Finite Group Databasebeta

← #13147008

Subgroups — #13147008

Order 1728.

391 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 44 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×#88611C211yesmaximal, characteristic
2C4×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#11720738C211yesmaximal, characteristic, complemented
4C3×#5394—33nomaximal
5C3×#5026—31nomaximal
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7#88611C412yescomplemented
8#88542—41nomaximal
9C2×C3×GL(2,𝔽3)—33no—
10C32×C4×Q8D311yescharacteristic, complemented
11C2×C3×#215—31no—
12C3×C4×D12—31no—
13C3×C4×SL(2,𝔽3)D313yescomplemented
14C3×#859—33no—
15C3×#835—31no—
16C3×#809—31no—
17C3×#807—31no—
18C3×#803—31no—
19C22×#272—41no—
20C2×C4×H(3,𝔽3)—41no—
21C2×#1084—41no—
22Q8⋊H(3,𝔽3)C2×C411yescharacteristic, complemented
23#5394—33no—
24C3×#440—91no—
25#5026—31no—
26C2×C32×Q8C2×D311yescharacteristic
27C2×C3×D12—31no—
28C2×C3×C4×D3—61no—
29C2×C3×SL(2,𝔽3)C2×D313yes—
30C3×GL(2,𝔽3)—36no—
31C32×C4⋊C4—31no—
32C32×C42—31no—
33C3×#215—32no—
34C3×#215—32no—
35C3×(C2×D3)⋊C4—61no—
36C3×(C2×D3)⋊C4—123no—
37C3×C4⋊Dic3—31no—
38C2×C3×C3⋊C8—31no—
39C2×C3×C3⋊C8—31no—
40C22×H(3,𝔽3)—41no—
41C2×#272—42no—
42C2×#272—81no—
43C2×#272—81no—
44C4×H(3,𝔽3)—81no—
45#1084—81no—
46C2×GL(2,𝔽3)—33no—
47C2×C3×SD16—91no—
48C3×C4×Q8—31no—
49C3×C4×Q8C3⋊D311yescharacteristic
50C3×C4×D4—91no—
51C2×#215—31no—
52C4×D12—31no—
53C4×SL(2,𝔽3)—33no—
54#859—33no—
55C3×#107—91no—
56C3×#103—91no—
57C3×#102—91no—
58C3×#97—91no—
59#835—31no—
60#809—31no—
61#807—31no—
62#803—31no—
63C22×C3×D3—61no—
64C22×C3×D3—123no—
65C32×Q8—31no—
66C32×Q8C4×D311yescharacteristic
67C2×C32×C4—31no—
68C2×C32×C4—31no—
69C2×C32×C4—43no—
70C2×C32×C4S411yescharacteristic
71C2×C3×Dic3—61no—
72C2×C3×Dic3—123no—
73C3×D12—32no—
74C3×D12—61no—
75C3×C4×D3—122no—
76C3×SL(2,𝔽3)C4×D313yescomplemented
77C3×C3⋊C8—32no—
78C3×C3⋊C8—61no—
79#440—91noSylow
80C2×H(3,𝔽3)—41no—
81C2×H(3,𝔽3)—41no—
82C2×H(3,𝔽3)—41no—
83#272—82no—
84C2×C3×Q8—31no—
85C2×C3×Q8C2×C3⋊D311yescharacteristic
86C2×C3×D4—91no—
87C22×C3×C4—181no—
88C2×D12—31no—
89C2×C4×D3—61no—
90C2×SL(2,𝔽3)—33no—
91GL(2,𝔽3)—36no—
92C3×SD16—92no—
93C3×SD16—92no—
94C2×C3×C8—91no—
95C2×C3×C8—91no—
96C3×C4⋊C4—31no—
97C3×C4⋊C4—31no—
98C3×C4⋊C4—61no—
99C3×C4⋊C4—91no—
100C3×C22⋊C4—181no—
101C3×C42—31no—
102C3×C42—31no—
103C3×C42—61no—
104#215—32no—
105#215—32no—
106(C2×D3)⋊C4—61no—
107(C2×D3)⋊C4—123no—
108C4⋊Dic3—31no—
109C2×C3⋊C8—31no—
110C2×C3⋊C8—31no—
111C22×C32—43no—
112C22×C32C2×S411yescharacteristic
113C2×C3×D3—62no—
114C2×C3×D3—126no—
115C2×C3×D3—121no—
116C2×C3×D3—121no—
117C2×C3×D3—243no—
118C2×C3×D3—243no—
119C32×C4—21no—
120C32×C4—31no—
121C32×C4—31no—
122C32×C4—61no—
123C32×C4—83no—
124C3×Dic3—121no—
125C3×Dic3—243no—
126C2×SD16—91no—
127C4×Q8#27211yescharacteristic, complemented
128C4×D4—91no—
129#107—91no—
