Finite Group Databasebeta

← C22×#88880 #13180061

Subgroups — C22×#88880 #13180061

Order 1728.

1098 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1098 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#88880C213yesmaximal, direct product factor
2C24×C3×A4C311yesmaximal, characteristic
3C22×C22⋊(C22⋊C9)C312yesmaximal, complemented
4C24×C22⋊C9C311yesmaximal, characteristic, complemented
5C22×#1094—41nomaximal
6C22×C9⋊A4—41nomaximal
7#88880C2211yescharacteristic, direct product factor
8C23×C3×A4—31no—
9C23×C3×A4—33no—
10C23×C3×A4C2×C313yes—
11C2×C22⋊(C22⋊C9)C2×C316yescomplemented
12C23×C22⋊C9—33no—
13C23×C22⋊C9—31no—
14C23×C22⋊C9C2×C313yescomplemented
15C2×#1094—43no—
16C2×C9⋊A4—43no—
17C26×C3C3211yescharacteristic
18C24×A4—31no—
19C24×C32—41no—
20C22×C3×A4—32no—
21C22×C3×A4—33no—
22C22×C3×A4—33no—
23C22×C3×A4—33no—
24C22×C3×A4A411yescharacteristic
25C22×C3×A4C22×C311yescharacteristic
26C24×C9—41no—
27C22⋊(C22⋊C9)C22×C312yescomplemented
28C22×C22⋊C9—32no—
29C22×C22⋊C9—33no—
30C22×C22⋊C9—33no—
31C22×C22⋊C9—33no—
32C22×C22⋊C9—42no—
33C22×C22⋊C9—42no—
34C22×C22⋊C9—122no—
35C22×C22⋊C9A411yescharacteristic, complemented
36C22×C22⋊C9C22×C311yescharacteristic, complemented
37C22×3−1+2—161no—
38#1094—41no—
39C9⋊A4—41no—
40C25×C3—33no—
41C25×C3—33no—
42C25×C3—31no—
43C25×C3—31no—
44C25×C3—91no—
45C25×C3—93no—
46C25×C3C2×C3213yes—
47C23×A4—39no—
48C23×A4—33no—
49C23×A4—33no—
50C23×C32—43no—
51C23×C32—121no—
52C23×C32—123no—
53C2×C3×A4—33no—
54C2×C3×A4—31no—
55C2×C3×A4C2×A413yes—
56C23×C9—43no—
57C23×C9—121no—
58C23×C9—123no—
59C2×C22⋊C9—33no—
60C2×C22⋊C9—31no—
61C2×C22⋊C9—46no—
62C2×C22⋊C9—46no—
63C2×C22⋊C9—126no—
64C2×C22⋊C9C2×A413yescomplemented
65C263−1+211yescharacteristic, complemented, Sylow
66C2×3−1+2—163no—
67C24×C3—32no—
68C24×C3—32no—
69C24×C3—32no—
70C24×C3—33no—
71C24×C3—33no—
72C24×C3—33no—
73C24×C3—33no—
74C24×C3—33no—
75C24×C3—33no—
76C24×C3—93no—
77C24×C3—91no—
78C24×C3—93no—
79C24×C3—93no—
80C24×C3—93no—
81C24×C3—93no—
82C24×C3—91no—
83C24×C3—91no—
84C24×C3—96no—
85C24×C3—93no—
86C24×C3—93no—
87C24×C3—96no—
88C24×C3—96no—
89C24×C3—96no—
90C24×C3—96no—
91C24×C3—96no—
92C24×C3—94no—
93C24×C3—121no—
94C24×C3C3×A411yescharacteristic
95C24×C3C3×A411yescharacteristic
96C24×C3C22×C3211yescharacteristic
97C22×A4—36no—
98C22×A4—39no—
99C22×A4—39no—
100C22×A4—39no—
101C22×A4—31no—
102C22×A4—31no—
103C22×C32—41no—
104C22×C32—41no—
105C22×C32—123no—
106C22×C32—123no—
107C22×C32—123no—
108C22×C32—122no—
109C3×A4C22×A411yescharacteristic
110C22×C9—41no—
111C22×C9—41no—
112C22×C9—123no—
113C22×C9—123no—
114C22×C9—123no—
115C22×C9—122no—
116C22×C9—162no—
117C22⋊C9—42no—
118C22⋊C9—42no—
119C22⋊C9—122no—
120C22⋊C9C22×A411yescharacteristic, complemented
121C25—33no—
122C25—33no—
123C25—31no—
124C25—31no—
125C25—91no—
126C25—93no—
127C25C2×3−1+213yescomplemented
1283−1+2—161noSylow
129C23×C3—33no—
130C23×C3—33no—
131C23×C3—31no—
132C23×C3—31no—
133C23×C3—36no—
134C23×C3—33no—
135C23×C3—31no—
136C23×C3—33no—
137C23×C3—31no—
138C23×C3—91no—
139C23×C3—93no—
140C23×C3—93no—
141C23×C3—93no—
142C23×C3—93no—
143C23×C3—93no—
144C23×C3—93no—
145C23×C3—93no—
146C23×C3—96no—
147C23×C3—93no—
148C23×C3—96no—
149C23×C3—93no—
150C23×C3—93no—
151C23×C3—93no—
152C23×C3—93no—
153C23×C3—92no—
154C23×C3—92no—
155C23×C3—96no—
156C23×C3—96no—
157C23×C3—96no—
158C23×C3—96no—
159C23×C3—96no—
160C23×C3—93no—
161C23×C3—96no—
162C23×C3—96no—
163C23×C3—96no—
164C23×C3—96no—
165C23×C3—96no—
166C23×C3—96no—
167C23×C3—924no—
168C23×C3—91no—
169C23×C3—129no—
170C23×C3—123no—
171C23×C3—123no—
172C23×C3C2×C3×A413yes—
173C23×C3C2×C3×A413yes—
174C2×A4—33no—
175C2×A4—39no—
176C2×A4—33no—
177C2×C32—43no—
