Finite Group Databasebeta

← #11783558

Subgroups — #11783558

Order 1152.

1121 conjugacy classes of subgroups; only the 236 normal subgroups are shown.

normal subgroups only

This group has 1121 conjugacy classes of subgroups — too many to draw. Only its 236 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
1C2×C42×C3⋊D3C211yesmaximal, characteristic
2C2×#63557C212yesmaximal
3C2×#63553C212yesmaximal, complemented
4C3×#4524C212yesmaximal, complemented
5#68599—32nomaximal
6C22×C4×C3⋊D3C412yes—
7C22×C4×C3⋊D3C2211yescharacteristic
8C2×C32×C42C2211yescharacteristic
9C2×C4×C3⋊Dic3C412yes—
10C2×C4×C3⋊Dic3C2211yescharacteristic
11C42×C3⋊D3—24no—
12C22×C3×C3⋊C8C2214yes—
13#63557C418yescomplemented
14#63553—24no—
15C2×C42×D3—32no—
16C2×C42×D3—61no—
17C2×(C2×D3)⋊C8—34no—
18C2×C8×Dic3—34no—
19C3×#336—32no—
20#4524D312yescomplemented
21C23×C3⋊D3C2×C411yescharacteristic
22C22×C32×C4C2×C412yes—
23C22×C32×C4C2311yescharacteristic
24C22×C3⋊Dic3C2×C411yescharacteristic
25C22×C3⋊Dic3C2×C412yes—
26C2×C4×C3⋊D3—22no—
27C2×C4×C3⋊D3—22no—
28C2×C4×C3⋊D3—24no—
29C2×C4×C3⋊D3—24no—
30C2×C4×C3⋊D3C818yescomplemented
31C2×C4×C3⋊D3C2×C414yes—
32C32×C42—22no—
33C4×C3⋊Dic3—24no—
34C4×C3⋊Dic3D414yes—
35C2×C3×C3⋊C8—24no—
36C2×C3×C3⋊C8—24no—
37C2×C3×C3⋊C8C2×C418yescomplemented
38C4×C22⋊C8—91noSylow
39C22×C4×D3—34no—
40C22×C4×D3—32no—
41C22×C4×D3—62no—
42C22×C4×D3—61no—
43C22×C3×C8—34no—
44C2×C3×C42—21no—
45C2×C3×C42C2×D312yes—
46C2×C4×Dic3—32no—
47C2×C4×Dic3—34no—
48C2×C4×Dic3—62no—
49C2×C4×Dic3—61no—
50C22×C3⋊C8C2×D314yes—
51C42×D3—68no—
52C42×D3—68no—
53(C2×D3)⋊C8—316no—
54C8×Dic3—68no—
55C23×C32C22×C411yescharacteristic
56C22×C3⋊D3—21no—
57C22×C3⋊D3—21no—
58C22×C3⋊D3—22no—
59C22×C3⋊D3C4212yes—
60C22×C3⋊D3C2×C814yes—
61C2×C32×C4—22no—
62C2×C32×C4—22no—
63C2×C32×C4C2×C814yes—
64C2×C32×C4C22×C412yes—
65C2×C32×C4C2×D412yes—
66C2×C32×C4#4112yes—
67C2×C3⋊Dic3—21no—
68C2×C3⋊Dic3—21no—
69C2×C3⋊Dic3C4212yes—
70C2×C3⋊Dic3C22⋊C414yes—
71C2×C3⋊Dic3C2×C814yes—
72C2×C3⋊Dic3M1614yes—
73C4×C3⋊D3—28no—
74C4×C3⋊D3—216no—
75C3×C3⋊C8—28no—
76C22×C42—91no—
77C2×C22⋊C8—92no—
78C2×C4×C8—92no—
79#336—32no—
80C23×D3—32no—
81C23×D3—61no—
82C22×C3×C4—22no—
83C22×C3×C4—21no—
84C22×C3×C4C4×D314yes—
85C22×C3×C4C22×D312yes—
86C22×Dic3—34no—
87C22×Dic3—32no—
88C22×Dic3—62no—
89C22×Dic3—61no—
90C2×C4×D3—316no—
91C2×C4×D3—38no—
92C2×C4×D3—68no—
93C2×C4×D3—64no—
94C2×C4×D3—68no—
95C2×C4×D3—68no—
96C2×C4×D3—68no—
97C2×C4×D3—64no—
98C2×C4×D3—68no—
99C2×C4×D3—64no—
100C2×C4×D3—64no—
101C2×C4×D3—62no—
102C2×C4×D3—62no—
103C2×C3×C8—38no—
104C2×C3×C8—64no—
105C2×C3×C8—64no—
106C3×C42—24no—
107C3×C42—24no—
108C4×Dic3—38no—
109C4×Dic3—64no—
110C4×Dic3—68no—
111C4×Dic3—68no—
112C2×C3⋊C8—24no—
113C2×C3⋊C8—24no—
114C2×C3⋊C8C4×D318yescomplemented
115C22×C32C2×C4211yescharacteristic
116C22×C32C2×C22⋊C411yescharacteristic
117C22×C32#11711yescharacteristic
118C22×C32C22×C812yes—
119C22×C32C2×M1612yes—
120C2×C3⋊D3—22no—
121C2×C3⋊D3—22no—
122C2×C3⋊D3—24no—
123C2×C3⋊D3—24no—
124C2×C3⋊D3C4×C814yescomplemented
125C32×C4—24no—
126C32×C4C4×D414yes—
127C3⋊Dic3—22no—
128C3⋊Dic3C22⋊C818yescomplemented
129C23×C4—92no—
130C23×C4—91no—
131C22×C8—34no—
132C2×C42—91no—
133C2×C42—92no—
134C2×C42—184no—
135C2×C42D3211yescharacteristic
136C22⋊C8—98no—
137C4×C8—184no—
138C23×C3—21no—
139C23×C3C2×C4×D312yes—
140C22×D3—34no—
141C22×D3—38no—
142C22×D3—62no—
143C22×D3—64no—
