Finite Group Databasebeta

← #13153896

Subgroups — #13153896

Order 1728.

1815 conjugacy classes of subgroups; only the 265 normal subgroups are shown.

normal subgroups only

This group has 1815 conjugacy classes of subgroups — too many to draw. Only its 265 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×#88740C211yesmaximal, characteristic, complemented
2C22×C4×#272C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
4#11721765C212yesmaximal, complemented
5#11721763C212yesmaximal, complemented
6#11721742C212yesmaximal, complemented
7#11721738C212yesmaximal, complemented
8C4×C4⋊#272C214yesmaximal, complemented
9C3×#5524—34nomaximal
10C23×#272C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235C2212yes—
13C22×#1084C2211yescharacteristic
14#88740—24no—
15C2×C4⋊#272C2214yes—
16C2×C4⋊#272C2211yescharacteristic, complemented
17C2×C4×#272—24no—
18C2×C4×#272C2212yes—
19C2×C4×#272C2212yes—
20C2×C4×#272C2212yescomplemented
21C2×#6229C2211yescharacteristic, complemented
22C4⋊C4×H(3,𝔽3)C2212yescomplemented
23C22⋊C4×H(3,𝔽3)C2212yes—
24C42×H(3,𝔽3)C2212yescomplemented
25#88542C2214yes—
26#88542C2214yescomplemented
27C4⋊#1084C2214yescomplemented
28C2×C3×#235—34no—
29C22×C3×C4×D3—34no—
30C32×#117D314yescomplemented
31C3×#897—38no—
32C3×#895—38no—
33C3×#886—38no—
34C3×#882—38no—
35C3×C4×D12—316no—
36C23×H(3,𝔽3)C2311yescharacteristic
37C22×#272—22no—
38C22×#272—24no—
39C22×#272C2312yes—
40C22×#272C2312yes—
41C2×C4×H(3,𝔽3)D414yescomplemented
42C2×C4×H(3,𝔽3)C2314yes—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#6235—24no—
46C2×#1084—22no—
47C2×#1084C2312yes—
48C2×#1084C2312yes—
49C4⋊#272—28no—
50C4⋊#272—22no—
51C4×#272—24no—
52C4×#272—24no—
53C4×#272—24no—
54#6229—22no—
55C3×#525—91no—
56#5524—34no—
57C23×C3×D3—34no—
58C22×C32×C4C2×D314yes—
59C2×C3×#67—38no—
60C22×C3×Dic3—34no—
61C3×#235—616no—
62C2×C3×D12—316no—
63C2×C3×D12—34no—
64C2×C3×C4×D3—38no—
65C2×C3×C4×D3—38no—
66C2×C3×C4×D3—38no—
67C2×C3×C4×D3—616no—
68C2×C3×Dic6—34no—
69C32×C4⋊C4C2×D318yescomplemented
70C32×C22⋊C4C2×D318yes—
71C32×C42C2×D318yescomplemented
72C3×(C2×D3)⋊C4—316no—
73C3×(C2×D3)⋊C4—316no—
74C3×C4⋊Dic3—316no—
75C22×H(3,𝔽3)—22no—
76C22×H(3,𝔽3)C2×D412yes—
77C22×H(3,𝔽3)C2411yescharacteristic
78C2×#272—28no—
79C2×#272—22no—
80C2×#272—22no—
81C2×#272—22no—
82C2×#272—24no—
83C2×#272#4114yescomplemented
84C4×H(3,𝔽3)—24no—
85C4×H(3,𝔽3)C2×D412yes—
86C4×H(3,𝔽3)C2×D412yes—
87#1084—22no—
88#1084—22no—
89C2×C3×#41—91no—
90C23×C3×C4—91no—
91C2×#235—34no—
92C22×C4×D3—34no—
93C3×#123—92no—
94C3×C22⋊Q8—92no—
95C3×#121—92no—
96C3×#120—92no—
97C3×C4×D4—94no—
98C3×#117—34no—
99C3×#117C3⋊D311yescharacteristic
100#897—38no—
101#895—38no—
102#886—38no—
103#882—38no—
104C4×D12—316no—
