Finite Group Databasebeta

← #13154304

Subgroups — #13154304

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)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#11721864C211yesmaximal, characteristic
3#11721863C212yesmaximal, complemented
4#11721861C211yesmaximal, characteristic, complemented
5#11721766C212yesmaximal, complemented
6#11721764C212yesmaximal, complemented
7#11721762C212yesmaximal
8#11721757C212yesmaximal, complemented
9#11721722C212yesmaximal, complemented
10C3×#5735—34nomaximal
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272C2212yes—
15C2×#6229C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)C2212yes—
17C4⋊C4×H(3,𝔽3)C2212yes—
18C42×H(3,𝔽3)C2211yescharacteristic
19#88542C2212yes—
20#88542C2212yes—
21#88542C2212yes—
22#88542C2212yes—
23#88542C2212yes—
24C4⋊#1084C2212yes—
25#88540C2212yes—
26#88540C2212yes—
27#88540C2212yes—
28#88540C2212yes—
29#88540C2212yes—
30C4×#1084C2211yescharacteristic
31C4×#1084C2211yescharacteristic
32C32×C4⋊Q8D314yescomplemented
33C3×#947—34no—
34C3×#946—38no—
35C3×#944—34no—
36C3×#898—38no—
37C3×#896—38no—
38C3×#894—38no—
39C3×#889—38no—
40C3×#876—38no—
41C22×#272C2312yes—
42Q8×H(3,𝔽3)—41no—
43Q8×H(3,𝔽3)—41no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×C4×H(3,𝔽3)C2312yes—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2312yes—
49C2×#1084C2312yes—
50C2×#1084C2312yes—
51C2×#1084C2312yes—
52C4⋊#272—41no—
53C4×#272—42no—
54#6229—41no—
55C3×#563—91no—
56#5735—34no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D314yes—
59C2×C3×D12—34no—
60C2×C3×C4×D3—38no—
61C2×C3×Dic6—34no—
62C32×C4⋊C4C2×D318yes—
63C32×C4⋊C4C2×D318yes—
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×Dic3⋊C4—38no—
72C3×Dic3⋊C4—38no—
73C3×Dic3⋊C4—38no—
74C3×Dic3⋊C4—38no—
75C3×Dic3⋊C4—38no—
76C3×C4×Dic3—34no—
77C3×C4×Dic3—34no—
78C22×H(3,𝔽3)C2411yescharacteristic
79C2×#272—22no—
80C2×#272—24no—
81C4×H(3,𝔽3)—21no—
82C4×H(3,𝔽3)—22no—
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)—21no—
85C4×H(3,𝔽3)—22no—
86#1084—22no—
87#1084—22no—
88#1084—22no—
89C3×C4⋊Q8—34no—
90C3×C4⋊Q8—91no—
91C3×C4⋊Q8C3⋊D311yescharacteristic
92C3×#126—92no—
93C3×#126—92no—
94C3×#125—92no—
95C3×#124—91no—
96C3×#123—92no—
97C3×C22⋊Q8—92no—
98C3×C22⋊Q8—92no—
99#947—34no—
100#946—38no—
101#944—34no—
102#898—38no—
103#896—38no—
104#894—38no—
105#889—38no—
106#876—38no—
107C22×C3×D3—38no—
108C32×Q8—44no—
109C32×Q8—44no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D318yes—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D318yes—
115C2×C3×Dic3—38no—
116C2×C3×Dic3—38no—
117C2×C3×Dic3—38no—
118C3×D12—124no—
119C3×C4×D3—128no—
120C3×Dic6—124no—
121#563—91noSylow
122C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
123C2×H(3,𝔽3)2−1+412yes—
124#272—42no—
125C2×C3×Q8—34no—
126C2×C3×Q8—34no—
127C2×C3×Q8—91no—
128C2×C3×Q8C2×C3⋊D311yescharacteristic
129C2×C3×Q8C2×C3⋊D311yescharacteristic
130C2×C3×D4—91no—
131C22×C3×C4—92no—
132C2×D12—34no—
133C2×C4×D3—38no—
134C2×Dic6—34no—
135C3×C4⋊C4—38no—
136C3×C4⋊C4—38no—
137C3×C4⋊C4—92no—
138C3×C4⋊C4—92no—
139C3×C4⋊C4—92no—
140C3×C4⋊C4—92no—
141C3×C4⋊C4—92no—
142C3×C4⋊C4—92no—
143C3×C4⋊C4C2×C3⋊D312yes—
144C3×C4⋊C4C2×C3⋊D312yes—
145C3×C22⋊C4—92no—
146C3×C22⋊C4—92no—
147C3×C22⋊C4—92no—
148C3×C22⋊C4—92no—
149C3×C22⋊C4—92no—
150C3×C42—34no—
151C3×C42—91no—
152C3×C42—91no—
153C3×C42C2×C3⋊D311yescharacteristic
