Finite Group Databasebeta

← #13166301

Subgroups — #13166301

Order 1728.

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

normal subgroups only

This group has 404 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
1#11723406C211yesmaximal, characteristic
2#11723104C211yesmaximal, characteristic
3C3×#63777C211yesmaximal, characteristic
4#11722732C211yesmaximal, characteristic
5Dic3⋊#622C211yesmaximal, characteristic
6#11722724C211yesmaximal, characteristic
7Dic3×#622C211yesmaximal, characteristic
8#10593754—31nomaximal
9C2×#6308C411yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C32×C4⋊Dic3C411yescharacteristic
16#88806C411yescharacteristic
17#88806C411yescharacteristic
18Dic3×C3⋊Dic3C2211yescharacteristic
19Dic3×C3⋊Dic3C2211yescharacteristic
20#64096—31no—
21#63844—62no—
22#63779—31no—
23#63779—31no—
24#63777—31no—
25#63777D311yescharacteristic
26C4⋊#622—31no—
27C4×#622—31no—
28C2×C33×C4C2×C411yescharacteristic
29C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
30C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
31#6308—22no—
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3C2×C411yescharacteristic
35C2×C32×Dic3C2×C411yescharacteristic
36C3⋊#622—21no—
37C3⋊#622—21no—
38C3×#622—21no—
39C3×#622Q812yescomplemented
40#4701—91nomaximal
41C2×#634—31no—
42C2×#634—62no—
43C2×#634—122no—
44C2×#622—31no—
45C2×#622—31no—
46C2×#622C2×D311yescharacteristic
47C2×#622C2×D311yescharacteristic
48C32×C4⋊C4—31no—
49#3750—31no—
50#3750—31no—
51C4×C3⋊Dic3—31no—
52C4×C3⋊Dic3—31no—
53C4×C3⋊Dic3C2×D311yescharacteristic
54C3×C4⋊Dic3—22no—
55C3×C4×Dic3—62no—
56#3724—62no—
57#3724—62no—
58Dic32—62no—
59Dic32—62no—
60C22×C33C22×C411yescharacteristic
61C33×C4—21no—
62C3⋊(C3⋊Dic3)—21no—
63C3⋊(C3⋊Dic3)M1612yes—
64C3×C3⋊Dic3—21no—
65C3×C3⋊Dic3C2×Q811yescharacteristic
66C3×C3⋊Dic3#4111yescharacteristic
67C32×Dic3—21no—
68C32×Dic3—21no—
69#887—182no—
70C4×Dic6—91no—
71C3×#97—91no—
72#815—91no—
73Dic3⋊C8—91no—
74Dic3⋊C8—91no—
75C8×Dic3—91no—
76C4⋊(C3⋊C8)—91no—
77C2×C32×C4—22no—
78C2×C32×C4—31no—
79C2×C32×C4—31no—
80C2×C32×C4—42no—
81C2×C32×C4C4×D311yescharacteristic
82C2×C3⋊Dic3—31no—
83C2×C3⋊Dic3—31no—
84C2×C3⋊Dic3—62no—
85C2×C3⋊Dic3—62no—
86C2×C3⋊Dic3—122no—
87C2×C3⋊Dic3—122no—
88C2×C3⋊Dic3C4×D311yescharacteristic
89C2×C3⋊Dic3C22×D311yescharacteristic
90#634—62no—
91#634—68no—
92#634—128no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97#622—21no—
98#622—61no—
99#622—61no—
100#622Dic612yescomplemented
101#446—271noSylow
102C2×C33C4×Q811yescharacteristic
103C2×C33C2×M1611yescharacteristic
104C2×C33#13111yescharacteristic
105C2×Dic6—91no—
106C2×Dic6—182no—
107C2×Dic6—362no—
108C2×C3×C8—91no—
109C2×C3×C8—91no—
110C3×C4⋊C4—62no—
111C3×C42—91no—
112C4⋊Dic3#15711yescharacteristic
113Dic3⋊C4—91no—
114Dic3⋊C4—91no—
115Dic3⋊C4—182no—
116Dic3⋊C4—182no—
117C4×Dic3—62no—
118C4×Dic3—91no—
119C4×Dic3—91no—
120C4×Dic3—182no—
121C4×Dic3—182no—
122C2×C3⋊C8—91no—
123C2×C3⋊C8—91no—
124C22×C32—22no—
125C22×C32—42no—
126C22×C32C2×C4×D311yescharacteristic
