Finite Group Databasebeta

← C22×#63739 #11857735

Subgroups — C22×#63739 #11857735

Order 1152.

1962 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 1962 conjugacy classes of subgroups — too many to draw. Only its 287 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
1C23×#625C211yesmaximal, characteristic
2C23×#622C211yesmaximal, characteristic, complemented
3C22×#3724C211yesmaximal, characteristic, complemented
4C2×#63739C2112yesmaximal, direct product factor
5C24×C3×D3—21no—
6C23×C3⋊Dic3C2211yescharacteristic
7C23×C3×Dic3—21no—
8C22×#625—24no—
9C22×#625—23no—
10C22×#625C418yescomplemented
11C22×#625C2216yes—
12C22×#622—24no—
13C22×#622C2216yescomplemented
14C2×#3724—23no—
15C2×#3724C2216yescomplemented
16#63739C22116yesdirect product factor
17C23×#67—61no—
18C22×Dic3⋊C4—61no—
19C24×C32D411yescharacteristic
20C23×C3×D3—28no—
21C23×C3×D3—26no—
22C23×C3×D3—44no—
23C23×C3×D3—44no—
24C22×C3⋊Dic3C2×C414yes—
25C22×C3⋊Dic3D413yes—
26C22×C3⋊Dic3D414yes—
27C22×C3⋊Dic3C2313yes—
28C22×C3×Dic3—26no—
29C22×C3×Dic3—44no—
30C2×#625—224no—
31C2×#625—26no—
32C2×#625—212no—
33C2×#625C2×C4124yescomplemented
34C2×#625C2314yes—
35C2×#622—212no—
36C2×#622C2314yescomplemented
37#3724—26no—
38#3724C2314yescomplemented
39C22×#102—91nomaximal, Sylow
40C25×C3—61no—
41C24×D3—21no—
42C23×C3×C4—61no—
43C22×#67—68no—
44C22×#67—66no—
45C22×#67—66no—
46C22×#67—68no—
47C23×Dic3—21no—
48C23×Dic3—61no—
49C23×Dic3—61no—
50C2×Dic3⋊C4—66no—
51C2×Dic3⋊C4—66no—
52C23×C32C22⋊C414yes—
53C23×C32D814yes—
54C23×C32SD1614yes—
55C23×C32C2×D413yes—
56C22×C3×D3—224no—
57C22×C3×D3—24no—
58C22×C3×D3—432no—
59C22×C3×D3—412no—
60C22×C3×D3—412no—
61C2×C3⋊Dic3C22⋊C4112yes—
62C2×C3⋊Dic3C22×C416yes—
63C2×C3⋊Dic3C2×D413yes—
64C2×C3⋊Dic3C2×D416yes—
65C2×C3⋊Dic3C2411yescharacteristic
66C2×C3×Dic3—24no—
67C2×C3×Dic3—412no—
68#625—224no—
69#625—24no—
70#625C22×C418yescomplemented
71#622—24no—
72C23×D4—91no—
73C23×C8—91no—
74C22×C4⋊C4—91no—
75C2×#102—912no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—66no—
79C24×C3—68no—
80C24×C3—124no—
81C24×C3—124no—
82C23×D3—26no—
83C23×D3—28no—
84C23×D3—44no—
85C23×D3—44no—
86C22×C3×C4—66no—
87C22×C3×C4—124no—
88C2×#67—624no—
89C2×#67—64no—
90C2×#67—648no—
91C2×#67—612no—
92C2×#67—624no—
93C22×Dic3—26no—
94C22×Dic3—44no—
95C22×Dic3—63no—
96C22×Dic3—63no—
97C22×Dic3—64no—
98C22×Dic3—63no—
99C22×Dic3—64no—
100C22×Dic3—63no—
101C22×Dic3—64no—
102C22×Dic3—64no—
103Dic3⋊C4—64no—
104Dic3⋊C4—612no—
105C22×C32#102116yescomplemented
106C22×C32C2×C22⋊C416yes—
107C22×C32C2×D816yes—
108C22×C32C2×SD1616yes—
109C22×C32C22×D411yescharacteristic
110C2×C3×D3—28no—
111C2×C3×D3—44no—
112C2×C3×D3—44no—
113C2×C3×D3—448no—
114C3⋊Dic3C2×C22⋊C416yes—
115C3⋊Dic3C23×C411yescharacteristic
116C3⋊Dic3C22×D411yescharacteristic
117C3×Dic3—44no—
118C25—61no—
119C22×D4—96no—
120C22×D4—98no—
121C22×D4—183no—
122C22×D4—184no—
123C23×C4—61no—
124C23×C4—91no—
125C22×C8—96no—
126C22×C8—184no—
127C2×C4⋊C4—96no—
128C2×C4⋊C4—183no—
129#102—916no—
130C23×C3—23no—
131C23×C3—23no—
132C23×C3—24no—
133C23×C3—24no—
134C23×C3—24no—
135C23×C3—24no—
136C23×C3—24no—
