Finite Group Databasebeta

← #29571731

Subgroups — #29571731

Order 15616.

802 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 151 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#21445815C211yesmaximal, characteristic, complemented
2#21444487C212yesmaximal
3#21444248C212yesmaximal, complemented
4#21444143C211yesmaximal, characteristic, complemented
5C2×C61×#559C211yesmaximal, characteristic
6#16839815C2212yes—
7#16839786C2212yes—
8C2×#14519696—21no—
9#16839531—21no—
10#16839506—24no—
11Q8⋊#92418—24no—
12#16839488—22no—
13C61×#559—24no—
14C61×#559—24no—
15C2×#127×C61C2211yescharacteristic
16C2×#124×C61—21no—
17C2×#124×C61C412yes—
18C2×#123×C61—22no—
19C2×C4×Q8×C61C2211yescharacteristic
20C2×C4×Q8×C61C2211yescharacteristic
21C2×C4×D4×C61—21no—
22C2×C4×D4×C61C412yes—
23C2×#11758194—22no—
24C4⋊#92418—22no—
25C4⋊#92418—22no—
26C4⋊#92418—22no—
27#14519696—42no—
28C22⋊#92418—22no—
29C4×#92418—22no—
30C22×Q8×C61C2×C411yescharacteristic
31C22×Q8×C61D411yescharacteristic
32C22×D4×C61—21no—
33C22×D4×C61—21no—
34C22×D4×C61C2×C412yes—
35C23×C4×C61—22no—
36#127×C61—22no—
37#127×C61D414yes—
38#124×C61—28no—
39#124×C61—24no—
40#124×C61—24no—
41#123×C61—216no—
42C4×Q8×C61—22no—
43C4×Q8×C61—22no—
44C4×Q8×C61—22no—
45C4×Q8×C61D414yes—
46C4×D4×C61—28no—
47C4×D4×C61—24no—
48C4×D4×C61—24no—
49C2×C4⋊C4×C61—21no—
50C2×C4⋊C4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61D412yes—
52C2×C4⋊C4×C61D411yescharacteristic
53C2×C22⋊C4×C61—22no—
54C2×C22⋊C4×C61—22no—
55C2×C22⋊C4×C61—22no—
56C2×C42×C61—21no—
57C2×C42×C61C2×C412yes—
58C2×C42×C61C2311yescharacteristic
59C24×C61—22no—
60C2×Q8×C61—21no—
61C2×Q8×C61—22no—
62C2×Q8×C61—22no—
63C2×Q8×C61—21no—
64C2×Q8×C61—21no—
65C2×Q8×C61—42no—
66C2×Q8×C61C22⋊C412yes—
67C2×D4×C61—22no—
68C2×D4×C61—24no—
69C2×D4×C61—24no—
70C2×D4×C61—44no—
71C2×D4×C61—42no—
72C2×D4×C61—41no—
73C2×D4×C61—41no—
74C2×D4×C61—82no—
75C2×D4×C61C22⋊C414yes—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—28no—
79C22×C4×C61—84no—
80C22×C4×C61C22⋊C412yes—
81C22×C4×C61C22⋊C412yes—
82C22×C4×C61C22×C411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C4⋊C4×C61—21no—
86C4⋊C4×C61—24no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—24no—
89C4⋊C4×C61C22⋊C412yes—
90C22⋊C4×C61—28no—
91C22⋊C4×C61—28no—
92C22⋊C4×C61—28no—
93C42×C61—22no—
94C42×C61—22no—
95C42×C61—24no—
96C42×C61C2×D412yes—
97C42×C61C2×D412yes—
98#11758194—42no—
99#11758194—42no—
100C2×#92418—42no—
101C2×#92418—42no—
102C23×C61—24no—
103C23×C61—44no—
104C23×C61—44no—
105C23×C61—42no—
106C23×C61—42no—
107C23×C61C2×C22⋊C411yescharacteristic
108Q8×C61—24no—
109Q8×C61—42no—
110Q8×C61—42no—
111Q8×C61—42no—
112D4×C61—44no—
113D4×C61—44no—
114D4×C61—44no—
115D4×C61—44no—
116D4×C61—84no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—22no—
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—88no—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61C2×C22⋊C411yescharacteristic
131C2×C4×C61#12012yes—
132C2×C4×C61#12012yes—
133C2×C4×C61#13512yes—
134C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C61#13711yescharacteristic
136#92418—82no—
137#92418—82no—
138#8386—611nomaximal, Sylow
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—44no—
142C22×C61—44no—
143C22×C61—44no—
144C22×C61—84no—
145C22×C61—84no—
146C22×C61#37911yescharacteristic
147C22×C61#41611yescharacteristic
148C22×C61#41711yescharacteristic
149C4×C61—22no—
150C4×C61—21no—
151C4×C61—42no—
152C4×C61—42no—
153C4×C61—42no—
154C4×C61—42no—
155C4×C61#41312yes—
156C4×C61#44912yes—
157C4×C61#45212yes—
158C2×#559D6111yescharacteristic
159#1801—611no—
