Finite Group Databasebeta

← C2×#11724055 #13180091

Subgroups — C2×#11724055 #13180091

Order 1728.

1153 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 1153 conjugacy classes of subgroups — too many to draw. Only its 41 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×C32⋊(C22⋊A4)C211yesmaximal, characteristic, complemented
2#11724055C212yesmaximal, direct product factor
3C2×C3×#1020—33nomaximal
4C2×C3×#953—31nomaximal
5C32⋊(C22⋊A4)C2211yescharacteristic, complemented
6C2×#6258—43nomaximal
7C25×C32D311yescharacteristic, complemented
8C2×C3×C22⋊A4D313yescomplemented
9C3×#1020—36no—
10C22×C3×#67—33no—
11C3×#953—36no—
12C3×#953—32no—
13C2×C3×#217—33no—
14C2×C22⋊H(3,𝔽3)—43no—
15C2×C22⋊H(3,𝔽3)—81no—
16#6258—46no—
17C2×#1020—33no—
18C2×C3×#120—91no—
19C2×#953—31no—
20C24×C32—33no—
21C24×C32—33no—
22C24×C32—61no—
23C24×C32—61no—
24C24×C32C2×D311yescharacteristic, complemented
25C23×C3×D3—31no—
26C3×C22⋊A4C2×D313yescomplemented
27C2×C3×S4—129no—
28C2×C3×#67—36no—
29C2×C3×#67—36no—
30C2×C3×#67—63no—
31C2×C3×#67—63no—
32C2×C3×#67—63no—
33C2×C3×#67—63no—
34C22×C3×Dic3—33no—
35C3×#217—36no—
36C3×#217—36no—
37C2×#272—161no—
38C22⋊H(3,𝔽3)—43no—
39C22⋊H(3,𝔽3)—81no—
40C25×C3—31no—
41C25×C3C3⋊D311yescharacteristic
42C2×C22⋊A4—33no—
43#1020—36no—
44C22×C3×D4—93no—
45C22×#67—33no—
46C3×#120—96no—
47C3×#120—92no—
48C2×C3×C22⋊C4—93no—
49#953—36no—
50#953—32no—
51C2×#217—33no—
52C23×C32—21no—
53C23×C32—31no—
54C23×C32—33no—
55C23×C32—33no—
56C23×C32—33no—
57C23×C32—33no—
58C23×C32—33no—
59C23×C32—61no—
60C23×C32—61no—
61C23×C32—66no—
62C23×C32—63no—
63C23×C32—63no—
64C23×C32—63no—
65C23×C32S413yescomplemented
66C22×C3×D3—32no—
67C22×C3×D3—63no—
68C22×C3×D3—63no—
69C2×C3×A4—49no—
70C2×C3×A4—83no—
71C3×S4—1218no—
72C3×#67—66no—
73C3×#67—66no—
74C3×#67—66no—
75C3×#67—66no—
76C2×C3×Dic3—36no—
77C2×C3×Dic3—63no—
78C2×C3×Dic3—63no—
79C2×#120—91noSylow
80C2×H(3,𝔽3)—161no—
81#272—162no—
82C24×C3—33no—
83C24×C3—33no—
84C24×C3—33no—
85C24×C3—33no—
86C24×C3—31no—
87C24×C3—63no—
88C24×C3—63no—
89C24×C3—61no—
90C24×C3—61no—
91C24×C3—63no—
92C24×C3—63no—
93C24×C3—91no—
94C24×C3C2×C3⋊D311yescharacteristic
95C23×D3—31no—
96C22⋊A4—33no—
97C2×S4—129no—
98C2×C3×D4—96no—
99C2×C3×D4—96no—
100C2×C3×D4—183no—
101C2×C3×D4—183no—
102C2×C3×D4—183no—
103C2×C3×D4—183no—
104C22×C3×C4—93no—
105C2×#67—36no—
106C2×#67—36no—
107C2×#67—63no—
108C2×#67—63no—
109C2×#67—63no—
110C2×#67—63no—
111C22×Dic3—33no—
112C3×C22⋊C4—96no—
113C3×C22⋊C4—96no—
114#217—36no—
115#217—36no—
116C22×C32—21no—
117C22×C32—33no—
118C22×C32—31no—
119C22×C32—33no—
120C22×C32—33no—
121C22×C32—33no—
122C22×C32—33no—
123C22×C32—63no—
124C22×C32—66no—
125C22×C32—63no—
126C22×C32—63no—
127C22×C32—61no—
128C22×C32—61no—
129C22×C32C2×S413yescomplemented
130C2×C3×D3—66no—
131C2×C3×D3—123no—
132C2×C3×D3—121no—
133C2×C3×D3—483no—
134C3×A4—49no—
135C3×A4—83no—
136C3×Dic3—66no—
137C25#27211yescharacteristic, complemented
138C22×D4—93no—
139#120—96no—
140#120—92no—
141C2×C22⋊C4—93no—
142H(3,𝔽3)—161noSylow
143C23×C3—21no—
144C23×C3—31no—
145C23×C3—31no—
146C23×C3—33no—
147C23×C3—33no—
148C23×C3—33no—
149C23×C3—33no—
150C23×C3—33no—
151C23×C3—33no—
152C23×C3—33no—
153C23×C3—33no—
154C23×C3—33no—
155C23×C3—33no—
156C23×C3—33no—
157C23×C3—61no—
158C23×C3—63no—
159C23×C3—63no—
160C23×C3—61no—
161C23×C3—63no—
162C23×C3—63no—
163C23×C3—63no—
164C23×C3—61no—
165C23×C3—61no—
166C23×C3—66no—
167C23×C3—63no—
168C23×C3—63no—
169C23×C3—63no—
170C23×C3—66no—
171C23×C3—66no—
172C23×C3—66no—
173C23×C3—66no—
174C23×C3—63no—
175C23×C3—63no—
