Finite Group Databasebeta

← #29572092

Subgroups — #29572092

Order 15616.

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

normal subgroups only

This group has 868 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#21446626C211yesmaximal, characteristic
2#21445824C212yesmaximal, complemented
3C2×#16839506C211yesmaximal, characteristic
4C2×Q8⋊#92418C212yesmaximal, complemented
5C2×C61×#559C211yesmaximal, characteristic
6C2×C4⋊#92418C2212yes—
7C2×#14519701—22no—
8C2×#14519698—22no—
9C2×C4×#92418C2211yescharacteristic
10#16839632C2211yescharacteristic
11#16839524—22no—
12#16839506—22no—
13Q8⋊#92418—24no—
14C61×#559—22no—
15C61×#559—24no—
16C61×#559C414yescomplemented
17C2×#127×C61C2211yescharacteristic
18C2×#124×C61—22no—
19C2×#123×C61—22no—
20C2×C4×Q8×C61C2212yes—
21C2×C4×D4×C61—22no—
22C22×#92418—21no—
23C22×#92418—22no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—22no—
26C4⋊#92418—22no—
27C4⋊#92418—22no—
28#14519701—44no—
29#14519698—44no—
30C4×#92418—22no—
31C22×Q8×C61D412yes—
32C22×D4×C61—22no—
33C22×D4×C61D412yes—
34C23×C4×C61—22no—
35#127×C61—22no—
36#127×C61C2×C412yes—
37#127×C61D412yes—
38#124×C61—28no—
39#124×C61—28no—
40#123×C61—28no—
41#123×C61—28no—
42C4×Q8×C61—22no—
43C4×Q8×C61—22no—
44C4×Q8×C61C2×C414yes—
45C4×Q8×C61D414yes—
46C4×D4×C61—28no—
47C4×D4×C61—28no—
48C2×C4⋊C4×C61—22no—
49C2×C4⋊C4×C61D412yes—
50C2×C22⋊C4×C61—24no—
51C2×C22⋊C4×C61—22no—
52C2×C42×C61—22no—
53C2×C42×C61C2311yescharacteristic
54C24×C61—22no—
55C2×Q8×C61—22no—
56C2×Q8×C61—22no—
57C2×Q8×C61—22no—
58C2×Q8×C61—22no—
59C2×Q8×C61—22no—
60C2×Q8×C61C22⋊C414yes—
61C2×D4×C61—24no—
62C2×D4×C61—24no—
63C2×D4×C61—22no—
64C2×D4×C61—42no—
65C2×D4×C61—42no—
66C2×D4×C61—42no—
67C2×D4×C61—42no—
68C2×D4×C61—42no—
69C2×D4×C61—42no—
70C2×D4×C61C22⋊C414yes—
71C22×C4×C61—22no—
72C22×C4×C61—22no—
73C22×C4×C61—28no—
74C22×C4×C61—82no—
75C22×C4×C61—82no—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D412yes—
78C22×Dic61—41no—
79C4⋊C4×C61—24no—
80C4⋊C4×C61—24no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61C22⋊C414yes—
83C22⋊C4×C61—216no—
84C22⋊C4×C61—28no—
85C42×C61—24no—
86C42×C61—24no—
87C42×C61C22×C411yescharacteristic
88C42×C61C2×D411yescharacteristic
89C42×C61C2×D412yes—
90C4⋊Dic61—42no—
91C4⋊Dic61—42no—
92C4⋊Dic61—42no—
93C2×#92418—22no—
94C2×#92418—41no—
95C2×#92418—41no—
96C2×#92418—42no—
97C2×#92418—42no—
98C2×#92418—42no—
99C23×C61—24no—
100C23×C61—44no—
101C23×C61—44no—
102C23×C61—42no—
103C23×C61—42no—
104C23×C61#12011yescharacteristic
105Q8×C61—24no—
106Q8×C61—24no—
107Q8×C61—42no—
108Q8×C61—42no—
109D4×C61—44no—
110D4×C61—44no—
111D4×C61—44no—
112D4×C61—44no—
113D4×C61—44no—
114D4×C61—82no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—24no—
118C2×C4×C61—24no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—88no—
124C2×C4×C61C2×C22⋊C411yescharacteristic
125C2×C4×C61C2×C22⋊C412yes—
126C2×C4×C61#12011yescharacteristic
127C2×C4×C61#12012yes—
128C2×C4×C61#13512yes—
129C2×C4×C61#13512yes—
130C2×C4×C61AGL(1,ℤ/8)12yes—
131C2×C4×C61#13712yes—
132C2×Dic61—82no—
133C2×Dic61—81no—
134#92418—42no—
135#92418—82no—
136#8445—611nomaximal, Sylow
137C22×C61—44no—
138C22×C61—48no—
139C22×C61—82no—
140C22×C61—82no—
141C22×C61—84no—
142C22×C61#37911yescharacteristic
143C22×C61#44912yes—
144C22×C61#45212yes—
145C22×C61#45311yescharacteristic
146C22×C61#45511yescharacteristic
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—44no—
150C4×C61—44no—
151C4×C61#41612yes—
152C4×C61#41712yes—
153C4×C61#44912yes—
154C4×C61#45212yes—
155Dic61—161no—
156C2×#559D6111yescharacteristic
