Finite Group Databasebeta

← C22×C3⋊#6258 #72220527

Subgroups — C22×C3⋊#6258 #72220527

Order 2592.

2478 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 2478 conjugacy classes of subgroups — too many to draw. Only its 201 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×C3×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3⋊#6258C216yesmaximal, direct product factor
3C22×C3×C3⋊S4—33nomaximal
4C22×C3×#638—31nomaximal
5C22×#6258—39nomaximal
6C22×C3⋊#272—41nomaximal
7C2×C3×C22⋊H(3,𝔽3)C2213yescomplemented
8C3⋊#6258C2214yesdirect product factor
9C24×C33D311yescharacteristic, complemented
10C23×C3×C3⋊D3—31no—
11C22×C32×A4D313yescomplemented
12C2×C3×C3⋊S4—318no—
13C2×C3×#638—36no—
14C2×C3×#638—36no—
15C22×C3×C3⋊Dic3—31no—
16C22×C22⋊H(3,𝔽3)D319yescomplemented
17C2×#6258—354no—
18C22×C3×H(3,𝔽3)—41no—
19C2×C3⋊#272—46no—
20C3×C22⋊H(3,𝔽3)C2311yescharacteristic, complemented
21C22×C3⋊S4—33no—
22C22×C3×S4—99no—
23C22×#638—31no—
24C22×C3×#67—91no—
25C22×C3×#67—93no—
26C23×C33—31no—
27C23×C33—33no—
28C23×C33C2×D313yescomplemented
29C22×C3×C3⋊D3—36no—
30C22×C3×C3⋊D3—63no—
31C22×C3×C3⋊D3—61no—
32C22×C3×C3⋊D3—123no—
33C2×C32×A4C2×D319yescomplemented
34C3×C3⋊S4—312no—
35C3×#638—312no—
36C3×#638—34no—
37C2×C3×C3⋊Dic3—36no—
38C22×#272—129no—
39C2×C22⋊H(3,𝔽3)C2×D3127yescomplemented
40#6258—336no—
41C2×C3×H(3,𝔽3)—43no—
42C3⋊#272—44no—
43C24×C32—31no—
44C24×C32—61no—
45C24×C32C3⋊D311yescharacteristic
46C24×C32C3⋊D313yescomplemented
47C23×C3⋊D3—31no—
48C23×C3×D3—91no—
49C23×C3×D3—93no—
50C22×C3×A4—33no—
51C22×C3×A4—63no—
52C22×C3×A4C3⋊D319yescomplemented
53C2×C3⋊S4—318no—
54C2×C3×S4—954no—
55C2×#638—36no—
56C2×#638—36no—
57C22×C3⋊Dic3—31no—
58C2×C3×#67—96no—
59C2×C3×#67—918no—
60C2×C3×#67—96no—
61C2×C3×#67—918no—
62C22×C3×Dic3—91no—
63C22×C3×Dic3—93no—
64C22×C33—33no—
65C22×C33—33no—
66C22×C33—33no—
67C22×C33—43no—
68C22×C33—61no—
69C22×C33S411yescharacteristic, complemented
70C22×C33C22×D311yescharacteristic, complemented
71C2×C3×C3⋊D3—34no—
72C2×C3×C3⋊D3—66no—
73C2×C3×C3⋊D3—66no—
74C2×C3×C3⋊D3—1218no—
75C32×A4C22×D313yescomplemented
76C3×C3⋊Dic3—34no—
77C22×H(3,𝔽3)—49no—
78C2×#272—1254no—
79C22⋊H(3,𝔽3)C22×D319yescomplemented
80C22×S4—99no—
81C22×C3×D4—271no—
82C22×#67—93no—
83C22×#67—91no—
84C3×H(3,𝔽3)—41noSylow
85C23×C32—31no—
86C23×C32—33no—
87C23×C32—33no—
88C23×C32—31no—
89C23×C32—33no—
90C23×C32—33no—
91C23×C32—39no—
92C23×C32—63no—
93C23×C32—61no—
94C23×C32—63no—
95C23×C32—63no—
96C23×C32—69no—
97C23×C32C2×C3⋊D313yes—
98C23×C32C2×C3⋊D319yescomplemented
99C22×C3⋊D3—36no—
100C22×C3⋊D3—63no—
101C22×C3⋊D3—61no—
102C22×C3⋊D3—123no—
103C22×C3×D3—96no—
104C22×C3×D3—918no—
105C22×C3×D3—183no—
106C22×C3×D3—189no—
107C22×C3×D3—181no—
108C22×C3×D3—183no—
109C22×C3×D3—369no—
110C2×C3×A4—39no—
111C2×C3×A4—69no—
112C2×C3×A4C2×C3⋊D3127yescomplemented
113C3⋊S4—312no—
114C3×S4—936no—
115#638—312no—
116#638—34no—
117C2×C3⋊Dic3—36no—
118C3×#67—912no—
119C3×#67—936no—
120C3×#67—94no—
121C3×#67—912no—