160#1545—612no—
161#1487—612no—
162#1461—611no—
163C2×C61—21no—
164C2×C61—21no—
165C2×C61—84no—
166C2×C61#178511yescharacteristic
167C2×C61#178811yescharacteristic
168C2×C61#179511yescharacteristic
169#559—24no—
170#559—24no—
171C2×#127C2×D6111yescharacteristic
172C2×#124—21no—
173C2×#124Dic6112yes—
174C2×#123—22no—
175C2×C4×Q8C2×D6111yescharacteristic
176C2×C4×Q8C2×D6111yescharacteristic
177C2×C4×D4—21no—
178C2×C4×D4Dic6112yes—
179#431—612no—
180#423—612no—
181C2×#101—1221no—
182#343—1221no—
183#331—1224no—
184Q8⋊C8—1224no—
185#324—1222no—
186C61#838611yescharacteristic, complemented, Sylow
187C22×Q8C2×Dic6111yescharacteristic
188C22×Q8C22⋊D6111yescharacteristic
189C22×D4—21no—
190C22×D4—21no—
191C22×D4C2×Dic6112yes—
192C23×C4—22no—
193C2×M16—1222no—
194#127—22no—
195#127C22⋊D6114yes—
196#124—28no—
197#124—24no—
198#124—24no—
199#123—216no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8C22⋊D6114yes—
204C4×D4—28no—
205C4×D4—24no—
206C4×D4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×Dic6111yescharacteristic
209C2×C4⋊C4C22⋊D6111yescharacteristic
210C2×C4⋊C4C22⋊D6112yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C42—21no—
215C2×C42C2×Dic6112yes—
216C2×C42C22×D6111yescharacteristic
217C4⋊C8—1222no—
218C4⋊C8—1222no—
219C4⋊C8—1222no—
220#101—2442no—
221C22⋊C8—1222no—
222C4×C8—1222no—
223C24—22no—
224C2×Q8—21no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—42no—
230C2×Q8C22⋊Dic6112yes—
231C2×D4—22no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—44no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—82no—
239C2×D4C22⋊Dic6114yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—28no—
243C22×C4—84no—
244C22×C4C22⋊Dic6112yes—
245C22×C4C22⋊Dic6112yes—
246C22×C4C22×Dic6111yescharacteristic
247C22×C4C2×C22⋊D6111yescharacteristic
248C22×C4C2×C22⋊D6111yescharacteristic
249M16—2442no—
250M16—2442no—
251C2×C8—2442no—
252C2×C8—2442no—
253C4⋊C4—21no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4C22⋊Dic6112yes—
258C22⋊C4—28no—
259C22⋊C4—28no—
260C22⋊C4—28no—
261C42—22no—
262C42—22no—
263C42—24no—
264C42C2×C22⋊D6112yes—
265C42C2×C22⋊D6112yes—
266C23—24no—
267C23—44no—
268C23—44no—
269C23—42no—
270C23—42no—
271C23C2×C22⋊Dic6111yescharacteristic
272Q8—24no—
273Q8—42no—
274Q8—42no—
275Q8—42no—
276D4—44no—
277D4—44no—
278D4—44no—
279D4—44no—
280D4—84no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—88no—
293C2×C4C2×C22⋊Dic6111yescharacteristic
294C2×C4C2×C22⋊Dic6111yescharacteristic
295C2×C4#1451974612yes—
296C2×C4#1451974612yes—
297C2×C4#1451979712yes—
298C2×C4#1451980211yescharacteristic
299C2×C4#1451980711yescharacteristic
300C8—4882no—
301C8—4882no—
302C22—21no—
303C22—21no—
304C22—44no—
305C22—44no—
306C22—44no—
307C22—84no—
308C22—84no—
309C22#1683961811yescharacteristic
310C22#1683975811yescharacteristic
311C22#1683976311yescharacteristic
312C4—21no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—42no—
318C4#1683974612yes—
319C4#1683990712yes—
320C4#1683992412yes—
321C2—21no—
322C2—21no—
323C2—84no—
324C2#2144574711yescharacteristic
325C2#2144575911yescharacteristic
326C2#2144578911yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#559C2
socleC22×C61#379
prime radicalC2×#127×C61C22
cosocleC2×C42×C61C23
hypercenterC2×#559D61
hypocenterC61#8386
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
1. #21445815C2semidirect
3. #21444248C2semidirect
4. #21444143C2semidirect
186. C61#8386semidirect

Maximal subgroups

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

subgroupindex
#214458152
#214444872
#214442482
#214441432
C2×C61×#5592
#838661