Finite Group Databasebeta

← #13166332

Subgroups — #13166332

Order 1728.

544 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 72 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×#6289C211yesmaximal, characteristic
2C4⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
3C3×C4⋊#622C211yesmaximal, characteristic, complemented
4#11722726C212yesmaximal
5#11722724C212yesmaximal, complemented
6#10593730—31nomaximal
7C2×C4×C3⋊(C3⋊D3)C411yescharacteristic
8C2×#6289C411yescharacteristic
9C2×#6289C2211yescharacteristic
10C2×C3⋊#622C2212yes—
11C2×C3×#622C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×C4×Dic3C411yescharacteristic
14Dic3×C3⋊Dic3C411yescharacteristic
15Dic3×C3⋊Dic3C2211yescharacteristic
16C42×C3⋊D3—31no—
17C4×#624—62no—
18#63778—32no—
19#63777—32no—
20C4⋊#622—31no—
21C4⋊#622D311yescharacteristic, complemented
22C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
23C2×C33×C4C2×C411yescharacteristic
24C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
25C4×C3⋊(C3⋊D3)—22no—
26C2×C3×C3⋊Dic3C2×C411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3C2×C411yescharacteristic
29C2×C32×Dic3C2×C411yescharacteristic
30#6289—22no—
31#6289—21no—
32#6289—21no—
33C3⋊#622—22no—
34C3×#622—22no—
35#4835—91nomaximal
36C2×C4×C3⋊D3—31no—
37C2×C4×C3⋊D3—31no—
38C2×C4×C3⋊D3—31no—
39C2×C4×C3⋊D3—62no—
40C2×C4×C3⋊D3—122no—
41C2×#624—62no—
42C2×#624—62no—
43C2×#622—32no—
44C2×#622C2×D312yes—
45C32×C42—31no—
46C4×C3⋊Dic3—31no—
47C4×C3⋊Dic3—31no—
48C4×C3⋊Dic3C2×D311yescharacteristic
49C3×C4×Dic3—22no—
50C3×C4×Dic3—62no—
51Dic32—62no—
52Dic32—62no—
53C22×C33C22×C411yescharacteristic
54C2×C3⋊(C3⋊D3)—21no—
55C2×C3⋊(C3⋊D3)—21no—
56C2×C3⋊(C3⋊D3)M1612yes—
57C33×C4M1612yes—
58C3⋊(C3⋊Dic3)—21no—
59C3×C3⋊Dic3—21no—
60C3×C3⋊Dic3#4111yescharacteristic
61C3×C3⋊Dic3#4111yescharacteristic
62C32×Dic3—21no—
63C32×Dic3—21no—
64C42×D3—91no—
65C42×D3—182no—
66C3×C4⋊C8—91no—
67(C2×D3)⋊C8—92no—
68#815—92no—
69C4⋊(C3⋊C8)—91no—
70C22×C3⋊D3—31no—
71C22×C3⋊D3—62no—
72C22×C3⋊D3—122no—
73C2×C32×C4—22no—
74C2×C32×C4—31no—
75C2×C32×C4—31no—
76C2×C32×C4—42no—
77C2×C32×C4C4×D311yescharacteristic
78C2×C3⋊Dic3—31no—
79C2×C3⋊Dic3—62no—
80C2×C3⋊Dic3—122no—
81C2×C3⋊Dic3C4×D311yescharacteristic
82C2×C3⋊Dic3C22×D311yescharacteristic
83C4×C3⋊D3—62no—
84C4×C3⋊D3—62no—
85C4×C3⋊D3—61no—
86C4×C3⋊D3—61no—
87C4×C3⋊D3—68no—
88C4×C3⋊D3—128no—
89C2×C3×Dic3—22no—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93#624—64no—
94#624—64no—
95#624—68no—
96#622—22no—
97#622—62no—
98#432—271noSylow
99C2×C33#11711yescharacteristic
100C2×C33C2×M1611yescharacteristic
101C2×C33C2×M1611yescharacteristic
102C3⋊(C3⋊D3)—22no—
103C2×C4×D3—91no—
104C2×C4×D3—91no—
105C2×C4×D3—91no—
106C2×C4×D3—182no—
107C2×C4×D3—182no—
108C2×C4×D3—182no—
109C2×C4×D3—362no—
110C2×C3×C8—92no—
111C3×C42—62no—
112C3×C42—91no—
113C4×Dic3—62no—
114C4×Dic3—91no—
115C4×Dic3—91no—
116C4×Dic3—182no—
117C4×Dic3—182no—
118C4×Dic3#15711yescharacteristic
119C2×C3⋊C8—92no—
120C22×C32—22no—
121C22×C32—42no—
122C22×C32C2×C4×D311yescharacteristic
123C2×C3⋊D3—32no—
124C2×C3⋊D3—64no—
125C2×C3⋊D3—61no—
126C2×C3⋊D3—61no—
127C2×C3⋊D3—64no—
128C2×C3⋊D3—64no—
