Finite Group Databasebeta

← #13153897

Subgroups — #13153897

Order 1728.

1837 conjugacy classes of subgroups; only the 261 normal subgroups are shown.

normal subgroups only

This group has 1837 conjugacy classes of subgroups — too many to draw. Only its 261 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#11721763C214yesmaximal, complemented
5#11721738C214yesmaximal, complemented
6#11721721C212yesmaximal, complemented
7#11721720C212yesmaximal, complemented
8C3×#5525—34nomaximal
9C23×#272C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235C2212yes—
12#88740—24no—
13C2×C4⋊#272—24no—
14C2×C4⋊#272C2212yes—
15C2×C4⋊#272C2212yes—
16C2×C4⋊#272C2211yescharacteristic, complemented
17C2×C4⋊#272C2218yescomplemented
18C2×C4×#272C2212yescomplemented
19C2×#6229C2211yescharacteristic, complemented
20C4⋊C4×H(3,𝔽3)C2212yescomplemented
21C22⋊C4×H(3,𝔽3)C2212yes—
22C42×H(3,𝔽3)C2212yescomplemented
23#88542C2218yescomplemented
24C2×C3×#235—34no—
25C22×C3×D12—34no—
26C32×#117D314yescomplemented
27C3×#895—316no—
28C3×#882—316no—
29C3×#875—38no—
30C3×#874—38no—
31C23×H(3,𝔽3)C2311yescharacteristic
32C22×#272—24no—
33C22×#272—44no—
34C22×#272C2314yes—
35C22×#272C2312yes—
36C2×C4×H(3,𝔽3)D414yescomplemented
37C2×C4×H(3,𝔽3)C2314yes—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40#6235—24no—
41C2×#1084C2312yes—
42C4⋊#272—24no—
43C4⋊#272—24no—
44C4⋊#272—22no—
45C4⋊#272—48no—
46C4×#272—24no—
47#6229—22no—
48C3×#535—91no—
49#5525—34no—
50C23×C3×D3—38no—
51C22×C32×C4C2×D314yes—
52C2×C3×#67—38no—
53C3×#235—616no—
54C2×C3×D12—38no—
55C2×C3×D12—38no—
56C2×C3×D12—34no—
57C2×C3×D12—332no—
58C2×C3×D12—616no—
59C2×C3×C4×D3—38no—
60C2×C3×Dic6—34no—
61C32×C4⋊C4C2×D318yescomplemented
62C32×C22⋊C4C2×D318yes—
63C32×C42C2×D318yescomplemented
64C3×(C2×D3)⋊C4—332no—
65C22×H(3,𝔽3)—22no—
66C22×H(3,𝔽3)C2×D412yes—
67C22×H(3,𝔽3)C2411yescharacteristic
68C2×#272—22no—
69C2×#272—24no—
70C2×#272—24no—
71C2×#272—28no—
72C2×#272—48no—
73C4×H(3,𝔽3)—24no—
74C4×H(3,𝔽3)C2×D412yes—
75C4×H(3,𝔽3)C2×D412yes—
76#1084—22no—
77C2×C3×#41—91no—
78C22×C3×D4—91no—
79C2×#235—34no—
80C22×D12—34no—
81C3×#127—92no—
82C3×#124—92no—
83C3×#121—94no—
84C3×#120—94no—
85C3×#117—34no—
86C3×#117C3⋊D311yescharacteristic
87#895—316no—
88#882—316no—
89#875—38no—
90#874—38no—
91C23×C32C22×D314yes—
92C22×C3×D3—316no—
93C22×C3×D3—38no—
94C22×C3×D3—616no—
95C22×C3×D3—1216no—
96C2×C32×C4D12116yescomplemented
97C2×C32×C4C22×D3116yes—
98C2×C32×C4C22×D314yes—
99C2×C32×C4C22×D314yes—
100C3×#67—616no—
101C2×C3×Dic3—38no—
102C3×D12—616no—
103C3×D12—616no—
104C3×D12—68no—
105C3×D12—1232no—
106C3×C4×D3—616no—
107C3×Dic6—68no—
108#535—91noSylow
109C2×H(3,𝔽3)—22no—
110C2×H(3,𝔽3)C22×D411yescharacteristic
111C2×H(3,𝔽3)H(5,𝔽2)12yes—
112#272—42no—
113#272—44no—
114C24×C3—92no—
115C23×D3—38no—
116C3×#41—184no—
117C2×C3×Q8—91no—
118C2×C3×D4—92no—
119C2×C3×D4—92no—
120C2×C3×D4—92no—
121C2×C3×D4—91no—
122C2×C3×D4—98no—
123C2×C3×D4—184no—
124C22×C3×C4—34no—
125C22×C3×C4—92no—
126C22×C3×C4C2×C3⋊D311yescharacteristic
127C2×#67—38no—
128#235—616no—
129C2×D12—34no—
130C2×D12—38no—
131C2×D12—38no—
132C2×D12—332no—
133C2×D12—616no—
134C2×C4×D3—38no—
135C2×Dic6—34no—
136C3×C4⋊C4—38no—
137C3×C4⋊C4C2×C3⋊D312yes—
138C3×C22⋊C4—38no—
139C3×C22⋊C4—98no—
140C3×C22⋊C4C2×C3⋊D312yes—
141C3×C42—38no—
142C3×C42C2×C3⋊D312yes—
143(C2×D3)⋊C4—332no—
144C22×C32—28no—
145C22×C32C2×D1218yes—
146C22×C32C23×D314yes—
147C2×C3×D3—68no—
148C2×C3×D3—616no—
149C2×C3×D3—616no—
150C2×C3×D3—632no—
151C2×C3×D3—1232no—
152C32×C4—216no—
153C32×C4C2×D1218yes—
