Finite Group Databasebeta

← #29658274

Subgroups — #29658274

Order 15616.

920 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 179 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#21456944C212yesmaximal, complemented
2#21456201C211yesmaximal, characteristic, complemented
3#21444970C218yesmaximal, complemented
4#21444484C211yesmaximal, characteristic, complemented
5#21444236C212yesmaximal, complemented
6C61×#3548C211yesmaximal, characteristic
7C2×#14519777C2211yescharacteristic
8#16840158—22no—
9#16840132—22no—
10#16840114C2214yes—
11#16840108—22no—
12#16840078—24no—
13#16840052C2218yes—
14#16839815C2212yes—
15#16839770—22no—
16C4×#11758194C2211yescharacteristic
17#16839508C2214yes—
18D4⋊#92418C2218yes—
19C61×#562—22no—
20C61×#554—22no—
21C61×#550—22no—
22C61×#547C2214yes—
23C61×#540—22no—
24C61×#537—24no—
25C61×#533C2211yescharacteristic
26C61×#532C2212yes—
27C22×Dic122—21no—
28C2×#11758194—21no—
29#14519777—22no—
30#14519777—21no—
31#14519777C2312yes—
32C2×C4⋊Dic61—22no—
33#14519708—28no—
34C4⋊#92418C2314yes—
35#14519700—28no—
36#14519698—28no—
37C22⋊#92418—22no—
38#14519688—21no—
39C4×#92418—22no—
40C4×#92418C2312yes—
41C2×#41×C61—24no—
42C2×#41×C61—22no—
43C4⋊Q8×C61D412yes—
44C4⋊Q8×C61C2312yes—
45#127×C61—22no—
46#127×C61D412yes—
47#125×C61—22no—
48#125×C61D412yes—
49#124×C61—22no—
50#123×C61—28no—
51C22⋊Q8×C61—28no—
52C22⋊Q8×C61—22no—
53C22⋊Q8×C61—24no—
54#121×C61—28no—
55#121×C61—28no—
56#121×C61—24no—
57#121×C61—22no—
58#121×C61D414yes—
59C4×Q8×C61—22no—
60C4×D4×C61—24no—
61C4×D4×C61—22no—
62C4×D4×C61C2314yes—
63#117×C61D412yes—
64C2×C4⋊C4×C61—24no—
65C2×C42×C61C2311yescharacteristic
66#41×C61—84no—
67#41×C61—88no—
68C2×Q8×C61—22no—
69C2×Q8×C61—22no—
70C2×D4×C61—28no—
71C2×D4×C61—24no—
72C2×D4×C61—22no—
73C2×D4×C61—22no—
74C2×D4×C61C2×D414yes—
75C22×C4×C61—28no—
76C22×C4×C61—24no—
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D412yes—
79C22×Dic61—41no—
80C2×Dic122—22no—
81C2×Dic122—44no—
82C2×Dic122—41no—
83C4⋊C4×C61—28no—
84C4⋊C4×C61—22no—
85C4⋊C4×C61—24no—
86C4⋊C4×C61—22no—
87C4⋊C4×C61C2×D414yes—
88C4⋊C4×C61C2×D414yes—
89C22⋊C4×C61—28no—
90C22⋊C4×C61—22no—
91C22⋊C4×C61—24no—
92C22⋊C4×C61—24no—
93C42×C61—22no—
94C42×C61C2×D412yes—
95C42×C61C2×D411yescharacteristic
96C42×C61C2411yescharacteristic
97C4⋊Dic61—24no—
98C4⋊Dic61—42no—
99#11758194—42no—
100C2×#92418—24no—
101C2×#92418—22no—
102C23×C61—24no—
103C23×C61#12011yescharacteristic
104Q8×C61—44no—
105Q8×C61—82no—
106D4×C61—28no—
107D4×C61—44no—
108D4×C61—82no—
109D4×C61—84no—
110D4×C61—88no—
111C2×C4×C61—24no—
112C2×C4×C61—22no—
113C2×C4×C61—24no—
114C2×C4×C61—24no—
115C2×C4×C61—48no—
116C2×C4×C61—88no—
117C2×C4×C61#12012yes—
118C2×C4×C61#12011yescharacteristic
119C2×C4×C61C22×D411yescharacteristic
120C2×C4×C61C22×D412yes—
121C2×Dic61—42no—
122C2×Dic61—82no—
123Dic122—44no—
124Dic122—82no—
125#92418—42no—
126#92418—44no—
127#18728—611nomaximal, Sylow
128C22×C61—22no—
129C22×C61—21no—
130C22×C61—44no—
131C22×C61—44no—
132C22×C61—44no—
133C22×C61C2×#12011yescharacteristic
134C4×C61—22no—
135C4×C61—21no—
136C4×C61—44no—
137C4×C61—44no—
138C4×C61#45412yes—
139C4×C61#57714yes—
140Dic61—82no—
141#3548D6111yescharacteristic
