Finite Group Databasebeta

← #11711050

Subgroups — #11711050

Order 768.

343 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#41×SL(2,𝔽3)C211yesmaximal, characteristic
2C4×#859C211yesmaximal, characteristic, complemented
3#73055C211yesmaximal, characteristic, complemented
4#8439—31nomaximal, Sylow
5Q8×SL(2,𝔽3)C411yescharacteristic, complemented
6D4×SL(2,𝔽3)—21no—
7D4×SL(2,𝔽3)C411yescharacteristic, complemented
8C2×C4×SL(2,𝔽3)—21no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859—21no—
11C2×#837—41nomaximal
12SL(2,𝔽3)⋊C8—21no—
13Q8×#41D311yescharacteristic
14#1806—31no—
15C4×#103—31no—
16#1503—31no—
17#1493—31no—
18#1487—31no—
19#1476—31no—
20C2×C3×#41—41no—
21C22×SL(2,𝔽3)—21no—
22C22×SL(2,𝔽3)D411yescharacteristic
23C2×C4×Dic3—41no—
24C2×#208—41no—
25C4×SL(2,𝔽3)—21no—
26C4×SL(2,𝔽3)C2×C411yescharacteristic
27C4×SL(2,𝔽3)D411yescharacteristic
28#859—22no—
29#837—44no—
30Q82Dic311yescharacteristic, complemented
31#557—31no—
32#554—31no—
33#554—61no—
34D4×Q8—21no—
35D4×Q8Dic311yescharacteristic, complemented
36#533—31no—
37#533—61no—
38C4×#41—31no—
39C2×C4×Q8—21no—
40C2×C4×Q8C2×D311yescharacteristic
41#431—31no—
42#423—31no—
43C2×#104—61no—
44C2×#103—61no—
45C2×C4×C8—31no—
46C4×C4⋊C4—31no—
47#340—61no—
48#339—61no—
49#328—62no—
50Q8⋊C8—61no—
51Q8⋊C8—61no—
52Q8⋊C8—62no—
53Q8⋊C8—61no—
54D4⋊C8—61no—
55C3×#41—42no—
56C3×#41—42no—
57C3×#41—81no—
58C3×#41—81no—
59C2×C3×Q8—41no—
60C2×C3×D4—41no—
61C2×C3×D4—81no—
62C22×C3×C4—41no—
63C22×C3×C4—81no—
64C22×Dic3—81no—
65C2×SL(2,𝔽3)—21no—
66C2×SL(2,𝔽3)—41no—
67C2×SL(2,𝔽3)C22⋊C411yescharacteristic
68C4×Dic3—42no—
69C4×Dic3—81no—
70#208—42no—
71#208—81no—
72C2×C3⋊C8—81no—
73C2×#41S411yescharacteristic
74C22×Q8—21no—
75C22×Q8#6711yescharacteristic
76C2×M16—61no—
77C22×C8—61no—
78C4⋊Q8—31no—
79C4⋊Q8—31no—
80C4⋊Q8—61no—
81C4⋊Q8—61no—
82#125—61no—
83#125—62no—
84C22⋊Q8—32no—
85C22⋊Q8—62no—
86C22⋊Q8—61no—
87C22⋊Q8—61no—
88C22⋊Q8—61no—
89C4×Q8—21no—
90C4×Q8—31no—
91C4×Q8—31no—
92C4×Q8—32no—
93C4×Q8—31no—
94C4×Q8—31no—
95C4×Q8—61no—
96C4×Q8—61no—
97C4×Q8C2×Dic311yescharacteristic
98C4×Q8#6711yescharacteristic
99C4×D4—31no—
100C4×D4—31no—
101C4×D4—61no—
102C4×D4—61no—
103#117—31no—
104#117—61no—
105C2×C4⋊C4—31no—
106C2×C4⋊C4—61no—
107C2×C4⋊C4—61no—
108C2×C42—31no—
109C2×C42—61no—
110C2×C42—61no—
111C4⋊C8—61no—
112C4⋊C8—61no—
113C4⋊C8—61no—
114#104—122no—
115#103—62no—
116#103—62no—
117C22⋊C8—61no—
118C4×C8—61no—
119C4×C8—61no—
120C4×C8—61no—
121#95—61no—
122C23×C3—41no—
123C23×C3—81no—
124C3×Q8—42no—
125C3×Q8—81no—
126C3×D4—42no—
127C3×D4—81no—
128C3×D4—82no—
129C3×D4—82no—
130C2×C3×C4—41no—
131C2×C3×C4—41no—
132C2×C3×C4—41no—
133C2×C3×C4—41no—
134C2×C3×C4—41no—
135C2×C3×C4—41no—
136C2×C3×C4—82no—
137C2×C3×C4—82no—
138C2×C3×C4—81no—
139C2×Dic3—82no—
140C2×Dic3—82no—
141C2×Dic3—82no—
142SL(2,𝔽3)#10411yescharacteristic, complemented
143C3⋊C8—82no—
144#41—21no—
145#41—21no—
146#41#22612yes—
147#41GL(2,𝔽3)12yes—
148C2×Q8—21no—
149C2×Q8—31no—
150C2×Q8—32no—
151C2×Q8—41no—
152C2×Q8—61no—
153C2×Q8—61no—
154C2×Q8—121no—
155C2×Q8#21711yescharacteristic