105C23×C32C22×D314yes—
106C22×C3×D3—38no—
107C22×C3×D3—38no—
108C22×C3×D3—616no—
109C22×C3×D3—68no—
110C2×C32×C4D12116yescomplemented
111C2×C32×C4C22×D3116yes—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D314yes—
114C3×#67—616no—
115C2×C3×Dic3—38no—
116C2×C3×Dic3—38no—
117C2×C3×Dic3—68no—
118C3×D12—632no—
119C3×D12—68no—
120C3×C4×D3—616no—
121C3×C4×D3—616no—
122C3×C4×D3—616no—
123C3×Dic6—68no—
124#525—91noSylow
125C2×H(3,𝔽3)—22no—
126C2×H(3,𝔽3)C22×D411yescharacteristic
127C2×H(3,𝔽3)C2×#4112yes—
128#272—24no—
129#272—42no—
130C24×C3—91no—
131C23×D3—34no—
132C3×#41—184no—
133C2×C3×Q8—91no—
134C2×C3×D4—92no—
135C2×C3×D4—94no—
136C2×C3×D4—91no—
137C22×C3×C4—34no—
138C22×C3×C4—91no—
139C22×C3×C4—92no—
140C22×C3×C4—92no—
141C22×C3×C4—92no—
142C22×C3×C4—184no—
143C22×C3×C4C2×C3⋊D311yescharacteristic
144C2×#67—38no—
145C22×Dic3—34no—
146#235—616no—
147C2×D12—316no—
148C2×D12—34no—
149C2×C4×D3—38no—
150C2×C4×D3—38no—
151C2×C4×D3—38no—
152C2×C4×D3—616no—
153C2×Dic6—34no—
154C3×C4⋊C4—38no—
155C3×C4⋊C4—94no—
156C3×C4⋊C4C2×C3⋊D312yes—
157C3×C22⋊C4—38no—
158C3×C22⋊C4—94no—
159C3×C22⋊C4—94no—
160C3×C22⋊C4C2×C3⋊D312yes—
161C3×C42—38no—
162C3×C42C2×C3⋊D312yes—
163(C2×D3)⋊C4—316no—
164(C2×D3)⋊C4—316no—
165C4⋊Dic3—316no—
166C22×C32—28no—
167C22×C32C2×D1218yes—
168C22×C32C23×D314yes—
169C2×C3×D3—316no—
170C2×C3×D3—632no—
171C2×C3×D3—68no—
172C2×C3×D3—68no—
173C2×C3×D3—68no—
174C2×C3×D3—616no—
175C32×C4—216no—
176C32×C4C2×D1218yes—
177C32×C4C2×D1218yes—
178C3×Dic3—68no—
179C3×Dic3—68no—
180C2×#41—91no—
181C23×C4—91no—
182#123—92no—
183C22⋊Q8—92no—
184#121—92no—
185#120—92no—
186C4×D4—94no—
187#117#27211yescharacteristic, complemented
188H(3,𝔽3)#52511yescharacteristic, complemented, Sylow
189C23×C3—34no—
190C23×C3—92no—
191C23×C3—92no—
192C23×C3—184no—
193C23×C3—182no—
194C23×C3C22×C3⋊D311yescharacteristic
195C22×D3—38no—
196C22×D3—38no—
197C22×D3—616no—
198C22×D3—68no—
199C3×Q8—182no—
200C3×D4—188no—
201C3×D4—184no—
202C3×D4—182no—
203C2×C3×C4—316no—
204C2×C3×C4—34no—
205C2×C3×C4—34no—
206C2×C3×C4—316no—
207C2×C3×C4—92no—
208C2×C3×C4—92no—
209C2×C3×C4—182no—
210C2×C3×C4—184no—
211C2×C3×C4—184no—
212C2×C3×C4—184no—
213C2×C3×C4C4⋊(C3⋊D3)14yes—
214C2×C3×C4C22×C3⋊D314yes—
215C2×C3×C4C22×C3⋊D311yescharacteristic
216C2×C3×C4C22×C3⋊D311yescharacteristic
217#67—616no—
218C2×Dic3—38no—
219C2×Dic3—38no—
220C2×Dic3—68no—
221D12—68no—
222D12—632no—
223C4×D3—616no—
224C4×D3—616no—
225C4×D3—616no—
226Dic6—68no—
227C2×C32—28no—
228C2×C32C22×D1214yes—
229C2×C32D3×#4118yes—
230C3×D3—616no—
231C3×D3—128no—
232C24—91no—
233#41—184no—
234C2×Q8—91no—
235C2×D4—92no—
236C2×D4—94no—
237C2×D4—91no—
238C22×C4—91no—
239C22×C4—92no—
240C22×C4—92no—
241C22×C4—92no—
242C22×C4—184no—