178C2×C32—121no—
179C2×C32—123no—
180C2×C9—43no—
181C2×C9—121no—
182C2×C9—123no—
183C2×C9—166no—
184C24—32no—
185C24—32no—
186C24—32no—
187C24—33no—
188C24—33no—
189C24—33no—
190C24—33no—
191C24—33no—
192C24—33no—
193C24—93no—
194C24—91no—
195C24—93no—
196C24—93no—
197C24—93no—
198C24—93no—
199C24—91no—
200C24—91no—
201C24—96no—
202C24—93no—
203C24—93no—
204C24—96no—
205C24—96no—
206C24—96no—
207C24—96no—
208C24—96no—
209C24—94no—
210C24C9⋊A411yescharacteristic, complemented
211C24#109411yescharacteristic, complemented
212C24C22×3−1+211yescharacteristic, complemented
213C22×C3—33no—
214C22×C3—33no—
215C22×C3—32no—
216C22×C3—32no—
217C22×C3—33no—
218C22×C3—33no—
219C22×C3—33no—
220C22×C3—33no—
221C22×C3—32no—
222C22×C3—91no—
223C22×C3—93no—
224C22×C3—93no—
225C22×C3—93no—
226C22×C3—96no—
227C22×C3—91no—
228C22×C3—93no—
229C22×C3—91no—
230C22×C3—93no—
231C22×C3—96no—
232C22×C3—96no—
233C22×C3—96no—
234C22×C3—96no—
235C22×C3—96no—
236C22×C3—94no—
237C22×C3—93no—
238C22×C3—93no—
239C22×C3—129no—
240C22×C3—126no—
241C22×C3—129no—
242C22×C3—129no—
243C22×C3—121no—
244C22×C3—121no—
245C22×C3A4211yescharacteristic
246C22×C3C22×C3×A411yescharacteristic
247C22×C3C22×C3×A411yescharacteristic
248A4—31no—
249C32—41no—
250C9—41no—
251C9—162no—
252C23—33no—
253C23—33no—
254C23—31no—
255C23—31no—
256C23—36no—
257C23—31no—
258C23—33no—
259C23—31no—
260C23—33no—
261C23—91no—
262C23—93no—
263C23—93no—
264C23—93no—
265C23—93no—
266C23—93no—
267C23—93no—
268C23—93no—
269C23—96no—
270C23—93no—
271C23—96no—
272C23—93no—
273C23—93no—
274C23—93no—
275C23—93no—
276C23—92no—
277C23—92no—
278C23—96no—
279C23—96no—
280C23—96no—
281C23—96no—
282C23—96no—
283C23—93no—
284C23—96no—
285C23—96no—
286C23—96no—
287C23—96no—
288C23—96no—
289C23—96no—
290C23—924no—
291C23—91no—
292C23C2×C9⋊A413yescomplemented
293C23C2×#109413yescomplemented
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—33no—
297C2×C3—33no—
298C2×C3—91no—
299C2×C3—93no—
300C2×C3—123no—
301C2×C3—129no—
302C2×C3—123no—
303C2×C3C2×A4213yes—
304C22—33no—
305C22—33no—
306C22—32no—
307C22—32no—
308C22—33no—
309C22—33no—
310C22—33no—
311C22—33no—
312C22—32no—
313C22—91no—
314C22—93no—
315C22—93no—
316C22—93no—
317C22—96no—
318C22—91no—
319C22—93no—
320C22—91no—
321C22—93no—
322C22—96no—
323C22—96no—
324C22—96no—
325C22—96no—
326C22—96no—
327C22—94no—
328C22—93no—
329C22—93no—
330C22#8888011yescharacteristic, direct product factor
331C22C22×C9⋊A411yescharacteristic, complemented
332C22C22×#109411yescharacteristic, complemented
333C3—121no—
334C3C22×A4211yescharacteristic
335C2—31no—
336C2—31no—
337C2—33no—
338C2—33no—
339C2—91no—
340C2—93no—
341C2C2×#8888013yesdirect product factor
subgroupG/H
center, hypercenterC22×C3A42
commutator, cosocleC24×C3C22×C32
FrattiniC3C22×A42
Fitting, socleC26×C3C32
prime radicalC24×C3×A4C3
hypocenterC24C22×3−1+2
LCSG ▷ C24×C3 ▷ C24
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×#88880C2direct (N × H)
3. C22×C22⋊(C22⋊C9)C3semidirect
4. C24×C22⋊C9C3semidirect
7. #88880C22direct (N × H)
11. C2×C22⋊(C22⋊C9)C2×C3semidirect
14. C23×C22⋊C9C2×C3semidirect
27. C22⋊(C22⋊C9)C22×C3semidirect
35. C22×C22⋊C9A4semidirect
36. C22×C22⋊C9C22×C3semidirect
64. C2×C22⋊C9C2×A4semidirect
65. C263−1+2semidirect
120. C22⋊C9C22×A4semidirect
127. C25C2×3−1+2semidirect
210. C24C9⋊A4semidirect
211. C24#1094semidirect
212. C24C22×3−1+2semidirect
292. C23C2×C9⋊A4semidirect
293. C23C2×#1094semidirect
330. C22#88880direct (N × H)
331. C22C22×C9⋊A4semidirect
332. C22C22×#1094semidirect
341. C2C2×#88880direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#888802
C24×C3×A43
C22×C22⋊(C22⋊C9)3
C24×C22⋊C93
C22×#10944
C22×C9⋊A44