Finite Group Databasebeta

← #13166579

Subgroups — #13166579

Order 1728.

560 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 82 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×Dic3×C3⋊Dic3C211yesmaximal, characteristic
2#11723112C211yesmaximal, characteristic
3C3×C22⋊#622C211yesmaximal, characteristic
4#11722732C212yesmaximal, complemented
5Dic3×#622C212yesmaximal, complemented
6#10593801—31nomaximal
7C22×C3⋊(C3⋊Dic3)C411yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C32×Dic3C411yescharacteristic
10C2×C3⋊#622C2212yes—
11C2×C3×#622C2212yes—
12Dic3×C3⋊Dic3C411yescharacteristic
13Dic3×C3⋊Dic3C411yescharacteristic
14Dic3×C3⋊Dic3C2211yescharacteristic
15Dic3×C3⋊Dic3C2211yescharacteristic
16C2×C4×C3⋊Dic3—31no—
17C2×Dic32—62no—
18C22⋊#622—31no—
19C22⋊#622D311yescharacteristic
20#63779—32no—
21C4×#622—32no—
22C23×C33C2×C411yescharacteristic
23C2×C3⋊(C3⋊Dic3)C814yescomplemented
24C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3—22no—
30C2×C32×Dic3C2×C411yescharacteristic
31C2×C32×Dic3C2×C411yescharacteristic
32C3⋊#622—22no—
33C3×#622—22no—
34#4718—91nomaximal
35C22×C32×C4—31no—
36C22×C3⋊Dic3—31no—
37C22×C3⋊Dic3—62no—
38C22×C3⋊Dic3—122no—
39C22×C3⋊Dic3C2×D311yescharacteristic
40C22×C3×Dic3—22no—
41C22×C3×Dic3—62no—
42C2×#622—32no—
43C2×#622C2×D312yes—
44C4×C3⋊Dic3—31no—
45C4×C3⋊Dic3—31no—
46C4×C3⋊Dic3—31no—
47C4×C3⋊Dic3—31no—
48Dic32—62no—
49Dic32—62no—
50Dic32—62no—
51Dic32—62no—
52C22×C33—21no—
53C22×C33—21no—
54C22×C33C2×C812yes—
55C22×C33C22×C411yescharacteristic
56C3⋊(C3⋊Dic3)C2×C812yes—
57C3⋊(C3⋊Dic3)C2×C812yes—
58C3×C3⋊Dic3—21no—
59C3×C3⋊Dic3#4112yes—
60C32×Dic3—21no—
61C32×Dic3M1612yes—
62C2×C4×Dic3—91no—
63C2×C4×Dic3—182no—
64C3×C22⋊C8—91no—
65#830—91no—
66Dic3⋊C8—92no—
67C8×Dic3—92no—
68C23×C32—22no—
69C23×C32—42no—
70C23×C32C4×D311yescharacteristic
71C2×C32×C4—31no—
72C2×C32×C4—31no—
73C2×C32×C4—62no—
74C2×C3⋊Dic3—22no—
75C2×C3⋊Dic3—34no—
76C2×C3⋊Dic3—31no—
77C2×C3⋊Dic3—31no—
78C2×C3⋊Dic3—62no—
79C2×C3⋊Dic3—62no—
80C2×C3⋊Dic3—68no—
81C2×C3⋊Dic3—128no—
82C2×C3⋊Dic3—122no—
83C2×C3⋊Dic3—122no—
84C2×C3⋊Dic3C4×D311yescharacteristic
85C2×C3⋊Dic3C22×D311yescharacteristic
86C2×C3×Dic3—28no—
87C2×C3×Dic3—22no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—68no—
92#622—22no—
93#622—62no—
94#431—271noSylow
95C2×C33—22no—
96C2×C33#11711yescharacteristic
97C2×C33C22×C811yescharacteristic
98C2×C33C2×M1611yescharacteristic
99C22×C3×C4—62no—
100C22×C3×C4—91no—
101C22×Dic3—62no—
102C22×Dic3—91no—
103C22×Dic3—182no—
104C22×Dic3—362no—
105C22×Dic3#15711yescharacteristic
106C2×C3×C8—92no—
107C4×Dic3—91no—
108C4×Dic3—91no—
109C4×Dic3—91no—
110C4×Dic3—91no—
111C4×Dic3—182no—
112C4×Dic3—182no—
113C4×Dic3—182no—
114C4×Dic3—182no—
115C2×C3⋊C8—92no—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32—24no—
119C22×C32—24no—
120C22×C32—22no—
121C22×C32—24no—
122C22×C32—44no—
123C22×C32—44no—
124C22×C32—44no—
125C22×C32—42no—
126C22×C32D3×C812yes—
127C22×C32C2×C4×D311yescharacteristic
128C32×C4—32no—
129C32×C4—61no—
130C3⋊Dic3—21no—
131C3⋊Dic3—32no—
