Finite Group Databasebeta

← #13147423

Subgroups — #13147423

Order 1728.

591 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 75 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×Q8×#272C211yesmaximal, characteristic
2C2×C4⋊#1088C211yesmaximal, characteristic
3C2×C4×#1088C211yesmaximal, characteristic
4#11720787C214yesmaximal
5C22×#1088C2212yes—
6C22×#1088C2212yes—
7C2×Q8×H(3,𝔽3)C411yescharacteristic, complemented
8Q8×#272—22no—
9Q8×#272—22no—
10C2×C4×#272C2212yes—
11C2×C4×#272C2211yescharacteristic
12C2×#6229C411yescharacteristic, complemented
13C2×#6229C412yescomplemented
14C4⋊#1088—22no—
15C4×#1088—22no—
16C2×C3×D3×Q8—62no—
17C22×#272C2311yescharacteristic
18C2×#1088—22no—
19C2×#1088—22no—
20C2×#1088—22no—
21C2×#1088—24no—
22C2×#1088Q814yescomplemented
23Q8×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)C2×C412yes—
25C2×C4×H(3,𝔽3)C2×C411yescharacteristic
26C2×#1084C2×C412yes—
27C2×#1084C2×C411yescharacteristic
28C4×#272—22no—
29C4×#272—22no—
30C4×#272D412yes—
31C4×#272D412yes—
32#6229—22no—
33#6229—24no—
34C3×#391—91nomaximal
35C2×C32×Q8—22no—
36C3×D3×Q8—68no—
37C3×D3×Q8—68no—
38C2×C3×C4×D3—62no—
39C2×C3×C4×D3—64no—
40C2×C3×Dic6—64no—
41C2×C3×Dic6—62no—
42C22×H(3,𝔽3)C22×C411yescharacteristic
43C2×#272C2×D411yescharacteristic
44C2×#272C2×D411yescharacteristic
45C2×#272C2×Q812yes—
46C2×#272#4112yes—
47#1088—24no—
48#1088—42no—
49C4×H(3,𝔽3)—22no—
50C4×H(3,𝔽3)C22⋊C412yes—
51#1084—22no—
52#1084C22⋊C412yes—
53C22×C3×Q8—91no—
54C2×D3×Q8—62no—
55C2×C3×C4⋊C4—91no—
56C2×C3×C42—91no—
57C3×#95—94no—
58C22×C3×D3—62no—
59C32×Q8—28no—
60C2×C32×C4—24no—
61C2×C32×C4—22no—
62C2×C3×Dic3—62no—
63C2×C3×Dic3—64no—
64C3×C4×D3—64no—
65C3×C4×D3—64no—
66C3×C4×D3—68no—
67C3×C4×D3—68no—
68C3×Dic6—616no—
69C3×Dic6—68no—
70#391—91noSylow
71C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
72C2×H(3,𝔽3)C4×Q812yes—
73#272C22⋊Q812yes—
74#272#12411yescharacteristic
75#272C4⋊Q811yescharacteristic
76C2×C3×Q8—62no—
77C2×C3×Q8—92no—
78C2×C3×Q8—91no—
79C2×C3×Q8—182no—
80C2×C3×Q8—182no—
81C2×C3×Q8#15711yescharacteristic
82C22×C3×C4—91no—
83C22×C3×C4—92no—
84C22×C3×C4—92no—
85C22×C3×C4—92no—
86D3×Q8—68no—
87D3×Q8—68no—
88C2×C4×D3—62no—
89C2×C4×D3—64no—
90C2×Dic6—64no—
91C2×Dic6—62no—
92C3×C4⋊C4—182no—
93C3×C42—182no—
94C22×C32—22no—
95C2×C3×D3—62no—
96C2×C3×D3—62no—
97C2×C3×D3—64no—
98C2×C3×D3—64no—
99C32×C4—28no—
100C32×C4—24no—
101C3×Dic3—64no—
102C3×Dic3—68no—
103C22×Q8—91no—
104C2×C4⋊C4—91no—