154C32×C4C2×D1218yes—
155C3×Dic3—68no—
156C2×#41—91no—
157C22×D4—91no—
158#127—92no—
159#124—92no—
160#121—94no—
161#120—94no—
162#117#27211yescharacteristic, complemented
163H(3,𝔽3)#53511yescharacteristic, complemented, Sylow
164C23×C3—34no—
165C23×C3—94no—
166C23×C3—92no—
167C23×C3—184no—
168C23×C3—364no—
169C23×C3C22×C3⋊D311yescharacteristic
170C22×D3—316no—
171C22×D3—38no—
172C22×D3—616no—
173C22×D3—1216no—
174C3×Q8—182no—
175C3×D4—184no—
176C3×D4—184no—
177C3×D4—184no—
178C3×D4—182no—
179C3×D4—368no—
180C2×C3×C4—316no—
181C2×C3×C4—34no—
182C2×C3×C4—34no—
183C2×C3×C4—316no—
184C2×C3×C4—92no—
185C2×C3×C4—184no—
186C2×C3×C4C4⋊(C3⋊D3)14yes—
187C2×C3×C4C22×C3⋊D314yes—
188C2×C3×C4C22×C3⋊D311yescharacteristic
189C2×C3×C4C22×C3⋊D311yescharacteristic
190#67—616no—
191C2×Dic3—38no—
192D12—616no—
193D12—616no—
194D12—68no—
195D12—1232no—
196C4×D3—616no—
197Dic6—68no—
198C2×C32—28no—
199C2×C32C22×D1214yes—
200C2×C32#100918yes—
201C3×D3—128no—
202C3×D3—1216no—
203C24—92no—
204#41—184no—
205C2×Q8—91no—
206C2×D4—92no—
207C2×D4—92no—
208C2×D4—92no—
209C2×D4—91no—
210C2×D4—98no—
211C2×D4—184no—
212C22×C4—92no—
213C22×C4C2×#27211yescharacteristic
214C4⋊C4C2×#27212yescomplemented
215C22⋊C4—98no—
216C22⋊C4C2×#27212yes—
217C42C2×#27212yescomplemented
218C22×C3—22no—
219C22×C3—34no—
220C22×C3—38no—
221C22×C3—68no—
222C22×C3—182no—
223C22×C3—184no—
224C22×C3—184no—
225C22×C3—188no—
226C22×C3—368no—
227C22×C3C2×C4⋊(C3⋊D3)12yes—
228C22×C3C23×C3⋊D311yescharacteristic
229C2×D3—68no—
230C2×D3—616no—
231C2×D3—616no—
232C2×D3—632no—
233C2×D3—1232no—
234C3×C4—24no—
235C3×C4—38no—
236C3×C4—38no—
237C3×C4—616no—
238C3×C4—182no—
239C3×C4C2×C4⋊(C3⋊D3)12yes—
240C3×C4C2×C4⋊(C3⋊D3)12yes—
241Dic3—68no—
242C32#552514yescomplemented
243C23—94no—
244C23—92no—
245C23—184no—
246C23—364no—
247C23C22×#27211yescharacteristic
248Q8—182no—
249D4—184no—
250D4—184no—
251D4—184no—
252D4—182no—
253D4—368no—
254C2×C4—92no—
255C2×C4—184no—
256C2×C4C4⋊#27214yescomplemented
257C2×C4C22×#27214yes—
258C2×C4C22×#27211yescharacteristic
259C2×C4C22×#27211yescharacteristic
260C2×C3—22no—
261C2×C3—34no—
262C2×C3—38no—
263C2×C3—68no—
264C2×C3—364no—
265C2×C3—362no—
266C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
267C2×C3#6436612yes—
268D3—1216no—
269D3—128no—
270C22—22no—
271C22—182no—
272C22—184no—
273C22—184no—
274C22—188no—
275C22—368no—
276C22C2×C4⋊#27212yes—
277C22C23×#27211yescharacteristic
278C4—24no—
279C4—182no—
280C4C2×C4⋊#27212yes—
281C4C2×C4⋊#27212yes—
282C3—34no—
283C3#1059599611yescharacteristic
284C2—22no—
285C2—364no—
286C2—362no—
287C2C22×C4⋊#27211yescharacteristic
288C2#1172267512yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#117C2
hypercenterC3×#117C3⋊D3
hypocenterH(3,𝔽3)#535
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ 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. #11721763C2semidirect
5. #11721738C2semidirect
6. #11721721C2semidirect
7. #11721720C2semidirect
16. C2×C4⋊#272C22semidirect
17. C2×C4⋊#272C22semidirect
18. C2×C4×#272C22semidirect
19. C2×#6229C22semidirect
20. C4⋊C4×H(3,𝔽3)C22semidirect
22. C42×H(3,𝔽3)C22semidirect
23. #88542C22semidirect
26. C32×#117D3semidirect
36. C2×C4×H(3,𝔽3)D4semidirect
61. C32×C4⋊C4C2×D3semidirect
63. C32×C42C2×D3semidirect
96. C2×C32×C4D12semidirect
162. #117#272semidirect
163. H(3,𝔽3)#535semidirect
214. C4⋊C4C2×#272semidirect
217. C42C2×#272semidirect
242. C32#5525semidirect
256. 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
#117217632
#117217382
#117217212
#117217202
C3×#55253