Finite Group Databasebeta

← #30599753

Subgroups — #30599753

Order 15616.

468 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462362C211yesmaximal, characteristic
2#21461949C211yesmaximal, characteristic
3#21461742C212yesmaximal, complemented
4#21460696C212yesmaximal, complemented
5#329×D61C211yesmaximal, characteristic
6C2×#14519840—21no—
7C2×#14519839—21no—
8#16840721C2211yescharacteristic
9(C2×#6923)⋊Q8—21no—
10#16840712—22no—
11#16840710—21no—
12#16840709—22no—
13#16840708C2211yescharacteristic
14#16840706C2212yes—
15#16840636—22no—
16#16840636—22no—
17#124×D61C2211yescharacteristic
18#97×D61C2212yes—
19#16839502C411yescharacteristic, complemented
20#16839501C412yescomplemented
21C61×#329C411yescharacteristic, complemented
22C23×#6923—21no—
23C2×D61⋊C8—21no—
24C2×C4⋊#6923—21no—
25C2×C4⋊#6923—22no—
26C2×C4⋊#6923—21no—
27C2×C4⋊#6923D412yes—
28C2×C4⋊#6923D412yes—
29C2×C22⋊#6923—22no—
30C2×C22⋊#6923—21no—
31C2×C4×#6923—21no—
32#14519840—42no—
33#14519839—42no—
34#14519832—22no—
35(C2×#6923)⋊C4—22no—
36(C2×#6923)⋊C4—22no—
37(C2×#6923)⋊C4—22no—
38C2×Q8×D61D411yescharacteristic
39C2×D4×D61D411yescharacteristic
40C2×C8×D61—22no—
41#14519766Q811yescharacteristic
42#14519765C2×C411yescharacteristic
43#14519764Q811yescharacteristic
44#14519763—22no—
45C22⋊C4×D61—22no—
46C42×D61C2311yescharacteristic
47#14519688C2×C412yescomplemented
48#14519687—22no—
49#124×C61C2×C411yescharacteristic
50#97×C61C2×C412yescomplemented
51C23×D61—21no—
52C22×#6923—22no—
53C22×#6923—22no—
54C22×#6923—41no—
55C22×#6923—81no—
56C22×#6923—82no—
57C2×Q8×C61C4⋊C411yescharacteristic
58C2×D4×C61C4⋊C411yescharacteristic
59C2×C22⋊D61—21no—
60Q8×D61—41no—
61Q8×D61—41no—
62D4×D61—41no—
63D4×D61—41no—
64C2×D244C4⋊C411yescharacteristic
65C2×C4×D61—21no—
66C2×C4×D61C2×D412yes—
67C2×C4×D61C2×D411yescharacteristic
68C2×Dic122—21no—
69C2×Dic122C4⋊C411yescharacteristic
70C22⋊#6923—81no—
71C22⋊#6923—82no—
72C2×#92420—41no—
73C4⋊#6923—42no—
74C4⋊#6923—42no—
75C4⋊#6923—41no—
76C4⋊#6923—41no—
77C4⋊#6923—44no—
78C4⋊#6923—81no—
79C4×#6923—81no—
80D61⋊C8—22no—
81D61⋊C8—41no—
82C2×C8×C61—22no—
83C22⋊C4×C61—22no—
84C42×C61C22×C411yescharacteristic
85C22⋊Dic61—22no—
86#11758198—22no—
87#11758198—22no—
88C4×Dic61C2×Q811yescharacteristic
89C4×Dic61#4112yes—
90C2×#92418—22no—
91C8×D61—42no—
92C8×D61—42no—
93C23×C61—21no—
94C22×D61—21no—
95C22×D61—41no—
96C22×D61—42no—
97C22×D61—41no—
98C22×D61—42no—
99C22×D61#12011yescharacteristic
100C2×#6923—42no—
101C2×#6923—42no—
102C2×#6923—42no—
103C2×#6923—42no—
104C2×#6923—42no—
105C2×#6923—42no—
106C2×#6923—84no—
107C2×#6923—81no—
108C2×#6923—82no—
109Q8×C61—41no—
110D4×C61—41no—
111C2×C4×C61—21no—
112C2×C4×C61C2×C4⋊C411yescharacteristic
113C2×C4×C61C4×D412yes—
114C22⋊D61—81no—
115C2×Dic61—21no—
116C2×Dic61C22⋊Q812yes—
117C2×Dic61#12311yescharacteristic
118D244—41no—
119C4×D61—21no—
120C4×D61—22no—
121C4×D61—22no—
122C4×D61—21no—
123C4×D61—41no—
124C4×D61—41no—
125Dic122—41no—
126Dic122—81no—
127#92420—42no—
128C8×C61—42no—
129#92418—42no—
130#8914—611nomaximal, Sylow
131C22×C61—41no—
132C22×C61—42no—
133C22×C61#38511yescharacteristic
134C2×D61—41no—
135C2×D61—42no—
136C2×D61—81no—
137C2×D61—81no—
138C2×D61—82no—
139C2×D61#45311yescharacteristic
140C2×D61#45412yes—
141C2×D61#45512yes—
142C2×D61#45511yescharacteristic
143#6923—82no—
144#6923—82no—
145#6923—161no—
146C4×C61—21no—
147C4×C61—22no—
148C4×C61—41no—
149Dic61—21no—
150Dic61—22no—
151Dic61—41no—
152#2379—611no—
153#1959—611no—
