Finite Group Databasebeta

← C3×#11709314 #72277099

Subgroups — C3×#11709314 #72277099

Order 2304.

2358 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 2358 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C3×#85645C213yesmaximal, complemented
2#11709314C313yesmaximal, direct product factor
3C3×#15947C311yesmaximal, characteristic, complemented
4C3×C22⋊(C22⋊A4)C2211yescharacteristic, complemented
5C22×C3×C22⋊A4—41nomaximal
6C22×C3×#120—31no—
7C22×C3×#120C2×C313yescomplemented
8#85645C2×C319yescomplemented
9C3×#1438—33no—
10C2×C3×C22⋊A4—43no—
11C3×#120⋊C3—86no—
12#15947C3211yescharacteristic, complemented, Sylow
13C26×C3A411yescharacteristic, complemented
14C26×C3C22×C311yescharacteristic, complemented
15C22⋊(C22⋊A4)C22×C313yescomplemented
16C22×C22⋊A4—43no—
17C23×C3×D4—33no—
18C2×C3×#120—66no—
19C2×C3×#120—66no—
20C2×C3×#120—66no—
21C2×C3×#120—66no—
22C22×C3×C22⋊C4—33no—
23C22×C3×C22⋊C4—33no—
24C2×C3×#95—32no—
25C3×C22⋊A4—41no—
26C3×C22⋊A4—86no—
27C3×C22⋊A4—162no—
28C22×C3×A4—162no—
29C22×C3×A4—163no—
30C22×#120—31no—
31C22×#120C2×C3213yescomplemented
32#1438—33no—
33C25×C3—31no—
34C25×C3—66no—
35C25×C3—63no—
36C25×C3—63no—
37C25×C3—122no—
38C25×C3—122no—
39C25×C3C2×A413yes—
40C2×C22⋊A4—49no—
41C22×C3×D4—36no—
42C22×C3×D4—36no—
43C22×C3×D4—612no—
44C22×C3×D4—66no—
45C22×C3×D4—66no—
46C22×C3×D4—126no—
47C23×C3×C4—33no—
48C3×#120—26no—
49C3×#120—66no—
50C3×#120—64no—
51C3×#120—612no—
52C3×#120—612no—
53C3×#120—64no—
54C3×#120—612no—
55C3×#120—66no—
56C3×#120—66no—
57C2×C3×C22⋊C4—612no—
58C2×C3×C22⋊C4—612no—
59C2×C3×C22⋊C4—63no—
60C2×C3×C22⋊C4—63no—
61C2×C3×C22⋊C4—66no—
62#120⋊C3—818no—
63C3×#95—122no—
64C2×C3×A4—166no—
65C2×C3×A4—166no—
66C2×C3×A4—163no—
67C26C3×A411yescharacteristic, complemented
68C26C22×C3211yescharacteristic, complemented
69C23×D4—33no—
70C2×#120—66no—
71C2×#120—66no—
72C2×#120—66no—
73C2×#120—66no—
74C22×C22⋊C4—33no—
75C22×C22⋊C4—33no—
76C2×#95—32no—
77C24×C3—26no—
78C24×C3—36no—
79C24×C3—33no—
80C24×C3—33no—
81C24×C3—42no—
82C24×C3—43no—
83C24×C3—42no—
84C24×C3—66no—
85C24×C3—66no—
86C24×C3—66no—
87C24×C3—66no—
88C24×C3—61no—
89C24×C3—63no—
90C24×C3—63no—
91C24×C3—61no—
92C24×C3—66no—
93C24×C3—123no—
94C24×C3—124no—
95C24×C3—122no—
96C24×C3—1224no—
97C24×C3—126no—
98C24×C3—1212no—
99C24×C3—126no—
100C24×C3—1212no—
101C24×C3—123no—
102C24×C3—123no—
103C24×C3—126no—
104C24×C3—122no—
105C24×C3C22×A411yescharacteristic
106C22⋊A4—43no—
107C22⋊A4—818no—
108C22⋊A4—166no—
109C22×A4—166no—
110C22×A4—169no—
111C2×C3×D4—612no—