130#103—91no—
131#102—91no—
132#97—91no—
133H(3,𝔽3)—41noSylow
134C23×C3—181no—
135C22×D3—61no—
136C22×D3—123no—
137C3×Q8—31no—
138C3×Q8—31no—
139C3×Q8—31no—
140C3×Q8—61no—
141C3×Q8C4×C3⋊D311yescharacteristic
142C3×D4—92no—
143C3×D4—181no—
144C2×C3×C4—31no—
145C2×C3×C4—31no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—61no—
150C2×C3×C4—61no—
151C2×C3×C4—123no—
152C2×C3×C4—181no—
153C2×C3×C4—362no—
154C2×C3×C4C3⋊S411yescharacteristic
155C2×Dic3—61no—
156C2×Dic3—123no—
157D12—32no—
158D12—61no—
159C4×D3—122no—
160SL(2,𝔽3)—33no—
161C3×C8—92no—
162C3×C8—181no—
163C3⋊C8—32no—
164C3⋊C8—61no—
165C2×C32—43no—
166C2×C32—43no—
167C2×C32—43no—
168C2×C32C4×S411yescharacteristic
169C2×C32#98311yescharacteristic
170C2×C32#98611yescharacteristic
171C3×D3—122no—
172C3×D3—246no—
173C2×Q8C2×#27211yescharacteristic
174C2×D4—91no—
175C22×C4—181no—
176SD16—92no—
177SD16—92no—
178C2×C8—91no—
179C2×C8—91no—
180C4⋊C4—31no—
181C4⋊C4—91no—
182C22⋊C4—181no—
183C42—31no—
184C22×C3—31no—
185C22×C3—123no—
186C22×C3—182no—
187C22×C3—361no—
188C22×C3—361no—
189C22×C3C2×C3⋊S411yescharacteristic
190C2×D3—62no—
191C2×D3—126no—
192C2×D3—121no—
193C2×D3—121no—
194C2×D3—243no—
195C2×D3—243no—
196C3×C4—21no—
197C3×C4—31no—
198C3×C4—31no—
199C3×C4—31no—
200C3×C4—31no—
201C3×C4—61no—
202C3×C4—61no—
203C3×C4—61no—
204C3×C4—61no—
205C3×C4—61no—
206C3×C4—121no—
207C3×C4—243no—
208C3×C4—361no—
209Dic3—121no—
210Dic3—243no—
211C32—43no—
212C32#539411yescharacteristic, complemented
213C23—181no—
214Q8—31no—
215Q8C4×#27211yescharacteristic
216D4—92no—
217D4—181no—
218C2×C4—31no—
219C2×C4—31no—
220C2×C4—181no—
221C2×C4—362no—
222C2×C4#625811yescharacteristic
223C8—92no—
224C8—181no—
225C2×C3—31no—
226C2×C3—31no—
227C2×C3—31no—
228C2×C3—123no—
229C2×C3—123no—
230C2×C3—123no—
231C2×C3—362no—
232C2×C3C4×C3⋊S411yescharacteristic
233C2×C3#6426411yescharacteristic
234C2×C3#6426711yescharacteristic
235D3—122no—
236D3—246no—
237C22—182no—
238C22—361no—
239C22—361no—
240C22C2×#625811yescharacteristic
241C4—21no—
242C4—31no—
243C4—31no—
244C4—61no—
245C4—361no—
246C3—31no—
247C3—123no—
248C3#1059362811yescharacteristic
249C2—362no—
250C2C4×#625811yescharacteristic
251C2#1172238811yescharacteristic
252C2#1172239411yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C2×C3⋊S4
commutator, hypocenterQ8⋊H(3,𝔽3)C2×C4
FittingC32×C4×Q8D3
prime radicalC2×#88611C2
cosocleC2×Q8⋊H(3,𝔽3)C22
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C4×Q8⋊H(3,𝔽3)C2semidirect
3. #11720738C2semidirect
7. #88611C4semidirect
10. C32×C4×Q8D3semidirect
13. C3×C4×SL(2,𝔽3)D3semidirect
22. Q8⋊H(3,𝔽3)C2×C4semidirect
76. C3×SL(2,𝔽3)C4×D3semidirect
127. C4×Q8#272semidirect
212. C32#5394semidirect

Maximal subgroups

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

subgroupindex
C2×#886112
C4×Q8⋊H(3,𝔽3)2
#117207382
C3×#53943
C3×#50263
#885424