Finite Group Databasebeta

← #11713722

Subgroups — #11713722

Order 768.

481 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 55 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×#557⋊C3C211yesmaximal, characteristic, complemented
2C2×#5395C211yesmaximal, characteristic
3#73065C212yesmaximal
4#73063C211yesmaximal, characteristic, complemented
5#73061C212yesmaximal, complemented
6#25941—31nomaximal, Sylow
7C22×GL(2,𝔽3)C2211yescharacteristic
8C22×#226C2211yescharacteristic
9#557⋊C3—22no—
10C2×C4×SL(2,𝔽3)C2212yes—
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×#859C2212yes—
13#5395—22no—
14#4992—41nomaximal
15C2×#557D311yescharacteristic, complemented
16C2×#489—31no—
17C2×#464—31no—
18#2592—31no—
19#2026—32no—
20#2025—32no—
21#1987—31no—
22#1985—31no—
23#1880—32no—
24#1878—32no—
25#1763—31no—
26C22×SL(2,𝔽3)C2311yescharacteristic
27C2×GL(2,𝔽3)—21no—
28C2×GL(2,𝔽3)—22no—
29C2×#226—21no—
30C2×#226—22no—
31C2×C3×C4⋊C4—41no—
32C2×(C2×D3)⋊C4—41no—
33C2×(C2×D3)⋊C4—42no—
34C4×SL(2,𝔽3)—22no—
35C4×SL(2,𝔽3)D412yes—
36#859—22no—
37C3⋊#95—42no—
38C3⋊#95—41no—
39C22×Q16—31no—
40C22×SD16—31no—
41#557—22no—
42#557—62no—
43C2×C4⋊Q8—31no—
44C2×#125—32no—
45C2×#124—31no—
46C2×C4×Q8—31no—
47C2×C4×Q8C2×D312yes—
48C2×C4×Q8C2×D311yescharacteristic
49#489—64no—
50#464—62no—
51#464—62no—
52C2×C4⋊C8—31no—
53C2×#103—32no—
54C2×#103—31no—
55C2×#103—32no—
56C2×#102—32no—
57C2×#102—31no—
58C2×#97—31no—
59#397—61no—
60#393—61no—
61#391—32no—
62#390—32no—
63#388—62no—
64C4×C4⋊C4—31no—
65#340—32no—
66#340—32no—
67#340—32no—
68#336—32no—
69#336—32no—
70C23×D3—41no—
71C22×C3×C4—41no—
72C22×C3×C4—42no—
73C22×Dic3—41no—
74C22×Dic3—42no—
75C2×SL(2,𝔽3)C2×D411yescharacteristic
76C2×SL(2,𝔽3)#4112yes—
77GL(2,𝔽3)—41no—
78#226—41no—
79C3×C4⋊C4—82no—
80(C2×D3)⋊C4—82no—
81(C2×D3)⋊C4—84no—
82C22×Q8—31no—
83C22×Q8—31no—
84C22×Q8C22×D311yescharacteristic
85C22×D4—31no—
86C2×Q16—62no—
87C2×Q16—61no—
88C2×Q16—62no—
89C2×Q16—61no—
90C2×SD16—62no—
91C2×SD16—61no—
92C2×SD16—62no—
93C2×SD16—61no—
94C22×C8—31no—
95C22×C8—31no—
96C22×C8—32no—
97C4⋊Q8—34no—
98C4⋊Q8—62no—
99#125—64no—
100#125—64no—
101#124—62no—
102#124—62no—
103C4×Q8—22no—
104C4×Q8—32no—
105C4×Q8—62no—
106C4×Q8—62no—
107C4×Q8—62no—
108C4×Q8#6712yes—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—61no—
114C2×C4⋊C4S411yescharacteristic
115C2×C22⋊C4—61no—
116C2×C22⋊C4—62no—
117C2×C22⋊C4—61no—
118C2×C22⋊C4—62no—
119C2×C42—31no—
120C2×C42—32no—
121C2×C42—61no—
122C4⋊C8—62no—
123#103—62no—
124#103—64no—
125#103—62no—
126#103—62no—
127#102—62no—
128#102—64no—
129#97—62no—
130#95—62no—
131#95—61no—
132#95—62no—
133C23×C3—41no—
134C22×D3—42no—
135C22×D3—81no—
136C22×D3—82no—
137C22×D3—82no—
138C22×D3—81no—
