Finite Group Databasebeta

← #13137421

Subgroups — #13137421

Order 1728.

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

normal subgroups only

This group has 612 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#11721720C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic, complemented
3C2×#88437C212yesmaximal, complemented
4C2×#88436C212yesmaximal, complemented
5#63630⋊C3C211yesmaximal, characteristic, complemented
6#10590371—31nomaximal
7#10590354—31nomaximal
8C2×C4⋊#272—22no—
9C2×C4⋊#272C2212yes—
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C42×H(3,𝔽3)C2211yescharacteristic
13#3752⋊C3—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88437—24no—
16#88436—24no—
17#64084D311yescharacteristic, complemented
18C3×#875—31no—
19C3×#874—31no—
20C3×#874—31no—
21C3×#874—61no—
22C2×#3716—32no—
23C2×#3715—32no—
24C2×#3714—32no—
25C2×#3713—32no—
26#63630D311yescharacteristic, complemented
27C3×#803—31no—
28C22×#272—22no—
29C22×(C3⋊D3)⋊C3—21no—
30C2×C4×H(3,𝔽3)D412yes—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C4⋊#272—22no—
33C4⋊#272—22no—
34C4⋊#272—44no—
35C2×(C3⋊Dic3)⋊C3—21no—
36C4⋊((C3⋊D3)⋊C3)—22no—
37#634⋊C3—22no—
38(C3⋊(C3⋊C8))⋊C3D414yescomplemented
39#5074—91no—
40#5058—91no—
41#4712—91no—
42C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
43C2×#634C2×D311yescharacteristic
44C2×C3×D12—31no—
45C2×C3×D12—32no—
46C2×C3×D12—32no—
47C2×C3×D12—62no—
48C2×C3×D12—64no—
49C2×C3×D12—62no—
50C2×C3×D12—62no—
51C2×C3×Dic6—31no—
52C32×C42—21no—
53C32×C42C2×D311yescharacteristic
54C32×C42C2×D311yescharacteristic
55#3752—22no—
56C2×C3⋊(C3⋊C8)C2×D312yes—
57C3×(C2×D3)⋊C4—62no—
58C2×C3×C3⋊C8—32no—
59#3716—64no—
60#3715—64no—
61#3714—64no—
62#3713—64no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—42no—
65C2×#272—44no—
66C2×(C3⋊D3)⋊C3—41no—
67C2×(C3⋊D3)⋊C3—42no—
68C4×H(3,𝔽3)—22no—
69C4×H(3,𝔽3)C2×D412yes—
70(C3⋊Dic3)⋊C3—41no—
71C3×#127—91no—
72C3×#124—91no—
73C2×#215—92no—
74C2×#214—92no—
75C2×#213—92no—
76C2×#213—92no—
77C2×D24—92no—
78C2×#204—92no—
79#875—31no—
80#875—91no—
81#874—31no—
82#874—31no—
83#874—61no—
84C3×#97—91no—
85#803—31no—
86#803—91no—
87C22×C3⋊D3—21no—
88C22×C3×D3—64no—
89C22×C3×D3—61no—
90C22×C3×D3—62no—
91C22×C3×D3—62no—
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4D1212yes—
95C2×C32×C4#6712yes—
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C22×D311yescharacteristic
98C2×C3⋊Dic3—21no—
99C4⋊(C3⋊D3)—22no—
100#634—22no—
101C2×C3×Dic3—61no—
102C3×D12—62no—
103C3×D12—62no—
104C3×D12—62no—
105C3×D12—62no—
106C3×D12—62no—
107C3×D12—122no—
108C3×D12—122no—
109C3×D12—128no—
110C3×D12—124no—
111C3×D12—124no—
112C3×Dic6—62no—
113C3⋊(C3⋊C8)D1214yescomplemented
114C3×C3⋊C8—34no—
115#496—271noSylow
116C2×H(3,𝔽3)#12711yescharacteristic
117C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
118#272—82no—
119(C3⋊D3)⋊C3—81no—
120C2×C3×Q8—91no—
121C2×C3×D4—91no—
122C2×C3×D4—92no—
123C2×C3×D4—182no—
124C2×D12—32no—
125C2×D12—32no—
126C2×D12—31no—
127C2×D12—62no—
128C2×D12—62no—
129C2×D12—62no—
130C2×D12—64no—
131C2×D12—91no—
132C2×Dic6—31no—
133C2×Dic6—91no—
134C2×C3×C8—92no—
135C3×C22⋊C4—182no—
136C3×C42—31no—
137C3×C42—31no—
138C3×C42—61no—
139C3×C42D3211yescharacteristic
140#215—184no—
141#214—184no—
142#213—184no—
143#213—184no—
144(C2×D3)⋊C4—62no—
145(C2×D3)⋊C4—182no—
146C2×C3⋊C8—32no—
147C2×C3⋊C8—92no—
148D24—184no—
149#204—184no—
150C22×C32—21no—
151C22×C32C2×D1211yescharacteristic