157#1957—611no—
158#1803—612no—
159C2×#331—611no—
160C2×Q8⋊C8—612no—
161C2×C61—84no—
162C2×C61#162011yescharacteristic
163C2×C61#163011yescharacteristic
164C2×C61#164012yes—
165C2×C61#178812yes—
166C2×C61#179211yescharacteristic
167#559—22no—
168#559—24no—
169#559Dic6114yescomplemented
170C2×#127C2×D6111yescharacteristic
171C2×#124—22no—
172C2×#123—22no—
173C2×C4×Q8C2×D6112yes—
174C2×C4×D4—22no—
175C2×C4⋊C8—612no—
176C2×#103—1222no—
177C2×#102—1222no—
178C2×C4×C8—611no—
179#384—611no—
180#340—1222no—
181#331—1222no—
182Q8⋊C8—1224no—
183C61#844511yescharacteristic, complemented, Sylow
184C22×Q8C22⋊D6112yes—
185C22×D4—22no—
186C22×D4C22⋊D6112yes—
187C23×C4—22no—
188C22×C8—1221no—
189C22×C8—1222no—
190#127—22no—
191#127C2×Dic6112yes—
192#127C22⋊D6112yes—
193#124—28no—
194#124—28no—
195#123—28no—
196#123—28no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8C2×Dic6114yes—
200C4×Q8C22⋊D6114yes—
201C4×D4—28no—
202C4×D4—28no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—1222no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4C22⋊D6112yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C42—22no—
210C2×C42C22×D6111yescharacteristic
211C4⋊C8—1222no—
212C4⋊C8—1222no—
213#103—2444no—
214#102—2444no—
215C4×C8—1222no—
216C24—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8C22⋊Dic6114yes—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4C22⋊Dic6114yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—28no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—2441no—
239C22×C4C2×C22⋊D6111yescharacteristic
240C22×C4C2×C22⋊D6112yes—
241C2×C8—1222no—
242C2×C8—2442no—
243C2×C8—2442no—
244C2×C8—2442no—
245C2×C8—2441no—
246C2×C8—2441no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—2442no—
251C4⋊C4—2442no—
252C4⋊C4—2442no—
253C4⋊C4C22⋊Dic6114yes—
254C22⋊C4—216no—
255C22⋊C4—28no—
256C42—24no—
257C42—24no—
258C42C22×Dic6111yescharacteristic
259C42C2×C22⋊D6112yes—
260C42C2×C22⋊D6111yescharacteristic
261C23—24no—
262C23—42no—
263C23—44no—
264C23—44no—
265C23—42no—
266C23#1451974611yescharacteristic
267Q8—24no—
268Q8—24no—
269Q8—42no—
270Q8—42no—
271D4—44no—
272D4—44no—
273D4—44no—
274D4—44no—
275D4—44no—
276D4—82no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—88no—
286C2×C4—4882no—
287C2×C4—4881no—
288C2×C4C2×C22⋊Dic6111yescharacteristic
289C2×C4C2×C22⋊Dic6112yes—
290C2×C4#1451974611yescharacteristic
291C2×C4#1451974612yes—
292C2×C4#1451979712yes—
293C2×C4#1451979712yes—
294C2×C4#1451980212yes—
295C2×C4#1451980712yes—
296C8—4882no—
297C8—2442no—
298C22—48no—
299C22—44no—
300C22—82no—
301C22—82no—
302C22—84no—
303C22#1683961811yescharacteristic
304C22#1683990712yes—
305C22#1683992412yes—
306C22#1683993011yescharacteristic
307C22#1683994011yescharacteristic
308C4—21no—
309C4—21no—
310C4—44no—
311C4—44no—
312C4—9761no—
313C4#1683975812yes—
314C4#1683976312yes—
315C4#1683990712yes—
316C4#1683992412yes—
317C2—84no—
318C2#2144487811yescharacteristic
319C2#2144494011yescharacteristic
320C2#2144500512yes—
321C2#2144575912yes—
322C2#2144577311yescharacteristic
subgroupG/H
centerC23#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#559C2
socleC23×C61#120
prime radicalC2×#127×C61C22
cosocleC2×C42×C61C23
hypercenterC2×#559D61
hypocenterC61#8445
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #21445824C2semidirect
4. C2×Q8⋊#92418C2semidirect
16. C61×#559C4semidirect
169. #559Dic61semidirect
183. C61#8445semidirect

Maximal subgroups

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

subgroupindex
#214466262
#214458242
C2×#168395062
C2×Q8⋊#924182
C2×C61×#5592
#844561