Finite Group Databasebeta

← #13143869

Subgroups — #13143869

Order 1728.

1089 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 1089 conjugacy classes of subgroups — too many to draw. Only its 99 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#11722664C211yesmaximal, characteristic, complemented
2#11721847C211yesmaximal, characteristic, complemented
3#11721738C212yesmaximal, complemented
4H(3,𝔽3)×#99C211yesmaximal, characteristic, complemented
5#11720524C212yesmaximal, complemented
6C3×#4741—34nomaximal
7C23×#272C2211yescharacteristic, complemented
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#6235—21no—
10C2×#6235—21no—
11C2×#6235C2212yes—
12#88744—22no—
13D4×#272—22no—
14#88740—21no—
15C2×C4⋊#272—22no—
16C22⋊C4×H(3,𝔽3)C2212yescomplemented
17#88552C2211yescharacteristic, complemented
18#88542—21no—
19#88542—22no—
20C3×#1004—34no—
21C3×#937—34no—
22C3×#882—38no—
23C32×#99D314yescomplemented
24C3×#806—38no—
25C23×H(3,𝔽3)D411yescharacteristic, complemented
26C23×H(3,𝔽3)C2311yescharacteristic
27C22×#272—22no—
28C22×#272—21no—
29C22×#272—21no—
30C22×#272—42no—
31C22×#272D412yescomplemented
32D4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)D411yescharacteristic, complemented
35#6235—22no—
36#6235—22no—
37#6235—22no—
38#6235—42no—
39C2×#1084—21no—
40C2×#1084—21no—
41C2×#1084D412yescomplemented
42C4⋊#272—21no—
43C4⋊#272—44no—
44C4×#272—22no—
45#6229—21no—
46C3×B(4,𝔽2)—91no—
47#4741—34no—
48C23×C3×D3—34no—
49C2×C32×D4C2×D314yes—
50C2×C3×#67—38no—
51C2×C3×#67—64no—
52C2×C3×#67—64no—
53C3×#237—68no—
54C3×D3×D4—68no—
55C3×#235—64no—
56C2×C3×D12—68no—
57C32×C22⋊C4C2×D318yescomplemented
58C3×#217—34no—
59C3×(C2×D3)⋊C4—68no—
60C3×(C2×D3)⋊C4—64no—
61C22×H(3,𝔽3)—21no—
62C22×H(3,𝔽3)—41no—
63C22×H(3,𝔽3)—41no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C22×H(3,𝔽3)C2×D412yes—
66C2×#272—24no—
67C2×#272—22no—
68C2×#272—22no—
69C2×#272—41no—
70C2×#272—42no—
71C2×#272—41no—
72C2×#272—42no—
73C2×#272—81no—
74C4×H(3,𝔽3)—21no—
75C4×H(3,𝔽3)—42no—
76#1084—22no—
77#1084—41no—
78C3×H(5,𝔽2)—91no—
79#1004—34no—
80C3×#120—91no—
81C3×#120—92no—
82#937—34no—
83#882—38no—
84C3×#99—34no—
85C3×#99—92no—
86C3×#99C3⋊D311yescharacteristic
87#806—38no—
88C23×C32D1214yescomplemented
89C23×C32C22×D314yes—
90C22×C3×D3—38no—
91C22×C3×D3—64no—
92C22×C3×D3—68no—
93C22×C3×D3—64no—
94C22×C3×D3—128no—
95C32×D4—28no—
96C2×C32×C4—28no—
97C2×C32×C4D1214yescomplemented
98C3×#67—68no—
99C3×#67—68no—
100C3×#67—68no—
101C3×#67—128no—
102C2×C3×Dic3—38no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—64no—
105C3×D12—64no—
106C3×D12—1216no—
107C3×C4×D3—68no—
108C3×Dic6—64no—
109B(4,𝔽2)—91noSylow
110C2×H(3,𝔽3)—21no—
111C2×H(3,𝔽3)—22no—
112C2×H(3,𝔽3)—41no—
113C2×H(3,𝔽3)#12011yescharacteristic
114#272—42no—
115#272—42no—
116C24×C3—91no—
117C23×D3—34no—
118C3×#41—182no—
119C3×#41—181no—
120C2×C3×D4—34no—
121C2×C3×D4—92no—
122C2×C3×D4—182no—
123C2×C3×D4—182no—
124C2×C3×D4—181no—
125C2×C3×D4—181no—
126C2×C3×D4C2×C3⋊D311yescharacteristic
127C2×#67—38no—
128C2×#67—64no—
129C2×#67—64no—
130#237—68no—
131D3×D4—68no—
132#235—64no—
133C2×D12—68no—
134C3×C22⋊C4—38no—
135C3×C22⋊C4—91no—
136C3×C22⋊C4—182no—
137C3×C22⋊C4—181no—
138C3×C22⋊C4C2×C3⋊D312yes—
139#217—34no—
140(C2×D3)⋊C4—68no—
141(C2×D3)⋊C4—64no—
142C22×C32—24no—
143C22×C32—44no—
144C22×C32—44no—
145C22×C32C2×D1214yes—
146C22×C32D3×D418yes—
147C2×C3×D3—616no—
148C2×C3×D3—68no—
149C2×C3×D3—68no—
150C2×C3×D3—124no—
151C2×C3×D3—128no—
152C2×C3×D3—124no—
153C2×C3×D3—128no—
154C2×C3×D3—244no—
155C32×C4—24no—
156C32×C4—48no—
157C3×Dic3—68no—
158C3×Dic3—124no—
159H(5,𝔽2)—91no—
160#120—91no—
161#120—92no—
162#99—92no—
163#99#27211yescharacteristic, complemented