127C32×C4—21no—
128C32×C4—24no—
129C32×C4—44no—
130C32×C4—61no—
131C32×C4—61no—
132C3⋊Dic3—21no—
133C3⋊Dic3—32no—
134C3⋊Dic3—64no—
135C3⋊Dic3—61no—
136C3⋊Dic3—64no—
137C3⋊Dic3—124no—
138C3⋊Dic3—124no—
139C3⋊Dic3C2×Dic611yescharacteristic
140C3⋊Dic3#23511yescharacteristic
141C3×Dic3—42no—
142C3×Dic3—42no—
143C3×Dic3—62no—
144C3×Dic3—62no—
145C3×Dic3—122no—
146C4×Q8—271no—
147C4⋊C8—271no—
148C4⋊C8—271no—
149C4⋊C8—271no—
150#97—91no—
151#97—271no—
152C4×C8—271no—
153C33#44611yescharacteristic, complemented, Sylow
154C2×C3×C4—22no—
155C2×C3×C4—42no—
156C2×C3×C4—62no—
157C2×C3×C4—62no—
158C2×C3×C4—91no—
159C2×C3×C4—91no—
160C2×C3×C4C2×#15711yescharacteristic
161C2×Dic3—62no—
162C2×Dic3—62no—
163C2×Dic3—91no—
164C2×Dic3—91no—
165C2×Dic3—182no—
166C2×Dic3—182no—
167C2×Dic3—362no—
168C2×Dic3—362no—
169C2×Dic3C2×#15711yescharacteristic
170C2×Dic3C2×#15711yescharacteristic
171Dic6—182no—
172Dic6—188no—
173Dic6—368no—
174C3×C8—92no—
175C3×C8—181no—
176C3⋊C8—181no—
177C3⋊C8—181no—
178C2×C32—22no—
179C2×C32—22no—
180C2×C32—22no—
181C2×C32—42no—
182C2×C32—42no—
183C2×C32—42no—
184C2×C32C4×Dic611yescharacteristic
185C2×C32D3×M1611yescharacteristic
186C2×C32#90711yescharacteristic
187C2×Q8—271no—
188C2×C8—91no—
189C2×C8—91no—
190C2×C8—271no—
191C2×C8—271no—
192C4⋊C4—31no—
193C4⋊C4—271no—
194C4⋊C4—271no—
195C42—91no—
196C42—271no—
197C42—271no—
198C22×C3—22no—
199C22×C3—42no—
200C22×C3C22×#15711yescharacteristic
201C3×C4—24no—
202C3×C4—21no—
203C3×C4—44no—
204C3×C4—91no—
205C3×C4—91no—
206C3×C4—122no—
207C3×C4—122no—
208C3×C4—181no—
209Dic3—21no—
210Dic3—21no—
211Dic3—62no—
212Dic3—62no—
213Dic3—92no—
214Dic3—122no—
215Dic3—184no—
216Dic3—184no—
217Dic3—181no—
218Dic3—364no—
219Dic3—364no—
220C32—22no—
221C32—42no—
222C32#470111yescharacteristic, complemented
223Q8—542no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—91no—
227C2×C4—91no—
228C2×C4—271no—
229C2×C4—271no—
230C2×C4D3×#15711yescharacteristic
231C8—92no—
232C8—181no—
233C8—541no—
234C8—541no—
235C2×C3—22no—
236C2×C3—22no—
237C2×C3—22no—
238C2×C3—42no—
239C2×C3—42no—
240C2×C3—42no—
241C2×C3#6428311yescharacteristic
242C2×C3#6428711yescharacteristic
243C2×C3Q8×#15711yescharacteristic
244C22C2×D3×#15711yescharacteristic
245C4—21no—
246C4—61no—
247C4—61no—
248C4—91no—
249C4—91no—
250C4—181no—
251C4—272no—
252C4—541no—
253C3—22no—
254C3—42no—
255C3#1059375411yescharacteristic, complemented
256C2Dic6×#15711yescharacteristic
257C2#1172425111yescharacteristic
258C2#1172426911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socle, prime radicalC22×C33C22×C4
FittingC2×C33×C4C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#446
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
39. C3×#622Q8semidirect
100. #622Dic6semidirect
153. C33#446semidirect
222. C32#4701semidirect
255. C3#10593754semidirect

Maximal subgroups

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

subgroupindex
#117234062
#117231042
C3×#637772
#117227322
Dic3⋊#6222
#117227242
Dic3×#6222
#105937543
#47019