Finite Group Databasebeta

← C2×(C2×D3)⋊#622 #13166584

Subgroups — C2×(C2×D3)⋊#622 #13166584

Order 1728.

984 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 130 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×D3×C3⋊Dic3C211yesmaximal, characteristic
2C22×C3⋊#622C211yesmaximal, characteristic, complemented
3C22×C3×#622C211yesmaximal, characteristic, complemented
4(C2×D3)⋊#622C214yesmaximal, direct product factor
5C2×C22⋊#622—31nomaximal
6C23×C32×D3C411yescharacteristic
7C22×C3⋊(C3⋊Dic3)C411yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×D3×C3⋊Dic3—22no—
10C2×D3×C3⋊Dic3—22no—
11C2×D3×C3⋊Dic3C412yes—
12C2×D3×C3⋊Dic3C2212yes—
13C2×C3⋊#622—22no—
14C2×C3⋊#622C2212yescomplemented
15C2×C3×#622—22no—
16C2×C3×#622C2212yescomplemented
17C23×C3⋊Dic3—31no—
18C22×D3×Dic3—62no—
19C22×#622—31no—
20C22×#622D311yescharacteristic, complemented
21C22⋊#622—34no—
22C23×C33C2×C411yescharacteristic
23C22×C32×D3—22no—
24C22×C32×D3—22no—
25C22×C32×D3C814yescomplemented
26C22×C32×D3C2×C412yes—
27C2×C3⋊(C3⋊Dic3)—22no—
28C2×C3⋊(C3⋊Dic3)C2×C412yes—
29C2×C3×C3⋊Dic3C2×C411yescharacteristic
30C2×C3×C3⋊Dic3D412yes—
31C2×C3×C3⋊Dic3D412yes—
32C2×C3×C3⋊Dic3C2311yescharacteristic
33D3×C3⋊Dic3—24no—
34D3×C3⋊Dic3—22no—
35D3×C3⋊Dic3—22no—
36C3⋊#622—22no—
37C3×#622—22no—
38C2×(C2×D3)⋊C8—91nomaximal
39C24×C32—31no—
40C23×C3×D3—22no—
41C22×C3⋊Dic3—31no—
42C22×C3⋊Dic3—32no—
43C22×C3⋊Dic3—32no—
44C22×C3⋊Dic3—62no—
45C22×C3⋊Dic3—62no—
46C22×C3⋊Dic3—62no—
47C22×C3⋊Dic3—122no—
48C22×C3⋊Dic3C2×D311yescharacteristic
49C22×C3×Dic3—62no—
50C2×D3×Dic3—64no—
51C2×D3×Dic3—64no—
52C2×D3×Dic3—68no—
53C2×D3×Dic3—68no—
54C2×#622—22no—
55C2×#622—32no—
56C2×#622—62no—
57C2×#622C2×D312yescomplemented
58C22×C33C22⋊C412yes—
59C22×C33C2×C812yes—
60C22×C33M1612yes—
61C22×C33C22×C411yescharacteristic
62C2×C32×D3—28no—
63C2×C32×D3—22no—
64C2×C32×D3—22no—
65C2×C32×D3C2×C814yescomplemented
66C3⋊(C3⋊Dic3)—22no—
67C3×C3⋊Dic3C22⋊C412yes—
68C3×C3⋊Dic3C2×D411yescharacteristic
69C3×C3⋊Dic3C2×D411yescharacteristic
70C23×Dic3—182no—
71C22×C4×D3—91no—
72C22×C3×C8—91no—
73C22×C3⋊C8—91no—
74(C2×D3)⋊C8—94no—
75C23×C32—22no—
76C23×C32—34no—
77C23×C32—32no—
78C23×C32—42no—
79C23×C32—62no—
80C23×C32—62no—
81C23×C32C4×D311yescharacteristic
82C22×C3×D3—28no—
83C22×C3×D3—28no—
84C22×C3×D3—28no—
85C22×C3×D3—24no—
86C2×C3⋊Dic3—32no—
87C2×C3⋊Dic3—64no—
88C2×C3⋊Dic3—62no—
89C2×C3⋊Dic3—64no—
90C2×C3⋊Dic3—62no—
91C2×C3⋊Dic3—62no—
92C2×C3⋊Dic3—68no—
93C2×C3⋊Dic3—128no—
94C2×C3⋊Dic3—124no—
95C2×C3⋊Dic3C4×D311yescharacteristic
96C2×C3⋊Dic3D1212yes—
