Finite Group Databasebeta

← #13154233

Subgroups — #13154233

Order 1728.

1078 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 191 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
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#11721766C212yesmaximal, complemented
3#11721766C212yesmaximal, complemented
4#11721765C212yesmaximal, complemented
5#11721764C212yesmaximal, complemented
6#11721762C212yesmaximal
7#11721756C212yesmaximal, complemented
8#11721721C211yesmaximal, characteristic
9#11721716C211yesmaximal, characteristic, complemented
10C3×#5699—34nomaximal
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2212yes—
13C2×#6229C2211yescharacteristic
14C2×#6229C2212yes—
15C4⋊C4×H(3,𝔽3)C2212yes—
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2212yes—
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C42×H(3,𝔽3)C2211yescharacteristic
20#88542C2212yes—
21#88542C2212yes—
22#88542C2212yes—
23#88542C2212yes—
24#88542C2212yes—
25C4⋊#1084C2212yes—
26C4⋊#1084C2211yescharacteristic
27C4⋊#1084C2211yescharacteristic
28#88540C2212yes—
29#88540C2212yes—
30#88540C2212yes—
31C4×#1084C2212yes—
32C32×#125D314yescomplemented
33C3×#898—38no—
34C3×#898—38no—
35C3×#897—38no—
36C3×#896—38no—
37C3×#894—38no—
38C3×#888—38no—
39C3×#875—34no—
40C3×#870—34no—
41C22×#272C2312yes—
42C2×C4×H(3,𝔽3)C2312yes—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×C4×H(3,𝔽3)C2312yes—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×#1084C2312yes—
48C2×#1084C2312yes—
49C2×#1084C2312yes—
50C4⋊#272—41no—
51C4×#272—42no—
52#6229—41no—
53#6229—42no—
54C3×#563—91no—
55#5699—34no—
56C2×C3×D12—34no—
57C2×C3×C4×D3—38no—
58C2×C3×Dic6—34no—
59C2×C3×Dic6—38no—
60C32×C4⋊C4C2×D318yes—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C2×D318yes—
63C32×C4⋊C4C2×D314yes—
64C32×C42C2×D314yes—
65C3×(C2×D3)⋊C4—38no—
66C3×(C2×D3)⋊C4—38no—
67C3×(C2×D3)⋊C4—38no—
68C3×(C2×D3)⋊C4—38no—
69C3×(C2×D3)⋊C4—38no—
70C3×C4⋊Dic3—38no—
71C3×C4⋊Dic3—34no—
72C3×C4⋊Dic3—34no—
73C3×Dic3⋊C4—38no—
74C3×Dic3⋊C4—38no—
75C3×Dic3⋊C4—38no—
76C3×C4×Dic3—38no—
77C22×H(3,𝔽3)C2411yescharacteristic
78C2×#272—22no—
79C2×#272—22no—
80C2×#272—22no—
81C4×H(3,𝔽3)—22no—
82C4×H(3,𝔽3)—21no—
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)—22no—
85C4×H(3,𝔽3)—21no—
86#1084—22no—
87#1084—22no—
88#1084—22no—
89C3×C4⋊Q8—92no—
90C3×#126—92no—
91C3×#126—92no—
92C3×#125—34no—
93C3×#125—91no—
94C3×#125C3⋊D311yescharacteristic
95C3×#124—91no—
96C3×#123—92no—
97C3×C22⋊Q8—92no—
98C3×C22⋊Q8—92no—
99#898—38no—
100#898—38no—
101#897—38no—
102#896—38no—
103#894—38no—
104#888—38no—
105#875—34no—
106#870—34no—
107C22×C3×D3—38no—
108C2×C32×C4C22×D318yes—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D318yes—
112C2×C32×C4C22×D314yes—
113C2×C3×Dic3—38no—
114C2×C3×Dic3—38no—
115C2×C3×Dic3—38no—
116C3×D12—124no—
117C3×C4×D3—128no—
118C3×Dic6—124no—
119C3×Dic6—128no—
120#563—91noSylow
121C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
122C2×H(3,𝔽3)2−1+411yescharacteristic
123C2×H(3,𝔽3)2−1+411yescharacteristic
124#272—42no—
125C2×C3×Q8—92no—
126C2×C3×Q8—91no—
127C2×C3×D4—91no—
128C22×C3×C4—92no—
129C2×D12—34no—
130C2×C4×D3—38no—
131C2×Dic6—34no—
132C2×Dic6—38no—
133C3×C4⋊C4—38no—
134C3×C4⋊C4—34no—
135C3×C4⋊C4—38no—
136C3×C4⋊C4—34no—
137C3×C4⋊C4—92no—
138C3×C4⋊C4—92no—
139C3×C4⋊C4—92no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—92no—
142C3×C4⋊C4—91no—
143C3×C4⋊C4C2×C3⋊D311yescharacteristic
144C3×C4⋊C4C2×C3⋊D312yes—
145C3×C4⋊C4C2×C3⋊D312yes—
146C3×C4⋊C4C2×C3⋊D311yescharacteristic
147C3×C22⋊C4—92no—
148C3×C22⋊C4—92no—
