Finite Group Databasebeta

← #29580625

Subgroups — #29580625

Order 15616.

716 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 83 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#21456594C211yesmaximal, characteristic, complemented
2#21455079C212yesmaximal
3#21444182C212yesmaximal, complemented
4#21443696C211yesmaximal, characteristic, complemented
5C61×#1780C211yesmaximal, characteristic
6C2×#14519807—22no—
7C2×#14519756C2212yes—
8#16840098—22no—
9#16840003—22no—
10#16839924—24no—
11#16839920—22no—
12#16839918—22no—
13#16839913—22no—
14#16839763—22no—
15#16839726C2212yes—
16#16839487—24no—
17C22×#41×C61C2211yescharacteristic
18C2×C22⋊C8×C61—22no—
19C2×C22⋊C8×C61C2211yescharacteristic
20C2×C22⋊C8×C61C2211yescharacteristic
21C22×Dic122D412yes—
22#14519807—48no—
23C2×Dic244—22no—
24C2×#11758202—24no—
25C2×#11758200—24no—
26C2×#11758194—22no—
27#14519756—22no—
28#14519756—22no—
29#14519756—22no—
30#14519756—22no—
31C2×C4⋊Dic61D412yes—
32C2×#11758196—22no—
33C2×C22⋊Dic61—22no—
34#14519711—22no—
35#14519702—24no—
36#14519701—24no—
37#14519698—24no—
38C22⋊#92418—22no—
39C22⋊#92418D414yes—
40C2×#41×C61—22no—
41C2×#41×C61—22no—
42C2×#41×C61—44no—
43C22×Q8×C61D411yescharacteristic
44C22×D4×C61—21no—
45C22×D4×C61D411yescharacteristic
46C23×C4×C61—21no—
47C23×C4×C61C2311yescharacteristic
48C22×C8×C61—21no—
49C22×C8×C61D412yes—
50C22⋊C8×C61—22no—
51C22⋊C8×C61—22no—
52C22⋊C8×C61—44no—
53C24×C61—21no—
54C24×C61#4111yescharacteristic
55#41×C61—28no—
56#41×C61—44no—
57#41×C61—44no—
58#41×C61—44no—
59C2×Q8×C61—22no—
60C2×Q8×C61—42no—
61C2×D4×C61—22no—
62C2×D4×C61—42no—
63C2×D4×C61—42no—
64C2×D4×C61—42no—
65C2×D4×C61—42no—
66C22×C4×C61—24no—
67C22×C4×C61—21no—
68C22×C4×C61—42no—
69C22×C4×C61—42no—
70C22×C4×C61—41no—
71C22×C4×C61—41no—
72C22×C4×C61C2×D412yes—
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D412yes—
76C22×Dic61—22no—
77C22×Dic61—22no—
78C2×Dic122—22no—
79C2×Dic122—22no—
80C2×Dic122—44no—
81C2×C8×C61—22no—
82C2×C8×C61—42no—
83C2×C8×C61—42no—
84C2×C8×C61—41no—
85C22⋊Dic61—42no—
86C22⋊Dic61—42no—
87#11758202—48no—
88#11758200—48no—
89C4⋊Dic61—22no—
90C4⋊Dic61—22no—
91#11758196—42no—
92#11758196—42no—
93#11758194—44no—
94C2×#92418—24no—
95Dic244—82no—
96C23×C61—22no—
97C23×C61—22no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61—42no—
102C23×C61—41no—
103C23×C61—42no—
104C23×C61#12112yes—
105C23×C61#12111yescharacteristic
106Q8×C61—24no—
107Q8×C61—42no—
108D4×C61—24no—
109D4×C61—42no—
110D4×C61—48no—
111C2×C4×C61—24no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—41no—
115C2×C4×C61—42no—
116C2×C4×C61—44no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—44no—
120C2×C4×C61#12012yes—
121C2×C4×C61#12011yescharacteristic
122C2×C4×C61#12711yescharacteristic
123C2×Dic61—44no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—82no—
127Dic122—44no—
128Dic122—44no—
129C8×C61—42no—
130C8×C61—81no—
131#92418—44no—
132#9976—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—22no—
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—44no—
139C22×C61—42no—
140C22×C61—82no—
141C22×C61—82no—
142C22×C61—81no—
143C22×C61—81no—
144C22×C61#39211yescharacteristic
145C22×C61#47312yes—
146C4×C61—22no—
147C4×C61—21no—
148C4×C61—42no—
149C4×C61B(4,𝔽2)12yes—
