Finite Group Databasebeta

← #30601151

Subgroups — #30601151

Order 15616.

512 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 78 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#21462296C211yesmaximal, characteristic
2#21461719C211yesmaximal, characteristic
3#21461229C212yesmaximal
4#21460667C212yesmaximal
5#21450303C211yesmaximal, characteristic
6#16840734C2212yes—
7#16840711—24no—
8#16840705C2212yes—
9#16840704—21no—
10C4×C22⋊#6923—21no—
11#16840642—21no—
12#16840642—21no—
13#16840616—24no—
14C2×#14519753C2211yescharacteristic
15C2×C4⋊C4×D61C2211yescharacteristic
16#16839703—21no—
17#16839703—21no—
18#16839669C411yescharacteristic, complemented
19#16839628C411yescharacteristic, complemented
20#16839627C411yescharacteristic, complemented
21#16839624—21no—
22#16839624—21no—
23C4×#11758198C2211yescharacteristic
24C61×#391C411yescharacteristic, complemented
25C2×#11758217—22no—
26C2×C4⋊#6923—22no—
27C2×C22⋊#6923—22no—
28C2×C4×#6923—22no—
29#14519834—22no—
30#14519834—22no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4D414yes—
35C22×C4×D61D411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61C2311yescharacteristic
38#14519753—22no—
39#14519753—22no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×#11758196—21no—
43C2×#11758196D411yescharacteristic
44C2×#11758196D411yescharacteristic
45C4⋊C4×D61—22no—
46C4⋊C4×D61—22no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C2×C4×Dic61C2×C411yescharacteristic
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C22×Q8×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C42×C61C2×C411yescharacteristic
60#95×C61—21no—
61#95×C61—21no—
62C23×D61C2×D411yescharacteristic
63C22×#6923—24no—
64C2×Q8×C61—22no—
65C2×Q8×C61—41no—
66C2×Q8×C61—41no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C2×D411yescharacteristic
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—41no—
82C2×C4×D61—41no—
83C2×C4×D61—42no—
84C2×C4×D61—42no—
85C2×C4×D61C2×D412yes—
86C2×C4×D61#4112yes—
87C22⋊#6923—42no—
88C22⋊#6923—42no—
89#11758217—42no—
90#11758217—42no—
91C2×#92420—42no—
92C4⋊#6923—82no—
93C4×#6923—82no—
94C4⋊C4×C61—22no—
95C42×C61—41no—
96#11758198—42no—
97#11758198—41no—
98#11758198—41no—
99C4⋊Dic61—22no—
100C4⋊Dic61—22no—
101#11758196—22no—
102#11758196—22no—
103#11758196—42no—
104C4×Dic61—41no—
105C23×C61C2×C22⋊C411yescharacteristic
106C22×D61—21no—
107C22×D61—21no—
108C22×D61—22no—
109C22×D61—41no—
110C22×D61#12011yescharacteristic
111C22×D61#12311yescharacteristic
112C2×#6923—44no—
113C2×#6923—44no—
114C2×#6923—44no—
115Q8×C61—82no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
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—41no—
127C2×C4×C61—41no—
128C2×C4×C61C4×D412yes—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—42no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61#12112yes—
141C4×D61—42no—
142C4×D61—42no—
143C4×D61—42no—
144C4×D61—42no—
145C4×D61—81no—
146C4×D61—81no—
147#92420—82no—
148#9460—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—21no—
151C22×C61#38611yescharacteristic
152C22×C61#41011yescharacteristic
153C22×C61#41011yescharacteristic
154C2×D61—21no—
155C2×D61—21no—
156C2×D61—21no—
157C2×D61—21no—
158C2×D61—21no—
159C2×D61—42no—
160C2×D61—42no—
161C2×D61#45512yes—
162#6923—84no—
163C4×C61—22no—
164C4×C61—41no—
165C4×C61—41no—
166C4×C61—81no—
167C4×C61—81no—
168Dic61—22no—
169Dic61—41no—
170Dic61—41no—
171Dic61—81no—
172Dic61—81no—
173#2503—611no—
174#2313—611no—
175#1821—611no—
176#1511—612no—
177#1284—612no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61#188511yescharacteristic
