Finite Group Databasebeta

← #11802910

Subgroups — #11802910

Order 1152.

1969 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3757C211yesmaximal, characteristic
2C22×C3×#217C212yesmaximal
3#10591720C214yesmaximal, complemented
4#10591693C218yesmaximal, complemented
5#82095—32nomaximal
6C25×C32C2211yescharacteristic
7C23×C3⋊Dic3C2212yes—
8C23×C3×Dic3C2214yes—
9C2×#3757C418yescomplemented
10C2×#3757C2214yes—
11C2×C3×#217—28no—
12C2×C3×#217C2214yes—
13C2×C3×#217C2214yes—
14C2×#3724C2218yescomplemented
15#63579C2218yescomplemented
16C22×C3×C22⋊C4—32no—
17C22×#217—32no—
18C22×#217—61no—
19C22×#217D312yes—
20#5210—38no—
21#4946—38no—
22#4942—38no—
23C24×C32—44no—
24C24×C32C2×C414yes—
25C24×C32D414yes—
26C24×C32Q814yes—
27C24×C32C2311yescharacteristic
28C24×C32C2312yes—
29C22×C3⋊Dic3—44no—
30C22×C3⋊Dic3C2×C418yescomplemented
31C22×C3⋊Dic3C2314yes—
32C22×C3×Dic3—28no—
33C22×C3×Dic3—48no—
34C22×C3×Dic3C2318yes—
35#3757C2×C4116yescomplemented
36C3×#217—28no—
37C3×#217—28no—
38#3724—48no—
39#2315—91noSylow
40C25×C3—21no—
41C25×C3C2×D312yes—
42C23×C3×C4—34no—
43C23×Dic3—34no—
44C23×Dic3—62no—
45C23×Dic3C2×D314yes—
46C2×C3×C22⋊C4—34no—
47C2×C3×C22⋊C4—34no—
48C2×C3×C22⋊C4—68no—
49C2×#217—28no—
50C2×#217—316no—
51C2×#217—34no—
52C2×#217—34no—
53C2×#217—68no—
54C2×#217—64no—
55C2×#217C2×D314yes—
56C2×#217C2×D314yes—
57C2×Dic3⋊C4—316no—
58C3⋊#95—316no—
59C23×C32—24no—
60C23×C32—28no—
61C23×C32—48no—
62C23×C32—48no—
63C23×C32—48no—
64C23×C32C4⋊C4116yescomplemented
65C23×C32C22×C414yes—
66C23×C32C22×C412yes—
67C23×C32C2×D412yes—
68C23×C32C2×D414yes—
69C23×C32C2×Q812yes—
70C23×C32C2×Q814yes—
71C23×C32C2411yescharacteristic
72C2×C3⋊Dic3—28no—
73C2×C3⋊Dic3—48no—
74C2×C3⋊Dic3C22×C418yes—
75C2×C3×Dic3—28no—
76C2×C3×Dic3—216no—
77C2×C3×Dic3—416no—
78C22×C22⋊C4—32no—
79C22×C22⋊C4—91no—
80#385—98no—
81#380—94no—
82C24×C3—24no—
83C24×C3—24no—
84C24×C3—24no—
85C24×C3—21no—
86C24×C3—22no—
87C24×C3—48no—
88C24×C3—48no—
89C24×C3Dic618yes—
90C24×C3C4×D318yes—
91C24×C3#6718yes—
92C24×C3C22×D312yes—
93C24×C3C22×D312yes—
94C24×C3C22×D312yes—
95C22×C3×C4—38no—
96C22×C3×C4—68no—
97C22×C3×C4—128no—
98C22×Dic3—28no—
99C22×Dic3—38no—
100C22×Dic3—316no—
101C22×Dic3—48no—
102C22×Dic3—68no—
103C22×Dic3—64no—
104C22×Dic3—128no—
105C22×Dic3—128no—
106C22×Dic3C22×D318yes—
107C3×C22⋊C4—68no—
108C3×C22⋊C4—68no—
109#217—28no—
110#217—28no—
111#217—316no—
112#217—316no—
113#217—616no—
114Dic3⋊C4—1216no—
115C22×C32—24no—
116C22×C32—24no—
117C22×C32—24no—
118C22×C32—28no—
119C22×C32—28no—
120C22×C32—28no—
121C22×C32—416no—
122C22×C32C2×C4⋊C414yes—
123C22×C32C2×C4⋊C418yes—
124C22×C32C23×C411yescharacteristic
125C22×C32C22×D411yescharacteristic
126C22×C32C22×Q811yescharacteristic
127C22×C32H(5,𝔽2)14yes—
128C3⋊Dic3—28no—
129C3×Dic3—48no—
130C25D3211yescharacteristic
131C23×C4—34no—
132C23×C4—92no—
133C2×C4⋊C4—98no—
134C2×C22⋊C4—34no—
135C2×C22⋊C4—34no—
136C2×C22⋊C4—68no—
137C2×C22⋊C4—98no—
138C2×C22⋊C4—94no—
139#95—98no—
140C23×C3—28no—
141C23×C3—28no—
142C23×C3—28no—
143C23×C3—28no—
144C23×C3—21no—
145C23×C3—216no—
146C23×C3—22no—
147C23×C3—24no—
148C23×C3—22no—
149C23×C3—24no—
150C23×C3—22no—
151C23×C3—24no—
152C23×C3—48no—
153C23×C3—416no—
154C23×C3—416no—
155C23×C3—416no—
156C23×C3—416no—
157C23×C3—416no—
158C23×C3—416no—
159C23×C3Dic3⋊C4132yescomplemented