122C2×C3×Dic3—96no—
123C2×C3×Dic3—918no—
124C2×C33—31no—
125C2×C33—33no—
126C2×C33—49no—
127C2×C33C2×S413yescomplemented
128C3×C3⋊D3—64no—
129C3×C3⋊D3—1212no—
130C2×H(3,𝔽3)—427no—
131#272—1236no—
132C24×C3—21no—
133C24×C3—33no—
134C24×C3—271no—
135C24×C3#27211yescharacteristic, complemented
136C24×C3C3⋊(C3⋊D3)11yescharacteristic
137C23×D3—93no—
138C23×D3—91no—
139C22×A4—39no—
140C2×S4—954no—
141C2×C3×D4—276no—
142C2×C3×D4—276no—
143C22×C3×C4—271no—
144C2×#67—96no—
145C2×#67—918no—
146C2×#67—918no—
147C2×#67—96no—
148C22×Dic3—93no—
149C22×Dic3—91no—
150C22×C32—33no—
151C22×C32—33no—
152C22×C32—33no—
153C22×C32—31no—
154C22×C32—31no—
155C22×C32—33no—
156C22×C32—39no—
157C22×C32—33no—
158C22×C32—33no—
159C22×C32—39no—
160C22×C32—39no—
161C22×C32—49no—
162C22×C32—61no—
163C22×C32—61no—
164C22×C32—63no—
165C22×C32—63no—
166C22×C32—63no—
167C22×C32—69no—
168C22×C32—69no—
169C22×C32—69no—
170C22×C32—61no—
171C22×C32—63no—
172C22×C32—63no—
173C22×C32—66no—
174C22×C32—123no—
175C22×C32—243no—
176C22×C32C3⋊S411yescharacteristic
177C22×C32C3⋊S413yescomplemented
178C22×C32C22×C3⋊D311yescharacteristic
179C22×C32C22×C3⋊D313yescomplemented
180C2×C3⋊D3—34no—
181C2×C3⋊D3—66no—
182C2×C3⋊D3—66no—
183C2×C3⋊D3—1218no—
184C2×C3×D3—94no—
185C2×C3×D3—912no—
186C2×C3×D3—186no—
187C2×C3×D3—1818no—
188C2×C3×D3—186no—
189C2×C3×D3—1818no—
190C2×C3×D3—3654no—
191C3×A4—33no—
192C3×A4—63no—
193C3×A4C22×C3⋊D319yescomplemented
194C3⋊Dic3—34no—
195C3×Dic3—94no—
196C3×Dic3—912no—
197C22×D4—271noSylow
198C33—43no—
199C33C22×S411yescharacteristic, complemented
200H(3,𝔽3)—49no—
201C23×C3—23no—
202C23×C3—33no—
203C23×C3—31no—
204C23×C3—31no—
205C23×C3—33no—
206C23×C3—33no—
207C23×C3—39no—
208C23×C3—39no—
209C23×C3—61no—
210C23×C3—63no—
211C23×C3—69no—
212C23×C3—63no—
213C23×C3—276no—
214C23×C3—543no—
215C23×C3—541no—
216C23×C3C2×#27213yescomplemented
217C23×C3C2×C3⋊(C3⋊D3)13yes—
218C22×D3—918no—
219C22×D3—96no—
220C22×D3—189no—
221C22×D3—183no—
222C22×D3—183no—
223C22×D3—181no—
224C22×D3—369no—
225C2×A4—327no—
226S4—936no—
227C3×D4—2712no—
228C3×D4—274no—
229C2×C3×C4—276no—
230#67—912no—
231#67—936no—
232#67—912no—
233#67—94no—
234C2×Dic3—918no—
235C2×Dic3—96no—
236C2×C32—31no—
237C2×C32—33no—
238C2×C32—33no—
239C2×C32—31no—
240C2×C32—33no—
241C2×C32—33no—
242C2×C32—39no—
243C2×C32—427no—
244C2×C32—63no—
245C2×C32—61no—
246C2×C32—63no—
247C2×C32—63no—
248C2×C32—69no—
249C2×C32—129no—
250C2×C32—249no—
251C2×C32C2×C3⋊S413yes—
252C2×C32C2×C3⋊S419yescomplemented
253C3⋊D3—64no—
254C3⋊D3—1212no—
255C3×D3—184no—
256C3×D3—1812no—
257C3×D3—3636no—
258C24—271no—
259C24C3⋊#27211yescharacteristic, complemented
260C2×D4—276no—
261C2×D4—276no—
262C22×C4—271no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—33no—
266C22×C3—33no—
267C22×C3—39no—
268C22×C3—39no—
269C22×C3—33no—
270C22×C3—33no—
271C22×C3—33no—
272C22×C3—33no—
273C22×C3—39no—
274C22×C3—33no—
275C22×C3—33no—
276C22×C3—63no—
277C22×C3—63no—
278C22×C3—69no—
279C22×C3—69no—
280C22×C3—69no—
