Finite Group Databasebeta

← #13147406

Subgroups — #13147406

Order 1728.

374 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 59 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×#88631C211yesmaximal, characteristic, complemented
2C22×#6188C211yesmaximal, characteristic, complemented
3D4×#1084C211yesmaximal, characteristic
4C22⋊#6188C211yesmaximal, characteristic, complemented
5C22⋊#6188C211yesmaximal, characteristic, complemented
6#11720785C211yesmaximal, characteristic, complemented
7#11720783C211yesmaximal, characteristic, complemented
8C2×D4×H(3,𝔽3)C411yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11#88631—22no—
12C2×#6188—22no—
13C2×#6188C2211yescharacteristic
14C2×#6188C2211yescharacteristic
15C2×#6188C2211yescharacteristic
16C2×#6188C2211yescharacteristic
17#88552C411yescharacteristic
18#88552C411yescharacteristic
19C4⋊#1084C411yescharacteristic
20C4×#1084C2211yescharacteristic
21C3×D4×Dic3—62no—
22C23×H(3,𝔽3)C2×C411yescharacteristic
23C23×H(3,𝔽3)C2×C411yescharacteristic
24D4×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)C2×C411yescharacteristic
26C2×#1084—21no—
27C2×#1084—21no—
28C2×#1084—21no—
29C2×#1084—21no—
30C2×#1084C2×C411yescharacteristic
31C2×#1084C2×C411yescharacteristic
32C2×#1084C2×C411yescharacteristic
33C2×#1084C2311yescharacteristic
34#6188—21no—
35#6188—21no—
36#6188—21no—
37#6188D412yescomplemented
38C3×#435—91nomaximal
39C2×C32×D4—22no—
40C22×C3×Dic3—62no—
41C22×C3×Dic3—62no—
42C3×#217—62no—
43C3×#217—62no—
44C3×C4⋊Dic3—62no—
45C3×C4×Dic3—62no—
46C22×H(3,𝔽3)—21no—
47C22×H(3,𝔽3)—21no—
48C22×H(3,𝔽3)—21no—
49C22×H(3,𝔽3)—21no—
50C22×H(3,𝔽3)—21no—
51C22×H(3,𝔽3)M1612yescomplemented
52C22×H(3,𝔽3)C22×C411yescharacteristic
53C4×H(3,𝔽3)—21no—
54#1084—21no—
55#1084—21no—
56#1084—21no—
57#1084C2×D411yescharacteristic
58#1084#4111yescharacteristic
59C2×C3×M16—91no—
60C22×C3×C8—91no—
61C3×C4×D4—91no—
62D4×Dic3—62no—
63C3×C4⋊C8—91no—
64C3×C22⋊C8—91no—
65C3×C22⋊C8—91no—
66C3×#97—91no—
67C23×C32—22no—
68C23×C32—22no—
69C32×D4—28no—
70C2×C32×C4—22no—
71C2×C3×Dic3—62no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—122no—
76C2×C3×Dic3—122no—
77C2×C3×Dic3—122no—
78C2×C3×Dic3—122no—
79#435—91noSylow
80C2×H(3,𝔽3)—22no—
81C2×H(3,𝔽3)—41no—
82C2×H(3,𝔽3)C4×D411yescharacteristic
83C2×H(3,𝔽3)C2×M1611yescharacteristic
84C2×H(3,𝔽3)#13111yescharacteristic
85C2×C3×D4—62no—
86C2×C3×D4#15711yescharacteristic
87C22×C3×C4—91no—
88C22×C3×C4—91no—
89C22×Dic3—62no—
90C22×Dic3—62no—
91C3×M16—182no—
92C2×C3×C8—91no—
93C2×C3×C8—91no—
94C2×C3×C8—91no—
95C2×C3×C8—91no—
96C2×C3×C8—182no—
97C3×C4⋊C4—91no—
98C3×C22⋊C4—91no—
99C3×C22⋊C4—91no—
100C3×C42—91no—
101#217—62no—
102#217—62no—
103C4⋊Dic3—62no—
104C4×Dic3—62no—
105C22×C32—24no—
106C22×C32—22no—
107C22×C32—24no—
108C22×C32—42no—
109C22×C32—42no—
110C22×C32—42no—
111C22×C32—42no—
112C32×C4—24no—
113C3×Dic3—62no—
114C3×Dic3—62no—
115C3×Dic3—122no—
116C3×Dic3—122no—
117C3×Dic3—122no—
118C2×M16—91no—