181C2×C61#189911yescharacteristic
182C2×C61#191211yescharacteristic
183D61—22no—
184D61—41no—
185C2×C22⋊Q8—611no—
186C22×C4⋊C4—611no—
187#407—612no—
188#400—1221no—
189#400—1221no—
190#391—611no—
191#391#692311yescharacteristic, complemented
192#387—1224no—
193#382—611no—
194#382—611no—
195#381—612no—
196#381—1221no—
197#381—1221no—
198#380—1221no—
199C4×C22⋊C4—611no—
200C4×C22⋊C4—1221no—
201#343—1221no—
202#343—1221no—
203#328—1224no—
204C61#946011yescharacteristic, complemented, Sylow
205C22×Q8C2×#692311yescharacteristic
206C23×C4—611no—
207C23×C4—611no—
208C23×C4—611no—
209C2×M16—1222no—
210C22⋊Q8—1222no—
211C22⋊Q8—1222no—
212C2×C4⋊C4—611no—
213C2×C4⋊C4—611no—
214C2×C4⋊C4—611no—
215C2×C4⋊C4—611no—
216C2×C4⋊C4—1222no—
217C2×C4⋊C4—1222no—
218C2×C4⋊C4—1222no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4C2×#692311yescharacteristic
221C2×C22⋊C4—1221no—
222C2×C22⋊C4—1221no—
223C2×C22⋊C4—1222no—
224C2×C42—611no—
225C2×C42—1222no—
226C2×C42C2×#692311yescharacteristic
227C22⋊C8—1222no—
228C22⋊C8—1222no—
229#95—21no—
230#95—21no—
231#95—614no—
232#95—1221no—
233#95—1221no—
234#95—1221no—
235#95—1221no—
236#95—1221no—
237#95—1222no—
238#95—1222no—
239#95—1222no—
240#95—1221no—
241#95—1221no—
242C24—611no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—611no—
249C22×C4—612no—
250C22×C4—612no—
251C22×C4—611no—
252C22×C4—611no—
253C22×C4—1221no—
254C22×C4—1221no—
255C22×C4—1221no—
256C22×C4—1221no—
257C22×C4—1224no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—2441no—
261C22×C4—2441no—
262C22×C4—2442no—
263C22×C4—2442no—
264C22×C4C22⋊#692311yescharacteristic
265C22×C4C22⋊#692311yescharacteristic
266C22×C4C22×#692311yescharacteristic
267M16—2442no—
268M16—2442no—
269C2×C8—2442no—
270C4⋊C4—22no—
271C4⋊C4—1222no—
272C4⋊C4—1222no—
273C4⋊C4—1222no—
274C4⋊C4—1222no—
275C4⋊C4—4882no—
276C4⋊C4—2442no—
277C22⋊C4—2441no—
278C22⋊C4—2441no—
279C22⋊C4—2442no—
280C22⋊C4—2442no—
281C22⋊C4—2442no—
282C42—41no—
283C42—4882no—
284C42—2441no—
285C23—611no—
286C23—611no—
287C23—1222no—
288C23—1221no—
289C23—1221no—
290C23—2441no—
291C23C2×C22⋊#692311yescharacteristic
292Q8—82no—
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—612no—
306C2×C4—1221no—
307C2×C4—1221no—
308C2×C4—1221no—
309C2×C4—1221no—
310C2×C4—4881no—
311C2×C4—4881no—
312C2×C4—2442no—
313C2×C4—2442no—
314C2×C4—2442no—
315C2×C4—2442no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2442no—
319C2×C4—2444no—
320C2×C4—2444no—
321C2×C4—2444no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4D4×#692312yes—
327C8—4882no—
328C22—21no—
329C22—21no—
330C22—612no—
331C22—1221no—
332C22—1221no—
333C22—1221no—
334C22—1221no—
335C22—1221no—
336C22—2442no—
337C22—2442no—
338C22#1684071011yescharacteristic
339C22#1684074111yescharacteristic
340C22#1684074211yescharacteristic
341C4—22no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—81no—
346C4—1222no—
347C4—4881no—
348C4—4884no—
349C4—4881no—
350C4—2441no—
351C4—2441no—
352C2—21no—
353C2—21no—
354C2—1222no—
355C2—2441no—
356C2#2146182411yescharacteristic
357C2#2146184211yescharacteristic
358C2#2146186611yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#391#6923
hypocenterC61#9460
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
18. #16839669C4semidirect
19. #16839628C4semidirect
20. #16839627C4semidirect
24. C61×#391C4semidirect
191. #391#6923semidirect
204. C61#9460semidirect

Maximal subgroups

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

subgroupindex
#214622962
#214617192
#214612292
#214606672
#214503032
#946061