150Dic61—82no—
151Dic61—82no—
152#3189—611no—
153#3007—612no—
154#1780D6111yescharacteristic
155#1472—612no—
156#1293—611no—
157C2×C61—22no—
158C2×C61—42no—
159C2×C61—82no—
160C2×C61#201711yescharacteristic
161C2×C61#202011yescharacteristic
162C2×C61#202111yescharacteristic
163C22×#41C2×D6111yescharacteristic
164C2×#137—1222no—
165C2×C22⋊Q8—612no—
166#484—1222no—
167#467—1222no—
168#452—1224no—
169#451—1222no—
170#451—1222no—
171#450—1222no—
172#417—1222no—
173#407—612no—
174C2×C22⋊C8—22no—
175C2×C22⋊C8C2×D6111yescharacteristic
176C2×C22⋊C8C2×D6111yescharacteristic
177#324—1224no—
178C61#997611yescharacteristic, complemented, Sylow
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—44no—
182C22×Q8—612no—
183C22×Q8D24411yescharacteristic
184C22×D4—21no—
185C22×D4D24411yescharacteristic
186C23×C4—21no—
187C23×C4C22×D6111yescharacteristic
188#137—2448no—
189C2×Q16—1222no—
190C2×Q16—1224no—
191C2×SD16—1224no—
192C2×M16—1222no—
193C22×C8—21no—
194C22×C8C22⋊D6112yes—
195C22⋊Q8—1222no—
196C22⋊Q8—1222no—
197C22⋊Q8—1222no—
198C22⋊Q8—1222no—
199C2×C4⋊C4—612no—
200C2×C4⋊C4—1222no—
201C2×C22⋊C4—1222no—
202#107—1222no—
203#103—1224no—
204#103—1224no—
205#102—1224no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—44no—
209C22⋊C8—614no—
210C22⋊C8—1222no—
211C24—21no—
212C24#1175822211yescharacteristic
213#41—28no—
214#41—44no—
215#41—44no—
216#41—44no—
217C2×Q8—22no—
218C2×Q8—42no—
219C2×Q8—1222no—
220C2×Q8—1222no—
221C2×Q8—2444no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—1222no—
234C22×C4—1222no—
235C22×C4C2×D24411yescharacteristic
236C22×C4D4×D6112yes—
237C22×C4D4×D6112yes—
238C22×C4C2×C22⋊D6111yescharacteristic
239Q16—4882no—
240Q16—2448no—
241SD16—2448no—
242M16—2444no—
243C2×C8—22no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—41no—
247C2×C8—1224no—
248C4⋊C4—1222no—
249C4⋊C4—1222no—
250C4⋊C4—2442no—
251C4⋊C4—2442no—
252C22⋊C4—2442no—
253C22⋊C4—2442no—
254C23—22no—
255C23—22no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—41no—
262C23#1451975012yes—
263C23#1451975111yescharacteristic
264Q8—24no—
265Q8—42no—
266Q8—2444no—
267Q8—2444no—
268D4—24no—
269D4—42no—
270D4—48no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—4882no—
281C2×C4—2444no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4#1451974511yescharacteristic
285C2×C4#1451974712yes—
286C2×C4#1451977311yescharacteristic
287C8—42no—
288C8—81no—
289C8—2444no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—44no—
296C22—42no—
297C22—82no—
298C22—82no—
299C22—81no—
300C22—81no—
301C22#1683967111yescharacteristic
302C22#1684003712yes—
303C4—21no—
304C4—22no—
305C4—42no—
306C4—4882no—
307C4—4882no—
308C4#1683994812yes—
309C2—22no—
310C2—42no—
311C2—82no—
312C2#2144692311yescharacteristic
313C2#2144694011yescharacteristic
314C2#2144694511yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1780C2
socleC22×C61#392
prime radicalC22×#41×C61C22
hypercenter#1780D61
hypocenterC61#9976
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
1. #21456594C2semidirect
3. #21444182C2semidirect
4. #21443696C2semidirect
178. C61#9976semidirect

Maximal subgroups

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

subgroupindex
#214565942
#214550792
#214441822
#214436962
C61×#17802
#997661