Finite Group Databasebeta

← #11713673

Subgroups — #11713673

Order 768.

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

normal subgroups only

This group has 397 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#3503⋊C3C211yesmaximal, characteristic
2#85339C211yesmaximal, characteristic
3#73076C212yesmaximal
4#73075C212yesmaximal
5#73067C211yesmaximal, characteristic
6#25832—31nomaximal, Sylow
7Q8×SL(2,𝔽3)—21no—
8#557⋊C3—21no—
9#520⋊C3C2212yes—
10C4×#41⋊C3C2211yescharacteristic
11#5426C2211yescharacteristic
12#5426C2211yescharacteristic
13C2×GU(2,𝔽3)C2212yes—
14#5389—21no—
15#5389—21no—
16#5081—41nomaximal
17#3503D311yescharacteristic
18#3166—31no—
19#3080—31no—
20#2660—31no—
21#2061—32no—
22#2060—32no—
23#1997—31no—
24#1995—31no—
25#1860—32no—
26#1859—32no—
27#1759—31no—
28C2×#41⋊C3C2311yescharacteristic
29C3×C4⋊Q8—41no—
30#870—41no—
31C4×SL(2,𝔽3)—22no—
32C4×SL(2,𝔽3)D411yescharacteristic
33C4×SL(2,𝔽3)D411yescharacteristic
34GU(2,𝔽3)Q814yescomplemented
35#859—21no—
36#859—21no—
37#808—42no—
38#807—42no—
39#803—41no—
40Q82—21no—
41#557—21no—
42#554—61no—
43D4×Q8—61no—
44#544—32no—
45#533—31no—
46C2×C4⋊Q8—31no—
47C2×#125—31no—
48#520C2×D312yes—
49C4×#41C2×D311yescharacteristic
50#501—61no—
51#501—61no—
52#499—62no—
53#479—61no—
54#479—61no—
55#478—61no—
56#478—61no—
57#428—31no—
58#428—31no—
59#428—32no—
60#428—32no—
61#423—31no—
62C2×#104—32no—
63#417—31no—
64#417—31no—
65C4×M16—31no—
66#398—62no—
67#395—62no—
68#384—32no—
69#382—32no—
70#376—31no—
71#343—32no—
72#343—32no—
73#339—32no—
74#339—32no—
75#339—32no—
76C2×C3×Q8—81no—
77#41⋊C3C2×Q812yes—
78C2×SL(2,𝔽3)C2×D411yescharacteristic
79C3×C4⋊C4—42no—
80C3×C4⋊C4—81no—
81C3×C42—41no—
82C4⋊Dic3—41no—
83C4⋊Dic3—41no—
84Dic3⋊C4—81no—
85Dic3⋊C4—81no—
86C2×C3⋊C8—42no—
87C2×#41C22×D311yescharacteristic
88C22×Q8—32no—
89C2×M16—32no—
90C2×M16—31no—
91C2×M16—31no—
92C4⋊Q8—31no—
93C4⋊Q8—61no—
94C4⋊Q8—61no—
95C4⋊Q8—61no—
96C4⋊Q8—62no—
97C4⋊Q8S411yescharacteristic
98#125—32no—
99#125—61no—
100#125—61no—
101#125—61no—
102#125—62no—
103#125—61no—
104C22⋊Q8—61no—
105C22⋊Q8—61no—
106C22⋊Q8—64no—
107C4×Q8—22no—
108C4×Q8—61no—
109C4×Q8—61no—
110C4×Q8#6711yescharacteristic
111C4×Q8#6711yescharacteristic
112C4×D4—31no—
113C4×D4—31no—
114C4×D4—62no—
115#117—31no—
116#117—32no—
117C2×C4⋊C4—32no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4—61no—
123C2×C4⋊C4—64no—
124C2×C42—31no—
125C2×C42—32no—
126#107—61no—
127#107—61no—
128#107—62no—
129#107—62no—
130#106—61no—
131#106—61no—
132#106—62no—
133#106—62no—
134C4⋊C8—61no—
135C4⋊C8—61no—
