Finite Group Databasebeta

← C2×2−1+4⋊H(3,𝔽3) #13164862

Subgroups — C2×2−1+4⋊H(3,𝔽3) #13164862

Order 1728.

1181 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1181 conjugacy classes of subgroups — too many to draw. Only its 169 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
12−1+4⋊H(3,𝔽3)C214yesmaximal, direct product factor
2C2×#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C22×Q8⋊H(3,𝔽3)C212yesmaximal, complemented
4C2×C32×2−1+4C311yesmaximal, characteristic, complemented
5C2×C3×2−1+4⋊C3C313yesmaximal, complemented
6C2×D4×H(3,𝔽3)—41nomaximal
7#41⋊H(3,𝔽3)C2212yescomplemented
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
9C2×Q8⋊H(3,𝔽3)C2214yescomplemented
10C32×2−1+4—34no—
11C32×2−1+4C2×C314yescomplemented
12C2×C32×#41—31no—
13C2×C32×#41—32no—
14C2×C32×#41C2×C311yescharacteristic, complemented
15C22×C32×Q8—31no—
16C22×C32×Q8C2×C312yescomplemented
17C3×2−1+4⋊C3C2×C3112yescomplemented
18C2×C3×#41⋊C3C2×C313yescomplemented
19C22×C3×SL(2,𝔽3)C2×C316yescomplemented
20C23×H(3,𝔽3)—42no—
21D4×H(3,𝔽3)—44no—
22C2×C4×H(3,𝔽3)—41no—
23Q8⋊H(3,𝔽3)C2311yescharacteristic
24C2×C3×2−1+4—31no—
25C2×C3×2−1+4C3211yescharacteristic
26C2×2−1+4⋊C3—33no—
27C32×#41—38no—
28C32×#41—34no—
29C32×#41—34no—
30C32×#41—38no—
31C32×#41—32no—
32C32×#41C22×C312yescomplemented
33C2×C32×Q8—34no—
34C2×C32×Q8—34no—
35C2×C32×Q8—31no—
36C2×C32×Q8—32no—
37C2×C32×Q8—34no—
38C2×C32×Q8C22×C311yescharacteristic
39C2×C32×Q8C22×C314yescomplemented
40C2×C32×D4—32no—
41C2×C32×D4—31no—
42C2×C32×D4—43no—
43C2×C32×D4A411yescharacteristic
44C22×C32×C4—32no—
45C22×C32×C4—32no—
46C22×C32×C4—31no—
47C3×#41⋊C3C22×C316yescomplemented
48C2×C3×SL(2,𝔽3)C22×C313yes—
49C2×C3×SL(2,𝔽3)C22×C3112yescomplemented
50C22×H(3,𝔽3)—41no—
51C22×H(3,𝔽3)—44no—
52C22×H(3,𝔽3)—84no—
53C4×H(3,𝔽3)—42no—
54C3×2−1+4—34no—
55C3×2−1+4—312no—
56C3×2−1+4—34no—
57C3×2−1+4C2×C3214yes—
58C2×C3×#41—33no—
59C2×C3×#41—36no—
60C2×C3×#41—31no—
61C2×C3×#41—32no—
62C2×C3×#41—31no—
63C2×C3×#41C2×C3211yescharacteristic
64C22×C3×Q8—31no—
65C22×C3×Q8—33no—
66C22×C3×Q8—32no—
67C22×C3×Q8C2×C3212yes—
682−1+4⋊C3—312no—
69C2×#41⋊C3—33no—
70C22×SL(2,𝔽3)—36no—
71C23×C32—31no—
72C23×C32—46no—
73C23×C32C2×A412yes—
74C32×Q8—31no—
75C32×Q8—34no—
76C32×Q8—34no—
77C32×Q8—34no—
78C32×Q8C23×C311yescharacteristic
79C32×D4—32no—
80C32×D4—32no—
81C32×D4—38no—
82C32×D4—412no—
83C32×D4C2×A414yes—
84C2×C32×C4—38no—
85C2×C32×C4—32no—
86C2×C32×C4—31no—
87C2×C32×C4—34no—
88C2×C32×C4—34no—
89C2×C32×C4—32no—
90C2×C32×C4—32no—
91C2×C32×C4—43no—
92C2×C32×C4C2×A411yescharacteristic
93C3×SL(2,𝔽3)C23×C313yes—
94C2×2−1+4H(3,𝔽3)11yescharacteristic, complemented, Sylow
95C2×H(3,𝔽3)—42no—
96C2×H(3,𝔽3)—41no—
97C2×H(3,𝔽3)—84no—
98C3×#41—324no—
99C3×#41—38no—
100C3×#41—312no—
101C3×#41—312no—
102C3×#41—324no—
103C3×#41—36no—
104C3×#41—34no—
105C3×#41—34no—
106C3×#41—38no—
107C3×#41—32no—
108C3×#41—32no—
109C3×#41C22×C3212yes—
110C2×C3×Q8—34no—
111C2×C3×Q8—33no—
112C2×C3×Q8—36no—
113C2×C3×Q8—312no—
114C2×C3×Q8—312no—
115C2×C3×Q8—34no—
116C2×C3×Q8—312no—
117C2×C3×Q8—31no—
118C2×C3×Q8—32no—
119C2×C3×Q8—34no—
120C2×C3×Q8—31no—
121C2×C3×Q8—34no—
122C2×C3×Q8C22×C3211yescharacteristic
123C2×C3×Q8C22×C3214yes—
124C2×C3×D4—31no—
125C2×C3×D4—33no—
126C2×C3×D4—36no—
127C2×C3×D4—32no—
128C2×C3×D4—31no—
129C2×C3×D4—123no—
130C2×C3×D4C3×A411yescharacteristic
131C22×C3×C4—36no—
132C22×C3×C4—32no—
133C22×C3×C4—36no—
134C22×C3×C4—33no—
135C22×C3×C4—32no—
