Finite Group Databasebeta

← #41×C22⋊H(3,𝔽3) #13164856

Subgroups — #41×C22⋊H(3,𝔽3) #13164856

Order 1728.

1250 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 170 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
1Q8×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2D4×C22⋊H(3,𝔽3)C213yesmaximal, complemented
3C2×C4×C22⋊H(3,𝔽3)C213yesmaximal, complemented
4C22×C32×#41C311yesmaximal, characteristic, complemented
5C3×A4×#41C313yesmaximal, complemented
6C22×C22⋊H(3,𝔽3)C2213yes—
7#41×H(3,𝔽3)—41nomaximal
8C4×C22⋊H(3,𝔽3)C2211yescharacteristic
9C4×C22⋊H(3,𝔽3)C2213yes—
10C2×C32×#41—33no—
11C2×C32×#41—33no—
12C2×C32×#41—31no—
13C2×C32×#41—31no—
14C22×C32×Q8C2×C311yescharacteristic, complemented
15C22×C32×D4C2×C313yescomplemented
16C23×C32×C4C2×C313yescomplemented
17C3×Q8×A4C2×C313yescomplemented
18C3×D4×A4C2×C319yescomplemented
19C2×C3×C4×A4C2×C319yescomplemented
20C2×C22⋊H(3,𝔽3)—23no—
21C2×C22⋊H(3,𝔽3)C2311yescharacteristic
22Q8×H(3,𝔽3)—41no—
23D4×H(3,𝔽3)—43no—
24C2×C4×H(3,𝔽3)—43no—
25C22×C3×#41—31no—
26C22×C3×#41C3211yescharacteristic
27A4×#41—33no—
28C24×C32C22×C313yes—
29C22×C3×A4C22×C319yes—
30C32×#41—36no—
31C32×#41—36no—
32C32×#41—32no—
33C32×#41—31no—
34C32×#41—33no—
35C32×#41—33no—
36C32×#41—43no—
37C32×#41A411yescharacteristic, complemented
38C2×C32×Q8—31no—
39C2×C32×Q8—33no—
40C2×C32×D4—33no—
41C2×C32×D4—33no—
42C2×C32×D4—36no—
43C22×C32×C4—33no—
44C22×C32×C4—33no—
45C22×C32×C4—33no—
46C22×C32×C4—33no—
47C22×C32×C4C22×C311yescharacteristic
48C22×C32×C4C22×C313yes—
49C3×C4×A4C22×C313yes—
50C3×C4×A4C22×C319yes—
51C22×H(3,𝔽3)—43no—
52C22⋊H(3,𝔽3)#4111yescharacteristic, direct product factor
53C4×H(3,𝔽3)—41no—
54C4×H(3,𝔽3)—43no—
55C2×C3×#41—33no—
56C2×C3×#41—33no—
57C2×C3×#41—31no—
58C2×C3×#41—39no—
59C2×C3×#41—39no—
60C2×C3×#41—33no—
61C2×C3×#41—33no—
62C2×C3×#41—31no—
63C22×C3×Q8—31no—
64C22×C3×Q8C2×C3211yescharacteristic
65C22×C3×D4—33no—
66C22×C3×D4C2×C3213yes—
67C23×C3×C4—33no—
68C23×C3×C4C2×C3213yes—
69Q8×A4—33no—
70D4×A4—39no—
71C2×C4×A4—39no—
72C23×C32—23no—
73C23×C32—33no—
74C23×C32—33no—
75C23×C32—63no—
76C23×C32C23×C311yescharacteristic
77C2×C3×A4—29no—
78C2×C3×A4C23×C313yes—
79C32×Q8—32no—
80C32×Q8—33no—
81C32×Q8—43no—
82C32×Q8C2×A411yescharacteristic, complemented
83C32×D4—36no—
84C32×D4—36no—
85C32×D4—33no—
86C32×D4—49no—
87C32×D4C2×A413yescomplemented
88C2×C32×C4—36no—
89C2×C32×C4—33no—
90C2×C32×C4—31no—
91C2×C32×C4—33no—
92C2×C32×C4—33no—
93C2×C32×C4—33no—
94C2×C32×C4—31no—
95C2×C32×C4—33no—
96C2×C32×C4—49no—
97C2×C32×C4C2×A413yescomplemented
98C22×#41H(3,𝔽3)11yescharacteristic, complemented, Sylow
99C2×H(3,𝔽3)—41no—
100C2×H(3,𝔽3)—83no—
101C24×C3—33no—
102C24×C3C22×C3213yes—
103C22×A4—39no—
104C3×#41—318no—
105C3×#41—318no—
106C3×#41—36no—
107C3×#41—36no—
108C3×#41—36no—
109C3×#41—32no—
110C3×#41—33no—
111C3×#41—39no—
112C3×#41—39no—
113C3×#41—31no—
114C3×#41—33no—
115C3×#41—33no—
116C3×#41—31no—
117C3×#41—123no—
118C3×#41C3×A411yescharacteristic
119C2×C3×Q8—33no—
120C2×C3×Q8—39no—
121C2×C3×Q8—33no—
122C2×C3×Q8—31no—
123C2×C3×D4—39no—
124C2×C3×D4—39no—
125C2×C3×D4—318no—
126C2×C3×D4—33no—
127C2×C3×D4—33no—
128C2×C3×D4—36no—
129C22×C3×C4—39no—
130C22×C3×C4—39no—
131C22×C3×C4—39no—
132C22×C3×C4—39no—
133C22×C3×C4—33no—
134C22×C3×C4—33no—
135C22×C3×C4—33no—
136C22×C3×C4—33no—
137C22×C3×C4—31no—
138C22×C3×C4—33no—
139C22×C3×C4C22×C3211yescharacteristic
140C22×C3×C4C22×C3213yes—
141C4×A4—33no—
142C4×A4—39no—
143C22×C32—31no—
144C22×C32—31no—
145C22×C32—33no—
146C22×C32—49no—
147C22×C32—63no—
148C22×C32—63no—
149C22×C32—63no—
150C22×C32—63no—
151C22×C32C3×#4111yescharacteristic, complemented
