Finite Group Databasebeta

← C22×#63740 #11857762

Subgroups — C22×#63740 #11857762

Order 1152.

1450 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 1450 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×#627C211yesmaximal, characteristic
2C23×#622C211yesmaximal, characteristic
3C22×#3724C211yesmaximal, characteristic
4C2×#63740C2112yesmaximal, direct product factor
5C23×C3⋊Dic3C2211yescharacteristic
6C23×C3×Dic3—21no—
7C23×C3×Dic3—21no—
8C22×#627—23no—
9C22×#627—24no—
10C22×#627C418yescomplemented
11C22×#627C2216yes—
12C22×#622—24no—
13C22×#622C2216yes—
14C2×#3724—23no—
15C2×#3724C2216yes—
16#63740C22116yesdirect product factor
17C23×Dic6—61no—
18C22×Dic3⋊C4—61no—
19C24×C32D411yescharacteristic
20C22×C3⋊Dic3C2×C414yes—
21C22×C3⋊Dic3D413yes—
22C22×C3⋊Dic3D414yes—
23C22×C3⋊Dic3C2313yes—
24C22×C3×Dic3—28no—
25C22×C3×Dic3—26no—
26C22×C3×Dic3—26no—
27C22×C3×Dic3—44no—
28C2×#627—224no—
29C2×#627—26no—
30C2×#627—212no—
31C2×#627C2×C4124yescomplemented
32C2×#627C2314yes—
33C2×#622—212no—
34C2×#622C2314yes—
35#3724—26no—
36#3724C2314yes—
37C22×#103—91nomaximal, Sylow
38C23×C3×C4—61no—
39C23×C3×C4—61no—
40C23×Dic3—21no—
41C23×Dic3—21no—
42C23×Dic3—61no—
43C23×Dic3—61no—
44C22×Dic6—68no—
45C22×Dic6—66no—
46C22×Dic6—66no—
47C22×Dic6—68no—
48C2×Dic3⋊C4—66no—
49C2×Dic3⋊C4—66no—
50C23×C32C22⋊C414yes—
51C23×C32SD1614yes—
52C23×C32Q1614yes—
53C23×C32C2×D413yes—
54C2×C3⋊Dic3C22⋊C4112yes—
55C2×C3⋊Dic3C22×C416yes—
56C2×C3⋊Dic3C2×D413yes—
57C2×C3⋊Dic3C2×D416yes—
58C2×C3⋊Dic3C2411yescharacteristic
59C2×C3×Dic3—224no—
60C2×C3×Dic3—24no—
61C2×C3×Dic3—24no—
62C2×C3×Dic3—412no—
63#627—224no—
64#627—24no—
65#627C22×C418yescomplemented
66#622—24no—
67C23×Q8—91no—
68C23×C8—91no—
69C22×C4⋊C4—91no—
70C2×#103—912no—
71C24×C3—21no—
72C24×C3—21no—
73C22×C3×C4—66no—
74C22×C3×C4—68no—
75C22×C3×C4—66no—
76C22×C3×C4—124no—
77C22×Dic3—26no—
78C22×Dic3—28no—
79C22×Dic3—26no—
80C22×Dic3—44no—
81C22×Dic3—63no—
82C22×Dic3—63no—
83C22×Dic3—64no—
84C22×Dic3—63no—
85C22×Dic3—64no—
86C22×Dic3—63no—
87C22×Dic3—64no—
88C22×Dic3—64no—
89C2×Dic6—624no—
90C2×Dic6—64no—
91C2×Dic6—648no—
92C2×Dic6—612no—
93C2×Dic6—624no—
94Dic3⋊C4—64no—
95Dic3⋊C4—612no—
96C22×C32#103116yescomplemented
97C22×C32C2×C22⋊C416yes—
98C22×C32C2×SD1616yes—
99C22×C32C2×Q1616yes—
100C22×C32C22×D411yescharacteristic
101C3⋊Dic3C2×C22⋊C416yes—
102C3⋊Dic3C23×C411yescharacteristic
103C3⋊Dic3C22×D411yescharacteristic
104C3×Dic3—28no—
105C3×Dic3—44no—
106C22×Q8—96no—
107C22×Q8—98no—
108C22×Q8—183no—
109C22×Q8—184no—
110C23×C4—61no—
111C23×C4—61no—
112C23×C4—91no—
113C22×C8—96no—
114C22×C8—184no—
115C2×C4⋊C4—96no—
116C2×C4⋊C4—183no—
117#103—916no—
118C23×C3—23no—
119C23×C3—23no—
