Finite Group Databasebeta

← #13154299

Subgroups — #13154299

Order 1728.

1254 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 227 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#11721863C214yesmaximal, complemented
3#11721765C212yesmaximal, complemented
4#11721761C212yesmaximal
5C4⋊C4×#272C212yesmaximal, complemented
6#11721758C212yesmaximal, complemented
7C4×C4⋊#272C211yesmaximal, characteristic
8C4×#6229C211yesmaximal, characteristic, complemented
9C3×#5730—34nomaximal
10C2×Q8×H(3,𝔽3)C2212yes—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2212yes—
13C2×C4×#272C2214yes—
14C2×#6229C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2212yes—
16C4⋊C4×H(3,𝔽3)C2212yes—
17C42×H(3,𝔽3)C2211yescharacteristic
18#88542C2214yes—
19#88542C2212yes—
20C4⋊#1084C2211yescharacteristic
21C4⋊#1084C2214yes—
22C4⋊#1084C2211yescharacteristic
23#88540C2214yes—
24#88540C2212yes—
25C4×#1084C2212yes—
26C32×C4⋊Q8D314yescomplemented
27C3×#946—316no—
28C3×#897—38no—
29C3×#893—38no—
30C3×D3×C4⋊C4—38no—
31C3×#890—38no—
32C3×C4×D12—34no—
33C3×C4×Dic6—34no—
34C22×#272C2312yes—
35Q8×H(3,𝔽3)—42no—
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⋊#272Q814yescomplemented
44C4×#272—24no—
45C4×#272—44no—
46#6229—22no—
47C3×#554—91no—
48#5730—34no—
49C2×C32×Q8C2×D318yes—
50C2×C3×D12—34no—
51C2×C3×C4×D3—38no—
52C2×C3×C4×D3—316no—
53C2×C3×Dic6—34no—
54C32×C4⋊C4C2×D318yes—
55C32×C4⋊C4C2×D318yes—
56C32×C42C2×D314yes—
57C3×(C2×D3)⋊C4—316no—
58C3×(C2×D3)⋊C4—38no—
59C3×C4⋊Dic3—34no—
60C3×C4⋊Dic3—316no—
61C3×C4⋊Dic3—34no—
62C3×Dic3⋊C4—316no—
63C3×Dic3⋊C4—38no—
64C3×C4×Dic3—38no—
65C22×H(3,𝔽3)C2411yescharacteristic
66C2×#272—22no—
67C2×#272—22no—
68C2×#272C2×Q814yes—
69C4×H(3,𝔽3)—24no—
70C4×H(3,𝔽3)—21no—
71C4×H(3,𝔽3)C2×Q812yes—
72C4×H(3,𝔽3)#4112yes—
73#1084—22no—
74#1084—22no—
75#1084—22no—
76C3×C4⋊Q8—34no—
77C3×C4⋊Q8C3⋊D311yescharacteristic
78C3×#125—92no—
79C3×C22⋊Q8—94no—
80C3×C22⋊Q8—92no—
81C3×C4×Q8—91no—
82C3×C4×D4—92no—
83C3×C4×D4—91no—
84C2×C3×C4⋊C4—92no—
85#946—316no—
86#897—38no—
87#893—38no—
88D3×C4⋊C4—38no—
89#890—38no—
90C4×D12—34no—
91C4×Dic6—34no—
92C22×C3×D3—38no—
93C32×Q8—48no—
94C2×C32×C4C22×D314yes—
95C2×C32×C4C22×D3116yes—
96C2×C32×C4C22×D314yes—
97C2×C32×C4C22×D314yes—
98C2×C3×Dic3—38no—
99C2×C3×Dic3—38no—
100C2×C3×Dic3—38no—
101C3×D12—316no—
102C3×C4×D3—616no—
103C3×C4×D3—1216no—
104C3×Dic6—68no—
105#554—91noSylow
106C2×H(3,𝔽3)C22×Q811yescharacteristic
107C2×H(3,𝔽3)C2×#4111yescharacteristic
108C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
109#272—24no—
110C2×C3×Q8—38no—
111C2×C3×Q8—91no—
112C2×C3×Q8C2×C3⋊D312yes—
113C2×C3×D4—91no—
114C22×C3×C4—92no—
115C22×C3×C4—94no—
116C2×D12—34no—
117C2×C4×D3—38no—
118C2×C4×D3—316no—
119C2×Dic6—34no—
120C3×C4⋊C4—38no—
121C3×C4⋊C4—38no—
122C3×C4⋊C4—94no—
123C3×C4⋊C4—92no—
124C3×C4⋊C4—91no—
125C3×C4⋊C4—91no—
126C3×C4⋊C4—94no—
127C3×C4⋊C4C2×C3⋊D312yes—
128C3×C4⋊C4C2×C3⋊D312yes—
129C3×C22⋊C4—94no—
130C3×C22⋊C4—92no—
131C3×C42—34no—
132C3×C42—92no—
133C3×C42C2×C3⋊D311yescharacteristic
134(C2×D3)⋊C4—38no—
135(C2×D3)⋊C4—316no—
136C4⋊Dic3—34no—
137C4⋊Dic3—34no—