154#1836—612no—
155C2×#329—611no—
156#1295—612no—
157C2×C61—81no—
158C2×C61#185611yescharacteristic
159C2×C61#186012yes—
160D61—81no—
161D61#165711yescharacteristic
162D61#165911yescharacteristic
163D61#166212yes—
164C2×#124—611no—
165C2×#103—1221no—
166C2×#102—1221no—
167C2×#97—612no—
168C2×#97—611no—
169#391—1221no—
170#388—1222no—
171#386—1221no—
172#385—1222no—
173#384—611no—
174#382—612no—
175#340—1222no—
176#340—1222no—
177#329—611no—
178#329—612no—
179#329#692311yescharacteristic, complemented
180C61#891411yescharacteristic, complemented, Sylow
181C22×Q8—611no—
182C22×D4—611no—
183C23×C4—1221no—
184C22×C8—1222no—
185C22×C8—1221no—
186#124—611no—
187#124—611no—
188#124—611no—
189#124—1222no—
190#124C2×#692311yescharacteristic
191C2×C4⋊C4—612no—
192C2×C4⋊C4—612no—
193C2×C4⋊C4—1221no—
194C2×C4⋊C4—1222no—
195C2×C4⋊C4—1221no—
196C2×C22⋊C4—1222no—
197C2×C22⋊C4—1222no—
198C2×C22⋊C4—1221no—
199C2×C42—611no—
200C2×C42—1221no—
201#103—2442no—
202#102—2442no—
203#97—612no—
204#97—1222no—
205#97—1222no—
206#97C2×#692312yescomplemented
207#95—1222no—
208#95—1222no—
209#95—1222no—
210C24—1221no—
211C2×Q8—611no—
212C2×Q8—1221no—
213C2×Q8—2441no—
214C2×Q8—2441no—
215C2×Q8C4⋊#692311yescharacteristic
216C2×D4—611no—
217C2×D4—1221no—
218C2×D4—2441no—
219C2×D4—2441no—
220C2×D4C4⋊#692311yescharacteristic
221C22×C4—611no—
222C22×C4—612no—
223C22×C4—1221no—
224C22×C4—1222no—
225C22×C4—1222no—
226C22×C4—4882no—
227C22×C4—4881no—
228C22×C4—2441no—
229C2×C8—22no—
230C2×C8—1222no—
231C2×C8—1222no—
232C2×C8—2442no—
233C2×C8—2442no—
234C2×C8—2441no—
235C2×C8—2441no—
236C4⋊C4—4881no—
237C4⋊C4—2441no—
238C4⋊C4—2441no—
239C4⋊C4—2444no—
240C4⋊C4—2442no—
241C4⋊C4—2442no—
242C22⋊C4—22no—
243C22⋊C4—1222no—
244C22⋊C4—1222no—
245C22⋊C4—1222no—
246C22⋊C4—4881no—
247C22⋊C4—4882no—
248C42—611no—
249C42—612no—
250C42—4881no—
251C42C22×#692311yescharacteristic
252C23—21no—
253C23—611no—
254C23—1221no—
255C23—2442no—
256C23—2441no—
257C23—2441no—
258C23—2442no—
259Q8—41no—
260Q8—4881no—
261Q8—2441no—
262D4—41no—
263D4—4881no—
264D4—2441no—
265C2×C4—21no—
266C2×C4—612no—
267C2×C4—611no—
268C2×C4—1221no—
269C2×C4—1222no—
270C2×C4—1222no—
271C2×C4—1221no—
272C2×C4—1221no—
273C2×C4—4881no—
274C2×C4—4882no—
275C2×C4—4884no—
276C2×C4—2441no—
277C2×C4—2441no—
278C2×C4—2442no—
279C2×C4—2442no—
280C2×C4—2442no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4C2×C4⋊#692311yescharacteristic
285C2×C4D4×#692312yes—
286C8—42no—
287C8—2442no—
288C8—2442no—
289C22—41no—
290C22—42no—
291C22—611no—
292C22—611no—
293C22—612no—
294C22—612no—
295C22—4881no—
296C22—4881no—
297C22—4882no—
298C22—2441no—
299C22—2442no—
300C22#1684070911yescharacteristic
301C4—21no—
302C4—22no—
303C4—41no—
304C4—1221no—
305C4—1222no—
306C4—9761no—
307C4—4882no—
308C4—4882no—
309C4—2441no—
310C2—81no—
311C2—611no—
312C2—611no—
313C2—612no—
314C2—4881no—
315C2#2146177411yescharacteristic
316C2#2146177912yes—
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#329C4
socleC22×C61#385
prime radicalC2×D4×D61D4
cosocleC42×D61C23
hypercenter#329#6923
hypocenterC61#8914
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
3. #21461742C2semidirect
4. #21460696C2semidirect
19. #16839502C4semidirect
20. #16839501C4semidirect
21. C61×#329C4semidirect
47. #14519688C2×C4semidirect
50. #97×C61C2×C4semidirect
179. #329#6923semidirect
180. C61#8914semidirect
206. #97C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623622
#214619492
#214617422
#214606962
#329×D612
#891461