Finite Group Databasebeta

← #11723037

Subgroups — #11723037

Order 864.

394 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 84 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×#6293C212yesmaximal
2#88815C211yesmaximal, characteristic
3Dic3×C3⋊Dic3C211yesmaximal, characteristic
4C3×#3724C211yesmaximal, characteristic
5C3×Dic3⋊Dic3C212yesmaximal
6#63919—31nomaximal
7#63840—32nomaximal
8C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C2×C3×C3⋊Dic3C2212yes—
10C2×C3×C3⋊Dic3C2211yescharacteristic
11C2×C32×Dic3C2212yes—
12C2×C32×Dic3C2211yescharacteristic
13#6293—24no—
14C2×#634—32no—
15C2×#627—32no—
16C2×#627—32no—
17C2×#627—62no—
18C4×C3⋊Dic3—31no—
19C3×C4⋊Dic3—32no—
20C3×Dic3⋊C4—32no—
21C3×Dic3⋊C4—32no—
22#3724—31no—
23#3724D311yescharacteristic
24Dic3⋊Dic3—32no—
25Dic3⋊Dic3D312yes—
26Dic32—32no—
27Dic32—61no—
28C22×C33C2311yescharacteristic
29C3⋊(C3⋊Dic3)D412yes—
30C3×C3⋊Dic3—22no—
31C3×C3⋊Dic3Q812yescomplemented
32C32×Dic3—22no—
33C32×Dic3Q812yescomplemented
34#888—91no—
35#888—92no—
36C2×C32×C4—31no—
37C2×C32×C4—32no—
38C2×C3⋊Dic3—32no—
39C2×C3⋊Dic3—31no—
40C2×C3⋊Dic3—61no—
41C2×C3⋊Dic3—62no—
42C2×C3⋊Dic3—121no—
43C2×C3⋊Dic3C2×D312yes—
44C2×C3⋊Dic3C2×D311yescharacteristic
45#634—64no—
46C2×C3×Dic3—21no—
47C2×C3×Dic3—22no—
48C2×C3×Dic3—32no—
49C2×C3×Dic3—32no—
50C2×C3×Dic3—32no—
51C2×C3×Dic3—61no—
52C2×C3×Dic3—62no—
53C2×C3×Dic3C2×D312yes—
54C2×C3×Dic3C2×D312yes—
55C2×C3×Dic3C2×D312yes—
56#627—64no—
57#627—64no—
58#627—68no—
59C2×C33C2×D411yescharacteristic
60C2×C33C2×Q811yescharacteristic
61C2×C33C2×Q811yescharacteristic
62C2×Dic6—92no—
63C2×Dic6—92no—
64C2×Dic6—92no—
65C2×Dic6—182no—
66C3×C4⋊C4—91no—
67C3×C4⋊C4—92no—
68C4⋊Dic3—32no—
69C4⋊Dic3—91no—
70Dic3⋊C4—32no—
71Dic3⋊C4—32no—
72Dic3⋊C4—92no—
73C4×Dic3—92no—
74C4×Dic3—91no—
75C4×Dic3—181no—
76C22×C32—22no—
77C22×C32—21no—
78C22×C32—41no—
79C22×C32C22×D311yescharacteristic
80C22×C32C22×D312yes—
81C32×C4—32no—
82C32×C4—62no—
83C3⋊Dic3—22no—
84C3⋊Dic3—32no—
85C3⋊Dic3—34no—
86C3⋊Dic3—64no—
87C3⋊Dic3—62no—
88C3⋊Dic3—122no—
89C3⋊Dic3Dic612yescomplemented
90C3×Dic3—24no—
91C3×Dic3—22no—
92C3×Dic3—22no—
93C3×Dic3—22no—
94C3×Dic3—34no—
95C3×Dic3—62no—
96C3×Dic3—64no—
97C3×Dic3—62no—
98C3×Dic3—62no—
99C3×Dic3Dic614yescomplemented
100C4⋊Q8—271noSylow
101C33C4⋊Q811yescharacteristic, complemented, Sylow
102C2×C3×C4—32no—
103C2×C3×C4—32no—
104C2×C3×C4—32no—
105C2×C3×C4—61no—
106C2×C3×C4—62no—
107C2×C3×C4—91no—
108C2×C3×C4—92no—
