Finite Group Databasebeta

← #13164418

Subgroups — #13164418

Order 1728.

658 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 70 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
2C4⋊((C3⋊S4)⋊C3)C213yesmaximal, complemented
3C4×(C3⋊S4)⋊C3C213yesmaximal, complemented
4#10596582C311yesmaximal, characteristic, complemented
5C3×#5817—31nomaximal
6C2×(C3⋊S4)⋊C3C2213yes—
7#88722—41nomaximal
8C4×C22⋊H(3,𝔽3)C2213yes—
9(C3⋊SL(2,ℤ/4))⋊C3C2211yescharacteristic
10C22×C32×Q8D311yescharacteristic, complemented
11C2×C3×#239—31no—
12C3×Q8×A4—21no—
13C3×Q8×A4C2×C311yescharacteristic, complemented
14C4⋊(C3⋊S4)C2×C313yescomplemented
15C4×C3⋊S4C2×C313yescomplemented
16C3×#947—33no—
17C3×#946—33no—
18C3×Q8×Dic3—31no—
19C3×#930—33no—
20C3×C4×#67—33no—
21C2×C22⋊H(3,𝔽3)C2311yescharacteristic
22(C3⋊S4)⋊C3—23no—
23Q8×H(3,𝔽3)—41no—
24C4⋊((C3⋊D3)⋊C3)—43no—
25C4×(C3⋊D3)⋊C3—43no—
26#5919—91no—
27C3×#548—91no—
28#5817—31no—
29C2×C3⋊S4C22×C313yes—
30C2×C32×Q8—31no—
31C2×C32×Q8—33no—
32C22×C32×C4C2×D313yes—
33#3832—41no—
34C2×C3×#67—33no—
35C3×#239—63no—
36C3×#239—61no—
37C2×C3×D12—33no—
38C2×C3×C4×D3—33no—
39C3×C4×A4—23no—
40C3×C4×A4C22×C313yes—
41C3⋊SL(2,ℤ/4)C22×C311yescharacteristic
42C3×#217—31no—
43C3×(C2×D3)⋊C4—33no—
44C3×(C2×D3)⋊C4—36no—
45C3×C4⋊Dic3—33no—
46C3×Dic3⋊C4—33no—
47C3×C4×Dic3—33no—
48C2×(C3⋊D3)⋊C3—43no—
49C22⋊H(3,𝔽3)#4111yescharacteristic, complemented
50C4×H(3,𝔽3)—43no—
51(C3⋊Dic3)⋊C3—41no—
52C2×C3×#41—91no—
53C22×C3×Q8—31no—
54C22×C3×Q8C3×D311yescharacteristic
55C2×#239—31no—
56Q8×A4—31no—
57Q8×A4—61no—
58C4⋊S4—93no—
59C4×S4—93no—
60C3×#124—93no—
61C3×C22⋊Q8—93no—
62C3×#121—93no—
63C3×C4×Q8—91no—
64C3×C4×D4—93no—
65#947—33no—
66#946—33no—
67Q8×Dic3—31no—
68#930—33no—
69C4×#67—33no—
70C23×C32C22×D311yescharacteristic
71C22×C3×D3—33no—
72C2×C3×A4—21no—
73C2×C3×A4C23×C311yescharacteristic
74C3⋊S4—23no—
75C32×Q8—33no—
76C32×Q8—41no—
77C32×Q8—61no—
78C32×Q8—81no—
79C32×Q8S411yescharacteristic, complemented
80C2×C32×C4—33no—
81C2×C32×C4—33no—
82C4⋊(C3⋊D3)—43no—
83C4×C3⋊D3—43no—
84C3×#67—63no—
85C3×#67—63no—
86C2×C3×Dic3—31no—
87C2×C3×Dic3—33no—
88C2×C3×Dic3—31no—
89C3×D12—63no—
90C3×D12—63no—
91C3×C4×D3—63no—
92C3×C4×D3—63no—
93#548—91noSylow
94C2×H(3,𝔽3)—41no—
95(C3⋊D3)⋊C3—83no—
96C2×S4—93no—
97C3×#41—183no—
98C3×#41—181no—
99C2×C3×Q8—31no—
100C2×C3×Q8—31no—
101C2×C3×Q8—33no—
102C2×C3×Q8—33no—
103C2×C3×Q8—61no—
104C2×C3×Q8—63no—
105C2×C3×D4—93no—
106C2×C3×D4—93no—
107C22×C3×C4—33no—
108C22×C3×C4—93no—
109C22×C3×C4C2×C3×D313yes—
110C2×#67—33no—
111#239—63no—
112#239—61no—
113#239—361no—
114C2×D12—33no—
115C2×C4×D3—33no—
116C4×A4—33no—
117C4×A4—63no—
118SL(2,ℤ/4)—91no—
119C3×C4⋊C4—93no—
120C3×C4⋊C4—93no—
121C3×C22⋊C4—93no—
122C3×C22⋊C4—96no—
123C3×C22⋊C4—91no—
124C3×C42—93no—
125#217—31no—
126(C2×D3)⋊C4—33no—
127(C2×D3)⋊C4—36no—
128C4⋊Dic3—33no—
129Dic3⋊C4—33no—
130C4×Dic3—33no—
131C22×C32—31no—
132C22×C32—31no—
133C22×C32#23911yescharacteristic, complemented
134C2×C3⋊D3—43no—
135C2×C3×D3—63no—
136C2×C3×D3—63no—
137C2×C3×D3—63no—
138C3×A4—21no—
139C3×A4C3×#4111yescharacteristic, complemented
140C32×C4—33no—
141C32×C4—43no—
142C32×C4—83no—
143C32×C4C2×S413yes—
144C3⋊Dic3—41no—
145C3×Dic3—61no—
146C3×Dic3—63no—
147C3×Dic3—61no—
148C2×#41—91no—
149C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
150#124—93no—
151C22⋊Q8—93no—
152#121—93no—