112C2×C3×D4—612no—
113C2×C3×D4—612no—
114C2×C3×D4—66no—
115C2×C3×D4—66no—
116C2×C3×D4—126no—
117C2×C3×D4—126no—
118C2×C3×D4—1224no—
119C2×C3×D4—1224no—
120C22×C3×C4—36no—
121C22×C3×C4—66no—
122C22×C3×C4—66no—
123C22×C3×C4—66no—
124C3×C22⋊C4—612no—
125C3×C22⋊C4—612no—
126C3×C22⋊C4—612no—
127C3×C22⋊C4—66no—
128C3×C22⋊C4—66no—
129C22×C32—641no—
130C3×A4—163no—
131C3×A4—162no—
132C3×A4—644no—
133C25—31no—
134C25—66no—
135C25—63no—
136C25—63no—
137C25—122no—
138C25—122no—
139C25C2×C3×A413yes—
140C22×D4—36no—
141C22×D4—36no—
142C22×D4—612no—
143C22×D4—66no—
144C22×D4—66no—
145C22×D4—126no—
146C23×C4—33no—
147#120—26no—
148#120—66no—
149#120—64no—
150#120—612no—
151#120—612no—
152#120—64no—
153#120—612no—
154#120—66no—
155#120—66no—
156C2×C22⋊C4—612no—
157C2×C22⋊C4—612no—
158C2×C22⋊C4—63no—
159C2×C22⋊C4—63no—
160C2×C22⋊C4—66no—
161#95—122no—
162C23×C3—33no—
163C23×C3—32no—
164C23×C3—34no—
165C23×C3—33no—
166C23×C3—46no—
167C23×C3—46no—
168C23×C3—63no—
169C23×C3—63no—
170C23×C3—63no—
171C23×C3—63no—
172C23×C3—66no—
173C23×C3—612no—
174C23×C3—612no—
175C23×C3—624no—
176C23×C3—66no—
177C23×C3—122no—
178C23×C3—126no—
179C23×C3—126no—
180C23×C3—126no—
181C23×C3—126no—
182C23×C3—1212no—
183C23×C3—123no—
184C23×C3—122no—
185C23×C3—128no—
186C23×C3—126no—
187C23×C3—123no—
188C23×C3—123no—
189C23×C3—126no—
190C23×C3—1212no—
191C23×C3—126no—
192C23×C3—126no—
193C23×C3—123no—
194C23×C3—1236no—
195C23×C3—126no—
196C23×C3—122no—
197C23×C3—1212no—
198C23×C3—126no—
199C23×C3—124no—
200C23×C3—1224no—
201C23×C3—122no—
202C23×C3#120⋊C313yescomplemented
203C2×A4—1618no—
204C2×A4—1618no—
205C2×A4—169no—
206C3×D4—1212no—
207C3×D4—1212no—
208C3×D4—1224no—
209C2×C3×C4—66no—
210C2×C3×C4—66no—
211C2×C3×C4—66no—
212C2×C3×C4—1212no—
213C2×C32—643no—
214C24—26no—
215C24—36no—
216C24—33no—
217C24—33no—
218C24—42no—
219C24—43no—
220C24—42no—
221C24—66no—
222C24—66no—
223C24—66no—
224C24—66no—
225C24—61no—
226C24—63no—
227C24—63no—
228C24—61no—
229C24—66no—
230C24—123no—
231C24—124no—
232C24—122no—
233C24—1224no—
234C24—126no—
235C24—1212no—
236C24—126no—
237C24—1212no—
238C24—123no—
239C24—123no—
240C24—126no—
241C24—122no—
242C24C22×C3×A411yescharacteristic
243C2×D4—612no—
244C2×D4—612no—
245C2×D4—612no—
246C2×D4—66no—
247C2×D4—66no—
248C2×D4—126no—
249C2×D4—126no—
250C2×D4—1224no—
251C2×D4—1224no—
252C22×C4—36no—
253C22×C4—66no—
254C22×C4—66no—
255C22×C4—66no—
256C22⋊C4—612no—
257C22⋊C4—612no—
258C22⋊C4—612no—
259C22⋊C4—66no—
260C22⋊C4—66no—
261C22×C3—31no—
262C22×C3—33no—
263C22×C3—36no—