119C22×C8—91no—
120C4×D4—91no—
121C4⋊C8—91no—
122C22⋊C8—91no—
123C22⋊C8—91no—
124#97—91no—
125H(3,𝔽3)#43511yescharacteristic, complemented, Sylow
126C23×C3—62no—
127C23×C3—62no—
128C23×C3C2×#15711yescharacteristic
129C23×C3C2×#15711yescharacteristic
130C3×D4—22no—
131C3×D4—68no—
132C2×C3×C4—62no—
133C2×C3×C4—91no—
134C2×C3×C4—91no—
135C2×C3×C4—91no—
136C2×C3×C4—91no—
137C2×C3×C4—181no—
138C2×C3×C4—181no—
139C2×C3×C4—181no—
140C2×C3×C4—181no—
141C2×C3×C4C2×#15711yescharacteristic
142C2×Dic3—62no—
143C2×Dic3—62no—
144C2×Dic3—62no—
145C2×Dic3—62no—
146C2×Dic3—122no—
147C2×Dic3—122no—
148C2×Dic3—122no—
149C2×Dic3—122no—
150C3×C8—92no—
151C3×C8—181no—
152C3×C8—181no—
153C3×C8—181no—
154C2×C32—22no—
155C2×C32—22no—
156C2×C32—22no—
157C2×C32—44no—
158C2×C32—44no—
159C2×D4#108811yescharacteristic
160C22×C4—91no—
161C22×C4—91no—
162M16—182no—
163C2×C8—91no—
164C2×C8—91no—
165C2×C8—91no—
166C2×C8—91no—
167C2×C8—182no—
168C4⋊C4—91no—
169C22⋊C4—91no—
170C22⋊C4—91no—
171C42—91no—
172C22×C3—21no—
173C22×C3—21no—
174C22×C3—21no—
175C22×C3—21no—
176C22×C3—21no—
177C22×C3—64no—
178C22×C3—64no—
179C22×C3—62no—
180C22×C3—122no—
181C22×C3—122no—
182C22×C3—122no—
183C22×C3—122no—
184C22×C3#379212yes—
185C22×C3C22×#15711yescharacteristic
186C3×C4—21no—
187C3×C4—64no—
188C3×C4—91no—
189C3×C4—91no—
190C3×C4—181no—
191C3×C4—181no—
192C3×C4—181no—
193Dic3—62no—
194Dic3—62no—
195Dic3—122no—
196Dic3—122no—
197Dic3—122no—
198C32—22no—
199C23C2×#108811yescharacteristic
200C23C2×#108811yescharacteristic
201D4—22no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—91no—
206C2×C4—181no—
207C2×C4—181no—
208C2×C4—181no—
209C2×C4—181no—
210C2×C4C2×#108811yescharacteristic
211C8—92no—
212C8—181no—
213C8—181no—
214C8—181no—
215C2×C3—22no—
216C2×C3—41no—
217C2×C3—62no—
218C2×C3—62no—
219C2×C3—62no—
220C2×C3—124no—
221C2×C3—124no—
222C2×C3#6428711yescharacteristic
223C2×C3D4×#15711yescharacteristic
224C2×C3C2×#379211yescharacteristic
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22#8863112yescomplemented
231C22C22×#108811yescharacteristic
232C4—21no—
233C4—91no—
234C4—91no—
235C4—181no—
236C4—181no—
237C4—181no—
238C3—62no—
239C3#1059382011yescharacteristic
240C2—22no—
241C2—41no—
242C2#1172244911yescharacteristic
243C2D4×#108811yescharacteristic
244C2C2×#8863111yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
Fitting, prime radicalC2×D4×H(3,𝔽3)C4
cosocleC2×#1084C23
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#435
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88631C2semidirect
2. C22×#6188C2semidirect
4. C22⋊#6188C2semidirect
5. C22⋊#6188C2semidirect
6. #11720785C2semidirect
7. #11720783C2semidirect
37. #6188D4semidirect
51. C22×H(3,𝔽3)M16semidirect
125. H(3,𝔽3)#435semidirect
230. C22#88631semidirect

Maximal subgroups

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

subgroupindex
C2×#886312
C22×#61882
D4×#10842
C22⋊#61882
#117207852
#117207832
C3×#4359