Finite Group Databasebeta

← #13166330

Subgroups — #13166330

Order 1728.

496 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×D3×C3⋊Dic3C211yesmaximal, characteristic
2C4⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
3C3×C4⋊#622C211yesmaximal, characteristic, complemented
4(C2×D3)⋊#622C212yesmaximal
5#11722724C212yesmaximal, complemented
6#10593806—31nomaximal
7C2×C32×C4×D3C411yescharacteristic
8C2×D3×C3⋊Dic3C411yescharacteristic
9C2×D3×C3⋊Dic3C2211yescharacteristic
10C2×C3⋊#622C2212yes—
11C2×C3×#622C2212yes—
12C4×C3⋊(C3⋊Dic3)C411yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14Dic3×C3⋊Dic3C411yescharacteristic
15Dic3×C3⋊Dic3C2211yescharacteristic
16C2×C4×C3⋊Dic3—31no—
17C4×D3×Dic3—62no—
18C22⋊#622—32no—
19#63777—32no—
20C4⋊#622—31no—
21C4⋊#622D311yescharacteristic, complemented
22C22×C32×D3C2×C411yescharacteristic
23C2×C33×C4C2×C411yescharacteristic
24C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2×C411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3C2×C411yescharacteristic
29C32×C4×D3—22no—
30D3×C3⋊Dic3—22no—
31D3×C3⋊Dic3—21no—
32D3×C3⋊Dic3—21no—
33C3⋊#622—22no—
34C3×#622—22no—
35#4835—91nomaximal
36C22×C32×C4—31no—
37C22×C3⋊Dic3—31no—
38C22×C3⋊Dic3—31no—
39C2×C3×C4×D3—22no—
40C2×D3×Dic3—62no—
41C2×D3×Dic3—62no—
42C2×#622—32no—
43C2×#622C2×D312yes—
44C4×C3⋊Dic3—31no—
45C4×C3⋊Dic3—31no—
46C4×C3⋊Dic3—31no—
47C4×C3⋊Dic3—62no—
48C4×C3⋊Dic3—122no—
49C4×C3⋊Dic3C2×D311yescharacteristic
50C3×C4×Dic3—62no—
51Dic32—62no—
52Dic32—62no—
53C22×C33C22×C411yescharacteristic
54C2×C32×D3—21no—
55C2×C32×D3—21no—
56C2×C32×D3M1612yes—
57C33×C4M1612yes—
58C3⋊(C3⋊Dic3)—21no—
59C3⋊(C3⋊Dic3)—21no—
60C3×C3⋊Dic3—21no—
61C3×C3⋊Dic3#4111yescharacteristic
62C3×C3⋊Dic3#4111yescharacteristic
63C32×Dic3—21no—
64C2×C4×Dic3—182no—
65C42×D3—91no—
66C3×C4⋊C8—91no—
67(C2×D3)⋊C8—92no—
68#815—92no—
69C4⋊(C3⋊C8)—91no—
70C23×C32—31no—
71C22×C3×D3—22no—
72C2×C32×C4—22no—
73C2×C32×C4—31no—
74C2×C32×C4—42no—
75C2×C32×C4—62no—
76C2×C32×C4C4×D311yescharacteristic
77C2×C3⋊Dic3—31no—
78C2×C3⋊Dic3—31no—
79C2×C3⋊Dic3—62no—
80C2×C3⋊Dic3—62no—
81C2×C3⋊Dic3—62no—
82C2×C3⋊Dic3—61no—
83C2×C3⋊Dic3—61no—
84C2×C3⋊Dic3—122no—
85C2×C3⋊Dic3—122no—
86C2×C3⋊Dic3C4×D311yescharacteristic
87C2×C3⋊Dic3C22×D311yescharacteristic
88C2×C3×Dic3—22no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C3×C4×D3—28no—
92D3×Dic3—64no—
93D3×Dic3—64no—
94D3×Dic3—68no—
95#622—22no—
96#622—62no—
97#432—271noSylow
98C2×C33#11711yescharacteristic
99C2×C33C2×M1611yescharacteristic
100C2×C33C2×M1611yescharacteristic
101C32×D3—22no—
102C22×C3×C4—62no—
103C22×Dic3—182no—
104C22×Dic3—182no—
105C2×C4×D3—91no—
106C2×C4×D3—91no—
107C2×C4×D3#15711yescharacteristic
108C2×C3×C8—92no—
109C3×C42—91no—
110C4×Dic3—62no—
111C4×Dic3—91no—
112C4×Dic3—91no—
113C4×Dic3—91no—
114C4×Dic3—182no—
115C4×Dic3—182no—
116C4×Dic3—182no—
117C4×Dic3—362no—
118C2×C3⋊C8—92no—
119C22×C32—22no—
120C22×C32—32no—
121C22×C32—42no—
122C22×C32—61no—
123C22×C32—61no—
124C22×C32C2×C4×D311yescharacteristic
125C2×C3×D3—24no—
126C2×C3×D3—24no—
127C2×C3×D3—24no—
128C32×C4—24no—
129C32×C4—44no—
