Finite Group Databasebeta

← C22×#63840 #11900727

Subgroups — C22×#63840 #11900727

Order 1152.

3673 conjugacy classes of subgroups; only the 972 normal subgroups are shown.

normal subgroups only

This group has 3673 conjugacy classes of subgroups — too many to draw. Only its 972 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×#634C211yesmaximal, characteristic
2C23×#627C211yesmaximal, characteristic
3C22×C3×Dic3⋊C4C212yesmaximal
4C22×Dic3⋊Dic3C212yesmaximal
5C22×Dic32C211yesmaximal, characteristic
6C2×#63840C2124yesmaximal, direct product factor
7C22×#888—32nomaximal
8C23×C32×C4C2211yescharacteristic
9C23×C3⋊Dic3C2211yescharacteristic
10C23×C3⋊Dic3C2211yescharacteristic
11C22×#634—28no—
12C22×#634C22112yes—
13C23×C3×Dic3C2212yes—
14C23×C3×Dic3C2212yes—
15C22×#627—28no—
16C22×#627C22112yes—
17C2×C3×Dic3⋊C4C22124yes—
18C2×Dic3⋊Dic3C22124yes—
19C2×Dic32C22112yes—
20#63840C22164yesdirect product factor
21C23×Dic6—32no—
22C23×Dic6—32no—
23C23×Dic6—61no—
24C22×C3×C4⋊C4—32no—
25C22×C4⋊Dic3—32no—
26C22×Dic3⋊C4—32no—
27C22×Dic3⋊C4D312yes—
28C22×C4×Dic3—32no—
29C2×#888—348no—
30C24×C32C2311yescharacteristic
31C22×C32×C4—24no—
32C22×C32×C4C2316yes—
33C22×C3⋊Dic3—24no—
34C22×C3⋊Dic3D418yes—
35C22×C3⋊Dic3C2316yes—
36C22×C3⋊Dic3C2316yes—
37C2×#634—248no—
38C2×#634C23116yes—
39C22×C3×Dic3—28no—
40C22×C3×Dic3Q8116yescomplemented
41C22×C3×Dic3C23112yes—
42C22×C3×Dic3C23112yes—
43C2×#627—248no—
44C2×#627C23116yes—
45C3×Dic3⋊C4C23132yes—
46Dic3⋊Dic3C23132yes—
47Dic32C23116yes—
48C22×C4⋊Q8—91noSylow
49C23×C3×C4—21no—
50C23×C3×C4—32no—
51C23×C3×C4—32no—
52C23×C3×C4C2×D312yes—
53C23×Dic3—32no—
54C23×Dic3—32no—
55C23×Dic3—61no—
56C23×Dic3—61no—
57C23×Dic3C2×D312yes—
58C23×Dic3C2×D312yes—
59C22×Dic6—324no—
60C22×Dic6—324no—
61C22×Dic6—612no—
62C22×Dic6—616no—
63C22×Dic6—616no—
64C22×Dic6—616no—
65C2×C3×C4⋊C4—324no—
66C2×C4⋊Dic3—324no—
67C2×Dic3⋊C4—324no—
68C2×Dic3⋊C4C2×D3124yes—
69C2×C4×Dic3—324no—
70#888—3128no—
71C23×C32C2×D414yes—
72C23×C32C2×Q818yes—
73C23×C32C2413yes—
74C2×C32×C4—212no—
75C2×C32×C4C2414yes—
76C2×C3⋊Dic3—212no—
77C2×C3⋊Dic3C2×D4124yes—
78C2×C3⋊Dic3C2414yes—
79C2×C3⋊Dic3C2414yes—
80#634—232no—
81C2×C3×Dic3—224no—
82C2×C3×Dic3C2×Q8148yescomplemented
83C2×C3×Dic3C2418yes—
84C2×C3×Dic3C2418yes—
85#627—232no—
86C23×Q8—91no—
87C23×Q8—91no—
88C2×C4⋊Q8—924no—
89C22×C4⋊C4—32no—
90C22×C4⋊C4—92no—
91C22×C42—91no—
92C24×C3—21no—
93C24×C3C22×D312yes—
94C22×C3×C4—28no—
95C22×C3×C4—28no—
96C22×C3×C4—26no—
97C22×C3×C4—316no—
98C22×C3×C4—312no—
99C22×C3×C4—312no—
100C22×C3×C4—68no—
101C22×C3×C4C22×D3112yes—
102C22×Dic3—28no—
103C22×Dic3—316no—
104C22×Dic3—312no—
105C22×Dic3—312no—
106C22×Dic3—68no—
107C22×Dic3—66no—
108C22×Dic3—66no—
109C22×Dic3—68no—
110C22×Dic3—68no—
111C22×Dic3Dic6116yescomplemented
112C22×Dic3C22×D3112yes—
113C22×Dic3C22×D3112yes—
114C2×Dic6—332no—
115C2×Dic6—332no—
116C2×Dic6—616no—
117C2×Dic6—696no—
118C2×Dic6—696no—
119C2×Dic6—696no—
120C3×C4⋊C4—332no—
121C4⋊Dic3—332no—
122Dic3⋊C4—332no—
123Dic3⋊C4C22×D3132yes—
124C4×Dic3—332no—
125C22×C32C4⋊Q8116yescomplemented
126C22×C32C22×D416yes—
127C22×C32C22×Q8112yes—
128C22×C32C2511yescharacteristic
129C32×C4—24no—
130C3⋊Dic3—24no—
131C3⋊Dic3C22×D418yes—
132C3×Dic3—28no—
