Finite Group Databasebeta

← #13173351

Subgroups — #13173351

Order 1728.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C3×#64096C211yesmaximal, characteristic
2#11723104C211yesmaximal, characteristic
3#11723104C211yesmaximal, characteristic
4#11723102C211yesmaximal, characteristic
5#11723102C211yesmaximal, characteristic
6C4⋊(C3⋊#622)C211yesmaximal, characteristic
7C4×C3⋊#622C211yesmaximal, characteristic
8#10593754—31nomaximal
9C2×C3×#634C411yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14C33×C4⋊C4C411yescharacteristic
15C3×#3750C411yescharacteristic
16C3×#3750C411yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20#64096D311yescharacteristic
21C3×#887—62no—
22#63779—31no—
23#63779—31no—
24#63777—31no—
25#63777—31no—
26C4⋊#622—31no—
27C4×#622—31no—
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C33×C4C2×C411yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C3×#634—22no—
36C3⋊#622—21no—
37C3⋊#622—21no—
38C3⋊#622—21no—
39C3⋊#622Q812yescomplemented
40#5024—91nomaximal
41C2×#634Dic311yescharacteristic
42C2×C3×Dic6—62no—
43C2×#622—31no—
44C2×#622—31no—
45C2×#622—31no—
46C2×#622—31no—
47C32×C4⋊C4—22no—
48C32×C4⋊C4—42no—
49C32×C4⋊C4Dic311yescharacteristic
50#3750Dic311yescharacteristic
51#3750Dic311yescharacteristic
52C4×C3⋊Dic3C2×D311yescharacteristic
53C4×C3⋊Dic3C2×D311yescharacteristic
54C4×C3⋊Dic3C2×D311yescharacteristic
55C3×Dic3⋊C4—62no—
56C3×Dic3⋊C4—62no—
57C3×C4×Dic3—62no—
58C3×C4×Dic3—62no—
59C3×C4×Dic3—62no—
60C22×C33C22×C411yescharacteristic
61C33×C4—21no—
62C33×C4—21no—
63C33×C4—21no—
64C3×C3⋊Dic3—21no—
65C3×C3⋊Dic3—21no—
66C3×C3⋊Dic3M1612yes—
67C3×C3⋊Dic3C2×Q811yescharacteristic
68C3×C3⋊Dic3#4111yescharacteristic
69C3×C4×Q8—91no—
70#887—62no—
71C4⋊(C3⋊C8)—91no—
72C4⋊(C3⋊C8)—91no—
73C4⋊(C3⋊C8)—91no—
74#803—91no—
75#803—91no—
76C4×C3⋊C8—91no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—42no—
81C2×C32×C4—42no—
82C2×C32×C4—42no—
83C2×C32×C4C2×Dic311yescharacteristic
84C2×C32×C4C2×Dic311yescharacteristic
85C2×C32×C4C2×Dic311yescharacteristic
86C2×C3⋊Dic3C2×Dic311yescharacteristic
87C2×C3⋊Dic3C2×Dic311yescharacteristic
88C2×C3⋊Dic3C2×Dic311yescharacteristic
89C2×C3⋊Dic3C22×D311yescharacteristic
90#634—22no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C3×Dic6—68no—
96#622—32no—
97#622—61no—
98#622—61no—
99#622—61no—
100#446—271noSylow
101C2×C33C4×Q811yescharacteristic
102C2×C33C2×M1611yescharacteristic
103C2×C33#13111yescharacteristic
104C2×C3×Q8—91no—
105C2×Dic6—62no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—42no—
108C3×C4⋊C4—91no—
109C3×C4⋊C4—91no—
110C3×C4⋊C4#15711yescharacteristic
111C3×C42—91no—
112C3×C42—91no—
113C3×C42—91no—
114Dic3⋊C4—62no—
115Dic3⋊C4—62no—
116C4×Dic3—62no—
117C4×Dic3—62no—
118C4×Dic3—62no—
119C2×C3⋊C8—91no—
120C2×C3⋊C8—91no—
121C2×C3⋊C8—91no—
122C2×C3⋊C8—91no—
123C22×C32—22no—
124C22×C32—42no—
