Finite Group Databasebeta

← #29575008

Subgroups — #29575008

Order 15616.

756 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 135 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#21448422C211yesmaximal, characteristic
2#21445754C212yesmaximal
3C2×#16839488C211yesmaximal, characteristic
4#21443698C212yesmaximal
5C2×C61×#395C211yesmaximal, characteristic
6C2×#14519701—24no—
7C2×C22⋊#92418C2212yes—
8#16839648—24no—
9#16839625—21no—
10#16839622C2212yes—
11C4×C22⋊Dic61—21no—
12#16839488—22no—
13C23×Q8×C61C2211yescharacteristic
14C22×C4⋊C4×C61—21no—
15C22×C4⋊C4×C61C2211yescharacteristic
16C61×#395—22no—
17C61×#395C414yescomplemented
18C2×#95×C61—21no—
19C2×#95×C61C2211yescharacteristic
20C22×#92418—22no—
21C2×C4⋊Dic61D414yes—
22C2×C22⋊Dic61—22no—
23C2×C4×Dic61—22no—
24#14519701—48no—
25C22⋊#92418—22no—
26C22⋊#92418—22no—
27#14519684—22no—
28#14519684—24no—
29#14519684—22no—
30C22×Q8×C61—21no—
31C22×Q8×C61—22no—
32C22×Q8×C61—22no—
33C22×Q8×C61—44no—
34C22×Q8×C61C2×C412yes—
35C23×C4×C61—21no—
36C23×C4×C61—21no—
37C23×C4×C61D411yescharacteristic
38C23×C4×C61D411yescharacteristic
39C23×C4×C61C2311yescharacteristic
40C2×C4⋊C4×C61—22no—
41C2×C4⋊C4×C61—22no—
42C2×C4⋊C4×C61—24no—
43C2×C4⋊C4×C61—44no—
44C2×C4⋊C4×C61C2×C412yes—
45C2×C4⋊C4×C61D412yes—
46#95×C61—24no—
47#95×C61—21no—
48#95×C61C2×C412yes—
49C24×C61C2×D411yescharacteristic
50C2×Q8×C61—28no—
51C2×Q8×C61—24no—
52C2×Q8×C61—44no—
53C2×Q8×C61—44no—
54C2×Q8×C61—44no—
55C2×Q8×C61—48no—
56C2×Q8×C61—42no—
57C22×C4×C61—24no—
58C22×C4×C61—24no—
59C22×C4×C61—22no—
60C22×C4×C61—22no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—22no—
65C22×C4×C61—22no—
66C22×C4×C61—21no—
67C22×C4×C61—42no—
68C22×C4×C61—42no—
69C22×C4×C61—42no—
70C22×C4×C61—41no—
71C22×C4×C61—41no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61C22⋊C412yes—
75C22×C4×C61C22⋊C412yes—
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D412yes—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61#4112yes—
80C22×Dic61—24no—
81C4⋊C4×C61—24no—
82C4⋊C4×C61—24no—
83C4⋊C4×C61—48no—
84C22⋊Dic61—42no—
85C22⋊Dic61—42no—
86C4⋊Dic61—44no—
87C4×Dic61—82no—
88C2×#92418—44no—
89C2×#92418—42no—
90C23×C61—22no—
91C23×C61—22no—
92C23×C61C2×C22⋊C411yescharacteristic
93C23×C61#12011yescharacteristic
94C23×C61#12311yescharacteristic
95C23×C61#13512yes—
96C23×C61#13712yes—
97Q8×C61—28no—
98Q8×C61—48no—
99Q8×C61—44no—
100C2×C4×C61—24no—
101C2×C4×C61—24no—
102C2×C4×C61—24no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—48no—
112C2×C4×C61—42no—
113C2×C4×C61—44no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—44no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—81no—
122C2×C4×C61C4×D412yes—
123C2×C4×C61#12112yes—
124C2×Dic61—44no—
125C2×Dic61—44no—
126C2×Dic61—44no—
127#92418—82no—
128#9014—611nomaximal, Sylow
129C22×C61—24no—
130C22×C61—24no—
131C22×C61—21no—
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—21no—
135C22×C61#38611yescharacteristic
136C22×C61#41611yescharacteristic
137C22×C61#41911yescharacteristic
138C22×C61#45212yes—
139C22×C61#45612yes—
140C22×C61B(4,𝔽2)12yes—
141C22×C61#48512yes—
142C4×C61—22no—
143C4×C61—22no—
144C4×C61—44no—
145C4×C61—42no—