149C3×C22⋊C4—92no—
150C3×C22⋊C4—92no—
151C3×C22⋊C4—92no—
152C3×C42—34no—
153C3×C42—92no—
154C3×C42C2×C3⋊D311yescharacteristic
155(C2×D3)⋊C4—38no—
156(C2×D3)⋊C4—38no—
157(C2×D3)⋊C4—38no—
158(C2×D3)⋊C4—38no—
159(C2×D3)⋊C4—38no—
160C4⋊Dic3—34no—
161C4⋊Dic3—34no—
162C4⋊Dic3—38no—
163Dic3⋊C4—38no—
164Dic3⋊C4—38no—
165Dic3⋊C4—38no—
166C4×Dic3—38no—
167C22×C32C23×D314yes—
168C2×C3×D3—68no—
169C2×C3×D3—68no—
170C2×C3×D3—68no—
171C32×C4—24no—
172C32×C4—28no—
173C32×C4—24no—
174C32×C4—28no—
175C32×C4—24no—
176C3×Dic3—68no—
177C3×Dic3—68no—
178C3×Dic3—68no—
179C4⋊Q8—92no—
180#126—92no—
181#126—92no—
182#125—91no—
183#125#27211yescharacteristic, complemented
184#124—91no—
185#123—92no—
186C22⋊Q8—92no—
187C22⋊Q8—92no—
188H(3,𝔽3)#56311yescharacteristic, complemented, Sylow
189C23×C3—92no—
190C22×D3—38no—
191C3×Q8—362no—
192C3×Q8—361no—
193C3×D4—361no—
194C2×C3×C4—38no—
195C2×C3×C4—34no—
196C2×C3×C4—34no—
197C2×C3×C4—38no—
198C2×C3×C4—34no—
199C2×C3×C4—92no—
200C2×C3×C4—92no—
201C2×C3×C4—92no—
202C2×C3×C4—362no—
203C2×C3×C4C22×C3⋊D312yes—
204C2×C3×C4C22×C3⋊D311yescharacteristic
205C2×C3×C4C22×C3⋊D311yescharacteristic
206C2×C3×C4C22×C3⋊D311yescharacteristic
207C2×C3×C4C22×C3⋊D312yes—
208C2×Dic3—38no—
209C2×Dic3—38no—
210C2×Dic3—38no—
211D12—124no—
212C4×D3—128no—
213Dic6—124no—
214Dic6—128no—
215C2×C32#100714yes—
216C2×C32#100914yes—
217C2×C32#101014yes—
218C3×D3—128no—
219C2×Q8—92no—
220C2×Q8—91no—
221C2×D4—91no—
222C22×C4—92no—
223C4⋊C4—91no—
224C4⋊C4—92no—
225C4⋊C4—92no—
226C4⋊C4—92no—
227C4⋊C4—92no—
228C4⋊C4—91no—
229C4⋊C4C2×#27211yescharacteristic
230C4⋊C4C2×#27212yes—
231C4⋊C4C2×#27212yes—
232C4⋊C4C2×#27211yescharacteristic
233C22⋊C4—92no—
234C22⋊C4—92no—
235C22⋊C4—92no—
236C22⋊C4—92no—
237C22⋊C4—92no—
238C42—92no—
239C42C2×#27211yescharacteristic
240C22×C3—34no—
241C22×C3—182no—
242C22×C3—182no—
243C22×C3—182no—
244C22×C3C23×C3⋊D311yescharacteristic
245C2×D3—68no—
246C2×D3—68no—
247C2×D3—68no—
248C3×C4—21no—
249C3×C4—22no—
250C3×C4—21no—
251C3×C4—22no—
252C3×C4—21no—
253C3×C4—68no—
254C3×C4—64no—
255C3×C4—64no—
256C3×C4—68no—
257C3×C4—64no—
258C3×C4—182no—
259C3×C4—182no—
260C3×C4—182no—
261Dic3—68no—
262Dic3—68no—
263Dic3—68no—
264C32#569914yescomplemented
265C23—92no—
266Q8—362no—
267Q8—361no—
268D4—361no—
269C2×C4—92no—
270C2×C4—92no—
271C2×C4—92no—
272C2×C4—362no—
273C2×C4C22×#27212yes—
274C2×C4C22×#27211yescharacteristic
275C2×C4C22×#27211yescharacteristic
276C2×C4C22×#27211yescharacteristic
277C2×C4C22×#27212yes—
278C2×C3—34no—
279C2×C3—34no—
280C2×C3—34no—
281C2×C3—362no—
282C2×C3#6436411yescharacteristic
283C2×C3#6436611yescharacteristic
284C2×C3#6436711yescharacteristic
285D3—128no—
286C22—182no—
287C22—182no—
288C22—182no—
289C22C23×#27211yescharacteristic
290C4—21no—
291C4—22no—
292C4—21no—
293C4—22no—
294C4—21no—
295C4—182no—
296C4—182no—
297C4—182no—
298C3—34no—
299C3#1059617011yescharacteristic
300C2—362no—
301C2#1172267011yescharacteristic
302C2#1172267511yescharacteristic
303C2#1172267611yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#125C2
prime radicalC2×C4⋊#272C22
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#563
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
2. #11721766C2semidirect
3. #11721766C2semidirect
4. #11721765C2semidirect
5. #11721764C2semidirect
7. #11721756C2semidirect
9. #11721716C2semidirect
32. C32×#125D3semidirect
183. #125#272semidirect
188. H(3,𝔽3)#563semidirect
264. C32#5699semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217662
#117217652
#117217642
#117217622
#117217562
#117217212
#117217162
C3×#56993