Finite Group Databasebeta

← #29576680

Subgroups — #29576680

Order 15616.

422 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 109 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#21456767C211yesmaximal, characteristic
2#21454759C211yesmaximal, characteristic
3#21444504C211yesmaximal, characteristic
4#21444382C211yesmaximal, characteristic
5#21443677C211yesmaximal, characteristic
6#21443650C211yesmaximal, characteristic
7C61×#1993C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840141C412yescomplemented
10#16840112C412yescomplemented
11#16839969—22no—
12#16839918—22no—
13#16839890—22no—
14#16839890—22no—
15#16839815C2211yescharacteristic
16#16839815C2211yescharacteristic
17#16839788—21no—
18#16839786C2211yescharacteristic
19#16839717—21no—
20#16839508—22no—
21#16839498—22no—
22C2×C4⋊Q8×C61C2211yescharacteristic
23C2×#97×C61C2211yescharacteristic
24C61×#382C2211yescharacteristic
25C61×#340—21no—
26C61×#340—21no—
27C2×#11758202—24no—
28C2×#11758194—21no—
29#14519777C2×C411yescharacteristic
30#14519777C2×C411yescharacteristic
31#14519769—21no—
32#14519756—21no—
33#14519756D412yes—
34C4×Dic122—21no—
35C4×Dic122D412yes—
36#14519736—21no—
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40C4⋊#92418D412yes—
41#14519706—24no—
42#14519701—24no—
43C22⋊#92418—21no—
44C22⋊#92418—21no—
45#14519687—22no—
46C4×#92418—21no—
47C4×#92418C2×C412yes—
48C8×Dic61—22no—
49C22×Q8×C61—21no—
50C22×C8×C61D411yescharacteristic
51C22×C8×C61D411yescharacteristic
52C4⋊Q8×C61—21no—
53C4⋊Q8×C61—21no—
54C4⋊Q8×C61—21no—
55C4⋊Q8×C61C2×C412yes—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61—21no—
59C2×C42×C61C2311yescharacteristic
60#97×C61—21no—
61#97×C61—21no—
62#95×C61—21no—
63#95×C61—21no—
64C2×Q8×C61—22no—
65C2×Q8×C61—41no—
66C2×Q8×C61—82no—
67C22×C4×C61—41no—
68C22×C4×C61—41no—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61#4111yescharacteristic
72C2×Dic122—21no—
73C2×Dic122C22⋊C412yes—
74C2×C8×C61—21no—
75C2×C8×C61—21no—
76C2×C8×C61—21no—
77C2×C8×C61—21no—
78C2×C8×C61—21no—
79C2×C8×C61Q1612yescomplemented
80C4⋊C4×C61—22no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—41no—
84C4⋊C4×C61—41no—
85C4⋊C4×C61—41no—
86C42×C61C22×C411yescharacteristic
87C42×C61C2×D411yescharacteristic
88C42×C61C2×D411yescharacteristic
89C42×C61#4111yescharacteristic
90C22⋊Dic61—22no—
91#11758202—84no—
92C4⋊Dic61—21no—
93C4⋊Dic61—21no—
94C4⋊Dic61—21no—
95C4⋊Dic61C22⋊C412yes—
96#11758196—22no—
97#11758196—22no—
98C4×Dic61—22no—
99#11758194—41no—
100#11758194—41no—
101C2×#92418—22no—
102C2×#92418—22no—
103C2×#92418—41no—
104C23×C61#12111yescharacteristic
105Q8×C61—44no—
106Q8×C61—82no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—42no—
110C2×C4×C61—82no—
111C2×C4×C61—81no—
112C2×C4×C61—81no—
113C2×C4×C61—81no—
114C2×C4×C61C2×C22⋊C411yescharacteristic
115C2×C4×C61C4×D411yescharacteristic
116C2×C4×C61C4×D411yescharacteristic
117C2×C4×C61#12011yescharacteristic
118C2×C4×C61#12111yescharacteristic
119C2×C4×C61#12311yescharacteristic
120C2×C4×C61C2×Q1611yescharacteristic
121C2×C4×C61#13511yescharacteristic
122C2×C4×C61#13711yescharacteristic
123C2×C4×C61#13711yescharacteristic
124C2×Dic61—22no—
125C2×Dic61—22no—
126Dic122—42no—
127Dic122—81no—
128C8×C61—22no—
129C8×C61—41no—
130#92418—42no—
131#92418—42no—
132#92418—81no—
133#9290—611nomaximal, Sylow
134C22×C61—21no—
135C22×C61—21no—
136C22×C61#38611yescharacteristic
137C22×C61#46211yescharacteristic
138C22×C61#46411yescharacteristic
139C4×C61—21no—
140C4×C61—21no—
141C4×C61—21no—
142C4×C61—21no—
143C4×C61—82no—
144C4×C61—161no—
145C4×C61C4×Q1611yescharacteristic
146C4×C61#44111yescharacteristic
147C4×C61#45111yescharacteristic
148C4×C61#45211yescharacteristic