136C22×C3×C4—31no—
137#41⋊C3—36no—
138C2×SL(2,𝔽3)—33no—
139C2×SL(2,𝔽3)—312no—
140C22×C32—24no—
141C22×C32—32no—
142C22×C32—412no—
143C22×C32—43no—
144C22×C32—62no—
145C22×C32—812no—
146C22×C32C22×A414yes—
147C22×C32C22×A411yescharacteristic
148C32×C4—34no—
149C32×C4—31no—
150C32×C4—31no—
151C32×C4—46no—
152C32×C4C22×A412yes—
1532−1+4—34no—
1542−1+4C2×H(3,𝔽3)14yescomplemented
155C2×#41—31no—
156C2×#41—32no—
157C2×#41C2×H(3,𝔽3)11yescharacteristic, complemented
158C22×Q8—31no—
159C22×Q8C2×H(3,𝔽3)12yescomplemented
160H(3,𝔽3)—41noSylow
161C23×C3—33no—
162C23×C3—31no—
163C23×C3—32no—
164C23×C3—126no—
165C23×C3C2×C3×A412yes—
166C3×Q8—31no—
167C3×Q8—34no—
168C3×Q8—33no—
169C3×Q8—312no—
170C3×Q8—312no—
171C3×Q8—34no—
172C3×Q8—312no—
173C3×Q8—34no—
174C3×Q8—31no—
175C3×Q8C23×C3211yescharacteristic
176C3×D4—32no—
177C3×D4—36no—
178C3×D4—36no—
179C3×D4—32no—
180C3×D4—324no—
181C3×D4—38no—
182C3×D4—34no—
183C3×D4—1212no—
184C3×D4C2×C3×A414yes—
185C2×C3×C4—33no—
186C2×C3×C4—36no—
187C2×C3×C4—324no—
188C2×C3×C4—38no—
189C2×C3×C4—312no—
190C2×C3×C4—312no—
191C2×C3×C4—36no—
192C2×C3×C4—36no—
193C2×C3×C4—34no—
194C2×C3×C4—34no—
195C2×C3×C4—32no—
196C2×C3×C4—31no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—31no—
200C2×C3×C4—123no—
201C2×C3×C4C2×C3×A411yescharacteristic
202SL(2,𝔽3)—33no—
203C2×C32—24no—
204C2×C32—46no—
205C2×C32—43no—
206C2×C32—62no—
207C2×C32—812no—
208C2×C322−1+4⋊C312yescomplemented
209C2×C32C23×A411yescharacteristic
210#41—38no—
211#41—34no—
212#41—34no—
213#41—38no—
214#41—32no—
215#41C22×H(3,𝔽3)12yescomplemented
216C2×Q8—34no—
217C2×Q8—34no—
218C2×Q8—31no—
219C2×Q8—32no—
220C2×Q8—34no—
221C2×Q8C22×H(3,𝔽3)11yescharacteristic
222C2×Q8C22×H(3,𝔽3)14yescomplemented
223C2×D4—31no—
224C2×D4—32no—
225C2×D4C22⋊H(3,𝔽3)11yescharacteristic
226C22×C4—32no—
227C22×C4—32no—
228C22×C4—31no—
229C22×C3—24no—
230C22×C3—36no—
231C22×C3—32no—
232C22×C3—31no—
233C22×C3—34no—
234C22×C3—64no—
235C22×C3—66no—
236C22×C3—62no—
237C22×C3—123no—
238C22×C3—1212no—
239C22×C3—2412no—
240C22×C3C22×C3×A414yes—
241C22×C3C22×C3×A411yescharacteristic
242C3×C4—33no—
243C3×C4—312no—
244C3×C4—33no—
245C3×C4—31no—
246C3×C4—34no—
247C3×C4—31no—
248C3×C4—32no—
249C3×C4—126no—
250C3×C4C22×C3×A412yes—
251C32—43no—
252C32C2×2−1+4⋊C311yescharacteristic, complemented
253C23—31no—
254C23C2×C22⋊H(3,𝔽3)12yes—
255Q8—31no—
256Q8—34no—
257Q8—34no—
258Q8—34no—
259Q8C23×H(3,𝔽3)11yescharacteristic
260D4—32no—
261D4—32no—
262D4—38no—
263D4C2×C22⋊H(3,𝔽3)14yes—
264C2×C4—38no—
265C2×C4—31no—
266C2×C4—32no—
267C2×C4—34no—
268C2×C4—34no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4C2×C22⋊H(3,𝔽3)11yescharacteristic
272C2×C3—24no—
273C2×C3—31no—
274C2×C3—32no—
275C2×C3—64no—
276C2×C3—66no—
277C2×C3—62no—
278C2×C3—123no—
279C2×C3—126no—
280C2×C3—2412no—
281C2×C3C3×2−1+4⋊C312yes—
282C2×C3C23×C3×A411yescharacteristic
283C22—24no—
284C22—32no—
285C22—62no—
286C22C22×C22⋊H(3,𝔽3)14yes—
287C22C22×C22⋊H(3,𝔽3)11yescharacteristic
288C4—31no—
289C4—34no—
290C4—31no—
291C4C22×C22⋊H(3,𝔽3)12yes—
292C3—31no—
293C3—123no—
294C3C2×C3×2−1+4⋊C311yescharacteristic
295C2—24no—
296C2—62no—
297C22−1+4⋊H(3,𝔽3)12yesdirect product factor
298C2C23×C22⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×C3×A4
commutator, cosocleC3×Q8C23×C32
FrattiniC2×C3C23×C3×A4
FittingC2×C32×2−1+4C3
hypercenterC2×C32×D4A4
hypocenterQ8C23×H(3,𝔽3)
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C2×C32×D4