120C23×C3—24no—
121C23×C3—24no—
122C23×C3—24no—
123C23×C3—24no—
124C23×C3—24no—
125C23×C3—24no—
126C2×C3×C4—64no—
127C2×C3×C4—624no—
128C2×C3×C4—64no—
129C2×C3×C4—1212no—
130C2×Dic3—24no—
131C2×Dic3—224no—
132C2×Dic3—24no—
133C2×Dic3—412no—
134C2×Dic3—612no—
135C2×Dic3—612no—
136C2×Dic3—66no—
137C2×Dic3—66no—
138C2×Dic3—63no—
139C2×Dic3—63no—
140C2×Dic3—66no—
141C2×Dic3—66no—
142C2×Dic3—61no—
143C2×Dic3—61no—
144Dic6—68no—
145Dic6—648no—
146Dic6—68no—
147C2×C32C2×#103112yescomplemented
148C2×C32C22×C22⋊C411yescharacteristic
149C2×C32C22×SD1611yescharacteristic
150C2×C32C22×Q1611yescharacteristic
151C24GO+(4,𝔽2)11yescharacteristic
152C2×Q8—94no—
153C2×Q8—924no—
154C2×Q8—186no—
155C2×Q8—1812no—
156C2×Q8—1824no—
157C22×C4—66no—
158C22×C4—68no—
159C22×C4—66no—
160C22×C4—93no—
161C22×C4—93no—
162C22×C4—94no—
163C22×C4—94no—
164C22×C4—124no—
165C2×C8—94no—
166C2×C8—1812no—
167C4⋊C4—94no—
168C4⋊C4—186no—
169C22×C3—21no—
170C22×C3—21no—
171C22×C3—26no—
172C22×C3—26no—
173C22×C3—26no—
174C22×C3—26no—
175C22×C3—26no—
176C22×C3—26no—
177C22×C3—216no—
178C22×C3—216no—
179C3×C4—68no—
180C3×C4—124no—
181Dic3—28no—
182Dic3—44no—
183Dic3—66no—
184Dic3—61no—
185Dic3—61no—
186Dic3—66no—
187Dic3—61no—
188Dic3—61no—
189C32C22×#10311yescharacteristic, complemented, Sylow
190C23D32⋊C414yes—
191C23#377514yes—
192C23#377614yes—
193C23C2×GO+(4,𝔽2)13yes—
194Q8—98no—
195Q8—184no—
196Q8—1824no—
197C2×C4—64no—
198C2×C4—624no—
199C2×C4—64no—
200C2×C4—91no—
201C2×C4—93no—
202C2×C4—96no—
203C2×C4—96no—
204C2×C4—912no—
205C2×C4—1212no—
206C8—184no—
207C2×C3—21no—
208C2×C3—21no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3—21no—
212C2×C3—21no—
213C2×C3—212no—
214C2×C3—212no—
215C22#63740116yesdirect product factor
216C22C2×D32⋊C416yes—
217C22C2×#377516yes—
218C22C2×#377616yes—
219C22C22×GO+(4,𝔽2)11yescharacteristic
220C4—68no—
221C4—91no—
222C4—91no—
223C4—96no—
224C4—124no—
225C3—21no—
226C3—21no—
227C2C2×#63740112yesdirect product factor
228C2C22×D32⋊C411yescharacteristic
229C2C22×#377511yescharacteristic
230C2C22×#377611yescharacteristic
subgroupG/H
center, hypercenterC24GO+(4,𝔽2)
commutatorC3⋊Dic3C23×C4
FrattiniC22C22×GO+(4,𝔽2)
Fitting, socle, prime radicalC24×C32D4
cosocleC2×C3⋊Dic3C24
hypocenterC32C22×#103
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×#63740C2direct (N × H)
10. C22×#627C4semidirect
16. #63740C22direct (N × H)
31. C2×#627C2×C4semidirect
65. #627C22×C4semidirect
96. C22×C32#103semidirect
147. C2×C32C2×#103semidirect
189. C32C22×#103semidirect
215. C22#63740direct (N × H)
227. C2C2×#63740direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#6272
C23×#6222
C22×#37242
C2×#637402
C22×#1039