Finite Group Databasebeta

← #72252792

Subgroups — #72252792

Order 2592.

506 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×C3⋊(C3⋊#622)C211yesmaximal, characteristic
2C3×#88815C212yesmaximal
3#11722778—32nomaximal
4C2×C32×C3⋊Dic3—22no—
5C2×C32×C3⋊Dic3C2211yescharacteristic
6C3⋊(C3⋊#622)C412yescomplemented
7C2×C3⋊#622—32no—
8C2×C3⋊#622—61no—
9#88815D312yes—
10C3×#3724—32no—
11C3×Dic3⋊Dic3—62no—
12C22×C34D411yescharacteristic
13C32×C3⋊Dic3—42no—
14C32×C3⋊Dic3C2×C411yescharacteristic
15C32×C3⋊Dic3Q811yescharacteristic
16#63742—91no—
17#63574—91nomaximal
18C2×C3×C3⋊Dic3—21no—
19C2×C3×C3⋊Dic3—22no—
20C2×C3×C3⋊Dic3—22no—
21C2×C3×C3⋊Dic3—42no—
22C2×C3×C3⋊Dic3C2×D312yes—
23C2×C32×Dic3—62no—
24C2×C32×Dic3—62no—
25C2×C32×Dic3—62no—
26C2×C32×Dic3—122no—
27C3⋊#622—34no—
28C3⋊#622—62no—
29C2×C34C4⋊C411yescharacteristic
30C2×C34SD1612yes—
31C2×#622—91no—
32C2×C3⋊(C3⋊C8)—91no—
33C3×C4⋊Dic3—92no—
34C3×Dic3⋊C4—182no—
35#3724—32no—
36Dic3⋊Dic3—62no—
37C22×C33—22no—
38C22×C33—21no—
39C22×C33—42no—
40C22×C33—42no—
41C22×C33—82no—
42C22×C33#6712yes—
43C3×C3⋊Dic3—21no—
44C3×C3⋊Dic3—21no—
45C3×C3⋊Dic3—42no—
46C3×C3⋊Dic3—44no—
47C3×C3⋊Dic3—42no—
48C3×C3⋊Dic3Dic612yes—
49C3×C3⋊Dic3C4×D312yes—
50C32×Dic3—62no—
51C32×Dic3—62no—
52C32×Dic3—122no—
53C32×Dic3—124no—
54C32×Dic3—122no—
55#808—272no—
56C34#10611yescharacteristic, complemented, Sylow
57C2×C32×C4—91no—
58C2×C32×C4—182no—
59C2×C3⋊Dic3—22no—
60C2×C3⋊Dic3D3211yescharacteristic
61C2×C3×Dic3—62no—
62C2×C3×Dic3—62no—
63C2×C3×Dic3—62no—
64C2×C3×Dic3—62no—
65C2×C3×Dic3—62no—
66C2×C3×Dic3—62no—
67C2×C3×Dic3—122no—
68C2×C3×Dic3—122no—
69C2×C3×Dic3—122no—
70C2×C3×Dic3—122no—
71C2×C3×Dic3—122no—
72C2×C3×Dic3—124no—
73#622—92no—
74C3⋊(C3⋊C8)—92no—
75C2×C33—22no—
76C2×C33—22no—
77C2×C33—22no—
78C2×C33—22no—
79C2×C33—21no—
80C2×C33—42no—
81C2×C33—42no—
82C2×C33—42no—
83C2×C33—42no—
84C2×C33—42no—
85C2×C33—42no—
86C2×C33—82no—
87C2×C33—82no—
88C2×C33—82no—
89C2×C33Dic3⋊C412yes—
90C2×C33#21412yes—
91C2×C33#21512yes—
92C3×C4⋊C4—272no—
93C4⋊Dic3—92no—
94Dic3⋊C4—182no—
95C2×C3⋊C8—272no—
96C2×C3⋊C8—541no—
97C22×C32—22no—
98C22×C32—22no—
99C22×C32—42no—
100C22×C32—42no—
101C22×C32—42no—
102C22×C32—44no—
103C22×C32—42no—
104C22×C32—42no—
105C22×C32—82no—
106C22×C32—82no—
107C22×C32—82no—
108C22×C32—82no—
109C22×C32#62511yescharacteristic
110C22×C32GO+(4,𝔽2)11yescharacteristic
111C32×C4—91no—
112C32×C4—91no—
113C32×C4—362no—
114C3⋊Dic3—42no—
115C3⋊Dic3#62411yescharacteristic
116C3⋊Dic3#62711yescharacteristic
117C3×Dic3—62no—
118C3×Dic3—62no—
119C3×Dic3—62no—
120C3×Dic3—62no—
121C3×Dic3—122no—
122C3×Dic3—122no—
123C3×Dic3—122no—
124C3×Dic3—122no—
125C3×Dic3—124no—
126C3×Dic3—124no—
