Finite Group Databasebeta

← #13165741

Subgroups — #13165741

Order 1728.

384 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 60 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#11723032C212yesmaximal
2#11722797C211yesmaximal, characteristic
3#11722779C212yesmaximal
4C3×#63743C211yesmaximal, characteristic
5Dic3×#622C211yesmaximal, characteristic
6#10593388—31nomaximal
7C2×C3×#627—22no—
8C2×C3⋊#622C2211yescharacteristic
9C2×C3×#622C2211yescharacteristic
10#88815C2212yes—
11#88806—22no—
12Dic3×C3⋊Dic3C2211yescharacteristic
13C3×#3724C2212yes—
14#63873—32no—
15#63839—62no—
16C4×#622—31no—
17#63743—31no—
18#63743—32no—
19#63743D311yescharacteristic
20C2×C3⋊(C3⋊Dic3)D411yescharacteristic
21C2×C3×C3⋊Dic3—22no—
22C2×C3×C3⋊Dic3C2311yescharacteristic
23C2×C32×Dic3—22no—
24C2×C32×Dic3D411yescharacteristic
25C3×#627—44no—
26C3⋊#622Q812yescomplemented
27C3×#622Q812yescomplemented
28#4874—91nomaximal
29C2×C3×Dic6—62no—
30C2×C3×Dic6—62no—
31C2×#627—22no—
32C2×#622—31no—
33C2×#622C2×D311yescharacteristic
34#3750—62no—
35C4×C3⋊Dic3—31no—
36C3×Dic3⋊C4—62no—
37#3724—32no—
38#3724—62no—
39#3724—62no—
40#3724—122no—
41#3724C2×D312yes—
42Dic3⋊Dic3—62no—
43Dic32—62no—
44C22×C33C2×D411yescharacteristic
45C3⋊(C3⋊Dic3)Q1612yes—
46C3×C3⋊Dic3—42no—
47C3×C3⋊Dic3C2×Q811yescharacteristic
48C3×C3⋊Dic3C2×Q811yescharacteristic
49C32×Dic3—42no—
50C32×Dic3D812yes—
51#944—182no—
52#888—92no—
53C3×#107—91no—
54#818—91no—
55C8×Dic3—91no—
56#807—92no—
57C2×C32×C4—31no—
58C2×C32×C4—62no—
59C2×C3⋊Dic3—22no—
60C2×C3⋊Dic3—31no—
61C2×C3⋊Dic3—62no—
62C2×C3⋊Dic3—122no—
63C2×C3⋊Dic3C22×D311yescharacteristic
64C2×C3×Dic3—22no—
65C2×C3×Dic3—22no—
66C2×C3×Dic3—22no—
67C2×C3×Dic3—42no—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—62no—
71C2×C3×Dic3—122no—
72C3×Dic6—124no—
73C3×Dic6—124no—
74#627—44no—
75#622—32no—
76#622Dic612yescomplemented
77#500—271noSylow
78C2×C33C4⋊Q811yescharacteristic
79C2×C33C2×D811yescharacteristic
80C2×C33C2×Q1611yescharacteristic
81C2×C3×Q8—182no—
82C2×Dic6—62no—
83C2×Dic6—62no—
84C2×C3×C8—91no—
85C3×C4⋊C4—92no—
86C4⋊Dic3—92no—
87Dic3⋊C4—62no—
88Dic3⋊C4—182no—
89Dic3⋊C4—182no—
90Dic3⋊C4—182no—
91Dic3⋊C4—182no—
92Dic3⋊C4—362no—
93C4×Dic3—91no—
94C4×Dic3—182no—
95C2×C3⋊C8—91no—
96C22×C32—22no—
97C22×C32—42no—
98C22×C32D3×D411yescharacteristic
99C32×C4—32no—
100C32×C4—122no—
101C3⋊Dic3—32no—
102C3⋊Dic3—42no—
103C3⋊Dic3—64no—
104C3⋊Dic3—124no—
105C3⋊Dic3C2×Dic611yescharacteristic
106C3⋊Dic3D3×Q811yescharacteristic
107C3×Dic3—24no—
108C3×Dic3—42no—
109C3×Dic3—42no—
110C3×Dic3—62no—
111C3×Dic3—62no—
112C3×Dic3—82no—
113C3×Dic3—122no—
114C3×Dic3—122no—
115C3×Dic3—242no—
116C4⋊Q8—272no—
117#107—91no—
118#107—272no—
119#107—271no—
120C4×C8—271no—
121C33#50011yescharacteristic, complemented, Sylow
122C3×Q8—364no—
123C2×C3×C4—62no—
124C2×C3×C4—62no—
125C2×C3×C4—62no—
126C2×C3×C4—91no—
127C2×C3×C4—122no—
128C2×C3×C4—182no—
129C2×Dic3—22no—
130C2×Dic3—62no—
131C2×Dic3—62no—
132C2×Dic3—62no—
133C2×Dic3—91no—
134C2×Dic3—122no—
135C2×Dic3—182no—
136C2×Dic3—362no—
137C2×Dic3GO+(4,𝔽2)11yescharacteristic
138Dic6—124no—
139Dic6—124no—
140C3×C8—92no—
141C3⋊C8—92no—
142C2×C32—22no—
143C2×C32—22no—
144C2×C32—22no—
145C2×C32—42no—
146C2×C32—42no—
147C2×C32—42no—
148C2×C32#88811yescharacteristic
149C2×C32D3×D811yescharacteristic
150C2×C32D3×Q1611yescharacteristic
151C2×Q8—182no—
152C2×C8—91no—
153C2×C8—271no—
154C4⋊C4—92no—
155C4⋊C4—272no—
156C4⋊C4—542no—
157C42—271no—
158C22×C3—22no—
159C22×C3—42no—
160C22×C3C2×GO+(4,𝔽2)11yescharacteristic
161C3×C4—64no—
162C3×C4—91no—
163C3×C4—91no—
164C3×C4—122no—
165C3×C4—122no—
166C3×C4—242no—
167C3×C4—362no—
168Dic3—42no—
169Dic3—62no—
170Dic3—62no—
171Dic3—92no—
172Dic3—122no—
173Dic3—122no—
174Dic3—184no—
175Dic3—242no—
176Dic3—364no—
177Dic3#377412yes—
178C32—22no—
179C32—42no—
180C32#487411yescharacteristic, complemented
181Q8—364no—
182C2×C4—31no—
183C2×C4—62no—
184C2×C4—91no—
185C2×C4—182no—
186C2×C4—271no—
187C8—92no—
188C8—272no—
189C2×C3—22no—
190C2×C3—22no—
191C2×C3—22no—
192C2×C3—42no—
193C2×C3—42no—
194C2×C3—42no—
195C2×C3#6422211yescharacteristic
196C2×C3#6422811yescharacteristic
197C2×C3C2×#377411yescharacteristic
198C22D3×GO+(4,𝔽2)11yescharacteristic
199C4—32no—
200C4—91no—
201C4—91no—
202C4—122no—
203C4—272no—
204C4—362no—
205C3—22no—
206C3—42no—
207C3#1059338811yescharacteristic, complemented
208C2#1172412511yescharacteristic
209C2D3×#377411yescharacteristic
210C2#1172413111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socle, prime radicalC22×C33C2×D4
hypocenterC33#500
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
26. C3⋊#622Q8semidirect
27. C3×#622Q8semidirect
76. #622Dic6semidirect
121. C33#500semidirect
180. C32#4874semidirect
207. C3#10593388semidirect

Maximal subgroups

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

subgroupindex
#117230322
#117227972
#117227792
C3×#637432
Dic3×#6222
#105933883
#48749