Finite Group Databasebeta

← #10592550⋊C3 #13142048

Subgroups — #10592550⋊C3 #13142048

Order 1728.

374 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 106 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
1#64078⋊C3C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic
4#63644⋊C3C212yesmaximal
5#63643⋊C3C212yesmaximal
6#10592550C311yesmaximal, characteristic, complemented
7C3×#4851—31nomaximal
8C2×#634⋊C3—21no—
9C2×#634⋊C3C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4⋊((C3⋊Dic3)⋊C3)—21no—
13C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#3750⋊C3—21no—
16C4×(C3⋊Dic3)⋊C3—21no—
17#64078C2×C311yescharacteristic
18C4×#634C2×C311yescharacteristic
19C3×#869—31no—
20C3×C4×Dic6—31no—
21C32×C4⋊C8—21no—
22C32×C4⋊C8D311yescharacteristic
23C32×C4⋊C8C2×C311yescharacteristic
24#63644C2×C312yes—
25#63643C2×C312yes—
26C3×#817—32no—
27C3×#816—32no—
28C2×C4×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×(C3⋊Dic3)⋊C3—21no—
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3—21no—
34#634⋊C3—21no—
35#634⋊C3—42no—
36#634⋊C3Q812yescomplemented
37C8×H(3,𝔽3)—22no—
38C3×#475—91no—
39#4851—31no—
40#4851—91no—
41C2×#634—21no—
42C2×#634C22×C311yescharacteristic
43C2×C3×Dic6—31no—
44C2×C3×Dic6—61no—
45C2×C32×C8—22no—
46C2×C32×C8C2×D312yes—
47C2×C32×C8C22×C312yes—
48C32×C42—21no—
49C32×C42C2×D311yescharacteristic
50C32×C42C22×C311yescharacteristic
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)C22×C311yescharacteristic
53C4⋊(C3⋊Dic3)C22×C312yes—
54#3750—21no—
55C4×C3⋊Dic3—21no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—61no—
59C3×Dic3⋊C4—61no—
60C3×C4×Dic3—61no—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C4×H(3,𝔽3)—21no—
63C4×H(3,𝔽3)SD1612yes—
64C4×H(3,𝔽3)C2×Q811yescharacteristic
65C4×H(3,𝔽3)#4111yescharacteristic
66(C3⋊Dic3)⋊C3—22no—
67(C3⋊Dic3)⋊C3—42no—
68(C3⋊Dic3)⋊C3—41no—
69C3×C4⋊Q8—91no—
70C3×C4×Q8—91no—
71#869—31no—
72#869—91no—
73C4×Dic6—31no—
74C4×Dic6—91no—
75C3×#106—92no—
76C3×C4⋊C8—31no—
77C3×C4⋊C8—31no—
78C3×C4⋊C8—61no—
79C3×C4⋊C8C3×D311yescharacteristic
80C3×#103—92no—
81#817—32no—
82#817—92no—
83#816—32no—
84#816—92no—
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4D1211yescharacteristic
89C2×C32×C4D1211yescharacteristic
90C2×C32×C4C3×D411yescharacteristic
91C2×C32×C4C3×D411yescharacteristic
92C2×C32×C4C22×D311yescharacteristic
93C2×C32×C4C23×C311yescharacteristic
94C2×C3⋊Dic3—21no—
95C2×C3⋊Dic3—21no—
96C2×C3⋊Dic3—22no—
97#634—21no—
98#634—42no—
99#634C3×Q812yescomplemented
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—62no—
103C3×Dic6—32no—
104C3×Dic6—61no—
105C3×Dic6—122no—
106C32×C8—22no—
107C32×C8—22no—
108C32×C8—42no—
109#475—91noSylow
110C2×H(3,𝔽3)C22⋊Q811yescharacteristic
111C2×H(3,𝔽3)C2×SD1611yescharacteristic
112C2×H(3,𝔽3)#13711yescharacteristic
113C2×C3×Q8—91no—
114C2×C3×Q8—181no—
115C2×Dic6—31no—
116C2×Dic6—61no—
117C2×Dic6—91no—
118C2×Dic6—181no—
119C2×C3×C8—32no—
120C2×C3×C8—32no—
121C2×C3×C8—62no—
122C2×C3×C8C2×C3×D312yes—
123C3×C4⋊C4—92no—
124C3×C4⋊C4—91no—
125C3×C4⋊C4—181no—
126C3×C4⋊C4—181no—
127C3×C42—31no—
128C3×C42—31no—
129C3×C42—61no—
130C3×C42—181no—
131C3×C42C2×C3×D311yescharacteristic
132C4⋊Dic3—32no—
133C4⋊Dic3—31no—
134C4⋊Dic3—61no—
135C4⋊Dic3—92no—
136C4⋊Dic3—91no—
137C4⋊Dic3—181no—
138Dic3⋊C4—61no—
139Dic3⋊C4—181no—
140C4×Dic3—61no—
141C4×Dic3—181no—
142C22×C32—21no—
143C22×C32C2×D1211yescharacteristic
144C22×C32C2×C3×D411yescharacteristic