97C2×C3⋊Dic3#6712yes—
98C2×C3⋊Dic3C22×D311yescharacteristic
99C2×C3×Dic3—62no—
100C2×C3×Dic3—64no—
101C2×C3×Dic3—64no—
102C2×C3×Dic3—62no—
103D3×Dic3—616no—
104D3×Dic3—68no—
105D3×Dic3—68no—
106#622—22no—
107#622—62no—
108C2×C22⋊C8—271noSylow
109C2×C33C22⋊C814yescomplemented
110C2×C33C2×C22⋊C411yescharacteristic
111C2×C33C22×C811yescharacteristic
112C2×C33C2×M1611yescharacteristic
113C32×D3—24no—
114C24×C3—62no—
115C23×D3#15711yescharacteristic
116C22×C3×C4—91no—
117C22×Dic3—62no—
118C22×Dic3—91no—
119C22×Dic3—184no—
120C22×Dic3—184no—
121C22×Dic3—182no—
122C22×Dic3—188no—
123C22×Dic3—188no—
124C22×Dic3—362no—
125C2×C4×D3—92no—
126C2×C4×D3—92no—
127C2×C4×D3—182no—
128C2×C4×D3—182no—
129C2×C3×C8—92no—
130C2×C3×C8—182no—
131C2×C3⋊C8—92no—
132C2×C3⋊C8—182no—
133C22×C32—22no—
134C22×C32—24no—
135C22×C32—24no—
136C22×C32—24no—
137C22×C32—34no—
138C22×C32—44no—
139C22×C32—44no—
140C22×C32—44no—
141C22×C32—42no—
142C22×C32—68no—
143C22×C32—62no—
144C22×C32—62no—
145C22×C32D3×C812yes—
146C22×C32ΓL(1,𝔽25)12yes—
147C22×C32(C2×D3)⋊C412yes—
148C22×C32C2×C4×D311yescharacteristic
149C2×C3×D3—28no—
150C2×C3×D3—28no—
151C2×C3×D3—232no—
152C2×C3×D3—28no—
153C3⋊Dic3—62no—
154C3⋊Dic3—68no—
155C3⋊Dic3—128no—
156C3⋊Dic3(C2×D3)⋊C412yes—
157C3⋊Dic3C2×D1211yescharacteristic
158C3⋊Dic3C2×#6711yescharacteristic
159C3×Dic3—64no—
160C3×Dic3—62no—
161C3×Dic3—62no—
162C23×C4—271no—
163C22×C8—91no—
164C22×C8—271no—
165C22⋊C8—274no—
166C33C2×C22⋊C811yescharacteristic, complemented, Sylow
167C23×C3—22no—
168C23×C3—42no—
169C23×C3—64no—
170C23×C3—68no—
171C23×C3—68no—
172C23×C3—68no—
173C23×C3C2×#15711yescharacteristic
174C22×D3—22no—
175C22×D3—22no—
176C22×D3#62214yescomplemented
177C22×D3C2×#15712yes—
178C2×C3×C4—91no—
179C2×C3×C4—91no—
180C2×C3×C4—92no—
181C2×C3×C4—92no—
182C2×Dic3—62no—
183C2×Dic3—64no—
184C2×Dic3—64no—
185C2×Dic3—62no—
186C2×Dic3—92no—
187C2×Dic3—184no—
188C2×Dic3—1816no—
189C2×Dic3—188no—
190C2×Dic3—188no—
191C2×Dic3—188no—
192C2×Dic3—182no—
193C2×Dic3—364no—
194C2×Dic3—368no—
195C4×D3—182no—
196C4×D3—182no—
197C4×D3—184no—
198C3×C8—182no—
199C3⋊C8—182no—
200C2×C32—22no—
201C2×C32—22no—
202C2×C32—22no—
203C2×C32—28no—
204C2×C32—48no—
205C2×C32—42no—
206C2×C32—42no—
207C2×C32—42no—
208C2×C32—64no—
209C2×C32(C2×D3)⋊C814yescomplemented
210C2×C32C2×D3×C811yescharacteristic
211C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
212C2×C32C2×(C2×D3)⋊C411yescharacteristic
213C3×D3—216no—
214C24—31no—
215C22×C4—91no—
216C22×C4—272no—
217C22×C4—272no—
218C22×C4—271no—
219C22×C4—542no—
220C22×C4—542no—
