
146Dic122—82no—
147#92420—82no—
148#11694—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—21no—
151C22×C61C2×#9911yescharacteristic
152C22×C61C2×#10211yescharacteristic
153C22×C61C2×#10411yescharacteristic
154C2×D61—21no—
155C2×D61—21no—
156C2×D61—41no—
157C2×D61—41no—
158C2×D61—41no—
159C2×D61—41no—
160C2×D61—81no—
161#6923—84no—
162C4×C61—22no—
163C4×C61—41no—
164C4×C61—41no—
165C4×C61—81no—
166C4×C61—81no—
167Dic61—41no—
168Dic61—41no—
169Dic61—81no—
170Dic61—81no—
171Dic61#32712yes—
172Dic61#32712yes—
173#2621—611no—
174#1740—611no—
175#1458—611no—
176#1344—612no—
177#1344—612no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61C2×#32711yescharacteristic
181C2×C61#212311yescharacteristic
182C2×C61#212911yescharacteristic
183D61—41no—
184D61—41no—
185C2×C4⋊Q8—611no—
186C2×C22⋊Q8—1221no—
187C2×#117—611no—
188C2×#117—1221no—
189#431—1221no—
190#398—611no—
191#398—1221no—
192#398#692311yescharacteristic, complemented
193#397—611no—
194#397—611no—
195#397—1221no—
196#391—1221no—
197#382—611no—
198#342—612no—
199#342—1221no—
200#324—1221no—
201#323—612no—
202C61#1169411yescharacteristic, complemented, Sylow
203C22×Q8—1221no—
204C23×C4—611no—
205C4⋊Q8—2441no—
206C4⋊Q8—2441no—
207C22⋊Q8—2442no—
208C22⋊Q8—2442no—
209#117—1224no—
210#117—2441no—
211#117—2441no—
212#117—2442no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—611no—
215C2×C4⋊C4—611no—
216C2×C4⋊C4—1221no—
217C2×C4⋊C4—1221no—
218C2×C4⋊C4—1221no—
219C2×C4⋊C4—2441no—
220C2×C4⋊C4C2×#692311yescharacteristic
221C2×C4⋊C4C2×#692311yescharacteristic
222C2×C22⋊C4—611no—
223C2×C22⋊C4—611no—
224C2×C22⋊C4—1222no—
225C2×C22⋊C4—1221no—
226C2×C22⋊C4—1221no—
227C2×C42—611no—
228C2×C42—611no—
229C2×C42—2441no—
230C4⋊C8—2441no—
231C22⋊C8—1222no—
232C4×C8—2441no—
233#95—21no—
234#95—611no—
235#95—1221no—
236#95—1221no—
237#95—1221no—
238#95—1221no—
239#95C2×#692311yescharacteristic
240C24—611no—
241C2×Q8—2442no—
242C2×Q8—2441no—
243C2×Q8—2441no—
244C2×Q8—2442no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—611no—
248C22×C4—612no—
249C22×C4—612no—
250C22×C4—611no—
251C22×C4—611no—
252C22×C4—1221no—
253C22×C4—1221no—
254C22×C4—1222no—
255C22×C4—1221no—
256C22×C4—1221no—
257C22×C4—2442no—
258C22×C4C22⋊#692311yescharacteristic
259C22×C4C22⋊#692311yescharacteristic
260C22×C4C22×#692311yescharacteristic
261C2×C8—2442no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—2441no—
266C4⋊C4—2441no—
267C4⋊C4—2442no—
268C4⋊C4—2442no—
269C4⋊C4—2444no—
270C4⋊C4—2442no—
271C22⋊C4—1224no—
272C22⋊C4—2441no—
273C22⋊C4—2441no—
274C22⋊C4—2441no—
275C22⋊C4—2441no—
276C22⋊C4—2441no—
277C22⋊C4—2441no—
278C22⋊C4—2442no—
279C42—1221no—
280C42—1221no—
281C42—1221no—
282C42—1221no—
283C42—2444no—
284C23—611no—
285C23—611no—
286C23—1221no—
287C23—1221no—
288C23—2441no—
289C23—2441no—
290C23C2×C22⋊#692311yescharacteristic
291Q8—4882no—
292Q8—4882no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—611no—
306C2×C4—611no—
307C2×C4—612no—
308C2×C4—612no—
309C2×C4—1222no—
310C2×C4—1222no—
311C2×C4—1222no—
312C2×C4—1222no—
313C2×C4—1221no—
314C2×C4—1221no—
315C2×C4—4882no—
316C2×C4—4882no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2444no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4#1451983912yes—
327C8—4882no—
328C22—21no—
329C22—21no—
330C22—1221no—
331C22—1221no—
332C22—4881no—
333C22—2441no—
334C22—2441no—
335C22—2441no—
336C22—2441no—
337C22D61⋊#9911yescharacteristic
338C22C2×#1451983911yescharacteristic
339C22D61⋊#10411yescharacteristic
340C4—22no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—612no—
346C4—612no—
347C4—4881no—
348C4—4884no—
349C4—4881no—
350C4—2441no—
351C4—2441no—
352C2—21no—
353C2—21no—
354C2—2441no—
355C2—2441no—
356C2D61⋊#32711yescharacteristic
357C2#2146208211yescharacteristic
358C2#2146210211yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#398C4
socleC22×C61C2×#99
prime radicalC23×D61C2×D4
cosocleC2×#11758198C23
hypercenter#398#6923
hypocenterC61#11694
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #398

Complements

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

NH ≅ G/Ntype
12. C2×#14519776C4semidirect
16. #16839692C4semidirect
21. #16839626C4semidirect
22. C61×#398C4semidirect
192. #398#6923semidirect
202. C61#11694semidirect

Maximal subgroups

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

subgroupindex
#214615192
#214610532
#214608002
#214607992
#214515872
#1169461