132C3⋊Dic3—32no—
133C3⋊Dic3—64no—
134C3⋊Dic3—64no—
135C3⋊Dic3—124no—
136C3⋊Dic3—124no—
137C3⋊Dic3#23712yes—
138C3×Dic3—24no—
139C3×Dic3—24no—
140C3×Dic3—64no—
141C3×Dic3—64no—
142C2×C42—271no—
143C4⋊C8—272no—
144C22⋊C8—91no—
145C22⋊C8—271no—
146C4×C8—272no—
147C33#43111yescharacteristic, complemented, Sylow
148C23×C3—22no—
149C23×C3—42no—
150C23×C3C2×#15711yescharacteristic
151C2×C3×C4—62no—
152C2×C3×C4—62no—
153C2×C3×C4—68no—
154C2×C3×C4—91no—
155C2×C3×C4—91no—
156C2×C3×C4—182no—
157C2×Dic3—22no—
158C2×Dic3—62no—
159C2×Dic3—62no—
160C2×Dic3—68no—
161C2×Dic3—91no—
162C2×Dic3—91no—
163C2×Dic3—94no—
164C2×Dic3—182no—
165C2×Dic3—182no—
166C2×Dic3—188no—
167C2×Dic3—362no—
168C2×Dic3—362no—
169C2×Dic3—368no—
170C2×Dic3C2×#15711yescharacteristic
171C2×Dic3C2×#15711yescharacteristic
172C3×C8—182no—
173C3⋊C8—182no—
174C2×C32—22no—
175C2×C32—28no—
176C2×C32—22no—
177C2×C32—22no—
178C2×C32—22no—
179C2×C32—48no—
180C2×C32—42no—
181C2×C32—42no—
182C2×C32—42no—
183C2×C32#87711yescharacteristic
184C2×C32C2×D3×C811yescharacteristic
185C2×C32D3×M1611yescharacteristic
186C22×C4—31no—
187C22×C4—91no—
188C22×C4—271no—
189C2×C8—92no—
190C2×C8—272no—
191C42—271no—
192C42—271no—
193C42—271no—
194C42—271no—
195C22×C3—24no—
196C22×C3—24no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—22no—
200C22×C3—24no—
201C22×C3—44no—
202C22×C3—44no—
203C22×C3—42no—
204C22×C3—44no—
205C22×C3#378712yes—
206C22×C3C22×#15711yescharacteristic
207C3×C4—64no—
208C3×C4—64no—
209C3×C4—92no—
210C3×C4—181no—
211Dic3—21no—
212Dic3—64no—
213Dic3—64no—
214Dic3—92no—
215Dic3—92no—
216Dic3—184no—
217Dic3—184no—
218Dic3—364no—
219Dic3—364no—
220Dic3#379212yes—
221C32—22no—
222C32—42no—
223C32#471811yescharacteristic, complemented
224C23D3×#15711yescharacteristic
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—62no—
228C2×C4—91no—
229C2×C4—91no—
230C2×C4—182no—
231C2×C4—271no—
232C2×C4—271no—
233C2×C4—274no—
234C8—182no—
235C8—542no—
236C2×C3—28no—
237C2×C3—22no—
238C2×C3—22no—
239C2×C3—22no—
240C2×C3—22no—
241C2×C3—48no—
242C2×C3—42no—
243C2×C3—42no—
244C2×C3—42no—
245C2×C3C2×#378711yescharacteristic
246C2×C3#6428611yescharacteristic
247C2×C3C2×#379211yescharacteristic
248C22—21no—
249C22—21no—
250C22#8892412yes—
251C22C2×D3×#15711yescharacteristic
252C4—32no—
253C4—61no—
254C4—92no—
255C4—181no—
256C4—272no—
257C4—272no—
258C3—22no—
259C3—42no—
260C3#1059380111yescharacteristic, complemented
261C2—22no—
262C2#1172426111yescharacteristic
263C2D3×#379211yescharacteristic
264C2C2×#8892411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutatorC2×C33C22×C8
Fitting, prime radicalC23×C33C2×C4
socleC22×C33C22×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D3×#157
hypocenterC33#431
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722732C2semidirect
5. Dic3×#622C2semidirect
23. C2×C3⋊(C3⋊Dic3)C8semidirect
147. C33#431semidirect
223. C32#4718semidirect
260. C3#10593801semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×C3⋊Dic32
#117231122
C3×C22⋊#6222
#117227322
Dic3×#6222
#105938013
#47189