221C2×C8—92no—
222C2×C8—182no—
223C2×C8—272no—
224C2×C8—542no—
225C22×C3—22no—
226C22×C3—24no—
227C22×C3—24no—
228C22×C3—24no—
229C22×C3—42no—
230C22×C3—44no—
231C22×C3—44no—
232C22×C3—44no—
233C22×C3—68no—
234C22×C3—68no—
235C22×C3—632no—
236C22×C3—68no—
237C22×C3C2×#62212yes—
238C22×C3#379212yes—
239C22×C3C22⋊#15712yes—
240C22×C3C22×#15711yescharacteristic
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—28no—
244C2×D3C2×#62214yescomplemented
245C3×C4—91no—
246C3×C4—91no—
247C3×C4—92no—
248Dic3—64no—
249Dic3—62no—
250Dic3—62no—
251Dic3—188no—
252Dic3—182no—
253Dic3—368no—
254C32—22no—
255C32—42no—
256C32C2×(C2×D3)⋊C811yescharacteristic, complemented
257C23—32no—
258C23—34no—
259C23—62no—
260C23—62no—
261C23D3×#15711yescharacteristic
262C2×C4—91no—
263C2×C4—91no—
264C2×C4—92no—
265C2×C4—92no—
266C2×C4—272no—
267C2×C4—542no—
268C2×C4—542no—
269C2×C4—544no—
270C2×C4—542no—
271C8—182no—
272C8—542no—
273C2×C3—28no—
274C2×C3—22no—
275C2×C3—22no—
276C2×C3—22no—
277C2×C3—48no—
278C2×C3—42no—
279C2×C3—42no—
280C2×C3—42no—
281C2×C3—616no—
282C2×C3C22⋊#62214yescomplemented
283C2×C3C22×#62211yescharacteristic
284C2×C3C2×#379211yescharacteristic
285C2×C3C2×C22⋊#15711yescharacteristic
286D3—24no—
287C22—34no—
288C22—68no—
289C22—62no—
290C22—62no—
291C22(C2×D3)⋊#15712yes—
292C22D3×#62212yes—
293C22#8892512yes—
294C22C2×D3×#15711yescharacteristic
295C4—91no—
296C4—91no—
297C4—92no—
298C4—542no—
299C3—22no—
300C3—42no—
301C3C2×C22⋊#62211yescharacteristic, complemented
302C2—64no—
303C2(C2×D3)⋊#62214yesdirect product factor
304C2C2×(C2×D3)⋊#15711yescharacteristic
305C2C2×D3×#62211yescharacteristic
306C2C2×#8892511yescharacteristic
subgroupG/H
center, hypercenterC23D3×#157
commutatorC2×C33C22×C8
FrattiniC22C2×D3×#157
Fitting, socleC23×C33C2×C4
prime radicalC23×C32×D3C4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C2×C22⋊C8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#622C2semidirect
3. C22×C3×#622C2semidirect
4. (C2×D3)⋊#622C2direct (N × H)
14. C2×C3⋊#622C22semidirect
16. C2×C3×#622C22semidirect
20. C22×#622D3semidirect
25. C22×C32×D3C8semidirect
57. C2×#622C2×D3semidirect
65. C2×C32×D3C2×C8semidirect
109. C2×C33C22⋊C8semidirect
166. C33C2×C22⋊C8semidirect
176. C22×D3#622semidirect
209. C2×C32(C2×D3)⋊C8semidirect
244. C2×D3C2×#622semidirect
256. C32C2×(C2×D3)⋊C8semidirect
282. C2×C3C22⋊#622semidirect
301. C3C2×C22⋊#622semidirect
303. C2(C2×D3)⋊#622direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C3⋊Dic32
C22×C3⋊#6222
C22×C3×#6222
(C2×D3)⋊#6222
C2×C22⋊#6223
C2×(C2×D3)⋊C89