139C2×C3×C4—42no—
140C2×C3×C4—82no—
141C2×C3×C4—82no—
142C2×C3×C4—82no—
143C2×C3×C4—82no—
144C2×Dic3—82no—
145C2×Dic3—81no—
146C2×Dic3—82no—
147C2×Dic3—82no—
148C2×Dic3—82no—
149SL(2,𝔽3)#12411yescharacteristic
150C24—61no—
151C2×Q8—32no—
152C2×Q8—31no—
153C2×Q8—61no—
154C2×Q8—61no—
155C2×Q8—62no—
156C2×Q8—62no—
157C2×Q8—62no—
158C2×Q8—122no—
159C2×Q8#23912yes—
160C2×Q8C2×#6711yescharacteristic
161C2×D4—61no—
162C2×D4—61no—
163C2×D4—62no—
164C2×D4—62no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—61no—
169C22×C4—62no—
170C22×C4C2×S412yes—
171C22×C4C2×S411yescharacteristic
172Q16—122no—
173Q16—121no—
174Q16—121no—
175SD16—122no—
176SD16—121no—
177SD16—121no—
178C2×C8—62no—
179C2×C8—61no—
180C2×C8—61no—
181C2×C8—62no—
182C2×C8—62no—
183C2×C8—62no—
184C2×C8—62no—
185C4⋊C4—22no—
186C4⋊C4—32no—
187C4⋊C4—34no—
188C4⋊C4—64no—
189C4⋊C4—61no—
190C4⋊C4—62no—
191C4⋊C4—64no—
192C4⋊C4—62no—
193C4⋊C4—62no—
194C22⋊C4—122no—
195C22⋊C4—124no—
196C22⋊C4—122no—
197C22⋊C4—242no—
198C42—32no—
199C42—32no—
200C42—62no—
201C42—62no—
202C42—241no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—41no—
208C2×D3—82no—
209C2×D3—84no—
210C2×D3—161no—
211C2×D3—162no—
212C2×D3—161no—
213C3×C4—82no—
214C3×C4—162no—
215Dic3—162no—
216Dic3—161no—
217C23—122no—
218C23—122no—
219C23—121no—
220C23—121no—
221C23—121no—
222C23C22×S411yescharacteristic
223Q8—32no—
224Q8—31no—
225Q8—61no—
226Q8—124no—
227Q8—121no—
228Q8—122no—
229Q8—121no—
230Q8#94711yescharacteristic
231D4—121no—
232D4—122no—
233D4—121no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—32no—
239C2×C4—32no—
240C2×C4—31no—
241C2×C4—31no—
242C2×C4—32no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—122no—
252C2×C4—121no—
253C2×C4GL(2,ℤ/4)12yes—
254C8—122no—
255C8—121no—
256C8—121no—
257C2×C3—41no—
258C2×C3—41no—
259C2×C3—41no—
260C2×C3—42no—
261C2×C3—42no—
262D3—162no—
263C22—241no—
264C22—241no—
265C22—241no—
266C22—242no—
267C22—242no—
268C22#591912yes—
269C22#592312yes—
270C22#592611yescharacteristic
271C22#592711yescharacteristic
272C22C2×GL(2,ℤ/4)11yescharacteristic
273C4—22no—
274C4—31no—
275C4—32no—
276C4—31no—
277C4—42no—
278C4—62no—
279C4—122no—
280C4—241no—
281C4—242no—
282C3—41noSylow
283C2—481no—
284C2#8542112yes—
285C2#8542512yes—
286C2#8550811yescharacteristic
287C2#8552211yescharacteristic
288C2#8553811yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#557D3
prime radicalC22×GL(2,𝔽3)C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#557⋊C3C2semidirect
4. #73063C2semidirect
5. #73061C2semidirect
15. C2×#557D3semidirect

Maximal subgroups

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

subgroupindex
C2×#557⋊C32
C2×#53952
#730652
#730632
#730612
#259413
#49924