176C23×C3—63no—
177C23×C3—63no—
178C23×C3—66no—
179C23×C3—63no—
180C23×C3—66no—
181C23×C3—92no—
182C23×C3—183no—
183C23×C3—183no—
184C23×C3C3⋊S413yes—
185C22×D3—32no—
186C22×D3—63no—
187C22×D3—63no—
188C2×A4—129no—
189C2×A4—243no—
190S4—1218no—
191C3×D4—186no—
192C3×D4—186no—
193C3×D4—186no—
194C3×D4—186no—
195C2×C3×C4—96no—
196C2×C3×C4—183no—
197C2×C3×C4—183no—
198#67—66no—
199#67—66no—
200#67—66no—
201#67—66no—
202C2×Dic3—36no—
203C2×Dic3—63no—
204C2×Dic3—63no—
205C2×C32—33no—
206C2×C32—33no—
207C2×C32—61no—
208C2×C32—61no—
209C2×C32—163no—
210C2×C32#102011yescharacteristic, complemented
211C3×D3—122no—
212C3×D3—486no—
213C24—33no—
214C24—33no—
215C24—61no—
216C24—61no—
217C24—91no—
218C24C2×#27211yescharacteristic, complemented
219C2×D4—96no—
220C2×D4—96no—
221C2×D4—183no—
222C2×D4—183no—
223C2×D4—183no—
224C2×D4—183no—
225C22×C4—93no—
226C22⋊C4—96no—
227C22⋊C4—96no—
228C22×C3—21no—
229C22×C3—33no—
230C22×C3—31no—
231C22×C3—33no—
232C22×C3—33no—
233C22×C3—31no—
234C22×C3—33no—
235C22×C3—33no—
236C22×C3—33no—
237C22×C3—33no—
238C22×C3—33no—
239C22×C3—33no—
240C22×C3—33no—
241C22×C3—33no—
242C22×C3—61no—
243C22×C3—63no—
244C22×C3—63no—
245C22×C3—61no—
246C22×C3—63no—
247C22×C3—63no—
248C22×C3—63no—
249C22×C3—63no—
250C22×C3—66no—
251C22×C3—63no—
252C22×C3—66no—
253C22×C3—66no—
254C22×C3—66no—
255C22×C3—66no—
256C22×C3—63no—
257C22×C3—63no—
258C22×C3—63no—
259C22×C3—61no—
260C22×C3—63no—
261C22×C3—66no—
262C22×C3—63no—
263C22×C3—66no—
264C22×C3—63no—
265C22×C3—61no—
266C22×C3—186no—
267C22×C3—363no—
268C22×C3—361no—
269C22×C3C2×C3⋊S413yes—
270C2×D3—66no—
271C2×D3—123no—
272C2×D3—121no—
273C2×D3—483no—
274A4—129no—
275A4—243no—
276C3×C4—186no—
277Dic3—66no—
278C32—163no—
279C32C2×#102011yescharacteristic, complemented
280C23—21no—
281C23—31no—
282C23—33no—
283C23—33no—
284C23—33no—
285C23—33no—
286C23—33no—
287C23—61no—
288C23—61no—
289C23—66no—
290C23—63no—
291C23—63no—
292C23—63no—
293C23—92no—
294C23—183no—
295C23—183no—
296C23#625813yescomplemented
297D4—186no—
298D4—186no—
299D4—186no—
300D4—186no—
301C2×C4—96no—
302C2×C4—183no—
303C2×C4—183no—
304C2×C3—33no—
305C2×C3—33no—
306C2×C3—33no—
307C2×C3—33no—
308C2×C3—31no—
309C2×C3—63no—
310C2×C3—63no—
311C2×C3—63no—
312C2×C3—63no—
313C2×C3—61no—
314C2×C3—61no—
315C2×C3—362no—
316C2×C3—483no—
317C2×C3C3⋊#102011yescharacteristic
318D3—122no—
319D3—486no—
320C22—21no—
321C22—33no—
322C22—31no—
323C22—33no—
324C22—33no—
325C22—33no—
326C22—33no—
327C22—63no—
328C22—66no—
329C22—63no—
330C22—63no—
331C22—61no—
332C22—61no—
333C22—186no—
334C22—363no—
335C22—361no—
336C22C2×#625813yescomplemented
337C4—186no—
338C3—31no—
339C3—483no—
340C3C2×C3⋊#102011yescharacteristic
341C2—33no—
342C2—33no—
343C2—61no—
344C2—61no—
345C2—362no—
346C2#1172405511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3C3⋊#1020
commutator, cosocle, hypocenterC32⋊(C22⋊A4)C22
FrattiniC3C2×C3⋊#1020
FittingC25×C32D3
socleC25×C3C3⋊D3
LCSG ▷ C32⋊(C22⋊A4)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C32⋊(C22⋊A4)C2semidirect
2. #11724055C2direct (N × H)
5. C32⋊(C22⋊A4)C22semidirect
7. C25×C32D3semidirect
8. C2×C3×C22⋊A4D3semidirect
24. C24×C32C2×D3semidirect
26. C3×C22⋊A4C2×D3semidirect
65. C23×C32S4semidirect
129. C22×C32C2×S4semidirect
137. C25#272semidirect
210. C2×C32#1020semidirect
218. C24C2×#272semidirect
279. C32C2×#1020semidirect
296. C23#6258semidirect
336. C22C2×#6258semidirect
346. C2#11724055direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32⋊(C22⋊A4)2
#117240552
C2×C3×#10203
C2×C3×#9533
C2×#62584