129C2×C3⋊D3—124no—
130C2×C3⋊D3—124no—
131C2×C3⋊D3—124no—
132C32×C4—24no—
133C32×C4—44no—
134C32×C4—61no—
135C32×C4—61no—
136C32×C4ΓL(1,𝔽25)12yes—
137C3⋊Dic3—21no—
138C3⋊Dic3—61no—
139C3⋊Dic3—64no—
140C3⋊Dic3—124no—
141C3⋊Dic3#23711yescharacteristic
142C3⋊Dic3#23911yescharacteristic
143C3×Dic3—24no—
144C3×Dic3—24no—
145C3×Dic3—62no—
146C3×Dic3—62no—
147C3×Dic3—64no—
148C2×C42—271no—
149C4⋊C8—91no—
150C4⋊C8—271no—
151C22⋊C8—272no—
152#97—272no—
153C33#43211yescharacteristic, complemented, Sylow
154C22×D3—91no—
155C22×D3—182no—
156C22×D3—362no—
157C2×C3×C4—22no—
158C2×C3×C4—42no—
159C2×C3×C4—62no—
160C2×C3×C4—62no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4C2×#15711yescharacteristic
164C2×Dic3—62no—
165C2×Dic3—62no—
166C2×Dic3—91no—
167C2×Dic3—182no—
168C2×Dic3—362no—
169C2×Dic3C2×#15711yescharacteristic
170C2×Dic3C2×#15711yescharacteristic
171C4×D3—181no—
172C4×D3—181no—
173C4×D3—182no—
174C4×D3—184no—
175C4×D3—184no—
176C4×D3—188no—
177C4×D3—188no—
178C4×D3—182no—
179C4×D3—368no—
180C3×C8—182no—
181C3⋊C8—182no—
182C2×C32—22no—
183C2×C32—22no—
184C2×C32—22no—
185C2×C32—42no—
186C2×C32—42no—
187C2×C32—42no—
188C2×C32#89211yescharacteristic
189C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
190C2×C32D3×M1611yescharacteristic
191C3⋊D3—62no—
192C3⋊D3—68no—
193C3⋊D3—128no—
194C22×C4—271no—
195C22×C4—271no—
196C22×C4—271no—
197C2×C8—92no—
198C2×C8—272no—
199C42—31no—
200C42—91no—
201C42—271no—
202C42—271no—
203C22×C3—22no—
204C22×C3—42no—
205C22×C3C22×#15711yescharacteristic
206C2×D3—92no—
207C2×D3—184no—
208C2×D3—184no—
209C2×D3—184no—
210C2×D3—181no—
211C2×D3—181no—
212C2×D3—364no—
213C2×D3—364no—
214C2×D3—364no—
215C3×C4—24no—
216C3×C4—44no—
217C3×C4—64no—
218C3×C4—64no—
219C3×C4—91no—
220C3×C4—91no—
221C3×C4—181no—
222C3×C4#378812yes—
223Dic3—21no—
224Dic3—21no—
225Dic3—62no—
226Dic3—62no—
227Dic3—64no—
228Dic3—184no—
229Dic3—181no—
230Dic3—364no—
231C32—22no—
232C32—42no—
233C32#483511yescharacteristic, complemented
234C23—271no—
235C2×C4—31no—
236C2×C4—31no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—271no—
240C2×C4—541no—
241C2×C4—541no—
242C2×C4—542no—
243C2×C4—542no—
244C2×C4D3×#15711yescharacteristic
245C8—182no—
246C8—542no—
247C2×C3—22no—
248C2×C3—22no—
249C2×C3—22no—
250C2×C3—42no—
251C2×C3—42no—
252C2×C3—42no—
253C2×C3C2×#378811yescharacteristic
254C2×C3#6428311yescharacteristic
255C2×C3#6428611yescharacteristic
256D3—188no—
257D3—182no—
258D3—368no—
259C22—272no—
260C22—541no—
261C22—541no—
262C22C2×D3×#15711yescharacteristic
263C4—61no—
264C4—61no—
265C4—91no—
266C4—91no—
267C4—181no—
268C4—541no—
269C4#8892612yes—
270C3—22no—
271C3—42no—
272C3#1059373011yescharacteristic, complemented
273C2—542no—
274C2#1172424711yescharacteristic
275C2#1172425511yescharacteristic
276C2C2×#8892611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×C3⋊(C3⋊D3)C2×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
153. C33#432semidirect
233. C32#4835semidirect
272. C3#10593730semidirect

Maximal subgroups

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

subgroupindex
C4×#62892
C4⋊(C3⋊#622)2
C3×C4⋊#6222
#117227262
#117227242
#105937303
#48359