154(C2×D3)⋊C4—38no—
155(C2×D3)⋊C4—38no—
156(C2×D3)⋊C4—38no—
157(C2×D3)⋊C4—38no—
158(C2×D3)⋊C4—38no—
159C4⋊Dic3—38no—
160Dic3⋊C4—38no—
161Dic3⋊C4—38no—
162Dic3⋊C4—38no—
163Dic3⋊C4—38no—
164Dic3⋊C4—38no—
165C4×Dic3—34no—
166C4×Dic3—34no—
167C22×C32C23×D314yes—
168C2×C3×D3—68no—
169C2×C3×D3—616no—
170C32×C4—24no—
171C32×C4—28no—
172C32×C4—24no—
173C32×C4—24no—
174C32×C4—28no—
175C3×Dic3—68no—
176C3×Dic3—68no—
177C3×Dic3—68no—
178C4⋊Q8—91no—
179C4⋊Q8#27211yescharacteristic, complemented
180#126—92no—
181#126—92no—
182#125—92no—
183#124—91no—
184#123—92no—
185C22⋊Q8—92no—
186C22⋊Q8—92no—
187H(3,𝔽3)#56311yescharacteristic, complemented, Sylow
188C23×C3—92no—
189C22×D3—38no—
190C3×Q8—41no—
191C3×Q8—41no—
192C3×Q8—124no—
193C3×Q8—124no—
194C3×Q8—361no—
195C3×D4—361no—
196C2×C3×C4—34no—
197C2×C3×C4—34no—
198C2×C3×C4—38no—
199C2×C3×C4—34no—
200C2×C3×C4—38no—
201C2×C3×C4—92no—
202C2×C3×C4—92no—
203C2×C3×C4—92no—
204C2×C3×C4—362no—
205C2×C3×C4C22×C3⋊D311yescharacteristic
206C2×C3×C4C22×C3⋊D311yescharacteristic
207C2×C3×C4C22×C3⋊D312yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209C2×C3×C4C22×C3⋊D312yes—
210C2×Dic3—38no—
211C2×Dic3—38no—
212C2×Dic3—38no—
213D12—124no—
214C4×D3—128no—
215Dic6—124no—
216C2×C32#100414yes—
217C2×C32#100718yes—
218C3×D3—128no—
219C2×Q8—91no—
220C2×Q8C2×#27211yescharacteristic
221C2×Q8C2×#27211yescharacteristic
222C2×D4—91no—
223C22×C4—92no—
224C4⋊C4—92no—
225C4⋊C4—92no—
226C4⋊C4—92no—
227C4⋊C4—92no—
228C4⋊C4—92no—
229C4⋊C4—92no—
230C4⋊C4C2×#27212yes—
231C4⋊C4C2×#27212yes—
232C22⋊C4—92no—
233C22⋊C4—92no—
234C22⋊C4—92no—
235C22⋊C4—92no—
236C22⋊C4—92no—
237C42—91no—
238C42—91no—
239C42C2×#27211yescharacteristic
240C22×C3—34no—
241C22×C3—182no—
242C22×C3—184no—
243C22×C3C23×C3⋊D311yescharacteristic
244C2×D3—68no—
245C2×D3—616no—
246C3×C4—22no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—22no—
251C3×C4—64no—
252C3×C4—68no—
253C3×C4—64no—
254C3×C4—64no—
255C3×C4—68no—
256C3×C4—182no—
257C3×C4—182no—
258C3×C4—182no—
259Dic3—68no—
260Dic3—68no—
261Dic3—68no—
262C32#573514yescomplemented
263C23—92no—
264Q8—41no—
265Q8—41no—
266Q8—361no—
267D4—361no—
268C2×C4—92no—
269C2×C4—92no—
270C2×C4—92no—
271C2×C4—362no—
272C2×C4C22×#27211yescharacteristic
273C2×C4C22×#27211yescharacteristic
274C2×C4C22×#27212yes—
275C2×C4C22×#27211yescharacteristic
276C2×C4C22×#27212yes—
277C2×C3—34no—
278C2×C3—38no—
279C2×C3—362no—
280C2×C3#6436111yescharacteristic
281C2×C3#6436412yes—
282D3—128no—
283C22—182no—
284C22—184no—
285C22C23×#27211yescharacteristic
286C4—22no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—22no—
291C4—182no—
292C4—182no—
293C4—182no—
294C3—34no—
295C3#1059620611yescharacteristic
296C2—362no—
297C2#1172266411yescharacteristic
298C2#1172267012yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC2×C4⋊#272C22
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#563
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C4⋊Q8C2semidirect
3. #11721863C2semidirect
4. #11721861C2semidirect
5. #11721766C2semidirect
6. #11721764C2semidirect
8. #11721757C2semidirect
9. #11721722C2semidirect
32. C32×C4⋊Q8D3semidirect
179. C4⋊Q8#272semidirect
187. H(3,𝔽3)#563semidirect
262. C32#5735semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#117218642
#117218632
#117218612
#117217662
#117217642
#117217622
#117217572
#117217222
C3×#57353