Finite Group Databasebeta

← #11753416

Subgroups — #11753416

Order 960.

584 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6765C212yesmaximal, complemented
2#91777C211yesmaximal, characteristic, complemented
3C5×D3×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#4041C211yesmaximal, characteristic, complemented
5#91456C211yesmaximal, characteristic, complemented
6#91452C212yesmaximal, complemented
7C4×#1201C211yesmaximal, characteristic, complemented
8#91434C211yesmaximal, characteristic, complemented
9#91431C212yesmaximal, complemented
10D5×C4⋊Dic3C211yesmaximal, characteristic, complemented
11#91407C212yesmaximal, complemented
12#65841—31nomaximal, Hall
13C2×Dic30C2212yes—
14C2×C4×C5×D3C2211yescharacteristic
15C2×C4×C5×D3C2212yes—
16C2×C3×D20C2212yes—
17C2×C3×D20C2211yescharacteristic
18C2×C3×C4×D5C2211yescharacteristic
19C2×#1201C2212yes—
20C2×#1201C2211yescharacteristic
21C2×D5×Dic3C2212yes—
22C2×D5×Dic3C2212yes—
23#6765—28no—
24C3×C5×C4⋊C4C2211yescharacteristic
25C4⋊Dic15C2211yescharacteristic
26#6712C2212yes—
27C4×Dic15C2211yescharacteristic
28C5×C4⋊Dic3C2211yescharacteristic
29C5×Dic3⋊C4C2212yes—
30C3×#685C2212yes—
31#6669C2211yescharacteristic
32#6668C2212yes—
33(C2×D3)⋊Dic5C2211yescharacteristic
34#6665C2211yescharacteristic
35#6665C2212yes—
36#6665C2212yes—
37#5611—51nomaximal, Hall
38C2×#709—32no—
39C2×C5×C4⋊C4—31no—
40#4076—31no—
41#4075—32no—
42C4×C22⋊D5—31no—
43#4042—32no—
44#4041D311yescharacteristic, complemented
45#4040—32no—
46#4039—31no—
47D5×C4⋊C4—31no—
48#4034—31no—
49C22×C5×D3C2311yescharacteristic
50C22×C3×D5C2312yes—
51C22×C3×D5C2311yescharacteristic
52C2×C3×C4×C5C2312yes—
53C2×C3×C4×C5C2311yescharacteristic
54C2×Dic15C2311yescharacteristic
55C2×Dic15C2312yes—
56C2×Dic15C2311yescharacteristic
57Dic30—24no—
58C2×C5×Dic3C2311yescharacteristic
59C2×C5×Dic3C2312yes—
60C4×C5×D3—24no—
61C4×C5×D3—22no—
62C2×C3×Dic5C2311yescharacteristic
63C3×D20—24no—
64C3×D20—22no—
65C3×C4×D5—22no—
66#1201—24no—
67#1201D414yescomplemented
68D5×Dic3—24no—
69D5×Dic3—24no—
70C2×#237—52no—
71C3×#121—51no—
72#938—51no—
73#935—52no—
74D4×Dic3—51no—
75C4×#67—51no—
76C2×C4⋊Dic3—51no—
77D3×C4⋊C4D511yescharacteristic, complemented
78#888—51no—
79#883—52no—
80#878—52no—
81C22×C4×C5—32no—
82C22×C4×C5—31no—
83C2×C22⋊D5—31no—
84C2×C22⋊D5—32no—
85#709—68no—
86C2×D20C2×D312yes—
87C2×D20C2×D311yescharacteristic
88C2×C4×D5—32no—
89C2×C4×D5—32no—
90C2×C4×D5C2×D311yescharacteristic
91C2×Dic10—32no—
92C5×C4⋊C4—32no—
93C5×C4⋊C4—31no—
94C5×C4⋊C4C2×D311yescharacteristic
95C22⋊Dic5—31no—
96#685—32no—
97#685—31no—
98#685—32no—
99#685C2×D312yes—
