Finite Group Databasebeta

← #29592115

Subgroups — #29592115

Order 15616.

476 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 71 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#21447747C212yesmaximal, complemented
2#21446817C211yesmaximal, characteristic
3#21443831C212yesmaximal, complemented
4#21443743C211yesmaximal, characteristic, complemented
5C61×#3189C211yesmaximal, characteristic, complemented
6C22×D488C2211yescharacteristic
7C2×#14519751—21no—
8C2×#14519720—22no—
9C2×#14519699—21no—
10#16839562—22no—
11C22⋊#11758185C2212yes—
12#16839525C2211yescharacteristic
13C2×C22⋊Q8×C61—21no—
14C61×#484—22no—
15C61×#467—21no—
16C61×#467C2212yes—
17C61×#451—22no—
18C2×C22⋊C8×C61C2211yescharacteristic
19C22×D244—21no—
20C22×C22⋊D61—41no—
21C2×D488—22no—
22C2×D488—42no—
23#14519751—42no—
24#14519751—42no—
25C2×C4⋊Dic61—21no—
26C2×#11758198—41no—
27#14519720—44no—
28C2×#11758185—22no—
29#14519699—42no—
30C22×Q8×C61—21no—
31C23×C4×C61D411yescharacteristic
32C2×Q16×C61—21no—
33C2×Q16×C61D412yescomplemented
34C22×C8×C61D411yescharacteristic
35C22×C8×C61C2311yescharacteristic
36C22⋊Q8×C61—22no—
37C22⋊Q8×C61—22no—
38C22⋊Q8×C61—41no—
39C22⋊Q8×C61—41no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61—41no—
42C2×C22⋊C4×C61—41no—
43#107×C61—21no—
44#107×C61D412yescomplemented
45#103×C61—22no—
46#103×C61—22no—
47C22⋊C8×C61—22no—
48C24×C61D811yescharacteristic
49C23×D61—41no—
50C2×Q8×C61—22no—
51C2×Q8×C61—41no—
52C2×Q8×C61—81no—
53C2×Q8×C61—81no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—41no—
57C22×C4×C61—41no—
58C22×C4×C61—41no—
59C22×C4×C61—41no—
60C22×C4×C61D811yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C2×C22⋊D61—44no—
63C2×C22⋊D61—82no—
64C2×C22⋊D61—82no—
65C22×Dic61—41no—
66C2×D244—22no—
67C2×D244—41no—
68C2×D244—41no—
69C2×D244—41no—
70C2×D244—41no—
71C2×D244—41no—
72Q16×C61—42no—
73Q16×C61—81no—
74C2×C8×C61—21no—
75C2×C8×C61—41no—
76C2×C8×C61—41no—
77C2×C8×C61C2×D412yes—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—41no—
80C4⋊C4×C61—42no—
81C4⋊C4×C61—42no—
82C22⋊C4×C61—42no—
83C22⋊C4×C61—42no—
84#11758198—44no—
85C4⋊Dic61—22no—
86C4⋊Dic61—41no—
87D488—24no—
88D488—42no—
89#11758185—42no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61SD3212yescomplemented
95C23×C61C2×D811yescharacteristic
96C22×D61—42no—
97C22×D61—81no—
98C22×D61—81no—
99C22×D61—81no—
100C22×D61—81no—
101C22×D61—81no—
102C22×D61—81no—
103Q8×C61—42no—
104Q8×C61—42no—
105Q8×C61—82no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—81no—
115C2×C4×C61—82no—
116C2×C4×C61—82no—
117C2×C4×C61#12011yescharacteristic
118C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
119C22⋊D61—88no—
120C2×Dic61—42no—
121C2×Dic61—81no—
122C2×Dic61—81no—
123D244—42no—
124D244—42no—
125D244—81no—
126D244—81no—
127C8×C61—22no—
128C8×C61—41no—
129#12103—611nomaximal, Sylow
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—22no—
133C22×C61—42no—
134C22×C61—41no—
135C22×C61—41no—
136C22×C61—81no—
137C22×C61#44711yescharacteristic
138C22×C61C2×SD3211yescharacteristic
139C22×C61#51011yescharacteristic
140C2×D61—82no—
141C2×D61—82no—
142C2×D61—82no—
143C2×D61—161no—
144C2×D61—161no—
145C2×D61—161no—
146C2×D61—161no—
147C4×C61—21no—
148C4×C61—41no—
149C4×C61—82no—
150C4×C61—82no—
151C4×C61#45311yescharacteristic
152C4×C61B(4,𝔽2)11yescharacteristic
153Dic61—82no—
154#3189D6111yescharacteristic, complemented
155#2187—612no—
156#1999—611no—
157#1348—612no—
158#1314—611no—
159C2×C61—21no—
160C2×C61—21no—
161C2×C61—42no—
162C2×C61#159511yescharacteristic
