Finite Group Databasebeta

← C22×#88938 #13181122

Subgroups — C22×#88938 #13181122

Order 1728.

936 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#6296C211yesmaximal, characteristic
2C22×C3×#627C211yesmaximal, characteristic
3C22×C3⋊#622C211yesmaximal, characteristic
4C2×#88938C2112yesmaximal, direct product factor
5C22×#3776—31nomaximal
6C22×C3×C3⋊Dic3—21no—
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C22×C32×Dic3—21no—
9C2×#6296—23no—
10C2×#6296C2216yes—
11C2×C3×#627—23no—
12C2×C3×#627C2216yes—
13C2×C3⋊#622C2216yes—
14#88938C22116yesdirect product factor
15C22×C3×Dic6—61no—
16C22×#627—31no—
17C22×#627—61no—
18C22×#627D311yescharacteristic
19C22×#622—31no—
20C2×#3776—312no—
21C23×C33D411yescharacteristic
22C2×C3×C3⋊Dic3—26no—
23C2×C3×C3⋊Dic3D413yes—
24C2×C3×C3⋊Dic3C2313yes—
25C2×C32×Dic3—26no—
26#6296—26no—
27#6296C2314yes—
28C3×#627—26no—
29C3×#627C2314yes—
30C3⋊#622C2314yes—
31C22×#216—91nomaximal
32C22×C32×C4—61no—
33C22×C3⋊Dic3—21no—
34C22×C3⋊Dic3C2×D311yescharacteristic
35C22×C3×Dic3—21no—
36C22×C3×Dic3—21no—
37C22×C3×Dic3—41no—
38C22×C3×Dic3—61no—
39C22×C3×Dic3—61no—
40C22×C3×Dic3—61no—
41C22×C3×Dic3—61no—
42C22×C3×Dic3—121no—
43C2×C3×Dic6—66no—
44C2×C3×Dic6—66no—
45C2×#627—23no—
46C2×#627—36no—
47C2×#627—66no—
48C2×#627—66no—
49C2×#627—63no—
50C2×#627C2×D316yes—
51C2×#622—36no—
52#3776—316no—
53C22×C33Q1614yescomplemented
54C22×C33C2×D413yes—
55C3×C3⋊Dic3—24no—
56C3×C3⋊Dic3C2×D413yes—
57C3×C3⋊Dic3C2411yescharacteristic
58C32×Dic3—24no—
59C22×C3×Q8—91no—
60C22×Dic6—61no—
61C22×Dic6—91no—
62C22×Dic6—181no—
63C2×#216—912no—
64C22×C3⋊C8—91no—
65C23×C32—21no—
66C23×C32—21no—
67C23×C32—41no—
68C23×C32—41no—
69C23×C32#6711yescharacteristic
70C2×C32×C4—66no—
71C2×C3⋊Dic3—26no—
72C2×C3⋊Dic3#6713yes—
73C2×C3⋊Dic3C22×D313yes—
74C2×C3×Dic3—26no—
75C2×C3×Dic3—26no—
76C2×C3×Dic3—46no—
77C2×C3×Dic3—63no—
78C2×C3×Dic3—63no—
79C2×C3×Dic3—63no—
80C2×C3×Dic3—63no—
81C2×C3×Dic3—66no—
82C2×C3×Dic3—66no—
83C2×C3×Dic3—126no—
84C3×Dic6—64no—
85C3×Dic6—612no—
86#627—26no—
87#627—34no—
88#627—64no—
89#627—612no—
90#627—66no—
91#627C22×D314yes—
92#622—34no—
93C22×Q16—271noSylow
94C2×C33C2×Q1616yescomplemented
95C2×C33C22×D411yescharacteristic
96C2×C3×Q8—96no—
97C2×C3×Q8—183no—
98C22×C3×C4—61no—
99C22×C3×C4—61no—
100C22×C3×C4—91no—
101C22×C3×C4—121no—
102C22×C3×C4—181no—
103C22×Dic3—21no—
104C22×Dic3—61no—
105C22×Dic3—61no—
106C22×Dic3—61no—
107C22×Dic3—61no—
108C22×Dic3—121no—
109C2×Dic6—66no—
110C2×Dic6—66no—
111C2×Dic6—96no—
112C2×Dic6—186no—
113C2×Dic6—183no—
114C2×Dic6—186no—
115#216—916no—
116C2×C3⋊C8—96no—
117C22×C32—23no—
118C22×C32—23no—
119C22×C32—24no—
120C22×C32—24no—
121C22×C32—44no—
122C22×C32—44no—
123C22×C32—43no—
124C22×C32—43no—
125C22×C32#21614yescomplemented
126C22×C32C2×#6713yes—
127C32×C4—64no—
128C3⋊Dic3—24no—
129C3⋊Dic3C2×#6713yes—
130C3⋊Dic3C23×D311yescharacteristic
131C3×Dic3—24no—