281C22×C3—63no—
282C22×C3—61no—
283C22×C3—61no—
284C22×C3—69no—
285C22×C3—62no—
286C22×C3—129no—
287C22×C3—274no—
288C22×C3—546no—
289C22×C3—546no—
290C22×C3#625811yescharacteristic, complemented
291C22×C3C22×#27211yescharacteristic, complemented
292C22×C3C3⋊(C3⋊S4)11yescharacteristic
293C22×C3C22×C3⋊(C3⋊D3)11yescharacteristic
294C2×D3—912no—
295C2×D3—94no—
296C2×D3—1818no—
297C2×D3—186no—
298C2×D3—1818no—
299C2×D3—186no—
300C2×D3—3654no—
301A4—39no—
302C3×C4—274no—
303Dic3—912no—
304Dic3—94no—
305C32—31no—
306C32—49no—
307C32—61no—
308C32—123no—
309C32—243no—
310C32C22×C3⋊S411yescharacteristic
311C32C22×C3⋊S413yescomplemented
312C23—31no—
313C23—33no—
314C23—276no—
315C23—543no—
316C23—541no—
317C23C2×C3⋊#27213yescomplemented
318D4—2712no—
319D4—274no—
320C2×C4—276no—
321C2×C3—23no—
322C2×C3—33no—
323C2×C3—39no—
324C2×C3—31no—
325C2×C3—33no—
326C2×C3—31no—
327C2×C3—33no—
328C2×C3—39no—
329C2×C3—61no—
330C2×C3—63no—
331C2×C3—63no—
332C2×C3—69no—
333C2×C3—1227no—
334C2×C3—544no—
335C2×C3C2×#625813yescomplemented
336C2×C3C2×C3⋊(C3⋊S4)13yes—
337D3—1812no—
338D3—184no—
339D3—3636no—
340C22—33no—
341C22—33no—
342C22—33no—
343C22—61no—
344C22—274no—
345C22—546no—
346C22—546no—
347C22C3⋊#625811yescharacteristic, direct product factor
348C22C22×C3⋊#27211yescharacteristic, complemented
349C4—274no—
350C3—21no—
351C3—33no—
352C3—129no—
353C3C22×#625811yescharacteristic, complemented
354C3C22×C3⋊(C3⋊S4)11yescharacteristic
355C2—31no—
356C2—33no—
357C2—544no—
358C2C2×C3⋊#625813yesdirect product factor
subgroupG/H
center, hypercenterC22×C3C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC3×C22⋊H(3,𝔽3)C23
FrattiniC3C22×C3⋊(C3⋊S4)
FittingC24×C33D3
socleC24×C32C3⋊D3
LCSG ▷ C3×C22⋊H(3,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊H(3,𝔽3)C2semidirect
2. C2×C3⋊#6258C2direct (N × H)
7. C2×C3×C22⋊H(3,𝔽3)C22semidirect
8. C3⋊#6258C22direct (N × H)
9. C24×C33D3semidirect
11. C22×C32×A4D3semidirect
16. C22×C22⋊H(3,𝔽3)D3semidirect
20. C3×C22⋊H(3,𝔽3)C23semidirect
28. C23×C33C2×D3semidirect
33. C2×C32×A4C2×D3semidirect
39. C2×C22⋊H(3,𝔽3)C2×D3semidirect
46. C24×C32C3⋊D3semidirect
52. C22×C3×A4C3⋊D3semidirect
69. C22×C33S4semidirect
70. C22×C33C22×D3semidirect
75. C32×A4C22×D3semidirect
79. C22⋊H(3,𝔽3)C22×D3semidirect
98. C23×C32C2×C3⋊D3semidirect
112. C2×C3×A4C2×C3⋊D3semidirect
127. C2×C33C2×S4semidirect
135. C24×C3#272semidirect
177. C22×C32C3⋊S4semidirect
179. C22×C32C22×C3⋊D3semidirect
193. C3×A4C22×C3⋊D3semidirect
199. C33C22×S4semidirect
216. C23×C3C2×#272semidirect
252. C2×C32C2×C3⋊S4semidirect
259. C24C3⋊#272semidirect
290. C22×C3#6258semidirect
291. C22×C3C22×#272semidirect
311. C32C22×C3⋊S4semidirect
317. C23C2×C3⋊#272semidirect
335. C2×C3C2×#6258semidirect
347. C22C3⋊#6258direct (N × H)
348. C22C22×C3⋊#272semidirect
353. C3C22×#6258semidirect
358. C2C2×C3⋊#6258direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊H(3,𝔽3)2
C2×C3⋊#62582
C22×C3×C3⋊S43
C22×C3×#6383
C22×#62583
C22×C3⋊#2724