163C2×C61#218511yescharacteristic
164C2×C61#219011yescharacteristic
165D61—162no—
166C22×D8—611no—
167C2×C22⋊Q8—21no—
168C2×#121—1221no—
169C2×SD32—1222no—
170#484—22no—
171#467—21no—
172#467C2×D6112yes—
173#451—22no—
174C2×#102—1221no—
175C2×C22⋊C8C2×D6111yescharacteristic
176#357—1222no—
177C22⋊C16—612no—
178#340—611no—
179C61#1210311yescharacteristic, complemented, Sylow
180C22×Q8—21no—
181C22×D4—1221no—
182C22×D4—2441no—
183C23×C4C22⋊D6111yescharacteristic
184C2×Q16—21no—
185C2×Q16C22⋊D6112yescomplemented
186C2×D8—1222no—
187C2×D8—2442no—
188C22×C8C22⋊D6111yescharacteristic
189C22×C8C22×D6111yescharacteristic
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194#121—2442no—
195#121—2442no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—1221no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—2441no—
201SD32—2444no—
202C2×C16—1222no—
203#107—21no—
204#107C22⋊D6112yescomplemented
205#103—22no—
206#103—22no—
207#102—2442no—
208C22⋊C8—22no—
209C24—2441no—
210C24#1175819911yescharacteristic
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—81no—
214C2×Q8—81no—
215C2×D4—1222no—
216C2×D4—4882no—
217C2×D4—4882no—
218C2×D4—2441no—
219C2×D4—2441no—
220C2×D4—2441no—
221C2×D4—2444no—
222C2×D4—2441no—
223C2×D4—2441no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—2441no—
231C22×C4#1175819911yescharacteristic
232C22×C4C2×C22⋊D6111yescharacteristic
233Q16—42no—
234Q16—81no—
235D8—1224no—
236D8—2442no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8C2×C22⋊D6112yes—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—1222no—
246C4⋊C4—2441no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—2444no—
250C16—2442no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—4881no—
256C23—4881no—
257C23—4881no—
258C23—4881no—
259C23—4881no—
260C23—4881no—
261C23—2442no—
262C23#1451972012yescomplemented
263C23C2×#1175819911yescharacteristic
264Q8—42no—
265Q8—42no—
266Q8—82no—
267D4—4881no—
268D4—4888no—
269D4—4881no—
270D4—2442no—
271D4—2442no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—4881no—
284C2×C4—4881no—
285C2×C4—2442no—
286C2×C4#1451974611yescharacteristic
287C2×C4#1451980311yescharacteristic
288C8—22no—
289C8—41no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—9761no—
298C22—9761no—
299C22—9761no—
300C22—9761no—
301C22—4882no—
302C22—4882no—
303C22—4882no—
304C22#1683989611yescharacteristic
305C22C2×#1451972011yescharacteristic
306C22#1684020311yescharacteristic
307C4—21no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4—4882no—
312C4#1683993011yescharacteristic
313C4#1683994911yescharacteristic
314C2—21no—
315C2—21no—
316C2—42no—
317C2—9762no—
318C2#2144473711yescharacteristic
319C2#2144773611yescharacteristic
320C2#2144776511yescharacteristic
subgroupG/H
centerC22#16839896
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#3189C2
socleC22×C61#447
prime radical#21446817C2
hypercenter#3189D61
hypocenterC61#12103
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3189

Complements

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

NH ≅ G/Ntype
1. #21447747C2semidirect
3. #21443831C2semidirect
4. #21443743C2semidirect
5. C61×#3189C2semidirect
33. C2×Q16×C61D4semidirect
44. #107×C61D4semidirect
94. C23×C61SD32semidirect
154. #3189D61semidirect
179. C61#12103semidirect
185. C2×Q16C22⋊D61semidirect
204. #107C22⋊D61semidirect
262. C23#14519720semidirect

Maximal subgroups

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

subgroupindex
#214477472
#214468172
#214438312
#214437432
C61×#31892
#1210361