264C22×C3—41no—
265C22×C3—44no—
266C22×C3—63no—
267C22×C3—612no—
268C22×C3—63no—
269C22×C3—66no—
270C22×C3—126no—
271C22×C3—126no—
272C22×C3—126no—
273C22×C3—126no—
274C22×C3—1212no—
275C22×C3—1212no—
276C22×C3—123no—
277C22×C3—123no—
278C22×C3—126no—
279C22×C3—1212no—
280C22×C3—128no—
281C22×C3—121no—
282C22×C3—122no—
283C22×C3—126no—
284C22×C3—643no—
285C22×C3C2×#120⋊C313yes—
286C22×C3#545012yes—
287A4—169no—
288A4—166no—
289A4—6412no—
290C3×C4—126no—
291C32—641noSylow
292C23—33no—
293C23—32no—
294C23—34no—
295C23—33no—
296C23—46no—
297C23—46no—
298C23—63no—
299C23—63no—
300C23—63no—
301C23—63no—
302C23—66no—
303C23—612no—
304C23—612no—
305C23—624no—
306C23—66no—
307C23—122no—
308C23—126no—
309C23—126no—
310C23—126no—
311C23—126no—
312C23—1212no—
313C23—123no—
314C23—122no—
315C23—128no—
316C23—126no—
317C23—123no—
318C23—123no—
319C23—126no—
320C23—1212no—
321C23—126no—
322C23—126no—
323C23—123no—
324C23—1236no—
325C23—126no—
326C23—122no—
327C23—1212no—
328C23—126no—
329C23—124no—
330C23—1224no—
331C23—122no—
332C23C3×#120⋊C313yescomplemented
333D4—1212no—
334D4—1212no—
335D4—1224no—
336C2×C4—66no—
337C2×C4—66no—
338C2×C4—66no—
339C2×C4—1212no—
340C2×C3—33no—
341C2×C3—32no—
342C2×C3—43no—
343C2×C3—124no—
344C2×C3—123no—
345C2×C3—649no—
346C22—31no—
347C22—33no—
348C22—36no—
349C22—41no—
350C22—44no—
351C22—63no—
352C22—612no—
353C22—63no—
354C22—66no—
355C22—126no—
356C22—126no—
357C22—126no—
358C22—126no—
359C22—1212no—
360C22—1212no—
361C22—123no—
362C22—123no—
363C22—126no—
364C22—1212no—
365C22—128no—
366C22—121no—
367C22—122no—
368C22—126no—
369C22C2×C3×#120⋊C313yes—
370C22C3×#545012yes—
371C4—126no—
372C3—643no—
373C3#1170931411yescharacteristic, direct product factor
374C2—33no—
375C2—32no—
376C2—43no—
377C2—124no—
378C2—123no—
subgroupG/H
center, hypercenterC3#11709314
commutator, cosocle, hypocenterC26C22×C32
FrattiniC24C22×C3×A4
FittingC3×#15947C3
socleC24×C3C22×A4
LCSG ▷ C26
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×#85645C2semidirect
2. #11709314C3direct (N × H)
3. C3×#15947C3semidirect
4. C3×C22⋊(C22⋊A4)C22semidirect
7. C22×C3×#120C2×C3semidirect
8. #85645C2×C3semidirect
12. #15947C32semidirect
13. C26×C3A4semidirect
14. C26×C3C22×C3semidirect
15. C22⋊(C22⋊A4)C22×C3semidirect
31. C22×#120C2×C32semidirect
67. C26C3×A4semidirect
68. C26C22×C32semidirect
202. C23×C3#120⋊C3semidirect
332. C23C3×#120⋊C3semidirect
373. C3#11709314direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#856452
#117093143
C3×#159473
C22×C3×C22⋊A44