138C4⋊Dic3—316no—
139Dic3⋊C4—38no—
140Dic3⋊C4—316no—
141C4×Dic3—38no—
142C22×C32C23×D314yes—
143C2×C3×D3—316no—
144C2×C3×D3—68no—
145C2×C3×D3—68no—
146C32×C4—216no—
147C32×C4—24no—
148C32×C4D3×Q818yes—
149C32×C4#23918yes—
150C3×Dic3—68no—
151C3×Dic3—68no—
152C3×Dic3—68no—
153C4⋊Q8#27211yescharacteristic, complemented
154#125—92no—
155C22⋊Q8—94no—
156C22⋊Q8—92no—
157C4×Q8—91no—
158C4×D4—92no—
159C4×D4—91no—
160C2×C4⋊C4—92no—
161H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
162C23×C3—92no—
163C22×D3—38no—
164C3×Q8—42no—
165C3×Q8—128no—
166C3×Q8—182no—
167C3×D4—94no—
168C2×C3×C4—34no—
169C2×C3×C4—34no—
170C2×C3×C4—34no—
171C2×C3×C4—316no—
172C2×C3×C4—92no—
173C2×C3×C4—92no—
174C2×C3×C4—92no—
175C2×C3×C4—184no—
176C2×C3×C4—364no—
177C2×C3×C4C22×C3⋊D311yescharacteristic
178C2×C3×C4C22×C3⋊D314yes—
179C2×C3×C4C22×C3⋊D311yescharacteristic
180C2×C3×C4C22×C3⋊D311yescharacteristic
181C2×Dic3—38no—
182C2×Dic3—38no—
183C2×Dic3—38no—
184D12—316no—
185C4×D3—616no—
186C4×D3—1216no—
187Dic6—68no—
188C2×C32#100414yes—
189C2×C32C2×D3×Q814yes—
190C2×C32C2×#23914yes—
191C3×D3—616no—
192C2×Q8—91no—
193C2×Q8C2×#27212yes—
194C2×D4—91no—
195C22×C4—92no—
196C22×C4—94no—
197C4⋊C4—91no—
198C4⋊C4—94no—
199C4⋊C4—91no—
200C4⋊C4—92no—
201C4⋊C4—94no—
202C4⋊C4C2×#27212yes—
203C4⋊C4C2×#27212yes—
204C22⋊C4—94no—
205C22⋊C4—92no—
206C42—92no—
207C42C2×#27211yescharacteristic
208C22×C3—34no—
209C22×C3—94no—
210C22×C3—182no—
211C22×C3—182no—
212C22×C3C23×C3⋊D311yescharacteristic
213C2×D3—316no—
214C2×D3—68no—
215C2×D3—68no—
216C3×C4—24no—
217C3×C4—21no—
218C3×C4—38no—
219C3×C4—38no—
220C3×C4—616no—
221C3×C4—64no—
222C3×C4—182no—
223C3×C4—182no—
224C3×C4—182no—
225C3×C4Q8×C3⋊D312yes—
226C3×C4#383212yes—
227Dic3—68no—
228Dic3—68no—
229Dic3—68no—
230C32#573014yescomplemented
231C23—92no—
232Q8—42no—
233Q8—182no—
234D4—94no—
235C2×C4—92no—
236C2×C4—92no—
237C2×C4—92no—
238C2×C4—184no—
239C2×C4—364no—
240C2×C4C22×#27211yescharacteristic
241C2×C4C22×#27214yes—
242C2×C4C22×#27211yescharacteristic
243C2×C4C22×#27211yescharacteristic
244C2×C3—34no—
245C2×C3—34no—
246C2×C3—34no—
247C2×C3—184no—
248C2×C3#6436111yescharacteristic
249C2×C3C2×Q8×C3⋊D311yescharacteristic
250C2×C3C2×#383211yescharacteristic
251D3—616no—
252C22—94no—
253C22—182no—
254C22—182no—
255C22C23×#27211yescharacteristic
256C4—24no—
257C4—21no—
258C4—182no—
259C4—182no—
260C4—182no—
261C4Q8×#27212yes—
262C4#8874812yes—
263C3—34no—
264C3#1059620111yescharacteristic
265C2—184no—
266C2#1172266411yescharacteristic
267C2C2×Q8×#27211yescharacteristic
268C2C2×#8874811yescharacteristic
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)#554
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
2. #11721863C2semidirect
3. #11721765C2semidirect
5. C4⋊C4×#272C2semidirect
6. #11721758C2semidirect
8. C4×#6229C2semidirect
26. C32×C4⋊Q8D3semidirect
43. C4⋊#272Q8semidirect
153. C4⋊Q8#272semidirect
161. H(3,𝔽3)#554semidirect
230. C32#5730semidirect

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
#117218632
#117217652
#117217612
C4⋊C4×#2722
#117217582
C4×C4⋊#2722
C4×#62292
C3×#57303