Complements

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

NH ≅ G/Ntype
1. 2−1+4⋊H(3,𝔽3)C2direct (N × H)
2. C2×#41⋊H(3,𝔽3)C2semidirect
3. C22×Q8⋊H(3,𝔽3)C2semidirect
4. C2×C32×2−1+4C3semidirect
5. C2×C3×2−1+4⋊C3C3semidirect
7. #41⋊H(3,𝔽3)C22semidirect
9. C2×Q8⋊H(3,𝔽3)C22semidirect
11. C32×2−1+4C2×C3semidirect
14. C2×C32×#41C2×C3semidirect
16. C22×C32×Q8C2×C3semidirect
17. C3×2−1+4⋊C3C2×C3semidirect
18. C2×C3×#41⋊C3C2×C3semidirect
19. C22×C3×SL(2,𝔽3)C2×C3semidirect
32. C32×#41C22×C3semidirect
39. C2×C32×Q8C22×C3semidirect
47. C3×#41⋊C3C22×C3semidirect
49. C2×C3×SL(2,𝔽3)C22×C3semidirect
94. C2×2−1+4H(3,𝔽3)semidirect
154. 2−1+4C2×H(3,𝔽3)semidirect
157. C2×#41C2×H(3,𝔽3)semidirect
159. C22×Q8C2×H(3,𝔽3)semidirect
208. C2×C322−1+4⋊C3semidirect
215. #41C22×H(3,𝔽3)semidirect
222. C2×Q8C22×H(3,𝔽3)semidirect
252. C32C2×2−1+4⋊C3semidirect
297. C22−1+4⋊H(3,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
2−1+4⋊H(3,𝔽3)2
C2×#41⋊H(3,𝔽3)2
C22×Q8⋊H(3,𝔽3)2
C2×C32×2−1+43
C2×C3×2−1+4⋊C33
C2×D4×H(3,𝔽3)4