Finite Group Databasebeta

← #13147416

Subgroups — #13147416

Order 1728.

565 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 57 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×#88748C211yesmaximal, characteristic
2#11722448C212yesmaximal, complemented
3#11720780C214yesmaximal, complemented
4C2×Q8×H(3,𝔽3)C411yescharacteristic, complemented
5#88748—22no—
6#88748—22no—
7C2×C4⋊#272C411yescharacteristic, complemented
8C2×C4⋊#272C2212yes—
9C2×C4×#272C412yescomplemented
10C2×C4×#272C2211yescharacteristic
11C22⋊#1088C2214yes—
12C4⋊#1088—22no—
13C4×#1088—22no—
14C2×C3×#239—62no—
15C22×#272C2×C412yes—
16C22×#272C2311yescharacteristic
17C2×#1088—24no—
18Q8×H(3,𝔽3)—22no—
19C2×C4×H(3,𝔽3)C2×C412yes—
20C2×C4×H(3,𝔽3)C2×C411yescharacteristic
21C2×#1084C2×C411yescharacteristic
22C4⋊#272—24no—
23C4⋊#272—22no—
24C4×#272—22no—
25C4×#272—22no—
26C4×#272D412yes—
27C4×#272D412yes—
28C3×#410—91nomaximal
29C2×C32×Q8—22no—
30C3×#239—68no—
31C3×#239—68no—
32C2×C3×D12—62no—
33C2×C3×D12—64no—
34C2×C3×C4×D3—64no—
35C2×C3×C4×D3—62no—
36C22×H(3,𝔽3)C22×C411yescharacteristic
37C2×#272—22no—
38C2×#272—41no—
39C2×#272—42no—
40C2×#272C2×D412yes—
41#1088—44no—
42C4×H(3,𝔽3)—22no—
43C4×H(3,𝔽3)C22⋊C412yes—
44#1084C22⋊C411yescharacteristic
45#1084C22⋊C411yescharacteristic
46C2×C3×#41—91no—
47C2×#239—62no—
48C3×#117—92no—
49C3×#99—94no—
50C22×C3×D3—64no—
51C22×C3×D3—62no—
52C32×Q8—28no—
53C2×C32×C4—24no—
54C2×C32×C4—22no—
55C2×C3×Dic3—62no—
56C3×D12—616no—
57C3×D12—68no—
58C3×C4×D3—64no—
59C3×C4×D3—64no—
60C3×C4×D3—68no—
61C3×C4×D3—68no—
62#410—91noSylow
63C2×H(3,𝔽3)—21no—
64C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
65#272—22no—
66#272—42no—
67C3×#41—182no—
68C3×#41—182no—
69C2×C3×Q8—62no—
70C2×C3×Q8#15711yescharacteristic
71C2×C3×D4—92no—
72C2×C3×D4—91no—
73C22×C3×C4—91no—
74C22×C3×C4—92no—
75#239—68no—
76#239—68no—
77C2×D12—64no—
78C2×D12—62no—
79C2×C4×D3—62no—
80C2×C4×D3—64no—
81C3×C4⋊C4—182no—
82C3×C22⋊C4—94no—
83C3×C42—182no—
84C22×C32—22no—
85C2×C3×D3—64no—
86C2×C3×D3—68no—
87C2×C3×D3—124no—
88C2×C3×D3—128no—
89C32×C4—28no—
90C32×C4—24no—
91C3×Dic3—62no—
92C3×Dic3—62no—
93C2×#41—91no—
94#117—92no—
95#99—94no—
96H(3,𝔽3)#41011yescharacteristic, complemented, Sylow
97C23×C3—91no—
98C23×C3—92no—
99C22×D3—62no—
100C22×D3—64no—
101C3×Q8—22no—
102C3×Q8—68no—
103C3×D4—184no—
104C3×D4—182no—
105C2×C3×C4—64no—
106C2×C3×C4—62no—
107C2×C3×C4—91no—
108C2×C3×C4—92no—
109C2×C3×C4—92no—
110C2×C3×C4—182no—
111C2×C3×C4—182no—
112C2×C3×C4—184no—
113C2×C3×C4C2×#15712yes—
114C2×C3×C4C2×#15711yescharacteristic
115C2×Dic3—62no—
116D12—616no—
117D12—68no—
118C4×D3—64no—
119C4×D3—64no—
120C4×D3—68no—
121C4×D3—68no—
122C2×C32—24no—
123C2×C32—22no—
124C3×D3—124no—
125C3×D3—128no—
126#41—182no—
127#41—182no—
128C2×Q8#108811yescharacteristic, complemented
129C2×D4—92no—
130C2×D4—91no—
131C22×C4—91no—
132C22×C4—92no—
133C4⋊C4—182no—
134C22⋊C4—94no—
135C42—182no—
136C22×C3—62no—
137C22×C3—92no—
138C22×C3—182no—
139C22×C3—361no—
140C22×C3—362no—
141C22×C3C22×#15711yescharacteristic
142C2×D3—64no—
143C2×D3—68no—
144C2×D3—124no—
145C2×D3—128no—
146C3×C4—22no—
147C3×C4—68no—
148C3×C4—64no—
149C3×C4—91no—
150C3×C4—91no—
151C3×C4—364no—
152C3×C4C22⋊#15712yes—
153Dic3—62no—
154Dic3—62no—
155C32—22no—
156C23—91no—
157C23—92no—
158Q8—22no—
159D4—184no—
160D4—182no—
161C2×C4—91no—
162C2×C4—92no—
163C2×C4—92no—
164C2×C4—182no—
165C2×C4—182no—
166C2×C4—184no—
167C2×C4C2×#108812yes—
168C2×C4C2×#108811yescharacteristic
169C2×C3—21no—
170C2×C3—64no—
171C2×C3—62no—
172C2×C3—182no—
173C2×C3—362no—
174C2×C3C2×C22⋊#15711yescharacteristic
175D3—124no—
176D3—128no—
177C22—92no—
178C22—182no—
179C22—361no—
180C22—362no—
181C22C22×#108811yescharacteristic
182C4—22no—
183C4—91no—
184C4—91no—
185C4—364no—
186C4C22⋊#108812yes—
187C3—62no—
188C3#1059383211yescharacteristic
189C2—21no—
190C2—182no—
191C2—362no—
192C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC2×C3C2×C22⋊#157
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC22×C3C22×#157
FittingC2×Q8×H(3,𝔽3)C4
prime radicalC2×#88748C2
cosocleC22×#272C23
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#410
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. #11722448C2semidirect
3. #11720780C2semidirect
4. C2×Q8×H(3,𝔽3)C4semidirect
7. C2×C4⋊#272C4semidirect
9. C2×C4×#272C4semidirect
96. H(3,𝔽3)#410semidirect
128. C2×Q8#1088semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
#117224482
#117207802
C3×#4109