Finite Group Databasebeta

← #13154301

Subgroups — #13154301

Order 1728.

1166 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 219 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#11721864C212yesmaximal
3Q8×#1084C212yesmaximal, complemented
4#11721766C214yesmaximal, complemented
5#11721765C212yesmaximal, complemented
6#11721760C212yesmaximal, complemented
7C4×C4⋊#272C211yesmaximal, characteristic
8C4×#6229C211yesmaximal, characteristic, complemented
9C3×#5732—34nomaximal
10C2×Q8×H(3,𝔽3)C2212yes—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2212yes—
13C2×#6229C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2212yes—
15C4⋊C4×H(3,𝔽3)C2212yes—
16C42×H(3,𝔽3)C2211yescharacteristic
17#88542C2214yes—
18#88542C2214yes—
19#88542C2212yes—
20C4⋊#1084C2211yescharacteristic
21C4⋊#1084C2214yes—
22C4⋊#1084C2211yescharacteristic
23#88540C2212yes—
24C4×#1084C2214yes—
25C4×#1084C2212yes—
26C32×C4⋊Q8D314yescomplemented
27C3×#947—38no—
28C3×Q8×Dic3—38no—
29C3×#898—316no—
30C3×#897—38no—
31C3×#892—38no—
32C3×C4×D12—34no—
33C3×C4×Dic6—34no—
34C22×#272C2312yes—
35Q8×H(3,𝔽3)—24no—
36C2×C4×H(3,𝔽3)C2314yes—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084C2312yes—
41C2×#1084C2312yes—
42C2×#1084C2312yes—
43C4⋊#272—22no—
44C4×#272—24no—
45#6229—22no—
46C3×#556—91no—
47#5732—34no—
48C2×C32×Q8C2×D318yes—
49C2×C3×D12—34no—
50C2×C3×C4×D3—38no—
51C2×C3×Dic6—34no—
52C32×C4⋊C4C2×D318yes—
53C32×C4⋊C4C2×D318yes—
54C32×C42C2×D314yes—
55C3×(C2×D3)⋊C4—316no—
56C3×(C2×D3)⋊C4—316no—
57C3×(C2×D3)⋊C4—38no—
58C3×C4⋊Dic3—34no—
59C3×C4⋊Dic3—316no—
60C3×C4⋊Dic3—34no—
61C3×Dic3⋊C4—38no—
62C3×C4×Dic3—316no—
63C3×C4×Dic3—38no—
64C22×H(3,𝔽3)C2411yescharacteristic
65C2×#272—22no—
66C2×#272—22no—
67C2×#272—22no—
68C4×H(3,𝔽3)—24no—
69C4×H(3,𝔽3)—21no—
70C4×H(3,𝔽3)#4112yes—
71C4×H(3,𝔽3)#4112yes—
72#1084—22no—
73#1084—22no—
74#1084—22no—
75C3×C4⋊Q8—34no—
76C3×C4⋊Q8C3⋊D311yescharacteristic
77C3×#126—94no—
78C3×#124—92no—
79C3×C22⋊Q8—92no—
80C3×C4×Q8—92no—
81C3×C4×Q8—91no—
82C3×C4×D4—91no—
83C3×#117—92no—
84#947—38no—
85Q8×Dic3—38no—
86#898—316no—
87#897—38no—
88#892—38no—
89C4×D12—34no—
90C4×Dic6—34no—
91C22×C3×D3—38no—
92C32×Q8—216no—
93C2×C32×C4C22×D3116yes—
94C2×C32×C4C22×D314yes—
95C2×C32×C4C22×D314yes—
96C2×C32×C4C22×D314yes—
97C2×C3×Dic3—38no—
98C2×C3×Dic3—38no—
99C2×C3×Dic3—38no—
100C3×D12—68no—
101C3×C4×D3—616no—
102C3×Dic6—68no—
103#556—91noSylow
104C2×H(3,𝔽3)C2×#4111yescharacteristic
105C2×H(3,𝔽3)C2×#4111yescharacteristic
106C2×H(3,𝔽3)2−1+411yescharacteristic
107#272—42no—
108C2×C3×Q8—38no—
109C2×C3×Q8—91no—
110C2×C3×Q8C2×C3⋊D312yes—
111C2×C3×D4—91no—
112C22×C3×C4—92no—
113C2×D12—34no—
114C2×C4×D3—38no—
115C2×Dic6—34no—
116C3×C4⋊C4—38no—
117C3×C4⋊C4—38no—
118C3×C4⋊C4—92no—
119C3×C4⋊C4—91no—
120C3×C4⋊C4—91no—
121C3×C4⋊C4—94no—
122C3×C4⋊C4C2×C3⋊D312yes—
123C3×C4⋊C4C2×C3⋊D312yes—
124C3×C22⋊C4—94no—
125C3×C22⋊C4—92no—
126C3×C22⋊C4—94no—
127C3×C42—34no—
128C3×C42—94no—
129C3×C42—92no—
130C3×C42C2×C3⋊D311yescharacteristic
131(C2×D3)⋊C4—38no—
132(C2×D3)⋊C4—316no—
133(C2×D3)⋊C4—316no—
134C4⋊Dic3—34no—