152C22×C32C2×#6711yescharacteristic
153C2×C3⋊D3—41no—
154C2×C3⋊D3—42no—
155C2×C3×D3—128no—
156C2×C3×D3—124no—
157C2×C3×D3—121no—
158C2×C3×D3—122no—
159C2×C3×D3—122no—
160C2×C3×D3—124no—
161C2×C3×D3—122no—
162C2×C3×D3—124no—
163C32×C4—24no—
164C32×C4—22no—
165C32×C4—22no—
166C32×C4—22no—
167C32×C4C2×D1212yes—
168C32×C4D3×D412yes—
169C3⋊Dic3—41no—
170C3×Dic3—121no—
171C2×SD16—272no—
172C2×D8—272no—
173#127—91no—
174#124—91no—
175#97—91no—
176H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
177C23×C3—181no—
178C23×C3—182no—
179C22×D3—62no—
180C22×D3—62no—
181C22×D3—64no—
182C22×D3—61no—
183C22×D3—181no—
184C3×Q8—182no—
185C3×D4—182no—
186C3×D4—182no—
187C3×D4—182no—
188C3×D4—364no—
189C2×C3×C4—32no—
190C2×C3×C4—32no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—62no—
194C2×C3×C4—61no—
195C2×C3×C4—181no—
196C2×C3×C4#62612yes—
197C2×C3×C4C2×D3211yescharacteristic
198C2×Dic3—61no—
199C2×Dic3—181no—
200D12—62no—
201D12—62no—
202D12—62no—
203D12—62no—
204D12—62no—
205D12—122no—
206D12—122no—
207D12—124no—
208D12—124no—
209D12—128no—
210D12—182no—
211Dic6—62no—
212Dic6—182no—
213C3×C8—94no—
214C3⋊C8—34no—
215C3⋊C8—94no—
216C2×C32—22no—
217C2×C32—21no—
218C2×C32#87411yescharacteristic
219C2×C32#90812yes—
220C2×C32#94011yescharacteristic
221C2×C32#94912yes—
222C3⋊D3—81no—
223C3×D3—244no—
224C3×D3—241no—
225C3×D3—242no—
226C3×D3—242no—
227C2×Q8—91no—
228C2×D4—91no—
229C2×D4—92no—
230C2×D4—182no—
231SD16—544no—
232D8—544no—
233C2×C8—92no—
234C22⋊C4—182no—
235C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—61no—
239C22×C3—361no—
240C22×C3—362no—
241C22×C3—364no—
242C22×C3—362no—
243C22×C3C2×#62611yescharacteristic
244C2×D3—122no—
245C2×D3—122no—
246C2×D3—124no—
247C2×D3—124no—
248C2×D3—128no—
249C2×D3—124no—
250C2×D3—122no—
251C2×D3—121no—
252C2×D3—361no—
253C2×D3—362no—
254C3×C4—22no—
255C3×C4—32no—
256C3×C4—32no—
257C3×C4—62no—
258C3×C4—62no—
259C3×C4—64no—
260C3×C4—62no—
261C3×C4—361no—
262C3×C4D3×D1212yes—
263Dic3—121no—
264Dic3—361no—
265C32—21no—
266C32#471211yescharacteristic, complemented
267C32#505811yescharacteristic, complemented
268C23—181no—
269C23—182no—
270Q8—182no—
271D4—182no—
272D4—182no—
273D4—182no—
274D4—364no—
275C2×C4—181no—
276C2×C4#620012yes—
277C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
278C8—94no—
279C2×C3—32no—
280C2×C3—32no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—62no—
284C2×C3—61no—
285C2×C3—721no—
286C2×C3—722no—
287C2×C3#6382512yes—
288C2×C3#6391111yescharacteristic
289D3—242no—
290D3—242no—
291D3—244no—
292D3—241no—
293D3—721no—
294C22—364no—
295C22—362no—
296C22—362no—
297C22—361no—
298C22C2×#620011yescharacteristic
299C4—22no—
300C4—361no—
301C4#8865412yes—
302C3—31no—
303C3—31no—
304C3—61no—
305C3#1059035111yescharacteristic
306C2—722no—
307C2—721no—
308C2#1172106812yes—
309C2#1172115811yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#496
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
1. #11721720C2semidirect
2. #64084⋊C3C2semidirect
3. C2×#88437C2semidirect
4. C2×#88436C2semidirect
5. #63630⋊C3C2semidirect
17. #64084D3semidirect
26. #63630D3semidirect
38. (C3⋊(C3⋊C8))⋊C3D4semidirect
113. C3⋊(C3⋊C8)D12semidirect
176. H(3,𝔽3)#496semidirect
266. C32#4712semidirect
267. C32#5058semidirect

Maximal subgroups

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

subgroupindex
#117217202
#64084⋊C32
C2×#884372
C2×#884362
#63630⋊C32
#105903713
#105903543