Finite Group Databasebeta

← #10596205⋊C3 #13153330

Subgroups — #10596205⋊C3 #13153330

Order 1728.

894 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 222 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
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64156⋊C3C214yesmaximal, complemented
3#64104⋊C3C214yesmaximal, complemented
4#64101⋊C3C214yesmaximal, complemented
5C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
6#10596205C311yesmaximal, characteristic, complemented
7C3×#5734—31nomaximal
8C2×Q8×H(3,𝔽3)C2212yes—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
11C2×C4×(C3⋊D3)⋊C3C2214yes—
12C4⋊C4×H(3,𝔽3)C2214yes—
13C42×H(3,𝔽3)C2211yescharacteristic
14#3752⋊C3C2214yes—
15#3752⋊C3C2218yes—
16C4⋊((C3⋊Dic3)⋊C3)C2212yes—
17C4×(C3⋊Dic3)⋊C3C2214yes—
18C32×C4⋊Q8—21no—
19C32×C4⋊Q8D311yescharacteristic, complemented
20C32×C4⋊Q8C2×C311yescharacteristic, complemented
21#64156C2×C314yescomplemented
22#64104C2×C314yescomplemented
23#64101C2×C314yescomplemented
24C4×C4⋊(C3⋊D3)C2×C312yescomplemented
25C3×#947—34no—
26C3×#895—34no—
27C3×#892—34no—
28C3×C4×D12—32no—
29C22×(C3⋊D3)⋊C3C2314yes—
30Q8×H(3,𝔽3)—42no—
31C2×C4×H(3,𝔽3)C2314yes—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×C4×H(3,𝔽3)C2312yes—
34C2×(C3⋊Dic3)⋊C3C2314yes—
35C4⋊((C3⋊D3)⋊C3)—24no—
36C4⋊((C3⋊D3)⋊C3)—44no—
37C4×(C3⋊D3)⋊C3—28no—
38C3×#555—91no—
39#5734—31no—
40#5734—91no—
41C2×C32×Q8—22no—
42C2×C32×Q8C2×D312yes—
43C2×C32×Q8C22×C312yes—
44C2×C4⋊(C3⋊D3)C22×C312yes—
45C2×C4⋊(C3⋊D3)C22×C314yes—
46C2×C4×C3⋊D3C22×C314yes—
47C2×C3×D12—32no—
48C2×C3×D12—34no—
49C2×C3×C4×D3—34no—
50C32×C4⋊C4—24no—
51C32×C4⋊C4C2×D314yes—
52C32×C4⋊C4C22×C314yes—
53C32×C42—21no—
54C32×C42C2×D311yescharacteristic
55C32×C42C22×C311yescharacteristic
56#3752C22×C314yes—
57#3752C22×C318yes—
58C4⋊(C3⋊Dic3)C22×C312yes—
59C4×C3⋊Dic3C22×C314yes—
60C3×(C2×D3)⋊C4—38no—
61C3×(C2×D3)⋊C4—34no—
62C3×C4⋊Dic3—32no—
63C3×C4×Dic3—34no—
64C22×H(3,𝔽3)C2411yescharacteristic
65C2×(C3⋊D3)⋊C3—24no—
66C2×(C3⋊D3)⋊C3—24no—
67C2×(C3⋊D3)⋊C3—24no—
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)—24no—
70C4×H(3,𝔽3)#4114yes—
71(C3⋊Dic3)⋊C3—24no—
72C3×C4⋊Q8—31no—
73C3×C4⋊Q8—31no—
74C3×C4⋊Q8—61no—
75C3×C4⋊Q8C3×D311yescharacteristic
76C3×#124—94no—
77C3×#121—94no—
78C3×C4×D4—92no—
79C3×#117—94no—
80#947—34no—
81#947—94no—
82#895—34no—
83#895—94no—
84#892—34no—
85#892—94no—
86C4×D12—32no—
87C4×D12—92no—
88C22×C3⋊D3C23×C314yes—
89C22×C3×D3—34no—
90C32×Q8—42no—
91C32×Q8—42no—
92C32×Q8—44no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—24no—
96C2×C32×C4C22×D314yes—
97C2×C32×C4C22×D311yescharacteristic
98C2×C32×C4C22×D312yes—
99C2×C32×C4C23×C314yes—
100C2×C32×C4C23×C311yescharacteristic
101C2×C32×C4C23×C312yes—
102C2×C3⋊Dic3C23×C314yes—
103C4⋊(C3⋊D3)—24no—
104C4⋊(C3⋊D3)—44no—
105C4×C3⋊D3—28no—
106C2×C3×Dic3—34no—
107C3×D12—64no—
108C3×D12—124no—
109C3×C4×D3—68no—
110#555—91noSylow
111C2×H(3,𝔽3)C2×#4112yes—
112C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
113(C3⋊D3)⋊C3—44no—
114C2×C3×Q8—32no—
115C2×C3×Q8—32no—
116C2×C3×Q8—62no—
117C2×C3×Q8C2×C3×D312yes—
118C2×C3×D4—92no—
119C2×C3×D4—94no—
120C22×C3×C4—94no—
121C2×D12—32no—
122C2×D12—34no—
123C2×D12—92no—
124C2×D12—94no—
125C2×C4×D3—34no—
126C2×C4×D3—94no—
127C3×C4⋊C4—34no—
128C3×C4⋊C4—34no—
129C3×C4⋊C4—64no—
130C3×C4⋊C4—92no—
131C3×C4⋊C4C2×C3×D314yes—
132C3×C22⋊C4—98no—
133C3×C22⋊C4—94no—
134C3×C42—31no—
135C3×C42—31no—
136C3×C42—61no—
137C3×C42—94no—
138C3×C42C2×C3×D311yescharacteristic
139(C2×D3)⋊C4—34no—
140(C2×D3)⋊C4—38no—
141(C2×D3)⋊C4—94no—
142(C2×D3)⋊C4—98no—
