Finite Group Databasebeta

← #13137865

Subgroups — #13137865

Order 1728.

472 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 71 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
1#11721761C211yesmaximal, characteristic
2#11721117C211yesmaximal, characteristic, complemented
3#63634⋊C3C212yesmaximal, complemented
4#11720324C212yesmaximal
5#11720304C211yesmaximal, characteristic, complemented
6#10590555—32nomaximal
7C2×C4⋊#272C2211yescharacteristic
8C2×C4×#272—21no—
9C2×#6196—21no—
10C4⋊C4×H(3,𝔽3)C2211yescharacteristic
11#88542—21no—
12C4×#1084C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16C3×#893—32no—
17C3×#893—61no—
18#63918—32no—
19#63634D312yescomplemented
20C3×#807—32no—
21Dic3⋊(C3⋊C8)—32no—
22#63563—32no—
23#63562—32no—
24C22×#272—21no—
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084D411yescharacteristic
28C2×#1084D411yescharacteristic
29C4⋊#272—21no—
30C4⋊#272Q812yescomplemented
31C4×#272—42no—
32C2×(C3⋊Dic3)⋊C3—22no—
33#6196—42no—
34(C3⋊(C3⋊C8))⋊C3—22no—
35#5007—91no—
36#4890—92no—
37C2×C3×D12—32no—
38C2×C3×D12—61no—
39C2×C3×C4×D3—62no—
40C2×C3×C4×D3—62no—
41C2×#627—62no—
42C32×C4⋊C4—21no—
43C32×C4⋊C4C2×D312yes—
44C4⋊(C3⋊Dic3)C2×D312yes—
45C2×C3⋊(C3⋊C8)C2×D312yes—
46C3×(C2×D3)⋊C4—62no—
47C3×(C2×D3)⋊C4—62no—
48C3×C4⋊Dic3—32no—
49C3×C4×Dic3—32no—
50C3×C4×Dic3—61no—
51C2×C3×C3⋊C8—32no—
52Dic3⋊Dic3—62no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C2×#272—22no—
55C2×#272—41no—
56C2×#272—41no—
57C4×H(3,𝔽3)—41no—
58C4×H(3,𝔽3)C2×Q811yescharacteristic
59C4×H(3,𝔽3)#4111yescharacteristic
60#1084—21no—
61#1084D812yes—
62(C3⋊Dic3)⋊C3—42no—
63C3×C4×D4—91no—
64#893—32no—
65#893—61no—
66#888—92no—
67#869—91no—
68C3×#107—92no—
69#832—92no—
70#821—92no—
71Dic3⋊C8—92no—
72#809—92no—
73#807—32no—
74#807—92no—
75C4⋊(C3⋊C8)—91no—
76C22×C3×D3—62no—
77C22×C3×D3—62no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—21no—
81C2×C32×C4C22×D312yes—
82C2×C3⋊Dic3—22no—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—62no—
88C3×D12—34no—
89C3×D12—62no—
90C3×D12—62no—
91C3×D12—62no—
92C3×C4×D3—124no—
93C3×C4×D3—124no—
94#627—124no—
95C3⋊(C3⋊C8)—22no—
96C3×C3⋊C8—62no—
97#474—271noSylow
98C2×H(3,𝔽3)C22⋊Q811yescharacteristic
99C2×H(3,𝔽3)C2×D811yescharacteristic
100C2×H(3,𝔽3)#13711yescharacteristic
101#272—42no—
102C2×C3×D4—91no—
103C22×C3×C4—181no—
104C2×D12—32no—
105C2×D12—61no—
106C2×C4×D3—62no—
107C2×C4×D3—62no—
108C2×Dic6—182no—
109C2×Dic6—181no—
110C2×C3×C8—92no—
111C3×C4⋊C4—32no—
112C3×C4⋊C4—61no—
113C3×C4⋊C4—92no—
114C3×C4⋊C4D3211yescharacteristic
115C3×C22⋊C4—181no—
116C3×C42—91no—
117(C2×D3)⋊C4—62no—
118(C2×D3)⋊C4—62no—
119C4⋊Dic3—32no—
120C4⋊Dic3—92no—
121C4⋊Dic3—181no—
122Dic3⋊C4—182no—
123C4×Dic3—32no—
124C4×Dic3—61no—
125C2×C3⋊C8—32no—
126C2×C3⋊C8—92no—
127C22×C32—21no—
128C22×C32D3×D412yes—
129C2×C3×D3—64no—
130C2×C3×D3—64no—
131C2×C3×D3—122no—
132C2×C3×D3—122no—
133C2×C3×D3—122no—
134C2×C3×D3—122no—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4—42no—
138C32×C4—42no—
139C32×C4#23512yes—