152C22×C32C22×A413yes—
153C3×A4C3×#4113yescomplemented
154C32×C4—31no—
155C32×C4—33no—
156C32×C4—49no—
157C32×C4—43no—
158C32×C4C22×A413yes—
159C32×C4C22×A411yescharacteristic
160C2×#41—33no—
161C2×#41—33no—
162C2×#41—31no—
163C2×#41—31no—
164C22×Q8C2×H(3,𝔽3)11yescharacteristic, complemented
165C22×D4C2×H(3,𝔽3)13yescomplemented
166C23×C4C2×H(3,𝔽3)13yescomplemented
167H(3,𝔽3)—41noSylow
168C23×C3—23no—
169C23×C3—39no—
170C23×C3—39no—
171C23×C3—33no—
172C23×C3—33no—
173C23×C3—31no—
174C23×C3—63no—
175C23×C3—63no—
176C23×C3—69no—
177C23×C3C23×C3211yescharacteristic
178C2×A4—33no—
179C2×A4—69no—
180C3×Q8—36no—
181C3×Q8—32no—
182C3×Q8—39no—
183C3×Q8—33no—
184C3×Q8—31no—
185C3×Q8—123no—
186C3×Q8C2×C3×A411yescharacteristic
187C3×D4—318no—
188C3×D4—36no—
189C3×D4—318no—
190C3×D4—39no—
191C3×D4—36no—
192C3×D4—33no—
193C3×D4—33no—
194C3×D4—129no—
195C3×D4C2×C3×A413yes—
196C2×C3×C4—33no—
197C2×C3×C4—33no—
198C2×C3×C4—39no—
199C2×C3×C4—39no—
200C2×C3×C4—318no—
201C2×C3×C4—36no—
202C2×C3×C4—39no—
203C2×C3×C4—39no—
204C2×C3×C4—39no—
205C2×C3×C4—33no—
206C2×C3×C4—33no—
207C2×C3×C4—33no—
208C2×C3×C4—31no—
209C2×C3×C4—33no—
210C2×C3×C4—31no—
211C2×C3×C4—33no—
212C2×C3×C4—33no—
213C2×C3×C4—129no—
214C2×C3×C4C2×C3×A413yes—
215C2×C32—23no—
216C2×C32—31no—
217C2×C32—31no—
218C2×C32—43no—
219C2×C32—63no—
220C2×C32—89no—
221C2×C32C23×A411yescharacteristic
222C24C22×H(3,𝔽3)13yes—
223#41—36no—
224#41—36no—
225#41—32no—
226#41—31no—
227#41—33no—
228#41—33no—
229#41C22⋊H(3,𝔽3)11yescharacteristic, direct product factor
230C2×Q8—33no—
231C2×Q8—31no—
232C2×D4—33no—
233C2×D4—33no—
234C2×D4—36no—
235C22×C4—33no—
236C22×C4—33no—
237C22×C4—33no—
238C22×C4—33no—
239C22×C4C22×H(3,𝔽3)11yescharacteristic
240C22×C4C22×H(3,𝔽3)13yes—
241C22×C3—31no—
242C22×C3—33no—
243C22×C3—33no—
244C22×C3—31no—
245C22×C3—39no—
246C22×C3—33no—
247C22×C3—33no—
248C22×C3—31no—
249C22×C3—63no—
250C22×C3—63no—
251C22×C3—63no—
252C22×C3—63no—
253C22×C3—69no—
254C22×C3—69no—
255C22×C3—69no—
256C22×C3—69no—
257C22×C3—129no—
258C22×C3C32×#4111yescharacteristic
259C22×C3C22×C3×A413yes—
260A4—33no—
261C3×C4—33no—
262C3×C4—39no—
263C3×C4—31no—
264C3×C4—33no—
265C3×C4—31no—
266C3×C4—33no—
267C3×C4—123no—
268C3×C4—129no—
269C3×C4C22×C3×A413yes—
270C3×C4C22×C3×A411yescharacteristic
271C32—43no—
272C32A4×#4111yescharacteristic, complemented
273C23—23no—
274C23—33no—
275C23—33no—
276C23—63no—
277C23C23×H(3,𝔽3)11yescharacteristic
278Q8—32no—
279Q8—33no—
280Q8C2×C22⋊H(3,𝔽3)11yescharacteristic, complemented
281D4—36no—
282D4—36no—
283D4—33no—
284D4C2×C22⋊H(3,𝔽3)13yescomplemented
285C2×C4—36no—
286C2×C4—33no—
287C2×C4—31no—
288C2×C4—33no—
289C2×C4—33no—
290C2×C4—33no—
291C2×C4—31no—
292C2×C4—33no—
293C2×C4C2×C22⋊H(3,𝔽3)13yescomplemented
294C2×C3—23no—
295C2×C3—33no—
296C2×C3—33no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—63no—
301C2×C3—69no—
302C2×C3—63no—
303C2×C3—123no—
304C2×C3—249no—
305C2×C3C23×C3×A411yescharacteristic
306C22—31no—
307C22—31no—
308C22—33no—
309C22—63no—
310C22—63no—
311C22—63no—
312C22—63no—
313C22#41×H(3,𝔽3)11yescharacteristic, complemented
314C22C22×C22⋊H(3,𝔽3)13yes—
315C4—31no—
316C4—33no—
317C4C22×C22⋊H(3,𝔽3)11yescharacteristic
318C4C22×C22⋊H(3,𝔽3)13yes—
319C3—31no—
320C3—123no—
321C3C3×A4×#4111yescharacteristic
322C2—23no—
323C2—31no—
324C2—31no—
325C2—63no—
326C2C23×C22⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC3×C4C22×C3×A4
commutator, socle, cosocleC23×C3C23×C32
FrattiniC2×C3C23×C3×A4
FittingC22×C32×#41C3
hypercenterC32×#41A4
hypocenterC22#41×H(3,𝔽3)
LCSG ▷ C23×C3 ▷ C22
UCSC1 ◁ C3×C4 ◁ C32×#41

