Finite Group Databasebeta

← #13147368

Subgroups — #13147368

Order 1728.

316 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 49 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
1Q8×#1088C211yesmaximal, characteristic, complemented
2D4×#1088C211yesmaximal, characteristic
3SD16×#272C211yesmaximal, characteristic
4#11720790C211yesmaximal, characteristic, complemented
5#11720788C211yesmaximal, characteristic
6#11720775C211yesmaximal, characteristic, complemented
7#11720773C211yesmaximal, characteristic, complemented
8C22×#1088—21no—
9Q8×#272C2211yescharacteristic
10D4×#272C2211yescharacteristic
11C22⋊#1088—21no—
12C4⋊#1088—21no—
13C4⋊#1088C2211yescharacteristic
14C4⋊#1088C2211yescharacteristic
15C4×#1088—21no—
16C4×#1088C2211yescharacteristic
17#88626C2211yescharacteristic
18SD16×H(3,𝔽3)C411yescharacteristic, complemented
19#88549C411yescharacteristic, complemented
20#88546C411yescharacteristic, complemented
21#88528C411yescharacteristic, complemented
22C8×#272C2211yescharacteristic
23C3×D3×SD16—62no—
24C22×#272—21no—
25C2×#1088—21no—
26C2×#1088—21no—
27C2×#1088—42no—
28C2×#1088D411yescharacteristic
29C2×#1088D411yescharacteristic
30Q8×H(3,𝔽3)C2×C411yescharacteristic, complemented
31D4×H(3,𝔽3)C2×C411yescharacteristic
32#6235—21no—
33C4⋊#272C2×C411yescharacteristic
34C4×#272—21no—
35C4×#272C2311yescharacteristic
36#6229—21no—
37#6229C2×C411yescharacteristic, complemented
38#6188—21no—
39C8×H(3,𝔽3)C2×C411yescharacteristic, complemented
40#6180C2×C411yescharacteristic, complemented
41C3×#440—91nomaximal
42C3×D3×Q8—62no—
43C3×D3×D4—62no—
44C32×SD16—22no—
45C3×#215—62no—
46C3×#214—62no—
47C3×#204—62no—
48C3×D3×C8—62no—
49C22×H(3,𝔽3)—21no—
50C2×#272—21no—
51C2×#272—41no—
52C2×#272—41no—
53C2×#272C2×D411yescharacteristic
54#1088—21no—
55#1088—21no—
56#1088—41no—
57#1088—41no—
58C4×H(3,𝔽3)—21no—
59C4×H(3,𝔽3)C22×C411yescharacteristic
60#1084—21no—
61#1084#4111yescharacteristic
62C2×C3×SD16—91no—
63C3×C4×Q8—91no—
64C3×C4×D4—91no—
65D3×SD16—62no—
66C3×#107—91no—
67C3×#103—91no—
68C3×#102—91no—
69C3×#97—91no—
70C22×C3×D3—122no—
71C32×Q8—22no—
72C32×D4—22no—
73C3×#67—122no—
74C3×D12—62no—
75C3×C4×D3—62no—
76C3×C4×D3—122no—
77C3×Dic6—62no—
78C3×Dic6—122no—
79C32×C8—22no—
80C3×C3⋊C8—62no—
81#440—91noSylow
82C2×H(3,𝔽3)—41no—
83C2×H(3,𝔽3)C4×D411yescharacteristic
84#272—41no—
85#272AGL(1,ℤ/8)11yescharacteristic
86#272#13711yescharacteristic
87C2×C3×Q8—91no—
88C2×C3×D4—91no—
89C22×C3×C4—181no—
90D3×Q8—62no—
91D3×D4—62no—
92C3×SD16—62no—
93C3×SD16—91no—
94C3×SD16—91no—
95C3×SD16—91no—
96C3×SD16#15711yescharacteristic
97C2×C3×C8—91no—
98C2×C3×C8—91no—
99C3×C4⋊C4—91no—
100C3×C4⋊C4—91no—
101C3×C4⋊C4—181no—
102C3×C22⋊C4—181no—
103C3×C42—91no—
104C3×C42—181no—
105#215—62no—
106#214—62no—
107#204—62no—
108D3×C8—62no—
109C22×C32—42no—
110C2×C3×D3—62no—
111C2×C3×D3—122no—
112C2×C3×D3—242no—
113C2×C3×D3—242no—
114C32×C4—22no—
115C32×C4—42no—
116C3×Dic3—62no—
117C3×Dic3—122no—
118C2×SD16—91no—
119C4×Q8—91no—
120C4×D4—91no—
121#107—91no—