125C22×C32C22×Dic311yescharacteristic
126C32×C4—21no—
127C32×C4—24no—
128C32×C4—21no—
129C32×C4—21no—
130C32×C4—44no—
131C32×C4—42no—
132C32×C4—42no—
133C32×C4—82no—
134C32×C4—82no—
135C3⋊Dic3—21no—
136C3⋊Dic3—21no—
137C3⋊Dic3#20812yes—
138C3⋊Dic3D3×Q811yescharacteristic
139C3⋊Dic3#23911yescharacteristic
140C3×Dic3—64no—
141C3×Dic3—62no—
142C3×Dic3—62no—
143C3×Dic3—64no—
144C3×Dic3—122no—
145C4×Q8—91no—
146C4⋊C8—271no—
147C4⋊C8—271no—
148C4⋊C8—271no—
149#97—271no—
150#97—271no—
151C4×C8—271no—
152C33#44611yescharacteristic, complemented, Sylow
153C3×Q8—182no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—91no—
164C2×C3×C4C2×#15711yescharacteristic
165C2×C3×C4C2×#15711yescharacteristic
166C2×C3×C4C2×#15711yescharacteristic
167C2×Dic3—62no—
168C2×Dic3—62no—
169C2×Dic3—62no—
170C2×Dic3—62no—
171Dic6—68no—
172C3⋊C8—92no—
173C3⋊C8—181no—
174C3⋊C8—181no—
175C3⋊C8—181no—
176C2×C32—22no—
177C2×C32—22no—
178C2×C32—22no—
179C2×C32—42no—
180C2×C32—42no—
181C2×C32—42no—
182C2×C32C2×#20811yescharacteristic
183C2×C32Q8×Dic311yescharacteristic
184C2×C32#94811yescharacteristic
185C2×Q8—91no—
186C2×C8—271no—
187C2×C8—271no—
188C2×C8—271no—
189C2×C8—271no—
190C4⋊C4—91no—
191C4⋊C4—91no—
192C4⋊C4C3⋊#15711yescharacteristic
193C42—91no—
194C42—91no—
195C42—91no—
196C22×C3—22no—
197C22×C3—42no—
198C22×C3C22×#15711yescharacteristic
199C3×C4—24no—
200C3×C4—21no—
201C3×C4—21no—
202C3×C4—21no—
203C3×C4—42no—
204C3×C4—42no—
205C3×C4—44no—
206C3×C4—82no—
207C3×C4—82no—
208C3×C4—92no—
209C3×C4—91no—
210C3×C4—91no—
211C3×C4—181no—
212C3×C4—181no—
213Dic3—64no—
214Dic3—62no—
215Dic3—62no—
216Dic3—64no—
217Dic3—122no—
218C32—22no—
219C32—42no—
220C32#502411yescharacteristic, complemented
221Q8—182no—
222C2×C4—91no—
223C2×C4—91no—
224C2×C4—91no—
225C2×C4—91no—
226C2×C4C2×C3⋊#15711yescharacteristic
227C2×C4C2×C3⋊#15711yescharacteristic
228C2×C4C2×C3⋊#15711yescharacteristic
229C8—272no—
230C8—541no—
231C8—541no—
232C8—541no—
233C2×C3—22no—
234C2×C3—22no—
235C2×C3—22no—
236C2×C3—42no—
237C2×C3—42no—
238C2×C3—42no—
239C2×C3#6428311yescharacteristic
240C2×C3#6428711yescharacteristic
241C2×C3Q8×#15711yescharacteristic
242C22C22×C3⋊#15711yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—92no—
247C4—91no—
248C4—91no—
249C4—181no—
250C4—181no—
251C3—22no—
252C3—42no—
253C3#1059375411yescharacteristic, complemented
254C2#1172451111yescharacteristic
255C2#1172451511yescharacteristic
256C2Q8×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socle, prime radicalC22×C33C22×C4
FittingC33×C4⋊C4C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC4⋊C4C3⋊#157
hypocenterC33#446
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
39. C3⋊#622Q8semidirect
152. C33#446semidirect
220. C32#5024semidirect
253. C3#10593754semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
#117231042
#117231022
C4⋊(C3⋊#622)2
C4×C3⋊#6222
#105937543
#50249