160C23×C3C2×Dic614yes—
161C23×C3C2×Dic614yes—
162C23×C3C2×Dic614yes—
163C23×C3C2×C4×D314yes—
164C23×C3C2×C4×D314yes—
165C23×C3C2×C4×D314yes—
166C23×C3C2×#6714yes—
167C23×C3C2×#6714yes—
168C23×C3C2×#6714yes—
169C23×C3C23×D312yes—
170C2×C3×C4—616no—
171C2×C3×C4—68no—
172C2×C3×C4—1216no—
173C2×Dic3—216no—
174C2×Dic3—28no—
175C2×Dic3—316no—
176C2×Dic3—416no—
177C2×Dic3—68no—
178C2×Dic3—616no—
179C2×Dic3—128no—
180C2×Dic3—1216no—
181C2×Dic3—1216no—
182C2×C32—24no—
183C2×C32—28no—
184C2×C32C22×C4⋊C411yescharacteristic
185C2×C32#51812yes—
186C2×C32#53412yes—
187C2×C32#54312yes—
188C24—44no—
189C24#62414yes—
190C24#62514yes—
191C24#62714yes—
192C24C2×D3211yescharacteristic
193C24C2×D3212yes—
194C22×C4—38no—
195C22×C4—68no—
196C22×C4—94no—
197C22×C4—98no—
198C22×C4—128no—
199C22×C4—364no—
200C4⋊C4—368no—
201C22⋊C4—68no—
202C22⋊C4—68no—
203C22⋊C4—916no—
204C22×C3—28no—
205C22×C3—28no—
206C22×C3—28no—
207C22×C3—28no—
208C22×C3—28no—
209C22×C3—28no—
210C22×C3—28no—
211C22×C3—28no—
212C22×C3—28no—
213C22×C3—28no—
214C22×C3—28no—
215C22×C3—28no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—24no—
220C22×C3—24no—
221C22×C3—28no—
222C22×C3—48no—
223C22×C3—48no—
224C22×C3—48no—
225C22×C3—432no—
226C22×C3—432no—
227C22×C3C2×Dic3⋊C418yes—
228C22×C3C2×Dic3⋊C418yes—
229C22×C3C2×Dic3⋊C418yes—
230C22×C3C22×Dic612yes—
231C22×C3C22×C4×D312yes—
232C22×C3#100418yes—
233C22×C3C22×#6712yes—
234C3×C4—128no—
235Dic3—48no—
236Dic3—616no—
237Dic3—128no—
238C32#231511yescharacteristic, complemented, Sylow
239C23—28no—
240C23—24no—
241C23—48no—
242C23—48no—
243C23—48no—
244C23#3724116yescomplemented
245C23C2×#62414yes—
246C23C2×#62412yes—
247C23C2×#62512yes—
248C23C2×#62514yes—
249C23C2×#62712yes—
250C23C2×#62714yes—
251C23C22×D3211yescharacteristic
252C2×C4—68no—
253C2×C4—616no—
254C2×C4—98no—
255C2×C4—1216no—
256C2×C4—188no—
257C2×C4—368no—
258C2×C3—28no—
259C2×C3—28no—
260C2×C3—28no—
261C2×C3—28no—
262C2×C3—21no—
263C2×C3—22no—
264C2×C3—22no—
265C2×C3—22no—
266C2×C3—48no—
267C2×C3#548314yes—
268C2×C3#548614yes—
269C2×C3C22×Dic3⋊C412yes—
270C2×C3#580414yes—
271C22—24no—
272C22—28no—
273C22—28no—
274C22—24no—
275C22—24no—
276C22—28no—
277C22—416no—
278C22C2×#372414yes—
279C22C2×#372418yes—
280C22C22×#62411yescharacteristic
281C22C22×#62511yescharacteristic
282C22C22×#62711yescharacteristic
283C22#6433014yes—
284C4—128no—
285C4—188no—
286C3—21no—
287C3#8209512yescomplemented
288C2—24no—
289C2—28no—
290C2C22×#372411yescharacteristic
291C2#1059551012yes—
292C2#1059551812yes—
293C2#1059552112yes—
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
Fitting, prime radicalC25×C32C22
socle, cosocleC23×C32C24
hypercenterC25D32
hypocenterC32#2315
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C25

Complements

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

NH ≅ G/Ntype
3. #10591720C2semidirect
4. #10591693C2semidirect
9. C2×#3757C4semidirect
14. C2×#3724C22semidirect
15. #63579C22semidirect
30. C22×C3⋊Dic3C2×C4semidirect
35. #3757C2×C4semidirect
64. C23×C32C4⋊C4semidirect
159. C23×C3Dic3⋊C4semidirect
238. C32#2315semidirect
244. C23#3724semidirect
287. C3#82095semidirect

Maximal subgroups

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

subgroupindex
C22×#37572
C22×C3×#2172
#105917202
#105916932
#820953