140C32×C4D3×Q812yes—
141C3⋊Dic3—42no—
142C3×Dic3—34no—
143C3×Dic3—62no—
144C3×Dic3—62no—
145C3×Dic3—121no—
146C3×Dic3—122no—
147C4⋊Q8—271no—
148C4×D4—91no—
149#107—92no—
150C4⋊C8—271no—
151#102—272no—
152H(3,𝔽3)#47411yescharacteristic, complemented, Sylow
153C23×C3—181no—
154C22×D3—62no—
155C22×D3—62no—
156C3×D4—92no—
157C3×D4—181no—
158C2×C3×C4—21no—
159C2×C3×C4—32no—
160C2×C3×C4—62no—
161C2×C3×C4—62no—
162C2×C3×C4—61no—
163C2×C3×C4—91no—
164C2×C3×C4—91no—
165C2×C3×C4—182no—
166C2×C3×C4—362no—
167C2×C3×C4C2×D3211yescharacteristic
168C2×Dic3—32no—
169C2×Dic3—32no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—62no—
173C2×Dic3—182no—
174D12—34no—
175D12—62no—
176D12—62no—
177D12—62no—
178C4×D3—124no—
179C4×D3—124no—
180Dic6—364no—
181Dic6—362no—
182C3×C8—182no—
183C3⋊C8—62no—
184C3⋊C8—182no—
185C2×C32—21no—
186C2×C32—21no—
187C2×C32—21no—
188C2×C32#89612yes—
189C2×C32D3×D812yes—
190C2×C32#91812yes—
191C3×D3—124no—
192C3×D3—124no—
193C2×Q8—541no—
194C2×D4—91no—
195C22×C4—181no—
196C2×C8—92no—
197C4⋊C4—92no—
198C4⋊C4—541no—
199C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
200C22⋊C4—181no—
201C42—91no—
202C22×C3—32no—
203C22×C3—61no—
204C22×C3—182no—
205C22×C3—361no—
206C22×C3—361no—
207C22×C3#381111yescharacteristic
208C2×D3—64no—
209C2×D3—64no—
210C2×D3—122no—
211C2×D3—122no—
212C2×D3—122no—
213C2×D3—122no—
214C3×C4—32no—
215C3×C4—32no—
216C3×C4—41no—
217C3×C4—61no—
218C3×C4—61no—
219C3×C4—92no—
220C3×C4—122no—
221C3×C4—122no—
222C3×C4—181no—
223C3×C4—362no—
224C3×C4#379711yescharacteristic
225C3×C4#379811yescharacteristic
226Dic3—34no—
227Dic3—62no—
228Dic3—62no—
229Dic3—121no—
230Dic3—122no—
231Dic3—362no—
232C32—21no—
233C32#489012yescomplemented
234C23—181no—
235Q8—1082no—
236D4—92no—
237D4—181no—
238C2×C4—21no—
239C2×C4—91no—
240C2×C4—91no—
241C2×C4—182no—
242C2×C4—362no—
243C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
244C8—182no—
245C2×C3—32no—
246C2×C3—32no—
247C2×C3—32no—
248C2×C3—61no—
249C2×C3—61no—
250C2×C3—61no—
251C2×C3—362no—
252C2×C3#6386511yescharacteristic
253C2×C3#6392611yescharacteristic
254C2×C3#6394511yescharacteristic
255D3—124no—
256D3—124no—
257C22—182no—
258C22—361no—
259C22—361no—
260C22#8867711yescharacteristic
261C4—41no—
262C4—92no—
263C4—181no—
264C4—362no—
265C4#8865011yescharacteristic
266C4#8865111yescharacteristic
267C3—32no—
268C3—61no—
269C3#1059043011yescharacteristic
270C2—362no—
271C2#1172109611yescharacteristic
272C2#1172119011yescharacteristic
273C2#1172122711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×C4⋊#272C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#474
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721117C2semidirect
3. #63634⋊C3C2semidirect
5. #11720304C2semidirect
19. #63634D3semidirect
30. C4⋊#272Q8semidirect
152. H(3,𝔽3)#474semidirect
233. C32#4890semidirect

Maximal subgroups

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

subgroupindex
#117217612
#117211172
#63634⋊C32
#117203242
#117203042
#105905553