143C4⋊Dic3—32no—
144C4⋊Dic3—92no—
145C4×Dic3—34no—
146C4×Dic3—94no—
147C22×C32—21no—
148C22×C32C23×D311yescharacteristic
149C22×C32C24×C311yescharacteristic
150C2×C3⋊D3—24no—
151C2×C3⋊D3—24no—
152C2×C3⋊D3—24no—
153C2×C3×D3—64no—
154C2×C3×D3—64no—
155C2×C3×D3—64no—
156C32×C4—22no—
157C32×C4—24no—
158C32×C4—21no—
159C32×C4—24no—
160C32×C4—21no—
161C32×C4—24no—
162C32×C4—44no—
163C32×C4#23914yes—
164C32×C4C3×#4114yes—
165C3⋊Dic3—24no—
166C3×Dic3—64no—
167C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
168#124—94no—
169#121—94no—
170C4×D4—92no—
171#117—94no—
172H(3,𝔽3)#55511yescharacteristic, complemented, Sylow
173C23×C3—94no—
174C22×D3—34no—
175C22×D3—94no—
176C3×Q8—42no—
177C3×Q8—122no—
178C3×Q8—122no—
179C3×Q8—124no—
180C3×D4—184no—
181C3×D4—364no—
182C2×C3×C4—32no—
183C2×C3×C4—32no—
184C2×C3×C4—31no—
185C2×C3×C4—34no—
186C2×C3×C4—34no—
187C2×C3×C4—31no—
188C2×C3×C4—62no—
189C2×C3×C4—61no—
190C2×C3×C4—64no—
191C2×C3×C4—94no—
192C2×C3×C4—188no—
193C2×C3×C4C22×C3×D314yes—
194C2×C3×C4C22×C3×D311yescharacteristic
195C2×C3×C4C22×C3×D312yes—
196C2×Dic3—34no—
197C2×Dic3—94no—
198D12—64no—
199D12—124no—
200D12—184no—
201D12—364no—
202C4×D3—68no—
203C4×D3—188no—
204C2×C32—21no—
205C2×C32—22no—
206C2×C32#100411yescharacteristic
207C2×C32C2×#23912yes—
208C2×C32C2×C3×#4112yes—
209C2×C32C3×H(5,𝔽2)11yescharacteristic
210C3⋊D3—44no—
211C3×D3—124no—
212C2×Q8C2×(C3⋊D3)⋊C312yes—
213C2×D4—92no—
214C2×D4—94no—
215C22×C4—94no—
216C4⋊C4—92no—
217C4⋊C4C2×(C3⋊D3)⋊C314yes—
218C22⋊C4—98no—
219C22⋊C4—94no—
220C42—94no—
221C42C2×(C3⋊D3)⋊C311yescharacteristic
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—61no—
225C22×C3—184no—
226C22×C3—184no—
227C22×C3—184no—
228C22×C3C23×C3×D311yescharacteristic
229C2×D3—64no—
230C2×D3—64no—
231C2×D3—64no—
232C2×D3—184no—
233C2×D3—184no—
234C2×D3—184no—
235C3×C4—24no—
236C3×C4—21no—
237C3×C4—34no—
238C3×C4—34no—
239C3×C4—64no—
240C3×C4—64no—
241C3×C4—61no—
242C3×C4—61no—
243C3×C4—62no—
244C3×C4—64no—
245C3×C4—124no—
246C3×C4—184no—
247C3×C4C3×#23914yes—
248Dic3—64no—
249Dic3—184no—
250C32—21no—
251C32#573411yescharacteristic, complemented
252C32C3×#55511yescharacteristic, complemented
253C23—94no—
254Q8—42no—
255D4—184no—
256D4—364no—
257C2×C4—94no—
258C2×C4—188no—
259C2×C4C22×(C3⋊D3)⋊C314yes—
260C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
261C2×C4C22×(C3⋊D3)⋊C312yes—
262C2×C3—31no—
263C2×C3—31no—
264C2×C3—32no—
265C2×C3—32no—
266C2×C3—61no—
267C2×C3—62no—
268C2×C3—364no—
269C2×C3C3×#100411yescharacteristic
270C2×C3C2×C3×#23912yes—
271D3—124no—
272D3—364no—
273C22—184no—
274C22—184no—
275C22—184no—
276C22C23×(C3⋊D3)⋊C311yescharacteristic
277C4—24no—
278C4—21no—
279C4—184no—
280C4#8872214yes—
281C3—31no—
282C3—31no—
283C3—61no—
284C3C3×#573411yescharacteristic
285C2—364no—
286C2#64361⋊C311yescharacteristic
287C2C2×#8872212yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#555
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C4⋊Q8C2semidirect
2. #64156⋊C3C2semidirect
3. #64104⋊C3C2semidirect
4. #64101⋊C3C2semidirect
5. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
6. #10596205C3semidirect
19. C32×C4⋊Q8D3semidirect
20. C32×C4⋊Q8C2×C3semidirect
21. #64156C2×C3semidirect
22. #64104C2×C3semidirect
23. #64101C2×C3semidirect
24. C4×C4⋊(C3⋊D3)C2×C3semidirect
167. C4⋊Q8(C3⋊D3)⋊C3semidirect
172. H(3,𝔽3)#555semidirect
251. C32#5734semidirect
252. C32C3×#555semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#64156⋊C32
#64104⋊C32
#64101⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#105962053
C3×#57343