156C2×Q8SL(2,ℤ/4)11yescharacteristic
157C2×D4—21no—
158C2×D4SL(2,ℤ/4)11yescharacteristic
159C22×C4—21no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—61no—
163C22×C4—61no—
164C22×C4—121no—
165C22×C4C2×S411yescharacteristic
166M16—121no—
167M16—121no—
168C2×C8—62no—
169C2×C8—121no—
170C2×C8—121no—
171C2×C8—121no—
172C2×C8—121no—
173C4⋊C4—31no—
174C4⋊C4—31no—
175C4⋊C4—32no—
176C4⋊C4—31no—
177C4⋊C4—31no—
178C4⋊C4—61no—
179C4⋊C4—61no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—62no—
183C4⋊C4—62no—
184C4⋊C4—62no—
185C4⋊C4—121no—
186C22⋊C4—61no—
187C22⋊C4—61no—
188C22⋊C4—61no—
189C42—31no—
190C42—31no—
191C42—31no—
192C42—31no—
193C42—61no—
194C42—61no—
195C42—61no—
196C42—121no—
197C42—121no—
198C22×C3—41no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—81no—
202C22×C3—81no—
203C22×C3—82no—
204C22×C3—161no—
205C22×C3—161no—
206C3×C4—41no—
207C3×C4—41no—
208C3×C4—41no—
209C3×C4—41no—
210C3×C4—82no—
211Dic3—84no—
212C23—21no—
213C23GL(2,ℤ/4)11yescharacteristic
214Q8—21no—
215Q8—31no—
216Q8—61no—
217Q8—61no—
218Q8—121no—
219Q8—121no—
220Q8—122no—
221Q8#83711yescharacteristic, complemented
222Q8#85912yescomplemented
223D4—22no—
224D4—22no—
225D4—21no—
226D4#85912yescomplemented
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—31no—
231C2×C4—31no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—61no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—121no—
239C2×C4—121no—
240C2×C4—121no—
241C2×C4—122no—
242C2×C4—241no—
243C2×C4—241no—
244C2×C4—241no—
245C2×C4C2×#22611yescharacteristic
246C2×C4C2×GL(2,𝔽3)11yescharacteristic
247C2×C4#98311yescharacteristic
248C2×C4#98311yescharacteristic
249C2×C4C2×SL(2,ℤ/4)11yescharacteristic
250C2×C4GL(2,ℤ/4)11yescharacteristic
251C8—122no—
252C8—241no—
253C8—241no—
254C2×C3—41no—
255C2×C3—41no—
256C2×C3—41no—
257C2×C3—81no—
258C2×C3—81no—
259C2×C3—162no—
260C22—21no—
261C22—21no—
262C22—22no—
263C22—41no—
264C22—41no—
265C22#541911yescharacteristic
266C22#542111yescharacteristic
267C22#543211yescharacteristic
268C4—22no—
269C4—31no—
270C4—31no—
271C4—61no—
272C4—61no—
273C4—61no—
274C4—121no—
275C4—121no—
276C4—242no—
277C4C2×#85911yescharacteristic
278C4#541911yescharacteristic
279C4#542011yescharacteristic
280C4#542111yescharacteristic
281C3—41noSylow
282C2—21no—
283C2—21no—
284C2—42no—
285C2#7314811yescharacteristic
286C2#7315011yescharacteristic
287C2#7315711yescharacteristic
subgroupG/H
centerC2×C4GL(2,ℤ/4)
commutatorC4×SL(2,𝔽3)C2×C4
FrattiniC22×C4C2×S4
FittingQ8×#41D3
socleC22#5432
prime radical#41×SL(2,𝔽3)C2
cosocleC2×C4×SL(2,𝔽3)C22
hypercenterC2×#41S4
hypocenterSL(2,𝔽3)#104
LCSG ▷ C4×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C4×#859C2semidirect
3. #73055C2semidirect
5. Q8×SL(2,𝔽3)C4semidirect
7. D4×SL(2,𝔽3)C4semidirect
30. Q82Dic3semidirect
35. D4×Q8Dic3semidirect
142. SL(2,𝔽3)#104semidirect
221. Q8#837semidirect
222. Q8#859semidirect
226. D4#859semidirect

Maximal subgroups

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

subgroupindex
#41×SL(2,𝔽3)2
C4×#8592
#730552
#84393
C2×#8374