144C22×D3—62no—
145C22×D3—64no—
146C22×D3—62no—
147C22×D3—62no—
148C22×D3—62no—
149C22×D3—64no—
150C2×C3×C4—24no—
151C2×C3×C4—24no—
152C2×C3×C4—24no—
153C2×C3×C4—24no—
154C2×C3×C4—24no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4D3×C818yes—
159C2×C3×C4C2×C4×D314yes—
160C2×C3×C4D3×D414yes—
161C2×C3×C4#23712yes—
162C2×C3×C4#23912yes—
163C2×Dic3—38no—
164C2×Dic3—38no—
165C2×Dic3—38no—
166C2×Dic3—34no—
167C2×Dic3—62no—
168C2×Dic3—64no—
169C2×Dic3—64no—
170C2×Dic3—64no—
171C2×Dic3—62no—
172C2×Dic3—62no—
173C2×Dic3—62no—
174C2×Dic3—62no—
175C4×D3—632no—
176C4×D3—632no—
177C4×D3—616no—
178C4×D3—68no—
179C4×D3—68no—
180C3×C8—68no—
181C3⋊C8—28no—
182C2×C32C4×C22⋊C411yescharacteristic
183C2×C32C2×C4×C811yescharacteristic
184C2×C32C4×M1611yescharacteristic
185C2×C32C2×C22⋊C812yes—
186C2×C32#43112yes—
187C3⋊D3—24no—
188C24—91no—
189C22×C4—92no—
190C22×C4—91no—
191C22×C4—98no—
192C22×C4—94no—
193C22×C4—184no—
194C22×C4—184no—
195C22×C4—182no—
196C22×C4—182no—
197C22×C4#62412yes—
198C22×C4C2×D3211yescharacteristic
199C2×C8—38no—
200C2×C8—64no—
201C2×C8—64no—
202C42—22no—
203C42—94no—
204C42—184no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3C42×D312yes—
211C22×C3D3×C22⋊C412yes—
212C22×C3#89212yes—
213C22×C3C2×D3×C814yes—
214C22×C3D3×M1614yes—
215C2×D3—38no—
216C2×D3—64no—
217C2×D3—64no—
218C2×D3—68no—
219C2×D3—64no—
220C2×D3—68no—
221C2×D3—64no—
222C2×D3—64no—
223C2×D3—68no—
224C2×D3—68no—
225C3×C4—28no—
226C3×C4—28no—
227C3×C4—24no—
228C3×C4#67⋊C414yes—
229C3×C4#89314yes—
230Dic3—316no—
231Dic3—68no—
232Dic3—64no—
233Dic3—64no—
234C32C4×C22⋊C811yescharacteristic, complemented, Sylow
235C23—94no—
236C23—92no—
237C23—182no—
238C23—181no—
239C23—181no—
240C23C2×#62411yescharacteristic
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—94no—
244C2×C4—94no—
245C2×C4—94no—
246C2×C4—92no—
247C2×C4—1816no—
248C2×C4—181no—
249C2×C4—181no—
250C2×C4—188no—
251C2×C4#371014yes—
252C2×C4#379612yes—
253C2×C4C4×D3212yes—
254C2×C4#381112yes—
255C8—68no—
256C2×C3—21no—
257C2×C3—21no—
258C2×C3—22no—
259C2×C3—21no—
260C2×C3—22no—
261C2×C3#466612yes—
262C2×C3C4×D3×C812yes—
263C2×C3#470712yes—
264C2×C3D3×C22⋊C814yes—
265C2×C3#483314yes—
266D3—68no—
267D3—68no—
268C22—94no—
269C22—182no—
270C22—182no—
271C22—184no—
272C22—184no—
273C22C2×#371012yes—
274C22#6381812yes—
275C22#6384911yescharacteristic
276C22C4×#62411yescharacteristic
277C22#6397411yescharacteristic
278C4—24no—
279C4—98no—
280C4—182no—
281C4#6388114yes—
282C3—21no—
283C3#6859912yescomplemented
284C2—184no—
285C2#1059024711yescharacteristic
286C2#1059025612yes—
287C2C4×#371011yescharacteristic
288C2#1059115212yes—
289C2#1059126011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C2×D32
commutatorC2×C32C2×C4×C8
FittingC2×C32×C42C22
socleC23×C32C22×C4
prime radicalC23×C3⋊D3C2×C4
cosocleC22×C32×C4C23
hypercenterC2×C42D32
hypocenterC32C4×C22⋊C8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
3. C2×#63553C2semidirect
4. C3×#4524C2semidirect
13. #63557C4semidirect
20. #4524D3semidirect
30. C2×C4×C3⋊D3C8semidirect
37. C2×C3×C3⋊C8C2×C4semidirect
114. C2×C3⋊C8C4×D3semidirect
124. C2×C3⋊D3C4×C8semidirect
128. C3⋊Dic3C22⋊C8semidirect
234. C32C4×C22⋊C8semidirect
283. C3#68599semidirect

Maximal subgroups

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

subgroupindex
C2×C42×C3⋊D32
C2×#635572
C2×#635532
C3×#45242
#685993