153C4×Q8—91no—
154C4×D4—93no—
155H(3,𝔽3)—41noSylow
156C23×C3—31no—
157C23×C3—93no—
158C23×C3C22×C3×D311yescharacteristic
159C22×D3—33no—
160C2×A4—31no—
161C2×A4—61no—
162S4—183no—
163C3×Q8—33no—
164C3×Q8—33no—
165C3×Q8—31no—
166C3×Q8—63no—
167C3×Q8—61no—
168C3×Q8—63no—
169C3×Q8—121no—
170C3×Q8—241no—
171C3×Q8C3×S411yescharacteristic
172C3×D4—183no—
173C3×D4—183no—
174C3×D4—183no—
175C3×D4—183no—
176C2×C3×C4—33no—
177C2×C3×C4—33no—
178C2×C3×C4—33no—
179C2×C3×C4—33no—
180C2×C3×C4—63no—
181C2×C3×C4—63no—
182C2×C3×C4—91no—
183C2×C3×C4—93no—
184C2×C3×C4—91no—
185C2×C3×C4—183no—
186C2×C3×C4—183no—
187#67—63no—
188#67—63no—
189C2×Dic3—31no—
190C2×Dic3—33no—
191C2×Dic3—31no—
192D12—63no—
193D12—63no—
194D12—363no—
195C4×D3—63no—
196C4×D3—63no—
197C4×D3—363no—
198C2×C32—31no—
199C2×C32—31no—
200C2×C32—41no—
201C2×C32—81no—
202C2×C32C22×S411yescharacteristic
203C3⋊D3—83no—
204C3×D3—123no—
205#41—183no—
206#41—181no—
207C2×Q8—31no—
208C2×Q8—33no—
209C2×D4—93no—
210C2×D4—93no—
211C22×C4—93no—
212C22×C4C2×(C3⋊D3)⋊C313yes—
213C4⋊C4—93no—
214C4⋊C4—93no—
215C22⋊C4—93no—
216C22⋊C4—96no—
217C22⋊C4—91no—
218C42—93no—
219C22×C3—31no—
220C22×C3—31no—
221C22×C3—31no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—61no—
225C22×C3—61no—
226C22×C3—183no—
227C22×C3—183no—
228C22×C3—183no—
229C22×C3C3×#23911yescharacteristic
230C2×D3—63no—
231C2×D3—63no—
232C2×D3—63no—
233C2×D3—363no—
234A4—31no—
235A4—61no—
236C3×C4—33no—
237C3×C4—33no—
238C3×C4—33no—
239C3×C4—63no—
240C3×C4—123no—
241C3×C4—181no—
242C3×C4—183no—
243C3×C4—181no—
244C3×C4—243no—
245C3×C4C2×C3×S413yes—
246Dic3—63no—
247Dic3—61no—
248Dic3—61no—
249Dic3—361no—
250C32—41no—
251C32—81no—
252C32#591911yescharacteristic, complemented
253C23—93no—
254C23C22×(C3⋊D3)⋊C311yescharacteristic
255Q8—33no—
256Q8—61no—
257Q8(C3⋊S4)⋊C311yescharacteristic, complemented
258D4—183no—
259D4—183no—
260D4—183no—
261D4—183no—
262C2×C4—33no—
263C2×C4—33no—
264C2×C4—91no—
265C2×C4—93no—
266C2×C4—91no—
267C2×C4—183no—
268C2×C4—183no—
269C2×C3—31no—
270C2×C3—31no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—61no—
275C2×C3—61no—
276C2×C3—121no—
277C2×C3—241no—
278C2×C3—363no—
279C2×C3C22×C3×S411yescharacteristic
280D3—123no—
281D3—723no—
282C22—31no—
283C22—31no—
284C22—183no—
285C22—183no—
286C22—183no—
287C22#8872211yescharacteristic, complemented
288C4—33no—
289C4—183no—
290C4—181no—
291C4—181no—
292C4C2×(C3⋊S4)⋊C313yes—
293C3—31no—
294C3—121no—
295C3—241no—
296C3C3×#591911yescharacteristic
297C2—31no—
298C2—31no—
299C2—363no—
300C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
centerC2C22×(C3⋊S4)⋊C3
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×Q8D3
socleC23×C3C22×C3×D3
hypercenterQ8(C3⋊S4)⋊C3
hypocenterC3×A4C3×#41
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊H(3,𝔽3)C2semidirect
2. C4⋊((C3⋊S4)⋊C3)C2semidirect
3. C4×(C3⋊S4)⋊C3C2semidirect
4. #10596582C3semidirect
10. C22×C32×Q8D3semidirect
13. C3×Q8×A4C2×C3semidirect
14. C4⋊(C3⋊S4)C2×C3semidirect
15. C4×C3⋊S4C2×C3semidirect
49. C22⋊H(3,𝔽3)#41semidirect
79. C32×Q8S4semidirect
133. C22×C32#239semidirect
139. C3×A4C3×#41semidirect
149. C22×Q8(C3⋊D3)⋊C3semidirect
252. C32#5919semidirect
257. Q8(C3⋊S4)⋊C3semidirect
287. C22#88722semidirect

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
C4⋊((C3⋊S4)⋊C3)2
C4×(C3⋊S4)⋊C32
#105965823
C3×#58173
#887224