127C3×Dic3—128no—
128C3×Dic3—124no—
129C3×Dic3—124no—
130C3×Dic3—122no—
131C3×Dic3—122no—
132#106—811noSylow
133C33—22no—
134C33—21no—
135C33—42no—
136C33—42no—
137C33—82no—
138C33#80812yescomplemented
139C2×C3×C4—92no—
140C2×C3×C4—181no—
141C2×C3×C4—182no—
142C2×C3×C4—182no—
143C2×C3×C4—362no—
144C2×Dic3—62no—
145C2×Dic3—62no—
146C2×Dic3—62no—
147C2×Dic3—122no—
148C3⋊C8—274no—
149C3⋊C8—542no—
150C2×C32—22no—
151C2×C32—22no—
152C2×C32—22no—
153C2×C32—22no—
154C2×C32—22no—
155C2×C32—22no—
156C2×C32—42no—
157C2×C32—42no—
158C2×C32—42no—
159C2×C32—42no—
160C2×C32—42no—
161C2×C32—42no—
162C2×C32—42no—
163C2×C32—42no—
164C2×C32—42no—
165C2×C32—44no—
166C2×C32—44no—
167C2×C32—44no—
168C2×C32—42no—
169C2×C32—42no—
170C2×C32—42no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32—42no—
174C2×C32—82no—
175C2×C32—82no—
176C2×C32—82no—
177C2×C32—82no—
178C2×C32—82no—
179C2×C32—82no—
180C2×C32—82no—
181C2×C32—82no—
182C2×C32—84no—
183C2×C32—84no—
184C2×C32#371512yes—
185C2×C32#372411yescharacteristic
186C2×C32#377311yescharacteristic
187C2×C32#377512yes—
188C2×C8—811no—
189C4⋊C4—272no—
190C22×C3—22no—
191C22×C3—21no—
192C22×C3—42no—
193C22×C3—42no—
194C22×C3—82no—
195C22×C3D32⋊D312yes—
196C3×C4—92no—
197C3×C4—92no—
198C3×C4—181no—
199C3×C4—181no—
200C3×C4—362no—
201C3×C4—364no—
202C3×C4—362no—
203Dic3—62no—
204Dic3—62no—
205Dic3—122no—
206Dic3—124no—
207Dic3—122no—
208C32—22no—
209C32—22no—
210C32—42no—
211C32—42no—
212C32—42no—
213C32—44no—
214C32—42no—
215C32—42no—
216C32—82no—
217C32—82no—
218C32—82no—
219C32—82no—
220C32#6357411yescharacteristic, complemented
221C32#6374211yescharacteristic, complemented
222C2×C4—91no—
223C2×C4—182no—
224C8—812no—
225C2×C3—22no—
226C2×C3—21no—
227C2×C3—22no—
228C2×C3—22no—
229C2×C3—22no—
230C2×C3—42no—
231C2×C3—42no—
232C2×C3—42no—
233C2×C3—42no—
234C2×C3—42no—
235C2×C3—42no—
236C2×C3—82no—
237C2×C3—82no—
238C2×C3—82no—
239C2×C3#8893412yes—
240C2×C3#8893612yes—
241C2×C3#8893712yes—
242C22#1062020311yescharacteristic
243C4—91no—
244C4—91no—
245C4—362no—
246C3—22no—
247C3—21no—
248C3—42no—
249C3—42no—
250C3—82no—
251C3#1172277812yescomplemented
252C2#1304943911yescharacteristic
253C2#1304944112yes—
subgroupG/H
center, Frattini, hypercenterC22#10620203
commutatorC32×C3⋊Dic3C2×C4
Fitting, socle, prime radicalC22×C34D4
cosocleC2×C32×C3⋊Dic3C22
hypocenterC34#106
LCSG ▷ C32×C3⋊Dic3 ▷ C2×C34 ▷ C34
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
6. C3⋊(C3⋊#622)C4semidirect
56. C34#106semidirect
138. C33#808semidirect
220. C32#63574semidirect
221. C32#63742semidirect
251. C3#11722778semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊#622)2
C3×#888152
#117227783
#635749