137C23×C3—24no—
138C23×C3—64no—
139C23×C3—624no—
140C23×C3—1212no—
141C23×C3—1212no—
142C23×C3—1232no—
143C22×D3—24no—
144C22×D3—224no—
145C22×D3—412no—
146C22×D3—412no—
147C22×D3—432no—
148C2×C3×C4—64no—
149C2×C3×C4—1212no—
150#67—68no—
151#67—648no—
152#67—68no—
153C2×Dic3—24no—
154C2×Dic3—412no—
155C2×Dic3—612no—
156C2×Dic3—612no—
157C2×Dic3—66no—
158C2×Dic3—66no—
159C2×Dic3—63no—
160C2×Dic3—66no—
161C2×Dic3—63no—
162C2×Dic3—66no—
163C2×Dic3—61no—
164C2×Dic3—61no—
165C2×C32C2×#102112yescomplemented
166C2×C32C22×C22⋊C411yescharacteristic
167C2×C32C22×D811yescharacteristic
168C2×C32C22×SD1611yescharacteristic
169C3×D3—48no—
170C24—66no—
171C24—68no—
172C24—124no—
173C24—124no—
174C24GO+(4,𝔽2)11yescharacteristic
175C2×D4—94no—
176C2×D4—924no—
177C2×D4—186no—
178C2×D4—1812no—
179C2×D4—1824no—
180C22×C4—66no—
181C22×C4—93no—
182C22×C4—93no—
183C22×C4—94no—
184C22×C4—94no—
185C22×C4—124no—
186C2×C8—94no—
187C2×C8—1812no—
188C4⋊C4—94no—
189C4⋊C4—186no—
190C22×C3—21no—
191C22×C3—21no—
192C22×C3—26no—
193C22×C3—26no—
194C22×C3—26no—
195C22×C3—26no—
196C22×C3—26no—
197C22×C3—26no—
198C22×C3—216no—
199C22×C3—216no—
200C22×C3—68no—
201C22×C3—124no—
202C22×C3—124no—
203C22×C3—1248no—
204C2×D3—28no—
205C2×D3—44no—
206C2×D3—44no—
207C2×D3—448no—
208C3×C4—124no—
209Dic3—44no—
210Dic3—66no—
211Dic3—61no—
212Dic3—61no—
213Dic3—66no—
214Dic3—61no—
215Dic3—61no—
216C32C22×#10211yescharacteristic, complemented, Sylow
217C23—64no—
218C23—624no—
219C23—1212no—
220C23—1212no—
221C23—1232no—
222C23D32⋊C414yes—
223C23#377414yes—
224C23#377514yes—
225C23C2×GO+(4,𝔽2)13yes—
226D4—98no—
227D4—184no—
228D4—1824no—
229C2×C4—64no—
230C2×C4—91no—
231C2×C4—93no—
232C2×C4—96no—
233C2×C4—96no—
234C2×C4—912no—
235C2×C4—1212no—
236C8—184no—
237C2×C3—21no—
238C2×C3—21no—
239C2×C3—21no—
240C2×C3—21no—
241C2×C3—21no—
242C2×C3—21no—
243C2×C3—212no—
244C2×C3—212no—
245C2×C3—128no—
246D3—48no—
247C22—68no—
248C22—124no—
249C22—124no—
250C22—1248no—
251C22#63739116yesdirect product factor
252C22C2×D32⋊C416yes—
253C22C2×#377416yes—
254C22C2×#377516yes—
255C22C22×GO+(4,𝔽2)11yescharacteristic
256C4—91no—
257C4—91no—
258C4—96no—
259C4—124no—
260C3—21no—
261C3—21no—
262C2—128no—
263C2C2×#63739112yesdirect product factor
264C2C22×D32⋊C411yescharacteristic
265C2C22×#377411yescharacteristic
266C2C22×#377511yescharacteristic
subgroupG/H
center, hypercenterC24GO+(4,𝔽2)
commutatorC3⋊Dic3C23×C4
FrattiniC22C22×GO+(4,𝔽2)
Fitting, socleC24×C32D4
prime radicalC23×#625C2
cosocleC2×C3⋊Dic3C24
hypocenterC32C22×#102
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×#622C2semidirect
3. C22×#3724C2semidirect
4. C2×#63739C2direct (N × H)
10. C22×#625C4semidirect
13. C22×#622C22semidirect
15. C2×#3724C22semidirect
16. #63739C22direct (N × H)
33. C2×#625C2×C4semidirect
36. C2×#622C23semidirect
38. #3724C23semidirect
70. #625C22×C4semidirect
105. C22×C32#102semidirect
165. C2×C32C2×#102semidirect
216. C32C22×#102semidirect
251. C22#63739direct (N × H)
263. C2C2×#63739direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#6252
C23×#6222
C22×#37242
C2×#637392
C22×#1029