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—22no—
148C32×C4—21no—
149C32×C4—21no—
150C32×C4—41no—
151C32×C4#20412yes—
152C32×C4C3×SD1612yes—
153C32×C4#23711yescharacteristic
154C32×C4D3×Q811yescharacteristic
155C32×C4C2×C3×Q811yescharacteristic
156C32×C4C3×#4111yescharacteristic
157C3⋊Dic3—22no—
158C3⋊Dic3—42no—
159C3⋊Dic3—41no—
160C3×Dic3—62no—
161C3×Dic3—122no—
162C3×Dic3—121no—
163C4⋊Q8—91no—
164C4×Q8—91no—
165#106—92no—
166C4⋊C8(C3⋊D3)⋊C311yescharacteristic
167#103—92no—
168H(3,𝔽3)#47511yescharacteristic, complemented, Sylow
169C3×Q8—92no—
170C3×Q8—181no—
171C3×Q8—362no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—181no—
182C2×C3×C4—181no—
183C2×C3×C4—182no—
184C2×C3×C4C3×D1211yescharacteristic
185C2×C3×C4C3×D1211yescharacteristic
186C2×C3×C4C22×C3×D311yescharacteristic
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3—62no—
190C2×Dic3—181no—
191C2×Dic3—181no—
192C2×Dic3—182no—
193Dic6—32no—
194Dic6—61no—
195Dic6—92no—
196Dic6—122no—
197Dic6—181no—
198Dic6—362no—
199C3×C8—22no—
200C3×C8—62no—
201C3×C8—62no—
202C3×C8—122no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32#89711yescharacteristic
207C2×C32C2×#20411yescharacteristic
208C2×C32#90911yescharacteristic
209C2×C32C3×C22⋊Q811yescharacteristic
210C2×C32C2×C3×SD1611yescharacteristic
211C2×C32C3×#13711yescharacteristic
212C2×Q8—91no—
213C2×Q8—181no—
214C2×C8C2×(C3⋊D3)⋊C312yes—
215C4⋊C4—92no—
216C4⋊C4—91no—
217C4⋊C4—181no—
218C4⋊C4—181no—
219C42—181no—
220C42C2×(C3⋊D3)⋊C311yescharacteristic
221C22×C3—31no—
222C22×C3—31no—
223C22×C3—61no—
224C22×C3C2×C3×D1211yescharacteristic
225C3×C4—21no—
226C3×C4—31no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—32no—
231C3×C4—32no—
232C3×C4—61no—
233C3×C4—61no—
234C3×C4—61no—
235C3×C4—61no—
236C3×C4—62no—
237C3×C4—121no—
238C3×C4—182no—
239C3×C4—362no—
240C3×C4—361no—
241C3×C4C3×#20412yes—
242C3×C4C3×#23711yescharacteristic
243C3×C4C3×D3×Q811yescharacteristic
244Dic3—62no—
245Dic3—122no—
246Dic3—121no—
247Dic3—182no—
248Dic3—362no—
249Dic3—361no—
250C32—21no—
251C32#485111yescharacteristic, complemented
252C32C3×#47511yescharacteristic, complemented
253Q8—92no—
254Q8—181no—
255Q8—362no—
256C2×C4—181no—
257C2×C4—181no—
258C2×C4—182no—
259C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
260C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
261C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
262C8—22no—
263C2×C3—31no—
264C2×C3—31no—
265C2×C3—31no—
266C2×C3—31no—
267C2×C3—31no—
268C2×C3—31no—
269C2×C3—61no—
270C2×C3—61no—
271C2×C3—61no—
272C2×C3C3×#89711yescharacteristic
273C2×C3C2×C3×#20411yescharacteristic
274C2×C3C3×#90911yescharacteristic
275C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
276C4—21no—
277C4—182no—
278C4—362no—
279C4—361no—
280C4#3744⋊C312yes—
281C4#3830⋊C311yescharacteristic
282C4Q8×(C3⋊D3)⋊C311yescharacteristic
283C3—31no—
284C3—31no—
285C3—61no—
286C3C3×#485111yescharacteristic
287C2#64106⋊C311yescharacteristic
288C2C2×#3744⋊C311yescharacteristic
289C2#64118⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#475
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
6. #10592550C3semidirect
36. #634⋊C3Q8semidirect
99. #634C3×Q8semidirect
168. H(3,𝔽3)#475semidirect
251. C32#4851semidirect
252. C32C3×#475semidirect

Maximal subgroups

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

subgroupindex
#64078⋊C32
C4×#634⋊C32
H(3,𝔽3)×C4⋊C82
#63644⋊C32
#63643⋊C32
#105925503
C3×#48513