164H(3,𝔽3)B(4,𝔽2)11yescharacteristic, complemented, Sylow
165C23×C3—34no—
166C23×C3—34no—
167C23×C3—92no—
168C23×C3—181no—
169C23×C3—181no—
170C23×C3—182no—
171C23×C3—362no—
172C23×C3C4⋊(C3⋊D3)11yescharacteristic
173C23×C3C22×C3⋊D311yescharacteristic
174C22×D3—38no—
175C22×D3—68no—
176C22×D3—64no—
177C22×D3—64no—
178C22×D3—128no—
179C3×Q8—181no—
180C3×D4—22no—
181C3×D4—68no—
182C3×D4—181no—
183C3×D4—182no—
184C3×D4—182no—
185C3×D4—182no—
186C3×D4—364no—
187C3×D4—362no—
188C2×C3×C4—22no—
189C2×C3×C4—34no—
190C2×C3×C4—68no—
191C2×C3×C4—92no—
192C2×C3×C4—182no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
196#67—68no—
197#67—68no—
198#67—68no—
199#67—128no—
200C2×Dic3—38no—
201C2×Dic3—64no—
202C2×Dic3—64no—
203D12—64no—
204D12—1216no—
205C4×D3—68no—
206Dic6—64no—
207C2×C32—24no—
208C2×C32—28no—
209C2×C32—44no—
210C2×C32#88214yes—
211C3×D3—128no—
212C3×D3—128no—
213C24—91no—
214#41—182no—
215#41—181no—
216C2×D4—92no—
217C2×D4—182no—
218C2×D4—182no—
219C2×D4—181no—
220C2×D4—181no—
221C2×D4C2×#27211yescharacteristic
222C22⋊C4—91no—
223C22⋊C4—182no—
224C22⋊C4—181no—
225C22⋊C4C2×#27212yescomplemented
226C22×C3—21no—
227C22×C3—34no—
228C22×C3—38no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—64no—
232C22×C3—124no—
233C22×C3—124no—
234C22×C3—184no—
235C22×C3—182no—
236C22×C3—182no—
237C22×C3—362no—
238C22×C3—362no—
239C22×C3—361no—
240C22×C3—361no—
241C22×C3—721no—
242C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
243C22×C3D4×C3⋊D312yes—
244C2×D3—68no—
245C2×D3—616no—
246C2×D3—68no—
247C2×D3—124no—
248C2×D3—124no—
249C2×D3—128no—
250C2×D3—128no—
251C2×D3—244no—
252C3×C4—21no—
253C3×C4—42no—
254C3×C4—64no—
255C3×C4—128no—
256C3×C4—182no—
257C3×C4—361no—
258Dic3—68no—
259Dic3—124no—
260C32#474114yescomplemented
261C23—92no—
262C23—181no—
263C23—181no—
264C23—182no—
265C23—362no—
266C23C4⋊#27211yescharacteristic, complemented
267C23C22×#27211yescharacteristic
268Q8—181no—
269D4—22no—
270D4—181no—
271D4—182no—
272D4—182no—
273D4—182no—
274D4—364no—
275D4—362no—
276C2×C4—22no—
277C2×C4—92no—
278C2×C4—182no—
279C2×C4—181no—
280C2×C4—181no—
281C2×C4C4⋊#27211yescharacteristic, complemented
282C2×C3—22no—
283C2×C3—21no—
284C2×C3—34no—
285C2×C3—41no—
286C2×C3—64no—
287C2×C3—68no—
288C2×C3—124no—
289C2×C3—362no—
290C2×C3—362no—
291C2×C3#6409111yescharacteristic
292D3—128no—
293D3—128no—
294C22—21no—
295C22—41no—
296C22—41no—
297C22—184no—
298C22—182no—
299C22—182no—
300C22—362no—
301C22—362no—
302C22—361no—
303C22—361no—
304C22—721no—
305C22C2×C4⋊#27211yescharacteristic
306C22D4×#27212yes—
307C4—21no—
308C4—42no—
309C4—182no—
310C4—361no—
311C3—34no—
312C3#1059244011yescharacteristic
313C2—22no—
314C2—21no—
315C2—41no—
316C2—362no—
317C2—362no—
318C2#1172173811yescharacteristic
subgroupG/H
center, socleC2×C3#64091
commutator, cosocleC23×H(3,𝔽3)C23
FrattiniC23×C3C22×C3⋊D3
FittingH(3,𝔽3)×#99C2
hypercenterC3×#99C3⋊D3
hypocenterH(3,𝔽3)B(4,𝔽2)
LCSG ▷ C23×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
1. #11722664C2semidirect
2. #11721847C2semidirect
3. #11721738C2semidirect
4. H(3,𝔽3)×#99C2semidirect
5. #11720524C2semidirect
7. C23×#272C22semidirect
16. C22⋊C4×H(3,𝔽3)C22semidirect
17. #88552C22semidirect
23. C32×#99D3semidirect
25. C23×H(3,𝔽3)D4semidirect
31. C22×#272D4semidirect
34. C2×C4×H(3,𝔽3)D4semidirect
41. C2×#1084D4semidirect
57. C32×C22⋊C4C2×D3semidirect
88. C23×C32D12semidirect
97. C2×C32×C4D12semidirect
163. #99#272semidirect
164. H(3,𝔽3)B(4,𝔽2)semidirect
225. C22⋊C4C2×#272semidirect
260. C32#4741semidirect
266. C23C4⋊#272semidirect
281. C2×C4C4⋊#272semidirect

Maximal subgroups

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

subgroupindex
#117226642
#117218472
#117217382
H(3,𝔽3)×#992
#117205242
C3×#47413