100C4⋊Dic5—31no—
101#683—32no—
102#683—32no—
103#683—31no—
104C4×Dic5—31no—
105#547—151noSylow
106C22×C3×C5C2411yescharacteristic
107C2×C5×D3—21no—
108C2×C5×D3—21no—
109C2×C5×D3C2×D412yes—
110C2×C3×D5—21no—
111C2×C3×D5—21no—
112C2×C3×D5—22no—
113C2×C3×D5—22no—
114C2×C3×D5—22no—
115C2×C3×D5C2×D412yes—
116C3×C4×C5—22no—
117C3×C4×C5#4112yescomplemented
118Dic15—22no—
119Dic15—21no—
120Dic15C2×D412yes—
121C3×Dic5—21no—
122C5×Dic3—22no—
123C5×Dic3—21no—
124C2×C3×D4—51no—
125C2×C3×D4—52no—
126C22×C3×C4—51no—
127C2×#67—51no—
128C2×#67—52no—
129C22×Dic3—52no—
130C22×Dic3—52no—
131#237—108no—
132C2×C4×D3C2×D512yes—
133C2×C4×D3C2×D511yescharacteristic
134C2×Dic6—52no—
135C3×C4⋊C4C2×D511yescharacteristic
136C3×C22⋊C4—52no—
137#217—52no—
138#217—52no—
139#217—51no—
140(C2×D3)⋊C4—51no—
141C4⋊Dic3—51no—
142C4⋊Dic3—52no—
143C4⋊Dic3C2×D511yescharacteristic
144Dic3⋊C4—52no—
145Dic3⋊C4—51no—
146Dic3⋊C4C2×D512yes—
147C4×Dic3—51no—
148C23×C5—31no—
149C22×D5C22×D312yes—
150C22×D5C22×D311yescharacteristic
151C2×C4×C5—31no—
152C2×C4×C5—32no—
153C2×C4×C5—64no—
154C2×C4×C5—62no—
155C2×C4×C5C22×D312yes—
156C2×C4×C5C22×D311yescharacteristic
157C22⋊D5—34no—
158C22⋊D5—64no—
159C2×Dic5—31no—
160C2×Dic5—31no—
161C2×Dic5—32no—
162C2×Dic5C22×D311yescharacteristic
163D20—24no—
164D20—22no—
165C4×D5—22no—
166C4×D5—64no—
167C4×D5—64no—
168Dic10—64no—
169C2×#41—152no—
170C4⋊Q8—151no—
171#123—152no—
172#123—152no—
173C22⋊Q8—152no—
174#121—51no—
175#121—151no—
176C4×D4—151no—
177C4×D4—151no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—151no—
180C2×C3×C5C22×D411yescharacteristic
181C2×C3×C5C2×#4111yescharacteristic
182C2×C3×C52−1+411yescharacteristic
183C3×D5—22no—
184C3×D5—42no—
185C5×D3—22no—
186C23×C3—51no—
187C23×C3—52no—
188C22×D3C22×D511yescharacteristic
189C3×D4—104no—
190C3×D4—102no—
191C2×C3×C4—51no—
192C2×C3×C4—102no—
193C2×C3×C4C22×D512yes—
194C2×C3×C4C22×D511yescharacteristic
195#67—54no—
196#67—104no—
197C2×Dic3—51no—
198C2×Dic3—51no—
199C2×Dic3—52no—
200C2×Dic3—104no—
201C2×Dic3—104no—
202C2×Dic3C22×D511yescharacteristic
203C2×Dic3C22×D512yes—
204C4×D3—24no—
205C4×D3—22no—
206Dic6—104no—
207C22×C5—32no—
208C22×C5—61no—
209C22×C5—61no—
210C22×C5C23×D311yescharacteristic
211C2×D5—21no—
212C2×D5—21no—
213C2×D5—22no—
214C2×D5—22no—
215C2×D5—22no—
216C2×D5D3×D412yes—
217C4×C5—22no—
218C4×C5—62no—
219C4×C5—61no—
220C4×C5#23712yescomplemented
221Dic5—21no—
222Dic5—32no—
223Dic5—62no—
224Dic5—61no—
225#41—308no—
226C2×Q8—152no—
227C2×D4—51no—
228C2×D4—52no—
229C2×D4—151no—
230C2×D4—152no—
231C22×C4—31no—
232C22×C4—32no—
233C22×C4—51no—
234C22×C4—152no—
235C22×C4—152no—