122#103—91no—
123#102—91no—
124#97—91no—
125H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
126C23×C3—181no—
127C22×D3—122no—
128C3×Q8—62no—
129C3×Q8—91no—
130C3×Q8—181no—
131C3×Q8C2×#15711yescharacteristic
132C3×D4—62no—
133C3×D4—91no—
134C3×D4—181no—
135C3×D4C2×#15711yescharacteristic
136C2×C3×C4—91no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—181no—
140C2×C3×C4—181no—
141C2×C3×C4—181no—
142C2×C3×C4—362no—
143#67—122no—
144D12—62no—
145C4×D3—62no—
146C4×D3—122no—
147Dic6—62no—
148Dic6—122no—
149C3×C8—62no—
150C3×C8—91no—
151C3×C8—181no—
152C3×C8C2×#15711yescharacteristic
153C3⋊C8—62no—
154C2×C32—22no—
155C2×C32—82no—
156C3×D3—62no—
157C3×D3—62no—
158C3×D3—242no—
159C2×Q8—91no—
160C2×D4—91no—
161C22×C4—181no—
162SD16—91no—
163SD16—91no—
164SD16—91no—
165SD16#108811yescharacteristic, complemented
166C2×C8—91no—
167C2×C8—91no—
168C4⋊C4—91no—
169C4⋊C4—91no—
170C4⋊C4—181no—
171C22⋊C4—181no—
172C42—91no—
173C42—181no—
174C22×C3—21no—
175C22×C3—91no—
176C22×C3—122no—
177C22×C3—181no—
178C22×C3—361no—
179C22×C3—361no—
180C2×D3—62no—
181C2×D3—122no—
182C2×D3—242no—
183C2×D3—242no—
184C3×C4—21no—
185C3×C4—62no—
186C3×C4—91no—
187C3×C4—122no—
188C3×C4—181no—
189C3×C4—181no—
190C3×C4—181no—
191C3×C4—361no—
192C3×C4—361no—
193C3×C4C22×#15711yescharacteristic
194Dic3—62no—
195Dic3—122no—
196C32—22no—
197C23—181no—
198Q8—91no—
199Q8—181no—
200Q8C2×#108811yescharacteristic, complemented
201D4—91no—
202D4—181no—
203D4C2×#108811yescharacteristic
204C2×C4—91no—
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—181no—
208C2×C4—181no—
209C2×C4—181no—
210C2×C4—362no—
211C8—91no—
212C8—181no—
213C8C2×#108811yescharacteristic, complemented
214C2×C3—41no—
215C2×C3—62no—
216C2×C3—91no—
217C2×C3—91no—
218C2×C3—242no—
219C2×C3—361no—
220C2×C3D4×#15711yescharacteristic
221D3—62no—
222D3—62no—
223D3—242no—
224C22—21no—
225C22—91no—
226C22—181no—
227C22—361no—
228C22—361no—
229C4—21no—
230C4—91no—
231C4—181no—
232C4—181no—
233C4—181no—
234C4—361no—
235C4—361no—
236C4C22×#108811yescharacteristic
237C3—62no—
238C3#1059378211yescharacteristic
239C2—41no—
240C2—91no—
241C2—91no—
242C2—361no—
243C2D4×#108811yescharacteristic
subgroupG/H
center, socleC2×C3D4×#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingSD16×H(3,𝔽3)C4
prime radicalD4×#272C22
cosocleC4×#272C23
hypercenterC3×SD16#157
hypocenterH(3,𝔽3)#440
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×#1088C2semidirect
4. #11720790C2semidirect
6. #11720775C2semidirect
7. #11720773C2semidirect
18. SD16×H(3,𝔽3)C4semidirect
19. #88549C4semidirect
20. #88546C4semidirect
21. #88528C4semidirect
30. Q8×H(3,𝔽3)C2×C4semidirect
37. #6229C2×C4semidirect
39. C8×H(3,𝔽3)C2×C4semidirect
40. #6180C2×C4semidirect
125. H(3,𝔽3)#440semidirect
165. SD16#1088semidirect
200. Q8C2×#1088semidirect
213. C8C2×#1088semidirect

Maximal subgroups

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

subgroupindex
Q8×#10882
D4×#10882
SD16×#2722
#117207902
#117207882
#117207752
#117207732
C3×#4409