135C4⋊Dic3—34no—
136C4⋊Dic3—316no—
137Dic3⋊C4—38no—
138C4×Dic3—38no—
139C4×Dic3—316no—
140C22×C32C23×D314yes—
141C2×C3×D3—68no—
142C2×C3×D3—68no—
143C2×C3×D3—68no—
144C32×C4—216no—
145C32×C4—24no—
146C32×C4#23718yes—
147C32×C4#23918yes—
148C3×Dic3—68no—
149C3×Dic3—68no—
150C3×Dic3—68no—
151C4⋊Q8#27211yescharacteristic, complemented
152#126—94no—
153#124—92no—
154C22⋊Q8—92no—
155C4×Q8—92no—
156C4×Q8—91no—
157C4×D4—91no—
158#117—92no—
159H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
160C23×C3—92no—
161C22×D3—38no—
162C3×Q8—24no—
163C3×Q8—616no—
164C3×Q8—182no—
165C3×D4—182no—
166C2×C3×C4—34no—
167C2×C3×C4—34no—
168C2×C3×C4—316no—
169C2×C3×C4—34no—
170C2×C3×C4—92no—
171C2×C3×C4—92no—
172C2×C3×C4—92no—
173C2×C3×C4—184no—
174C2×C3×C4C22×C3⋊D314yes—
175C2×C3×C4C22×C3⋊D311yescharacteristic
176C2×C3×C4C22×C3⋊D311yescharacteristic
177C2×C3×C4C22×C3⋊D311yescharacteristic
178C2×Dic3—38no—
179C2×Dic3—38no—
180C2×Dic3—38no—
181D12—68no—
182C4×D3—616no—
183Dic6—68no—
184C2×C32C2×#23714yes—
185C2×C32C2×#23914yes—
186C2×C32#100714yes—
187C3×D3—128no—
188C2×Q8—91no—
189C2×Q8C2×#27212yes—
190C2×D4—91no—
191C22×C4—92no—
192C4⋊C4—91no—
193C4⋊C4—91no—
194C4⋊C4—94no—
195C4⋊C4—92no—
196C4⋊C4C2×#27212yes—
197C4⋊C4C2×#27212yes—
198C22⋊C4—94no—
199C22⋊C4—92no—
200C22⋊C4—94no—
201C42—94no—
202C42—92no—
203C42C2×#27211yescharacteristic
204C22×C3—34no—
205C22×C3—182no—
206C22×C3—182no—
207C22×C3—182no—
208C22×C3C23×C3⋊D311yescharacteristic
209C2×D3—68no—
210C2×D3—68no—
211C2×D3—68no—
212C3×C4—24no—
213C3×C4—21no—
214C3×C4—38no—
215C3×C4—38no—
216C3×C4—616no—
217C3×C4—64no—
218C3×C4—182no—
219C3×C4—182no—
220C3×C4—182no—
221C3×C4#383012yes—
222C3×C4#383212yes—
223Dic3—68no—
224Dic3—68no—
225Dic3—68no—
226C32#573214yescomplemented
227C23—92no—
228Q8—24no—
229Q8—182no—
230D4—182no—
231C2×C4—92no—
232C2×C4—92no—
233C2×C4—92no—
234C2×C4—184no—
235C2×C4C22×#27214yes—
236C2×C4C22×#27211yescharacteristic
237C2×C4C22×#27211yescharacteristic
238C2×C4C22×#27211yescharacteristic
239C2×C3—34no—
240C2×C3—34no—
241C2×C3—34no—
242C2×C3—362no—
243C2×C3C2×#383011yescharacteristic
244C2×C3C2×#383211yescharacteristic
245C2×C3#6436411yescharacteristic
246D3—128no—
247C22—182no—
248C22—182no—
249C22—182no—
250C22C23×#27211yescharacteristic
251C4—24no—
252C4—21no—
253C4—182no—
254C4—182no—
255C4—182no—
256C4#8874412yes—
257C4#8874812yes—
258C3—34no—
259C3#1059620311yescharacteristic
260C2—362no—
261C2C2×#8874411yescharacteristic
262C2C2×#8874811yescharacteristic
263C2#1172267011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC2×C4⋊#272C22
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#556
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C4⋊Q8C2semidirect
3. Q8×#1084C2semidirect
4. #11721766C2semidirect
5. #11721765C2semidirect
6. #11721760C2semidirect
8. C4×#6229C2semidirect
26. C32×C4⋊Q8D3semidirect
151. C4⋊Q8#272semidirect
159. H(3,𝔽3)#556semidirect
226. C32#5732semidirect

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
#117218642
Q8×#10842
#117217662
#117217652
#117217602
C4×C4⋊#2722
C4×#62292
C3×#57323