Finite Group Databasebeta

← #13166349

Subgroups — #13166349

Order 1728.

544 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 68 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
2#11723102C211yesmaximal, characteristic, complemented
3C3×#63777C211yesmaximal, characteristic, complemented
4Dic3⋊#622C212yesmaximal, complemented
5#11722726C212yesmaximal
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—31no—
20#63777D311yescharacteristic, complemented
21C4⋊#622—32no—
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—22no—
32C3⋊#622—22no—
33C3×#622—22no—
34#4705—91nomaximal
35C2×C4×C3⋊D3—31no—
36C2×C4×C3⋊D3—31no—
37C2×C4×C3⋊D3—31no—
38C2×C4×C3⋊D3—62no—
39C2×C4×C3⋊D3—122no—
40C2×#624—62no—
41C2×#624—62no—
42C2×#622—32no—
43C2×#622C2×D312yes—
44C32×C42—31no—
45C4×C3⋊Dic3—31no—
46C4×C3⋊Dic3—31no—
47C4×C3⋊Dic3C2×D311yescharacteristic
48C3×C4×Dic3—22no—
49C3×C4×Dic3—62no—
50Dic32—62no—
51Dic32—62no—
52C22×C33C22×C411yescharacteristic
53C2×C3⋊(C3⋊D3)—21no—
54C2×C3⋊(C3⋊D3)—21no—
55C2×C3⋊(C3⋊D3)M1612yes—
56C33×C4—21no—
57C3⋊(C3⋊Dic3)—21no—
58C3×C3⋊Dic3—21no—
59C3×C3⋊Dic3#4112yes—
60C32×Dic3—21no—
61C32×Dic3M1612yes—
62C42×D3—91no—
63C42×D3—182no—
64C3×#97—91no—
65(C2×D3)⋊C8—92no—
66Dic3⋊C8—92no—
67#803—91no—
68C22×C3⋊D3—31no—
69C22×C3⋊D3—62no—
70C22×C3⋊D3—122no—
71C2×C32×C4—22no—
72C2×C32×C4—31no—
73C2×C32×C4—31no—
74C2×C32×C4—42no—
75C2×C32×C4C4×D311yescharacteristic
76C2×C3⋊Dic3—31no—
77C2×C3⋊Dic3—62no—
78C2×C3⋊Dic3—122no—
79C2×C3⋊Dic3C4×D311yescharacteristic
80C2×C3⋊Dic3C22×D311yescharacteristic
81C4×C3⋊D3—62no—
82C4×C3⋊D3—62no—
83C4×C3⋊D3—62no—
84C4×C3⋊D3—68no—
85C4×C3⋊D3—128no—
86C2×C3×Dic3—22no—
87C2×C3×Dic3—22no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90#624—68no—
91#624—68no—
92#622—22no—
93#622—62no—
94#432—271noSylow
95C2×C33#11711yescharacteristic
96C2×C33C2×M1611yescharacteristic
97C2×C33C2×M1611yescharacteristic
98C3⋊(C3⋊D3)—22no—
99C2×C4×D3—91no—
100C2×C4×D3—91no—
101C2×C4×D3—91no—
102C2×C4×D3—182no—
103C2×C4×D3—182no—
104C2×C4×D3—182no—
105C2×C4×D3—362no—
106C2×C3×C8—92no—
107C3×C42—62no—
108C3×C42—91no—
109C4×Dic3—62no—
110C4×Dic3—91no—
111C4×Dic3—91no—
112C4×Dic3—182no—
113C4×Dic3—182no—
114C4×Dic3#15711yescharacteristic
115C2×C3⋊C8—92no—
116C22×C32—22no—
117C22×C32—42no—
118C22×C32C2×C4×D311yescharacteristic
119C2×C3⋊D3—32no—
120C2×C3⋊D3—64no—
121C2×C3⋊D3—61no—
122C2×C3⋊D3—61no—
123C2×C3⋊D3—64no—
124C2×C3⋊D3—64no—
125C2×C3⋊D3—124no—
126C2×C3⋊D3—124no—
127C2×C3⋊D3—124no—
128C32×C4—21no—
129C32×C4—24no—
130C32×C4—32no—
131C32×C4—44no—
132C32×C4—61no—
133C3⋊Dic3—21no—
134C3⋊Dic3—61no—
135C3⋊Dic3—64no—
136C3⋊Dic3—124no—
137C3⋊Dic3#23512yes—
138C3×Dic3—24no—
139C3×Dic3—24no—
140C3×Dic3—64no—