243C22×C4C2×#27211yescharacteristic
244C4⋊C4—94no—
245C4⋊C4C2×#27212yescomplemented
246C22⋊C4—94no—
247C22⋊C4—94no—
248C22⋊C4C2×#27212yes—
249C42C2×#27212yescomplemented
250C22×C3—22no—
251C22×C3—34no—
252C22×C3—38no—
253C22×C3—68no—
254C22×C3—94no—
255C22×C3—188no—
256C22×C3—182no—
257C22×C3—182no—
258C22×C3—182no—
259C22×C3—184no—
260C22×C3C2×C4⋊(C3⋊D3)12yes—
261C22×C3C23×C3⋊D311yescharacteristic
262C2×D3—316no—
263C2×D3—68no—
264C2×D3—68no—
265C2×D3—632no—
266C2×D3—68no—
267C2×D3—616no—
268C3×C4—24no—
269C3×C4—38no—
270C3×C4—38no—
271C3×C4—616no—
272C3×C4—182no—
273C3×C4—182no—
274C3×C4C2×C4⋊(C3⋊D3)12yes—
275C3×C4C2×C4⋊(C3⋊D3)12yes—
276Dic3—68no—
277Dic3—68no—
278C32#552414yescomplemented
279C23—92no—
280C23—92no—
281C23—184no—
282C23—182no—
283C23C22×#27211yescharacteristic
284Q8—182no—
285D4—188no—
286D4—184no—
287D4—182no—
288C2×C4—92no—
289C2×C4—92no—
290C2×C4—182no—
291C2×C4—184no—
292C2×C4—184no—
293C2×C4—184no—
294C2×C4C4⋊#27214yescomplemented
295C2×C4C22×#27214yes—
296C2×C4C22×#27211yescharacteristic
297C2×C4C22×#27211yescharacteristic
298C2×C3—22no—
299C2×C3—34no—
300C2×C3—38no—
301C2×C3—68no—
302C2×C3—184no—
303C2×C3—362no—
304C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
305C2×C3#41×C3⋊D312yes—
306D3—616no—
307D3—128no—
308C22—22no—
309C22—94no—
310C22—188no—
311C22—182no—
312C22—182no—
313C22—182no—
314C22—184no—
315C22C2×C4⋊#27212yes—
316C22C23×#27211yescharacteristic
317C4—24no—
318C4—182no—
319C4—182no—
320C4C2×C4⋊#27212yes—
321C4C2×C4⋊#27212yes—
322C3—34no—
323C3#1059599511yescharacteristic
324C2—22no—
325C2—184no—
326C2—362no—
327C2C22×C4⋊#27211yescharacteristic
328C2#41×#27212yes—
subgroupG/H
centerC2×C3×C4C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C23×C3⋊D3
FittingH(3,𝔽3)×#117C2
hypercenterC3×#117C3⋊D3
hypocenterH(3,𝔽3)#525
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
1. C2×#88740C2semidirect
2. C22×C4×#272C2semidirect
3. H(3,𝔽3)×#117C2semidirect
4. #11721765C2semidirect
5. #11721763C2semidirect
6. #11721742C2semidirect
7. #11721738C2semidirect
8. C4×C4⋊#272C2semidirect
16. C2×C4⋊#272C22semidirect
20. C2×C4×#272C22semidirect
21. C2×#6229C22semidirect
22. C4⋊C4×H(3,𝔽3)C22semidirect
24. C42×H(3,𝔽3)C22semidirect
26. #88542C22semidirect
27. C4⋊#1084C22semidirect
30. C32×#117D3semidirect
41. C2×C4×H(3,𝔽3)D4semidirect
69. C32×C4⋊C4C2×D3semidirect
71. C32×C42C2×D3semidirect
83. C2×#272#41semidirect
110. C2×C32×C4D12semidirect
187. #117#272semidirect
188. H(3,𝔽3)#525semidirect
245. C4⋊C4C2×#272semidirect
249. C42C2×#272semidirect
278. C32#5524semidirect
294. C2×C4C4⋊#272semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
C22×C4×#2722
H(3,𝔽3)×#1172
#117217652
#117217632
#117217422
#117217382
C4×C4⋊#2722
C3×#55243