133C3×Dic3C22×Q8116yescomplemented
134C22×Q8—912no—
135C22×Q8—912no—
136C22×Q8—188no—
137C22×Q8—188no—
138C23×C4—32no—
139C23×C4—32no—
140C23×C4—91no—
141C23×C4—91no—
142C23×C4D3211yescharacteristic
143C4⋊Q8—964no—
144C2×C4⋊C4—324no—
145C2×C4⋊C4—924no—
146C2×C42—912no—
147C23×C3—23no—
148C23×C3—28no—
149C23×C3—24no—
150C23×C3C2×Dic618yes—
151C23×C3D3×D418yes—
152C23×C3D3×Q818yes—
153C23×C3C23×D316yes—
154C2×C3×C4—224no—
155C2×C3×C4—224no—
156C2×C3×C4—24no—
157C2×C3×C4—348no—
158C2×C3×C4—38no—
159C2×C3×C4—38no—
160C2×C3×C4—624no—
161C2×C3×C4C23×D318yes—
162C2×Dic3—224no—
163C2×Dic3—348no—
164C2×Dic3—38no—
165C2×Dic3—38no—
166C2×Dic3—624no—
167C2×Dic3—64no—
168C2×Dic3—64no—
169C2×Dic3—624no—
170C2×Dic3—624no—
171C2×Dic3C2×Dic6148yescomplemented
172C2×Dic3C23×D318yes—
173C2×Dic3C23×D318yes—
174Dic6—664no—
175Dic6—664no—
176Dic6—664no—
177C2×C32C2×C4⋊Q8112yescomplemented
178C2×C32C23×D411yescharacteristic
179C2×C32C23×Q812yes—
180C24C2×D3211yescharacteristic
181C2×Q8—916no—
182C2×Q8—916no—
183C2×Q8—1848no—
184C2×Q8—1848no—
185C22×C4—24no—
186C22×C4—312no—
187C22×C4—316no—
188C22×C4—312no—
189C22×C4—68no—
190C22×C4—96no—
191C22×C4—98no—
192C22×C4—96no—
193C22×C4—184no—
194C22×C4C2×D3216yes—
195C4⋊C4—332no—
196C4⋊C4—932no—
197C42—916no—
198C22×C3—212no—
199C22×C3—26no—
200C22×C3—216no—
201C22×C3—21no—
202C22×C3#888132yescomplemented
203C22×C3C22×Dic6112yes—
204C22×C3C2×D3×D4112yes—
205C22×C3C2×D3×Q8112yes—
206C22×C3C24×D312yes—
207C3×C4—28no—
208C3×C4—28no—
209C3×C4—316no—
210C3×C4—68no—
211Dic3—28no—
212Dic3—316no—
213Dic3—68no—
214Dic3—68no—
215Dic3—68no—
216Dic3C22×Dic6116yescomplemented
217C32C22×C4⋊Q811yescharacteristic, complemented, Sylow
218C23D3×Dic618yes—
219C23#381114yes—
220C23C22×D3213yes—
221Q8—1832no—
222Q8—1832no—
223C2×C4—212no—
224C2×C4—348no—
225C2×C4—38no—
226C2×C4—38no—
227C2×C4—624no—
228C2×C4—924no—
229C2×C4—94no—
230C2×C4—94no—
231C2×C4—1812no—
232C2×C4C22×D3214yes—
233C2×C3—22no—
234C2×C3—21no—
235C2×C3—212no—
236C2×C3C2×#888124yescomplemented
237C2×C3C23×Dic612yes—
238C2×C3C22×D3×D412yes—
239C2×C3C22×D3×Q812yes—
240C22#63840116yesdirect product factor
241C22C2×D3×Dic6112yes—
242C22C2×#381116yes—
243C22C23×D3211yescharacteristic
244C4—24no—
245C4—316no—
246C4—68no—
247C4—98no—
248C4—184no—
249C3—21no—
250C3C22×#88812yescomplemented
251C2C2×#63840112yesdirect product factor
252C2C22×D3×Dic612yes—
253C2C22×#381111yescharacteristic
subgroupG/H
centerC24C2×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC23×C32×C4C22
socle, prime radicalC24×C32C23
hypercenterC23×C4D32
hypocenterC32C22×C4⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
6. C2×#63840C2direct (N × H)
20. #63840C22direct (N × H)
40. C22×C3×Dic3Q8semidirect
82. C2×C3×Dic3C2×Q8semidirect
111. C22×Dic3Dic6semidirect
125. C22×C32C4⋊Q8semidirect
133. C3×Dic3C22×Q8semidirect
171. C2×Dic3C2×Dic6semidirect
177. C2×C32C2×C4⋊Q8semidirect
202. C22×C3#888semidirect
216. Dic3C22×Dic6semidirect
217. C32C22×C4⋊Q8semidirect
236. C2×C3C2×#888semidirect
240. C22#63840direct (N × H)
250. C3C22×#888semidirect
251. C2C2×#63840direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#6342
C23×#6272
C22×C3×Dic3⋊C42
C22×Dic3⋊Dic32
C22×Dic322
C2×#638402
C22×#8883