130C32×C4—61no—
131C32×C4ΓL(1,𝔽25)12yes—
132C3⋊Dic3—21no—
133C3⋊Dic3—61no—
134C3⋊Dic3—61no—
135C3⋊Dic3—64no—
136C3⋊Dic3—64no—
137C3⋊Dic3—124no—
138C3⋊Dic3—124no—
139C3⋊Dic3#23711yescharacteristic
140C3⋊Dic3#23911yescharacteristic
141C3×Dic3—24no—
142C3×Dic3—62no—
143C3×Dic3—62no—
144C3×Dic3—64no—
145C2×C42—271no—
146C4⋊C8—91no—
147C4⋊C8—271no—
148C22⋊C8—272no—
149#97—272no—
150C33#43211yescharacteristic, complemented, Sylow
151C23×C3—62no—
152C22×D3C2×#15711yescharacteristic
153C2×C3×C4—22no—
154C2×C3×C4—42no—
155C2×C3×C4—62no—
156C2×C3×C4—68no—
157C2×C3×C4—91no—
158C2×C3×C4—91no—
159C2×C3×C4C2×#15711yescharacteristic
160C2×Dic3—62no—
161C2×Dic3—62no—
162C2×Dic3—91no—
163C2×Dic3—91no—
164C2×Dic3—182no—
165C2×Dic3—182no—
166C2×Dic3—184no—
167C2×Dic3—184no—
168C2×Dic3—188no—
169C2×Dic3—362no—
170C2×Dic3—362no—
171C2×Dic3C2×#15711yescharacteristic
172C4×D3—22no—
173C4×D3—181no—
174C4×D3—181no—
175C4×D3—182no—
176C3×C8—182no—
177C3⋊C8—182no—
178C2×C32—22no—
179C2×C32—22no—
180C2×C32—22no—
181C2×C32—42no—
182C2×C32—42no—
183C2×C32—42no—
184C2×C32—62no—
185C2×C32#89211yescharacteristic
186C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
187C2×C32D3×M1611yescharacteristic
188C3×D3—28no—
189C22×C4—31no—
190C22×C4—271no—
191C22×C4—271no—
192C2×C8—92no—
193C2×C8—272no—
194C42—91no—
195C42—271no—
196C42—271no—
197C42—271no—
198C22×C3—22no—
199C22×C3—42no—
200C22×C3—64no—
201C22×C3—64no—
202C22×C3—64no—
203C22×C3C22×#15711yescharacteristic
204C2×D3—21no—
205C2×D3—21no—
206C2×D3#378812yes—
207C3×C4—24no—
208C3×C4—44no—
209C3×C4—64no—
210C3×C4—91no—
211C3×C4—91no—
212C3×C4—181no—
213C3×C4#379212yes—
214Dic3—21no—
215Dic3—62no—
216Dic3—62no—
217Dic3—64no—
218Dic3—184no—
219Dic3—184no—
220Dic3—181no—
221Dic3—181no—
222Dic3—364no—
223Dic3—364no—
224C32—22no—
225C32—42no—
226C32#483511yescharacteristic, complemented
227C23—31no—
228C2×C4—31no—
229C2×C4—62no—
230C2×C4—91no—
231C2×C4—91no—
232C2×C4—271no—
233C2×C4—271no—
234C2×C4—541no—
235C2×C4—541no—
236C2×C4—542no—
237C2×C4D3×#15711yescharacteristic
238C8—182no—
239C8—542no—
240C2×C3—22no—
241C2×C3—22no—
242C2×C3—22no—
243C2×C3—42no—
244C2×C3—42no—
245C2×C3—42no—
246C2×C3—68no—
247C2×C3C2×#378811yescharacteristic
248C2×C3#6428611yescharacteristic
249C2×C3C2×#379211yescharacteristic
250D3—22no—
251C22—32no—
252C22—61no—
253C22—61no—
254C22C2×D3×#15711yescharacteristic
255C4—61no—
256C4—91no—
257C4—91no—
258C4—181no—
259C4—541no—
260C4—541no—
261C4#8892512yes—
262C3—22no—
263C3—42no—
264C3#1059380611yescharacteristic, complemented
265C2—62no—
266C2#1172424711yescharacteristic
267C2D3×#378811yescharacteristic
268C2C2×#8892511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#432
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4⋊(C3⋊#622)C2semidirect
3. C3×C4⋊#622C2semidirect
5. #11722724C2semidirect
21. C4⋊#622D3semidirect
150. C33#432semidirect
226. C32#4835semidirect
264. C3#10593806semidirect

Maximal subgroups

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

subgroupindex
C4×D3×C3⋊Dic32
C4⋊(C3⋊#622)2
C3×C4⋊#6222
(C2×D3)⋊#6222
#117227242
#105938063
#48359