Finite Group Databasebeta

← #13137315

Subgroups — #13137315

Order 1728.

383 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721715C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic
3#63630⋊C3C211yesmaximal, characteristic
4#11720307C212yesmaximal
5#11720306C212yesmaximal
6#10590315—31nomaximal
7#10590305—31nomaximal
8C2×#6229—21no—
9C42×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084—21no—
11C4⋊#1084C2212yes—
12C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
14#3750⋊C3—21no—
15#3750⋊C3—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#64079D311yescharacteristic
18C3×#870—31no—
19C3×#869—31no—
20C3×#869—31no—
21C3×#869—61no—
22#63630D311yescharacteristic
23C3×#803—31no—
24#63576—32no—
25#63575—32no—
26#63574—32no—
27#63573—32no—
28C2×C4×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084—22no—
32#6229—42no—
33C2×(C3⋊Dic3)⋊C3—21no—
34C2×(C3⋊Dic3)⋊C3—21no—
35(C3⋊(C3⋊C8))⋊C3Q814yescomplemented
36#5081—91no—
37#5064—91no—
38#4702—91no—
39C2×C3×Dic6—62no—
40C2×C3×Dic6—61no—
41C2×C3×Dic6—61no—
42C32×C42—21no—
43C32×C42C2×D311yescharacteristic
44C32×C42C2×D311yescharacteristic
45C4⋊(C3⋊Dic3)C2×D311yescharacteristic
46C4⋊(C3⋊Dic3)C2×D311yescharacteristic
47#3750—21no—
48#3750—21no—
49C2×C3⋊(C3⋊C8)C2×D312yes—
50C3×C4⋊Dic3—32no—
51C3×C4⋊Dic3—31no—
52C3×C4⋊Dic3—32no—
53C3×C4⋊Dic3—31no—
54C3×C4⋊Dic3—62no—
55C3×C4⋊Dic3—62no—
56C3×C4⋊Dic3—61no—
57C3×C4⋊Dic3—61no—
58C3×Dic3⋊C4—61no—
59C3×Dic3⋊C4—61no—
60C2×C3×C3⋊C8—32no—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C4×H(3,𝔽3)—21no—
63C4×H(3,𝔽3)—21no—
64C4×H(3,𝔽3)C2×Q812yes—
65#1084—42no—
66(C3⋊Dic3)⋊C3—41no—
67(C3⋊Dic3)⋊C3—41no—
68C3×C4⋊Q8—91no—
69C3×#125—91no—
70#870—31no—
71#870—91no—
72#869—31no—
73#869—31no—
74#869—61no—
75C3×#97—91no—
76#818—92no—
77#817—92no—
78#808—92no—
79#808—92no—
80#807—92no—
81#807—92no—
82#803—31no—
83#803—91no—
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4D1211yescharacteristic
88C2×C32×C4D1211yescharacteristic
89C2×C32×C4#6711yescharacteristic
90C2×C32×C4#6711yescharacteristic
91C2×C32×C4C22×D311yescharacteristic
92C2×C32×C4C22×D311yescharacteristic
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—21no—
95C2×C3×Dic3—64no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C3×Dic6—124no—
101C3×Dic6—122no—
102C3×Dic6—122no—
103C3⋊(C3⋊C8)Dic614yescomplemented
104C3×C3⋊C8—34no—
105#501—271noSylow
106C2×H(3,𝔽3)C4⋊Q811yescharacteristic
107C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
108C2×H(3,𝔽3)#13711yescharacteristic
109C2×C3×Q8—181no—
110C2×Dic6—61no—
111C2×Dic6—61no—
112C2×Dic6—62no—
113C2×C3×C8—92no—
114C3×C4⋊C4—91no—
115C3×C4⋊C4—91no—
116C3×C4⋊C4—92no—
117C3×C4⋊C4—181no—
118C3×C4⋊C4—181no—
119C3×C4⋊C4—181no—
120C3×C42—31no—
121C3×C42—31no—
122C3×C42—61no—
123C3×C42D3211yescharacteristic
124C4⋊Dic3—32no—
125C4⋊Dic3—32no—
126C4⋊Dic3—31no—
127C4⋊Dic3—31no—
128C4⋊Dic3—62no—
129C4⋊Dic3—61no—
130C4⋊Dic3—61no—
131C4⋊Dic3—62no—
132C4⋊Dic3—91no—
133C4⋊Dic3—91no—
134Dic3⋊C4—61no—
135Dic3⋊C4—61no—
136Dic3⋊C4—181no—
137Dic3⋊C4—181no—
138C2×C3⋊C8—32no—
139C2×C3⋊C8—92no—