149Dic61—42no—
150Dic61—42no—
151#3209—611no—
152#2961—611no—
153#1993D6111yescharacteristic
154#1549—611no—
155#1518—611no—
156#1286—611no—
157#1275—611no—
158C2×C61—21no—
159C2×C61—21no—
160C2×C61#187811yescharacteristic
161C2×C61#190211yescharacteristic
162C2×C61#191011yescharacteristic
163#533—611no—
164C2×C4⋊Q8C2×D6111yescharacteristic
165#494—612no—
166#487—612no—
167#462—1222no—
168#451—1222no—
169#446—1222no—
170#446—1222no—
171#431—611no—
172#431—611no—
173#424—1221no—
174#423—611no—
175#405—1221no—
176C2×#97C2×D6111yescharacteristic
177#382C2×D6111yescharacteristic
178#340—21no—
179#340—21no—
180#332—1222no—
181#328—1222no—
182C61#929011yescharacteristic, complemented, Sylow
183C22×Q8—21no—
184C2×Q16—1224no—
185C2×M16—1221no—
186C22×C8C22⋊D6111yescharacteristic
187C22×C8C22⋊D6111yescharacteristic
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—611no—
192C4⋊Q8—611no—
193C4⋊Q8C4×D6112yes—
194#125—1221no—
195C22⋊Q8—612no—
196C22⋊Q8—1221no—
197C4×Q8—612no—
198C4×Q8—1221no—
199#117—1221no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C42C22×D6111yescharacteristic
204C4⋊C8—612no—
205C4⋊C8—1224no—
206C4⋊C8—1221no—
207C4⋊C8—1221no—
208C4⋊C8—1221no—
209#103—1224no—
210C22⋊C8—1221no—
211C22⋊C8—1221no—
212#97—21no—
213#97—21no—
214#97—1222no—
215C4×C8—612no—
216C4×C8—1222no—
217C4×C8—1221no—
218#95—21no—
219#95—21no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—612no—
224C2×Q8—1221no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4D4×D6111yescharacteristic
228C22×C4#1175822411yescharacteristic
229C22×C4C2×C22⋊D6111yescharacteristic
230Q16—4884no—
231M16—2441no—
232M16—2441no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—1222no—
239C2×C8—1222no—
240C2×C8—2441no—
241C2×C8#1175820212yescomplemented
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—612no—
249C4⋊C4—1221no—
250C4⋊C4—1221no—
251C4⋊C4—1221no—
252C4⋊C4—1222no—
253C4⋊C4—1222no—
254C22⋊C4—1222no—
255C42—1222no—
256C42C2×C4×D6111yescharacteristic
257C42#1175822211yescharacteristic
258C42D4×D6111yescharacteristic
259C42D4×D6111yescharacteristic
260C23#1451974811yescharacteristic
261Q8—44no—
262Q8—82no—
263Q8—4881no—
264Q8—2442no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—1222no—
273C2×C4—1222no—
274C2×C4C22⋊C4×D6111yescharacteristic
275C2×C4#1451973711yescharacteristic
276C2×C4C4×C22⋊D6111yescharacteristic
277C2×C4#1451974511yescharacteristic
278C2×C4#1451975011yescharacteristic
279C2×C4#1451975911yescharacteristic
280C2×C4C2×#1175820211yescharacteristic
281C2×C4#1451979711yescharacteristic
282C2×C4#1451980711yescharacteristic
283C2×C4#1451980811yescharacteristic
284C8—22no—
285C8—41no—
286C8—4881no—
287C8—2442no—
288C8—2442no—
289C22—21no—
290C22—21no—
291C22#1683963811yescharacteristic
292C22#1683996911yescharacteristic
293C22#1683998111yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—161no—
300C4—2442no—
301C4—2442no—
302C4C4×#1175820211yescharacteristic
303C4#1683986711yescharacteristic
304C4#1683991711yescharacteristic
305C4#1683992211yescharacteristic
306C2—21no—
307C2—21no—
308C2#2144616311yescharacteristic
309C2#2144628911yescharacteristic
310C2#2144634311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1993C2
socleC22×C61#386
prime radicalC23×C61#121
cosocleC2×C42×C61C23
hypercenter#1993D61
hypocenterC61#9290
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
9. #16840141C4semidirect
10. #16840112C4semidirect
79. C2×C8×C61Q16semidirect
182. C61#9290semidirect
241. C2×C8#11758202semidirect

Maximal subgroups

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

subgroupindex
#214567672
#214547592
#214445042
#214443822
#214436772
#214436502
C61×#19932
#929061