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊H(3,𝔽3)C2semidirect
2. D4×C22⋊H(3,𝔽3)C2semidirect
3. C2×C4×C22⋊H(3,𝔽3)C2semidirect
4. C22×C32×#41C3semidirect
5. C3×A4×#41C3semidirect
14. C22×C32×Q8C2×C3semidirect
15. C22×C32×D4C2×C3semidirect
16. C23×C32×C4C2×C3semidirect
17. C3×Q8×A4C2×C3semidirect
18. C3×D4×A4C2×C3semidirect
19. C2×C3×C4×A4C2×C3semidirect
37. C32×#41A4semidirect
52. C22⋊H(3,𝔽3)#41direct (N × H)
82. C32×Q8C2×A4semidirect
87. C32×D4C2×A4semidirect
97. C2×C32×C4C2×A4semidirect
98. C22×#41H(3,𝔽3)semidirect
151. C22×C32C3×#41semidirect
153. C3×A4C3×#41semidirect
164. C22×Q8C2×H(3,𝔽3)semidirect
165. C22×D4C2×H(3,𝔽3)semidirect
166. C23×C4C2×H(3,𝔽3)semidirect
229. #41C22⋊H(3,𝔽3)direct (N × H)
272. C32A4×#41semidirect
280. Q8C2×C22⋊H(3,𝔽3)semidirect
284. D4C2×C22⋊H(3,𝔽3)semidirect
293. C2×C4C2×C22⋊H(3,𝔽3)semidirect
313. C22#41×H(3,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊H(3,𝔽3)2
D4×C22⋊H(3,𝔽3)2
C2×C4×C22⋊H(3,𝔽3)2
C22×C32×#413
C3×A4×#413
#41×H(3,𝔽3)4