136#104—34no—
137#104—34no—
138#103—61no—
139#103—61no—
140#102—61no—
141#102—61no—
142#97—61no—
143C4×C8—61no—
144#95—61no—
145#95—61no—
146#95—64no—
147C3×Q8—162no—
148C2×C3×C4—41no—
149C2×C3×C4—41no—
150C2×C3×C4—41no—
151C2×C3×C4—82no—
152C2×Dic3—81no—
153C2×Dic3—81no—
154SL(2,𝔽3)C4⋊Q811yescharacteristic
155C3⋊C8—44no—
156#41—32no—
157#41D3×Q812yes—
158C2×Q8—21no—
159C2×Q8—62no—
160C2×Q8—61no—
161C2×Q8—242no—
162C2×Q8C2×#6711yescharacteristic
163C2×D4—31no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—34no—
168C22×C4—62no—
169C22×C4—61no—
170C22×C4—61no—
171M16—34no—
172M16—62no—
173M16—62no—
174M16—62no—
175C2×C8—62no—
176C2×C8—61no—
177C2×C8—61no—
178C4⋊C4—21no—
179C4⋊C4—31no—
180C4⋊C4—31no—
181C4⋊C4—32no—
182C4⋊C4—62no—
183C4⋊C4—62no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C4⋊C4—61no—
187C4⋊C4—61no—
188C4⋊C4—61no—
189C4⋊C4—61no—
190C4⋊C4—61no—
191C4⋊C4—121no—
192C4⋊C4—121no—
193C4⋊C4—121no—
194C4⋊C4—121no—
195C4⋊C4—244no—
196C4⋊C4C2×S412yes—
197C22⋊C4—62no—
198C22⋊C4—61no—
199C42—31no—
200C42—31no—
201C42—31no—
202C42—34no—
203C42—62no—
204C42—62no—
205C42C2×S411yescharacteristic
206C22×C3—41no—
207C3×C4—42no—
208C3×C4—81no—
209C3×C4—81no—
210C3×C4—162no—
211Dic3—161no—
212Dic3—161no—
213C23—31no—
214Q8—31no—
215Q8—42no—
216Q8—122no—
217Q8—242no—
218Q8#94411yescharacteristic
219D4—32no—
220D4—61no—
221C2×C4—22no—
222C2×C4—31no—
223C2×C4—32no—
224C2×C4—32no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—64no—
228C2×C4—62no—
229C2×C4—121no—
230C2×C4—121no—
231C2×C4—122no—
232C2×C4—122no—
233C2×C4—124no—
234C2×C4—122no—
235C2×C4—122no—
236C2×C4—121no—
237C2×C4—121no—
238C2×C4GL(2,ℤ/4)11yescharacteristic
239C2×C4GL(2,ℤ/4)11yescharacteristic
240C2×C4C22×S411yescharacteristic
241C8—64no—
242C8—121no—
243C8—121no—
244C2×C3—41no—
245C2×C3—41no—
246C2×C3—41no—
247C22—32no—
248C22—61no—
249C22—61no—
250C22C2×GL(2,ℤ/4)11yescharacteristic
251C4—21no—
252C4—21no—
253C4—31no—
254C4—31no—
255C4—42no—
256C4—61no—
257C4—61no—
258C4—124no—
259C4—122no—
260C4—241no—
261C4—241no—
262C4Q8×S412yes—
263C3—41noSylow
264C2—62no—
265C2#8550211yescharacteristic
266C2#8552611yescharacteristic
267C2#8553311yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, prime radical, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3503D3
hypercenterC4⋊Q8S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
34. GU(2,𝔽3)Q8semidirect

Maximal subgroups

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

subgroupindex
#3503⋊C32
#853392
#730762
#730752
#730672
#258323
#50814