140C22×C32—21no—
141C22×C32C2×D1211yescharacteristic
142C22×C32C2×#6711yescharacteristic
143C32×C4—22no—
144C32×C4—22no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—21no—
149C32×C4—22no—
150C32×C4C2×Dic612yes—
151C32×C4D3×Q812yes—
152C3⋊Dic3—41no—
153C3⋊Dic3—41no—
154C3×Dic3—124no—
155C3×Dic3—121no—
156C3×Dic3—122no—
157C3×Dic3—121no—
158C3×Dic3—122no—
159C4⋊Q8—91no—
160#125—91no—
161#107—272no—
162#106—272no—
163#97—91no—
164H(3,𝔽3)#50111yescharacteristic, complemented, Sylow
165C3×Q8—362no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—61no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—181no—
176C2×C3×C4—181no—
177C2×C3×C4—182no—
178C2×C3×C4#62611yescharacteristic
179C2×C3×C4#62611yescharacteristic
180C2×C3×C4C2×D3211yescharacteristic
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—64no—
184C2×Dic3—61no—
185C2×Dic3—61no—
186C2×Dic3—181no—
187C2×Dic3—181no—
188Dic6—122no—
189Dic6—122no—
190Dic6—124no—
191C3×C8—94no—
192C3⋊C8—34no—
193C3⋊C8—94no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32#86911yescharacteristic
198C2×C32#90811yescharacteristic
199C2×C32#90911yescharacteristic
200C2×C32#94411yescharacteristic
201C2×C32#94911yescharacteristic
202C2×C32#95111yescharacteristic
203C2×Q8—181no—
204C2×C8—92no—
205C4⋊C4—91no—
206C4⋊C4—92no—
207C4⋊C4—91no—
208C4⋊C4—181no—
209C4⋊C4—181no—
210C4⋊C4—181no—
211C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
212C22×C3—31no—
213C22×C3—31no—
214C22×C3—61no—
215C22×C3C2×#62611yescharacteristic
216C3×C4—21no—
217C3×C4—21no—
218C3×C4—32no—
219C3×C4—32no—
220C3×C4—61no—
221C3×C4—61no—
222C3×C4—61no—
223C3×C4—61no—
224C3×C4—62no—
225C3×C4—62no—
226C3×C4—62no—
227C3×C4—361no—
228C3×C4—361no—
229C3×C4—362no—
230C3×C4D3×Dic612yes—
231Dic3—122no—
232Dic3—122no—
233Dic3—124no—
234Dic3—121no—
235Dic3—121no—
236Dic3—361no—
237Dic3—361no—
238C32—21no—
239C32#470211yescharacteristic, complemented
240C32#506411yescharacteristic, complemented
241Q8—362no—
242C2×C4—181no—
243C2×C4—181no—
244C2×C4—182no—
245C2×C4#620011yescharacteristic
246C2×C4#620011yescharacteristic
247C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
248C8—94no—
249C2×C3—31no—
250C2×C3—31no—
251C2×C3—31no—
252C2×C3—31no—
253C2×C3—31no—
254C2×C3—31no—
255C2×C3—61no—
256C2×C3—61no—
257C2×C3—61no—
258C2×C3#6382511yescharacteristic
259C2×C3#6383111yescharacteristic
260C2×C3#6386611yescharacteristic
261C22C2×#620011yescharacteristic
262C4—21no—
263C4—21no—
264C4—362no—
265C4—361no—
266C4—361no—
267C4((C3⋊D3)⋊C3)⋊Q812yes—
268C3—31no—
269C3—31no—
270C3—61no—
271C3#1059030611yescharacteristic
272C2#1172106811yescharacteristic
273C2#1172107411yescharacteristic
274C2#1172109011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#501
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
35. (C3⋊(C3⋊C8))⋊C3Q8semidirect
103. C3⋊(C3⋊C8)Dic6semidirect
164. H(3,𝔽3)#501semidirect
239. C32#4702semidirect
240. C32#5064semidirect

Maximal subgroups

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

subgroupindex
#117217152
#64079⋊C32
#63630⋊C32
#117203072
#117203062
#105903153
#105903053