146C4×C61—44no—
147C4×C61—42no—
148Dic61—84no—
149C2×#395D6111yescharacteristic
150#2313—611no—
151#1787—612no—
152C2×#324—611no—
153#1294—612no—
154C2×C61—21no—
155C2×C61—21no—
156C2×C61—22no—
157C2×C61#160314yes—
158C2×C61#187311yescharacteristic
159C2×C61#188611yescharacteristic
160C2×C61#188911yescharacteristic
161C23×Q8C2×D6111yescharacteristic
162C22×C4⋊C4—21no—
163C22×C4⋊C4C2×D6111yescharacteristic
164C2×#103—1224no—
165C2×C22⋊C8—612no—
166#395—22no—
167#395Dic6114yescomplemented
168#387—1224no—
169#381—1221no—
170#380—612no—
171C4×C22⋊C4—1221no—
172C2×#95—21no—
173C2×#95C2×D6111yescharacteristic
174#324—1222no—
175C61#901411yescharacteristic, complemented, Sylow
176C22×Q8—21no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—44no—
180C22×Q8C2×Dic6112yes—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C22⋊D6111yescharacteristic
184C23×C4C22⋊D6111yescharacteristic
185C23×C4C22×D6111yescharacteristic
186C22×C8—1222no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—614no—
192C2×C4⋊C4C2×Dic6112yes—
193C2×C4⋊C4C22⋊D6112yes—
194C2×C22⋊C4—1222no—
195C2×C42—1222no—
196#103—2448no—
197C22⋊C8—1222no—
198C22⋊C8—1222no—
199#95—24no—
200#95—21no—
201#95—1222no—
202#95—1224no—
203#95—1222no—
204#95C2×Dic6112yes—
205C24C2×C22⋊D6111yescharacteristic
206C2×Q8—28no—
207C2×Q8—24no—
208C2×Q8—44no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—48no—
212C2×Q8—42no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—1224no—
231C22×C4C22⋊Dic6112yes—
232C22×C4C22⋊Dic6112yes—
233C22×C4D4×D6112yes—
234C22×C4#1175822412yes—
235C22×C4C22×Dic6111yescharacteristic
236C22×C4C2×C22⋊D6111yescharacteristic
237C2×C8—2444no—
238C2×C8—2442no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—48no—
242C4⋊C4—2444no—
243C22⋊C4—2442no—
244C22⋊C4—2442no—
245C42—4882no—
246C23—22no—
247C23—22no—
248C23C2×C22⋊Dic6111yescharacteristic
249C23#1451974511yescharacteristic
250C23#1451975811yescharacteristic
251C23#1451979712yes—
252C23#1451980812yes—
253Q8—28no—
254Q8—48no—
255Q8—44no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—48no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—2444no—
279C2×C4—2444no—
280C2×C4—2444no—
281C2×C4D4×Dic6112yes—
282C2×C4#1451974912yes—
283C8—4882no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—24no—
289C22—24no—
290C22#1683964211yescharacteristic
291C22#1683975811yescharacteristic
292C22#1683977111yescharacteristic
293C22#1683992212yes—
294C22#1683994412yes—
295C22#1683994812yes—
296C22#1684010012yes—
297C4—22no—
298C4—22no—
299C4—44no—
300C4—44no—
301C4—42no—
302C4—42no—
303C4—4884no—
304C2—21no—
305C2—21no—
306C2—22no—
307C2#2144478014yes—
308C2#2144613211yescharacteristic
309C2#2144621311yescharacteristic
310C2#2144622811yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#395C2
socleC23×C61#120
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC2×#395D61
hypocenterC61#9014
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
17. C61×#395C4semidirect
167. #395Dic61semidirect
175. C61#9014semidirect

Maximal subgroups

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

subgroupindex
#214484222
#214457542
C2×#168394882
#214436982
C2×C61×#3952
#901461