Finite Group Databasebeta

← #13146676

Subgroups — #13146676

Order 1728.

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

normal subgroups only

This group has 597 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
1C22×#6198C211yesmaximal, characteristic
2C2×#88631C211yesmaximal, characteristic, complemented
3#11721298C211yesmaximal, characteristic, complemented
4#11720705C214yesmaximal, complemented
5C23×(C3⋊D3)⋊C3—21no—
6C22×#1084C2211yescharacteristic
7C2×#6198—21no—
8C2×#6198—22no—
9C2×#6198C414yescomplemented
10C2×#6198C2212yes—
11#88631—22no—
12C2×#6188C2212yescomplemented
13#3757⋊C3—21no—
14#88449C2212yescomplemented
15#88447—21no—
16C22×#626—61no—
17#63955—61no—
18C23×H(3,𝔽3)D411yescharacteristic
19C22×(C3⋊D3)⋊C3—24no—
20C22×(C3⋊D3)⋊C3—22no—
21C22×(C3⋊D3)⋊C3—44no—
22C2×#1084C2×C412yes—
23C2×#1084D411yescharacteristic
24C2×#1084D412yes—
25C2×#1084C2311yescharacteristic
26C2×(C3⋊Dic3)⋊C3—22no—
27#6198—24no—
28#6198—22no—
29#6198C2×C414yes—
30#6188—22no—
31#5131—91nomaximal
32C23×C3⋊D3—21no—
33C23×C3×D3—61no—
34C22×C3×Dic3—61no—
35C22×C3×Dic3—61no—
36C2×#626—64no—
37C2×#626—62no—
38C2×#626—62no—
39C2×#626—64no—
40#3757—21no—
41C3×#217—61no—
42Dic3⋊Dic3—62no—
43Dic32—62no—
44C22×H(3,𝔽3)—22no—
45C22×H(3,𝔽3)C22⋊C412yes—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C2×(C3⋊D3)⋊C3—24no—
48C2×(C3⋊D3)⋊C3—48no—
49C2×(C3⋊D3)⋊C3—44no—
50#1084C22⋊C412yes—
51#1084C22×C411yescharacteristic
52#1084C2×D411yescharacteristic
53(C3⋊Dic3)⋊C3—42no—
54C22×#67—181no—
55C22×D12—91no—
56C2×C3×M16—91no—
57#926—91no—
58#877—181no—
59#821—94no—
60C23×C32—21no—
61C23×C32—21no—
62C22×C3⋊D3—22no—
63C22×C3⋊D3—24no—
64C22×C3⋊D3—44no—
65C22×C3×D3—62no—
66C22×C3×D3—64no—
67C22×C3×D3—124no—
68C2×C3⋊Dic3—22no—
69C2×C3×Dic3—62no—
70C2×C3×Dic3—62no—
71C2×C3×Dic3—61no—
72C2×C3×Dic3—61no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—62no—
78#626—64no—
79#626—68no—
80#626—64no—
81#418—271noSylow
82C2×H(3,𝔽3)—22no—
83C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
84C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
85(C3⋊D3)⋊C3—44no—
86C24×C3—181no—
87C23×D3—61no—
88C23×D3—181no—
89C22×C3×C4—91no—
90C2×#67—182no—
91C2×#67—184no—
92C2×#67—184no—
93C2×#67—182no—
94C22×Dic3—61no—
95C22×Dic3—61no—
96C2×D12—92no—
97C2×D12—94no—
98C2×D12—181no—
99C2×D12—182no—
100C3×M16—182no—
101C2×C3×C8—92no—
102C3×C22⋊C4—181no—
103#217—61no—
104#217—181no—
105C4⋊Dic3—92no—
106Dic3⋊C4—182no—
107C4×Dic3—181no—
108C4×Dic3—182no—
109C22×C32—24no—
110C22×C32—21no—
111C22×C32—21no—
112C22×C32—22no—
113C22×C32—22no—
114C22×C32—42no—
115C2×C3⋊D3—24no—
116C2×C3⋊D3—48no—
117C2×C3⋊D3—44no—
118C2×C3×D3—64no—
119C2×C3×D3—128no—
120C2×C3×D3—124no—
121C3⋊Dic3—42no—
122C3×Dic3—62no—
123C3×Dic3—62no—
124C3×Dic3—61no—
125C3×Dic3—61no—
126C3×Dic3—61no—
127C3×Dic3—61no—
128C3×Dic3—122no—
