Finite Group Databasebeta

← #11713723

Subgroups — #11713723

Order 768.

513 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 59 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#5395C211yesmaximal, characteristic
3#73065C212yesmaximal
4#73063C211yesmaximal, characteristic, complemented
5#73061C212yesmaximal, complemented
6#25942—31nomaximal, Sylow
7C22×GL(2,𝔽3)C2211yescharacteristic
8C22×#226C2211yescharacteristic
9Q8×SL(2,𝔽3)—22no—
10C2×C4×SL(2,𝔽3)C2212yes—
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×#859C2212yes—
13#5395—22no—
14#5231—41nomaximal
15C2×Q82D311yescharacteristic, complemented
16C2×#489—31no—
17C2×#464—31no—
18#2592—31no—
19#2024—32no—
20#2023—32no—
21#1988—31no—
22#1985—31no—
23#1880—32no—
24#1878—32no—
25#1763—31no—
26C22×C3×Q8—41no—
27C22×SL(2,𝔽3)C2311yescharacteristic
28C2×GL(2,𝔽3)—21no—
29C2×GL(2,𝔽3)—22no—
30C2×#226—21no—
31C2×#226—22no—
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—
41Q82—22no—
42Q82—62no—
43C2×C4⋊Q8—31no—
44C2×C4⋊Q8—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—
71C2×C3×Q8—82no—
72C2×C3×Q8—84no—
73C22×C3×C4—42no—
74C22×C3×C4—41no—
75C22×Dic3—41no—
76C22×Dic3—42no—
77C2×SL(2,𝔽3)C2×D411yescharacteristic
78C2×SL(2,𝔽3)#4112yes—
79GL(2,𝔽3)—41no—
80#226—41no—
81(C2×D3)⋊C4—82no—
82(C2×D3)⋊C4—84no—
83C22×Q8—31no—
84C22×Q8—61no—
85C22×Q8S411yescharacteristic
86C22×Q8C22×D311yescharacteristic
87C22×D4—31no—
88C2×Q16—62no—
89C2×Q16—61no—
90C2×Q16—62no—
91C2×Q16—61no—
92C2×SD16—62no—
93C2×SD16—61no—
94C2×SD16—62no—
95C2×SD16—61no—
96C22×C8—31no—
97C22×C8—31no—
98C22×C8—32no—
99C4⋊Q8—34no—
100C4⋊Q8—62no—
101C4⋊Q8—64no—
102C4⋊Q8—64no—
103#124—62no—
104#124—62no—
105C4×Q8—22no—
106C4×Q8—32no—
107C4×Q8—62no—
108C4×Q8—62no—
109C4×Q8—62no—
110C4×Q8#6712yes—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—32no—
116C2×C22⋊C4—61no—
117C2×C22⋊C4—61no—
118C2×C22⋊C4—62no—
119C2×C22⋊C4—62no—
120C2×C42—31no—
121C2×C42—32no—
122C2×C42—61no—
123C4⋊C8—62no—
124#103—62no—
125#103—64no—
126#103—62no—
127#103—62no—
128#102—62no—
129#102—64no—
130#97—62no—
131#95—62no—
132#95—61no—
133#95—62no—
134C23×C3—41no—
135C22×D3—42no—
136C22×D3—81no—
137C22×D3—82no—
138C22×D3—82no—
139C22×D3—81no—
140C3×Q8—88no—
141C2×C3×C4—44no—
142C2×C3×C4—42no—
143C2×C3×C4—82no—
144C2×C3×C4—82no—
145C2×C3×C4—82no—
146C2×Dic3—82no—
147C2×Dic3—81no—
148C2×Dic3—82no—
149C2×Dic3—82no—
150C2×Dic3—82no—
151SL(2,𝔽3)#12411yescharacteristic
152C24—61no—
153C2×Q8—24no—
154C2×Q8—22no—
155C2×Q8—32no—
156C2×Q8—31no—
157C2×Q8—64no—
158C2×Q8—61no—
159C2×Q8—61no—
160C2×Q8—62no—
161C2×Q8—62no—
162C2×Q8—241no—
163C2×Q8—241no—
164C2×Q8#23912yes—
165C2×Q8C2×#6711yescharacteristic
166C2×D4—61no—
167C2×D4—61no—
168C2×D4—62no—
169C2×D4—62no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—32no—
173C22×C4—61no—
174C22×C4—62no—
175C22×C4C2×S412yes—
176C22×C4C2×S411yescharacteristic
177Q16—122no—
178Q16—121no—
179Q16—121no—
180SD16—122no—
181SD16—121no—
182SD16—121no—
183C2×C8—62no—
184C2×C8—61no—
185C2×C8—61no—
186C2×C8—62no—
187C2×C8—62no—
188C2×C8—62no—
189C2×C8—62no—
190C4⋊C4—32no—
191C4⋊C4—34no—
192C4⋊C4—64no—
193C4⋊C4—61no—
194C4⋊C4—62no—
195C4⋊C4—62no—
196C4⋊C4—62no—
197C4⋊C4—62no—
198C22⋊C4—122no—
199C22⋊C4—124no—
200C22⋊C4—122no—
201C22⋊C4—242no—
202C42—32no—
203C42—32no—
204C42—62no—
205C42—62no—
206C42—241no—
207C22×C3—42no—
208C22×C3—42no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—41no—
212C2×D3—82no—
213C2×D3—84no—
214C2×D3—161no—
215C2×D3—162no—
216C2×D3—161no—
217C3×C4—84no—
218C3×C4—82no—
219Dic3—162no—
220Dic3—161no—
221C23—122no—
222C23—122no—
223C23—121no—
224C23—121no—
225C23—121no—
226C23C22×S411yescharacteristic
227Q8—28no—
228Q8—32no—
229Q8—31no—
230Q8—61no—
231Q8—121no—
232Q8—122no—
233Q8—121no—
234Q8—244no—
235Q8#94711yescharacteristic
236D4—121no—
237D4—122no—
238D4—121no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—32no—
243C2×C4—32no—
244C2×C4—31no—
245C2×C4—31no—
246C2×C4—32no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—122no—
253C2×C4—122no—
254C2×C4—122no—
255C2×C4—122no—
256C2×C4—121no—
257C2×C4#98314yes—
258C2×C4GL(2,ℤ/4)12yes—
259C8—122no—
260C8—121no—
261C8—121no—
262C2×C3—41no—
263C2×C3—41no—
264C2×C3—41no—
265C2×C3—42no—
266C2×C3—42no—
267D3—162no—
268C22—241no—
269C22—241no—
270C22—241no—
271C22—242no—
272C22—242no—
273C22C2×#98312yes—
274C22#591912yes—
275C22#592411yescharacteristic
276C22#592511yescharacteristic
277C22C2×GL(2,ℤ/4)11yescharacteristic
278C4—24no—
279C4—22no—
280C4—31no—
281C4—32no—
282C4—31no—
283C4—62no—
284C4—122no—
285C4—241no—
286C4—242no—
287C3—41noSylow
288C2—481no—
289C2#8543212yes—
290C2#8543312yes—
291C2#8550711yescharacteristic
292C2#8550811yescharacteristic
293C2#8552911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×Q82D3
prime radicalC22×GL(2,𝔽3)C22
hypercenterC22×Q8S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×SL(2,𝔽3)C2semidirect
4. #73063C2semidirect
5. #73061C2semidirect
15. C2×Q82D3semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×SL(2,𝔽3)2
C2×#53952
#730652
#730632
#730612
#259423
#52314