132C3×Dic3—24no—
133C3×Dic3—44no—
134C3×Dic3—63no—
135C3×Dic3—63no—
136C3×Dic3—61no—
137C3×Dic3—61no—
138C3×Dic3—64no—
139C3×Dic3—64no—
140C3×Dic3—124no—
141C22×Q8—91no—
142C22×Q8—271no—
143C2×Q16—2712no—
144C22×C8—271no—
145C33C22×Q1611yescharacteristic, complemented, Sylow
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—41no—
149C23×C3—41no—
150C23×C3GO+(4,𝔽2)11yescharacteristic
151C3×Q8—94no—
152C3×Q8—186no—
153C2×C3×C4—66no—
154C2×C3×C4—66no—
155C2×C3×C4—93no—
156C2×C3×C4—93no—
157C2×C3×C4—126no—
158C2×C3×C4—186no—
159C2×Dic3—26no—
160C2×Dic3—63no—
161C2×Dic3—63no—
162C2×Dic3—63no—
163C2×Dic3—63no—
164C2×Dic3—66no—
165C2×Dic3—66no—
166C2×Dic3—126no—
167Dic6—64no—
168Dic6—612no—
169Dic6—94no—
170Dic6—184no—
171Dic6—186no—
172Dic6—1812no—
173C3⋊C8—94no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32—26no—
177C2×C32—26no—
178C2×C32—46no—
179C2×C32—46no—
180C2×C32—41no—
181C2×C32—41no—
182C2×C32C2×#21616yescomplemented
183C2×C32C22×#6711yescharacteristic
184C2×Q8—96no—
185C2×Q8—183no—
186C2×Q8—276no—
187C2×Q8—543no—
188C22×C4—61no—
189C22×C4—91no—
190C22×C4—181no—
191Q16—2716no—
192C2×C8—276no—
193C22×C3—23no—
194C22×C3—23no—
195C22×C3—24no—
196C22×C3—24no—
197C22×C3—43no—
198C22×C3—43no—
199C22×C3—44no—
200C22×C3—44no—
201C22×C3#377614yescomplemented
202C22×C3C2×GO+(4,𝔽2)13yes—
203C3×C4—64no—
204C3×C4—64no—
205C3×C4—91no—
206C3×C4—93no—
207C3×C4—124no—
208C3×C4—184no—
209Dic3—24no—
210Dic3—63no—
211Dic3—61no—
212Dic3—63no—
213Dic3—61no—
214Dic3—64no—
215Dic3—64no—
216Dic3—124no—
217C32—21no—
218C32—21no—
219C32—41no—
220C32—41no—
221C32C22×#21611yescharacteristic, complemented
222C23D32⋊D311yescharacteristic
223Q8—94no—
224Q8—186no—
225Q8—274no—
226Q8—546no—
227C2×C4—66no—
228C2×C4—93no—
229C2×C4—93no—
230C2×C4—186no—
231C8—274no—
232C2×C3—26no—
233C2×C3—26no—
234C2×C3—21no—
235C2×C3—21no—
236C2×C3—46no—
237C2×C3—46no—
238C2×C3—41no—
239C2×C3—41no—
240C2×C3C2×#377616yescomplemented
241C2×C3C22×GO+(4,𝔽2)11yescharacteristic
242C22#8893814yesdirect product factor
243C22C2×D32⋊D313yes—
244C4—64no—
245C4—91no—
246C4—93no—
247C4—184no—
248C3—21no—
249C3—21no—
250C3—41no—
251C3—41no—
252C3C22×#377611yescharacteristic, complemented
253C2C2×#8893816yesdirect product factor
254C2C22×D32⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×D32⋊D3
Fitting, socle, prime radicalC23×C33D4
hypocenterC33C22×Q16
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. C2×#88938C2direct (N × H)
14. #88938C22direct (N × H)
53. C22×C33Q16semidirect
94. C2×C33C2×Q16semidirect
125. C22×C32#216semidirect
145. C33C22×Q16semidirect
182. C2×C32C2×#216semidirect
201. C22×C3#3776semidirect
221. C32C22×#216semidirect
240. C2×C3C2×#3776semidirect
242. C22#88938direct (N × H)
252. C3C22×#3776semidirect
253. C2C2×#88938direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62962
C22×C3×#6272
C22×C3⋊#6222
C2×#889382
C22×#37763
C22×#2169