Finite Group Databasebeta

← #11721094

Subgroups — #11721094

Order 864.

313 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 62 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×C4×#272C211yesmaximal, characteristic
2C2×#634⋊C3C211yesmaximal, characteristic, complemented
3C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4#88451C212yesmaximal
5#88448C212yesmaximal, complemented
6#63857—31nomaximal
7#63856—31nomaximal
8C22×#272C2211yescharacteristic
9C2×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#1084C2211yescharacteristic
11C4×#272—22no—
12C2×(C3⋊Dic3)⋊C3C2212yes—
13C2×(C3⋊Dic3)⋊C3C2212yes—
14#634⋊C3—22no—
15C2×#634D311yescharacteristic, complemented
16C2×C3×C4×D3—31no—
17C2×C3×C4×D3—31no—
18C2×C3×C4×D3—61no—
19C2×C3×Dic6—31no—
20C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
21C3×C4⋊Dic3—31no—
22#3724—32no—
23Dic3⋊Dic3—32no—
24(C2×D3)⋊Dic3—32no—
25(C2×D3)⋊Dic3—32no—
26C22×H(3,𝔽3)C2311yescharacteristic
27C2×#272—21no—
28C2×#272—21no—
29C2×#272Q812yescomplemented
30C4×H(3,𝔽3)D412yes—
31#1084—21no—
32(C3⋊Dic3)⋊C3—22no—
33(C3⋊Dic3)⋊C3—22no—
34#946—91no—
35#924—91no—
36#897—91no—
37C22×C3×D3—31no—
38C22×C3×D3—31no—
39C22×C3×D3—61no—
40C2×C32×C4—21no—
41C2×C32×C4C2×D311yescharacteristic
42C2×C32×C4C2×D311yescharacteristic
43C2×C3⋊Dic3C2×D312yes—
44C2×C3⋊Dic3C2×D312yes—
45#634—22no—
46C2×C3×Dic3—32no—
47C2×C3×Dic3—32no—
48C2×C3×Dic3—31no—
49C2×C3×Dic3—31no—
50C2×C3×Dic3—61no—
51C3×C4×D3—64no—
52C3×C4×D3—62no—
53C3×C4×D3—62no—
54C3×Dic6—62no—
55C2×H(3,𝔽3)C2×D411yescharacteristic
56C2×H(3,𝔽3)C2×Q811yescharacteristic
57C2×H(3,𝔽3)#4111yescharacteristic
58#272—22no—
59C2×C3×Q8—91no—
60C22×C3×C4—91no—
61C2×C4×D3—31no—
62C2×C4×D3—31no—
63C2×C4×D3—61no—
64C2×Dic6—31no—
65C2×Dic6—91no—
66C3×C4⋊C4—91no—
67#217—92no—
68(C2×D3)⋊C4—92no—
69(C2×D3)⋊C4—92no—
70C4⋊Dic3—31no—
71C4⋊Dic3—92no—
72C4⋊Dic3—181no—
73Dic3⋊C4—92no—
74Dic3⋊C4—92no—
75C22×C32—21no—
76C22×C32C22×D311yescharacteristic
77C22×C32C22×D311yescharacteristic
78C2×C3×D3—32no—
79C2×C3×D3—32no—
80C2×C3×D3—62no—
81C2×C3×D3—62no—
82C2×C3×D3—62no—
83C2×C3×D3—61no—
84C2×C3×D3—61no—
85C2×C3×D3—61no—
86C2×C3×D3—61no—
87C32×C4—22no—
88C32×C4D1212yes—
89C32×C4#6712yes—
90C3⋊Dic3—22no—
91C3⋊Dic3—22no—
92C3×Dic3—62no—
93C3×Dic3—62no—
94C3×Dic3—61no—
95C3×Dic3—62no—
96C3×Dic3—61no—
97C22⋊Q8—271noSylow
98H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
99C23×C3—91no—
100C22×D3—31no—
101C22×D3—31no—
102C22×D3—61no—
103C3×Q8—182no—
104C2×C3×C4—31no—
105C2×C3×C4—61no—
106C2×C3×C4—92no—
107C2×C3×C4—92no—
108C2×C3×C4—91no—
109C2×C3×C4—121no—
110C2×C3×C4—182no—
111C2×C3×C4D3211yescharacteristic
112C2×Dic3—31no—
113C2×Dic3—31no—
114C2×Dic3—32no—
115C2×Dic3—32no—
116C2×Dic3—61no—
117C2×Dic3—92no—
118C2×Dic3—182no—
119C4×D3—62no—
120C4×D3—62no—
121C4×D3—64no—
122Dic6—62no—
123Dic6—182no—
124C2×C32—21no—
125C2×C32—21no—
126C2×C32—21no—
127C2×C32C2×D1211yescharacteristic
128C2×C32#23711yescharacteristic
129C2×C32D3×Q811yescharacteristic
130C2×C32D3×Q811yescharacteristic
131C2×C32#23911yescharacteristic
132C2×C32C2×#6711yescharacteristic
133C3×D3—64no—
134C3×D3—62no—
135C3×D3—62no—
136C2×Q8—91no—
137C22×C4—91no—
138C4⋊C4—91no—
139C4⋊C4—272no—
140C22⋊C4—272no—
141C22×C3—31no—
142C22×C3—61no—
143C22×C3—92no—
144C22×C3—121no—
145C22×C3—181no—
146C22×C3—181no—
147C22×C3C2×D3211yescharacteristic
148C2×D3—32no—
149C2×D3—32no—
150C2×D3—62no—
151C2×D3—61no—
152C2×D3—61no—
153C2×D3—61no—
154C2×D3—61no—
155C2×D3—62no—
156C2×D3—62no—
157C3×C4—32no—
158C3×C4—62no—
159C3×C4—122no—
160C3×C4—182no—
161C3×C4—182no—
162C3×C4—181no—
163C3×C4#62612yes—
164Dic3—61no—
165Dic3—61no—
166Dic3—62no—
167Dic3—62no—
168Dic3—62no—
169Dic3—182no—
170Dic3—362no—
171C32—21no—
172C32#89711yescharacteristic, complemented
173C32#94611yescharacteristic, complemented
174C23—91no—
175Q8—182no—
176C2×C4—92no—
177C2×C4—92no—
178C2×C4—91no—
179C2×C4—182no—
180C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
181C2×C3—31no—
182C2×C3—31no—
183C2×C3—31no—
184C2×C3—61no—
185C2×C3—61no—
186C2×C3—61no—
187C2×C3—121no—
188C2×C3—121no—
189C2×C3—121no—
190C2×C3—182no—
191C2×C3#379611yescharacteristic
192C2×C3#379711yescharacteristic
193C2×C3C2×#62611yescharacteristic
194D3—62no—
195D3—62no—
196D3—64no—
197C22—92no—
198C22—181no—
199C22—181no—
200C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
201C4—182no—
202C4—181no—
203C4—182no—
204C4#620012yes—
205C3—31no—
206C3—61no—
207C3—121no—
208C3#6387111yescharacteristic
209C2—182no—
210C2#8864911yescharacteristic
211C2#8865111yescharacteristic
212C2C2×#620011yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC22×#272C22
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#634⋊C3C2semidirect
3. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. #88448C2semidirect
15. C2×#634D3semidirect
20. C4⋊(C3⋊Dic3)D3semidirect
29. C2×#272Q8semidirect
98. H(3,𝔽3)C22⋊Q8semidirect
172. C32#897semidirect
173. C32#946semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2722
C2×#634⋊C32
C4⋊((C3⋊Dic3)⋊C3)2
#884512
#884482
#638573
#638563