105C2×C42—91no—
106#95—94no—
107H(3,𝔽3)#39111yescharacteristic, complemented, Sylow
108C23×C3—91no—
109C22×D3—62no—
110C3×Q8—22no—
111C3×Q8—68no—
112C3×Q8—184no—
113C3×Q8—182no—
114C2×C3×C4—64no—
115C2×C3×C4—62no—
116C2×C3×C4—91no—
117C2×C3×C4—92no—
118C2×C3×C4—92no—
119C2×C3×C4—94no—
120C2×C3×C4—92no—
121C2×C3×C4—182no—
122C2×C3×C4—182no—
123C2×C3×C4—182no—
124C2×C3×C4—182no—
125C2×C3×C4—182no—
126C2×C3×C4—184no—
127C2×C3×C4C2×#15712yes—
128C2×C3×C4C2×#15711yescharacteristic
129C2×Dic3—62no—
130C2×Dic3—64no—
131C4×D3—64no—
132C4×D3—64no—
133C4×D3—68no—
134C4×D3—68no—
135Dic6—616no—
136Dic6—68no—
137C2×C32—24no—
138C2×C32—22no—
139C3×D3—64no—
140C3×D3—62no—
141C3×D3—62no—
142C2×Q8—92no—
143C2×Q8—91no—
144C2×Q8—182no—
145C2×Q8—182no—
146C2×Q8#108811yescharacteristic, complemented
147C22×C4—91no—
148C22×C4—92no—
149C22×C4—92no—
150C22×C4—92no—
151C4⋊C4—182no—
152C42—182no—
153C22×C3—62no—
154C22×C3—92no—
155C22×C3—92no—
156C22×C3—91no—
157C22×C3—91no—
158C22×C3C22×#15711yescharacteristic
159C2×D3—62no—
160C2×D3—62no—
161C2×D3—64no—
162C2×D3—64no—
163C3×C4—22no—
164C3×C4—68no—
165C3×C4—64no—
166C3×C4—92no—
167C3×C4—184no—
168C3×C4—182no—
169C3×C4—362no—
170C3×C4C22⋊#15712yes—
171Dic3—64no—
172Dic3—68no—
173C32—22no—
174C23—91no—
175Q8—22no—
176Q8—184no—
177Q8—182no—
178C2×C4—91no—
179C2×C4—92no—
180C2×C4—92no—
181C2×C4—94no—
182C2×C4—92no—
183C2×C4—182no—
184C2×C4—182no—
185C2×C4—182no—
186C2×C4—182no—
187C2×C4—182no—
188C2×C4—184no—
189C2×C4C2×#108812yes—
190C2×C4C2×#108811yescharacteristic
191C2×C3—64no—
192C2×C3—62no—
193C2×C3—92no—
194C2×C3—91no—
195C2×C3—91no—
196C2×C3Q8×#15712yes—
197C2×C3C2×C22⋊#15711yescharacteristic
198D3—64no—
199D3—62no—
200D3—62no—
201C22—92no—
202C22—92no—
203C22—91no—
204C22—91no—
205C22C22×#108811yescharacteristic
206C4—22no—
207C4—92no—
208C4—184no—
209C4—182no—
210C4—362no—
211C4C22⋊#108812yes—
212C3—62no—
213C3#1059383711yescharacteristic
214C2—92no—
215C2—91no—
216C2—91no—
217C2Q8×#108812yes—
218C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×Q8×H(3,𝔽3)C4
prime radical, cosocleC22×#272C23
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#391
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
7. C2×Q8×H(3,𝔽3)C4semidirect
12. C2×#6229C4semidirect
13. C2×#6229C4semidirect
22. C2×#1088Q8semidirect
107. H(3,𝔽3)#391semidirect
146. C2×Q8#1088semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
C2×C4⋊#10882
C2×C4×#10882
#117207872
C3×#3919