109C2×Dic3—32no—
110C2×Dic3—32no—
111C2×Dic3—32no—
112C2×Dic3—61no—
113C2×Dic3—62no—
114C2×Dic3—91no—
115C2×Dic3—92no—
116C2×Dic3—181no—
117C2×Dic3—182no—
118C2×Dic3—361no—
119C2×Dic3D3212yes—
120C2×Dic3D3211yescharacteristic
121Dic6—184no—
122Dic6—184no—
123Dic6—184no—
124Dic6—188no—
125C2×C32—22no—
126C2×C32—22no—
127C2×C32—21no—
128C2×C32—21no—
129C2×C32—21no—
130C2×C32—22no—
131C2×C32—41no—
132C2×C32—41no—
133C2×C32—41no—
134C2×C32C2×Dic611yescharacteristic
135C2×C32C2×Dic612yes—
136C2×C32D3×D412yes—
137C2×C32D3×D411yescharacteristic
138C2×C32D3×Q811yescharacteristic
139C2×C32D3×Q812yes—
140C2×Q8—272no—
141C4⋊C4—91no—
142C4⋊C4—92no—
143C4⋊C4—271no—
144C42—271no—
145C22×C3—22no—
146C22×C3—21no—
147C22×C3—41no—
148C22×C3C2×D3212yes—
149C22×C3C2×D3211yescharacteristic
150C3×C4—34no—
151C3×C4—62no—
152C3×C4—64no—
153C3×C4—62no—
154C3×C4—62no—
155C3×C4—92no—
156C3×C4—182no—
157Dic3—22no—
158Dic3—34no—
159Dic3—62no—
160Dic3—64no—
161Dic3—62no—
162Dic3—62no—
163Dic3—92no—
164Dic3—94no—
165Dic3—184no—
166Dic3—182no—
167Dic3—362no—
168Dic3#62712yescomplemented
169C32—22no—
170C32—21no—
171C32—41no—
172C32#88811yescharacteristic, complemented
173C32#88812yescomplemented
174Q8—544no—
175C2×C4—31no—
176C2×C4—32no—
177C2×C4—91no—
178C2×C4—92no—
179C2×C4—271no—
180C2×C3—22no—
181C2×C3—21no—
182C2×C3—21no—
183C2×C3—22no—
184C2×C3—21no—
185C2×C3—22no—
186C2×C3—41no—
187C2×C3—41no—
188C2×C3—41no—
189C2×C3D3×Dic612yes—
190C2×C3D3×Dic612yes—
191C2×C3#379711yescharacteristic
192C2×C3#380211yescharacteristic
193C2×C3C2×#62711yescharacteristic
194C2×C3#381112yes—
195C22D3311yescharacteristic
196C4—32no—
197C4—62no—
198C4—92no—
199C4—182no—
200C4—272no—
201C3—22no—
202C3—21no—
203C3—41no—
204C3#6384012yescomplemented
205C3#6391911yescharacteristic, complemented
206C2#8895411yescharacteristic
207C2D3×#62711yescharacteristic
208C2#8895711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, prime radical, cosocleC22×C33C23
hypocenterC33C4⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
31. C3×C3⋊Dic3Q8semidirect
33. C32×Dic3Q8semidirect
89. C3⋊Dic3Dic6semidirect
99. C3×Dic3Dic6semidirect
101. C33C4⋊Q8semidirect
168. Dic3#627semidirect
172. C32#888semidirect
173. C32#888semidirect
204. C3#63840semidirect
205. C3#63919semidirect

Maximal subgroups

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

subgroupindex
C2×#62932
#888152
Dic3×C3⋊Dic32
C3×#37242
C3×Dic3⋊Dic32
#639193
#638403