129C22×D4—271no—
130C2×M16—91no—
131#117—271no—
132#102—274no—
133H(3,𝔽3)#41811yescharacteristic, complemented, Sylow
134C23×C3—61no—
135C23×C3—61no—
136C23×C3—184no—
137C23×C3—182no—
138C23×C3—364no—
139C23×C3GO+(4,𝔽2)11yescharacteristic
140C22×D3—62no—
141C22×D3—64no—
142C22×D3—124no—
143C22×D3—184no—
144C22×D3—182no—
145C22×D3—364no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—92no—
149C2×C3×C4—92no—
150C2×C3×C4—182no—
151#67—184no—
152#67—184no—
153#67—188no—
154C2×Dic3—61no—
155C2×Dic3—61no—
156C2×Dic3—62no—
157C2×Dic3—62no—
158C2×Dic3—62no—
159C2×Dic3—62no—
160C2×Dic3—61no—
161C2×Dic3—61no—
162C2×Dic3—62no—
163C2×Dic3—182no—
164D12—94no—
165D12—182no—
166D12—184no—
167C3×C8—182no—
168C2×C32—24no—
169C2×C32—21no—
170C2×C32—21no—
171C2×C32—22no—
172C2×C32—22no—
173C2×C32—42no—
174C3⋊D3—44no—
175C3×D3—124no—
176C24—181no—
177C2×D4—272no—
178C2×D4—274no—
179C2×D4—542no—
180C2×D4—541no—
181C22×C4—91no—
182M16—182no—
183C2×C8—92no—
184C4⋊C4—272no—
185C22⋊C4—181no—
186C42—541no—
187C22×C3—22no—
188C22×C3—64no—
189C22×C3—62no—
190C22×C3—62no—
191C22×C3—61no—
192C22×C3—61no—
193C22×C3—122no—
194C22×C3—184no—
195C22×C3—368no—
196C22×C3—364no—
197C22×C3D32⋊C412yes—
198C22×C3C2×GO+(4,𝔽2)11yescharacteristic
199C2×D3—64no—
200C2×D3—128no—
201C2×D3—124no—
202C2×D3—184no—
203C2×D3—368no—
204C2×D3—364no—
205C3×C4—91no—
206C3×C4—91no—
207C3×C4—92no—
208C3×C4—362no—
209Dic3—61no—
210Dic3—61no—
211Dic3—62no—
212Dic3—61no—
213Dic3—61no—
214Dic3—62no—
215Dic3—122no—
216Dic3—362no—
217C32—21no—
218C32—21no—
219C23—184no—
220C23—182no—
221C23—364no—
222C23#1088⋊C211yescharacteristic
223D4—274no—
224D4—542no—
225D4—544no—
226C2×C4—92no—
227C2×C4—92no—
228C2×C4—91no—
229C2×C4—91no—
230C2×C4—182no—
231C8—182no—
232C2×C3—22no—
233C2×C3—64no—
234C2×C3—62no—
235C2×C3—62no—
236C2×C3—61no—
237C2×C3—61no—
238C2×C3—122no—
239C2×C3—364no—
240C2×C3C2×D32⋊C411yescharacteristic
241C2×C3#6423612yes—
242D3—124no—
243D3—364no—
244C22—22no—
245C22—184no—
246C22—368no—
247C22—364no—
248C22(((C3⋊D3)⋊C3)⋊C2)⋊C412yes—
249C22C2×#1088⋊C211yescharacteristic
250C4—91no—
251C4—91no—
252C4—92no—
253C4—362no—
254C3—61no—
255C3—61no—
256C3#1059349311yescharacteristic
257C2—22no—
258C2—364no—
259C2#1172230912yes—
260C2C2×(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator#1084C22×C4
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC23×H(3,𝔽3)D4
prime radicalC22×#6198C2
cosocleC2×#1084C23
hypercenterC23#1088⋊C2
hypocenterH(3,𝔽3)#418
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88631C2semidirect
3. #11721298C2semidirect
4. #11720705C2semidirect
9. C2×#6198C4semidirect
12. C2×#6188C22semidirect
14. #88449C22semidirect
133. H(3,𝔽3)#418semidirect

Maximal subgroups

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

subgroupindex
C22×#61982
C2×#886312
#117212982
#117207052
#51319