142#3227—612no—
143#3139—611no—
144#1635—618no—
145#1544—611no—
146#1484—612no—
147C2×C61—41no—
148C2×C61—84no—
149C2×C61#300912yes—
150C2×C61C2×#45411yescharacteristic
151#562—22no—
152#554—22no—
153#550—22no—
154#547C2×D6114yes—
155#540—22no—
156#537—24no—
157#533C2×D6111yescharacteristic
158#532C2×D6112yes—
159C2×C4⋊Q8—611no—
160#498—1222no—
161#491—1222no—
162#487—614no—
163#486—1222no—
164#481—1224no—
165#476—618no—
166#431—612no—
167#419—1222no—
168C4×M16—611no—
169#332—614no—
170D4⋊C8—618no—
171C61#1872811yescharacteristic, complemented, Sylow
172C2×#41—24no—
173C2×#41—22no—
174C22×Q8—1221no—
175C2×M16—1221no—
176C4⋊Q8—612no—
177C4⋊Q8—1222no—
178C4⋊Q8—1221no—
179C4⋊Q8C22⋊D6112yes—
180C4⋊Q8C22×D6112yes—
181#127—22no—
182#127C22⋊D6112yes—
183#125—22no—
184#125C22⋊D6112yes—
185#124—22no—
186#123—28no—
187C22⋊Q8—28no—
188C22⋊Q8—22no—
189C22⋊Q8—24no—
190#121—28no—
191#121—28no—
192#121—24no—
193#121—22no—
194#121C22⋊D6114yes—
195C4×Q8—22no—
196C4×D4—24no—
197C4×D4—22no—
198C4×D4C22×D6114yes—
199#117C22⋊D6112yes—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—1222no—
202C2×C42C22×D6111yescharacteristic
203#106—1228no—
204C4⋊C8—614no—
205#103—1228no—
206#102—1228no—
207C22⋊C8—1222no—
208#97—1221no—
209C4×C8—612no—
210C4×C8—1222no—
211#41—84no—
212#41—88no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—1222no—
216C2×Q8—2444no—
217C2×Q8—2441no—
218C2×D4—28no—
219C2×D4—24no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4C2×C22⋊D6114yes—
223C22×C4—28no—
224C22×C4—24no—
225C22×C4—2441no—
226C22×C4C2×C22⋊D6112yes—
227C22×C4C2×C22⋊D6111yescharacteristic
228M16—2442no—
229C2×C8—1222no—
230C2×C8—1224no—
231C4⋊C4—28no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—24no—
235C4⋊C4—1224no—
236C4⋊C4—2442no—
237C4⋊C4C2×C22⋊D6114yes—
238C4⋊C4C2×C22⋊D6114yes—
239C22⋊C4—28no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C42—22no—
244C42C2×C22⋊D6112yes—
245C42C2×C22⋊D6111yescharacteristic
246C42C23×D6111yescharacteristic
247C23—24no—
248C23#1451974611yescharacteristic
249Q8—44no—
250Q8—82no—
251Q8—4882no—
252Q8—2444no—
253D4—28no—
254D4—44no—
255D4—82no—
256D4—84no—
257D4—88no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—48no—
263C2×C4—88no—
264C2×C4—4882no—
265C2×C4—2442no—
266C2×C4#1451974612yes—
267C2×C4#1451974611yescharacteristic
268C2×C4C22×C22⋊D6111yescharacteristic
269C2×C4C22×C22⋊D6112yes—
270C8—2442no—
271C8—2444no—
272C22—21no—
273C22—22no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22C2×#1451974611yescharacteristic
278C4—21no—
279C4—22no—
280C4—44no—
281C4—44no—
282C4—4882no—
283C4#1683993512yes—
284C4#1684057314yes—
285C2—41no—
286C2—84no—
287C2#2145509612yes—
288C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3548C2
socleC22×C61C2×#120
hypercenter#3548D61
hypocenterC61#18728
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #21456944C2semidirect
2. #21456201C2semidirect
3. #21444970C2semidirect
4. #21444484C2semidirect
5. #21444236C2semidirect
171. C61#18728semidirect

Maximal subgroups

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

subgroupindex
#214569442
#214562012
#214449702
#214444842
#214442362
C61×#35482
#1872861