236C4⋊C4—31no—
237C4⋊C4—32no—
238C4⋊C4—152no—
239C4⋊C4—151no—
240C4⋊C4—151no—
241C4⋊C4—152no—
242C4⋊C4D3×D511yescharacteristic
243C22⋊C4—52no—
244C22⋊C4—151no—
245C22⋊C4—152no—
246C22⋊C4—152no—
247C22⋊C4—151no—
248C42—151no—
249C3×C5#54711yescharacteristic, complemented, Hall
250C22×C3—52no—
251C22×C3—101no—
252C22×C3—101no—
253C22×C3—102no—
254C22×C3—102no—
255C22×C3—102no—
256C22×C3C23×D511yescharacteristic
257C2×D3—21no—
258C2×D3—21no—
259C2×D3D4×D512yes—
260C3×C4—22no—
261C3×C4—101no—
262C3×C4#70912yescomplemented
263Dic3—21no—
264Dic3—22no—
265Dic3—52no—
266Dic3—102no—
267Dic3—101no—
268C2×C5—62no—
269C2×C5C2×D3×D411yescharacteristic
270C2×C5C2×#23711yescharacteristic
271C2×C5#101011yescharacteristic
272D5—22no—
273D5—42no—
274C23—31no—
275C23—51no—
276C23—52no—
277Q8—304no—
278D4—104no—
279D4—102no—
280D4—154no—
281D4—304no—
282C2×C4—31no—
283C2×C4—32no—
284C2×C4—51no—
285C2×C4—64no—
286C2×C4—62no—
287C2×C4—102no—
288C2×C4—151no—
289C2×C4—151no—
290C2×C4—152no—
291C2×C4—304no—
292C2×C4—304no—
293C2×C4C2×D3×D512yes—
294C2×C4C2×D3×D511yescharacteristic
295C2×C3—102no—
296C2×C3—202no—
297C2×C3C2×#70911yescharacteristic
298C2×C3C2×D4×D511yescharacteristic
299C2×C3#414711yescharacteristic
300D3—22no—
301C5#561111yescharacteristic, complemented, Sylow
302C22—32no—
303C22—52no—
304C22—61no—
305C22—61no—
306C22—101no—
307C22—101no—
308C22—102no—
309C22—102no—
310C22—102no—
311C22C22×D3×D511yescharacteristic
312C4—22no—
313C4—61no—
314C4—62no—
315C4—101no—
316C4—152no—
317C4—302no—
318C4—301no—
319C4#676512yescomplemented
320C3#6584111yescharacteristic, complemented, Sylow
321C2—62no—
322C2—102no—
323C2—202no—
324C2C2×#676511yescharacteristic
325C2D3×D4×D511yescharacteristic
326C2#9202811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C4⋊C4C22
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#547
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#6765C2semidirect
2. #91777C2semidirect
3. C5×D3×C4⋊C4C2semidirect
4. C3×#4041C2semidirect
5. #91456C2semidirect
6. #91452C2semidirect
7. C4×#1201C2semidirect
8. #91434C2semidirect
9. #91431C2semidirect
10. D5×C4⋊Dic3C2semidirect
11. #91407C2semidirect
44. #4041D3semidirect
67. #1201D4semidirect
77. D3×C4⋊C4D5semidirect
117. C3×C4×C5#41semidirect
220. C4×C5#237semidirect
249. C3×C5#547semidirect
262. C3×C4#709semidirect
301. C5#5611semidirect
319. C4#6765semidirect
320. C3#65841semidirect

Maximal subgroups

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

subgroupindex
C2×#67652
#917772
C5×D3×C4⋊C42
C3×#40412
#914562
#914522
C4×#12012
#914342
#914312
D5×C4⋊Dic32
#914072
#658413
#56115