141C3×Dic3—64no—
142C2×C42—271no—
143C4⋊C8—272no—
144C22⋊C8—272no—
145#97—91no—
146#97—271no—
147C33#43211yescharacteristic, complemented, Sylow
148C22×D3—91no—
149C22×D3—182no—
150C22×D3—362no—
151C2×C3×C4—22no—
152C2×C3×C4—42no—
153C2×C3×C4—62no—
154C2×C3×C4—62no—
155C2×C3×C4—91no—
156C2×C3×C4—91no—
157C2×C3×C4C2×#15711yescharacteristic
158C2×Dic3—62no—
159C2×Dic3—62no—
160C2×Dic3—91no—
161C2×Dic3—182no—
162C2×Dic3—362no—
163C2×Dic3C2×#15711yescharacteristic
164C2×Dic3C2×#15711yescharacteristic
165C4×D3—182no—
166C4×D3—188no—
167C4×D3—188no—
168C4×D3—182no—
169C4×D3—188no—
170C4×D3—182no—
171C4×D3—368no—
172C3×C8—182no—
173C3⋊C8—182no—
174C2×C32—22no—
175C2×C32—22no—
176C2×C32—22no—
177C2×C32—42no—
178C2×C32—42no—
179C2×C32—42no—
180C2×C32#87211yescharacteristic
181C2×C32D3×M1611yescharacteristic
182C2×C32D3×M1611yescharacteristic
183C3⋊D3—62no—
184C3⋊D3—68no—
185C3⋊D3—128no—
186C22×C4—271no—
187C22×C4—271no—
188C22×C4—271no—
189C2×C8—92no—
190C2×C8—272no—
191C42—31no—
192C42—91no—
193C42—271no—
194C42—271no—
195C22×C3—22no—
196C22×C3—42no—
197C22×C3C22×#15711yescharacteristic
198C2×D3—92no—
199C2×D3—184no—
200C2×D3—184no—
201C2×D3—184no—
202C2×D3—181no—
203C2×D3—181no—
204C2×D3—364no—
205C2×D3—364no—
206C2×D3—364no—
207C3×C4—24no—
208C3×C4—21no—
209C3×C4—44no—
210C3×C4—64no—
211C3×C4—64no—
212C3×C4—92no—
213C3×C4—181no—
214Dic3—21no—
215Dic3—64no—
216Dic3—64no—
217Dic3—184no—
218Dic3—181no—
219Dic3—364no—
220Dic3#378812yes—
221C32—22no—
222C32—42no—
223C32#470511yescharacteristic, complemented
224C23—271no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—91no—
228C2×C4—91no—
229C2×C4—271no—
230C2×C4—542no—
231C2×C4—542no—
232C2×C4—542no—
233C2×C4D3×#15711yescharacteristic
234C8—182no—
235C8—542no—
236C2×C3—22no—
237C2×C3—22no—
238C2×C3—22no—
239C2×C3—42no—
240C2×C3—42no—
241C2×C3—42no—
242C2×C3C2×#378811yescharacteristic
243C2×C3#6428311yescharacteristic
244C2×C3#6428611yescharacteristic
245D3—188no—
246D3—182no—
247D3—368no—
248C22—272no—
249C22—541no—
250C22—541no—
251C22C2×D3×#15711yescharacteristic
252C4—21no—
253C4—32no—
254C4—61no—
255C4—92no—
256C4—181no—
257C4—541no—
258C3—22no—
259C3—42no—
260C3#1059373011yescharacteristic, complemented
261C2—542no—
262C2#1172424611yescharacteristic
263C2D3×#378811yescharacteristic
264C2#1172426911yescharacteristic
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. #11723102C2semidirect
3. C3×#63777C2semidirect
4. Dic3⋊#622C2semidirect
20. #63777D3semidirect
147. C33#432semidirect
223. C32#4705semidirect
260. C3#10593730semidirect

Maximal subgroups

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

subgroupindex
C4×#62892
#117231022
C3×#637772
Dic3⋊#6222
#117227262
#105937303
#47059