Finite Group Databasebeta

← #11713014

Subgroups — #11713014

Order 768.

497 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 71 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×#5388C212yesmaximal
2C2×C4×#226C211yesmaximal, characteristic
3#2351⋊C3C211yesmaximal, characteristic
4#73064C211yesmaximal, characteristic
5#73062C212yesmaximal
6#22675—31nomaximal, Sylow
7C22×#226C2211yescharacteristic
8C2×C4×SL(2,𝔽3)C2212yes—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2212yes—
11C2×#859C2211yescharacteristic
12#5388—24no—
13C4×#226C414yescomplemented
14#4650—41nomaximal
15C2×#441—32no—
16C2×C4×Q16—31no—
17#2351—31no—
18#2351D311yescharacteristic
19#1943—32no—
20#1863—32no—
21#1863—32no—
22#1846—31no—
23#1805—31no—
24#1805—31no—
25#1770—31no—
26C22×SL(2,𝔽3)C2311yescharacteristic
27C2×#226C2×C412yes—
28C2×#226D412yes—
29C2×#226Q812yes—
30C2×Dic3⋊C4—42no—
31C2×C4×Dic3—41no—
32C4×SL(2,𝔽3)—22no—
33C4×SL(2,𝔽3)C2×C412yes—
34#859—22no—
35#859C2×C412yes—
36C3×#95—41no—
37C3⋊#95—42no—
38C3⋊#95—41no—
39C22×Q16—31no—
40C2×C4×Q8—31no—
41C2×C4×Q8—32no—
42C2×C4×Q8C2×D312yes—
43C2×C4×Q8C2×D311yescharacteristic
44#441—64no—
45#441—64no—
46C4×Q16—34no—
47C4×Q16—34no—
48C2×#107—31no—
49C2×#107—32no—
50C2×#106—32no—
51C2×C4⋊C8—31no—
52C2×#103—31no—
53C2×#103—31no—
54C2×#103—32no—
55C2×#103—32no—
56C2×#97—32no—
57C2×C4×C8—31no—
58#389—32no—
59#389—32no—
60#389—62no—
61#382—31no—
62#382—31no—
63#382—61no—
64C4×C4⋊C4—31no—
65C4×C4⋊C4—61no—
66#340—32no—
67#340—32no—
68#336—32no—
69C22×C3×C4—41no—
70C22×C3×C4—42no—
71C22×Dic3—41no—
72C22×Dic3—41no—
73C22×Dic3—42no—
74C2×SL(2,𝔽3)C22×C411yescharacteristic
75C2×SL(2,𝔽3)C2×D411yescharacteristic
76C2×SL(2,𝔽3)C2×Q811yescharacteristic
77#226C4⋊C414yescomplemented
78Dic3⋊C4—84no—
79C4×Dic3—44no—
80C22×Q8—31no—
81C22×Q8C22×D311yescharacteristic
82C2×Q16—32no—
83C2×Q16—32no—
84C2×Q16—32no—
85C2×Q16—32no—
86C2×Q16—32no—
87C2×Q16—32no—
88C22×C8—31no—
89C22×C8—31no—
90C22×C8—32no—
91C4×Q8—22no—
92C4×Q8—32no—
93C4×Q8—34no—
94C4×Q8—62no—
95C4×Q8—62no—
96C4×Q8—64no—
97C4×Q8—122no—
98C4×Q8C4×D312yes—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4—61no—
105C2×C4⋊C4—62no—
106C2×C4⋊C4—62no—
107C2×C42—31no—
108C2×C42—32no—
109C2×C42—61no—
110C2×C42—62no—
111#107—34no—
112#107—122no—
113#106—64no—
114C4⋊C8—62no—
115#103—34no—
116#103—32no—
117#103—32no—
118#103—64no—
119#103—62no—
120#103—62no—
121#97—64no—
122C4×C8—34no—
123#95—31no—
124#95—61no—
125#95—61no—
126#95—61no—
127#95—62no—
128#95S411yescharacteristic
129C23×C3—41no—
130C2×C3×C4—42no—
131C2×C3×C4—82no—
132C2×C3×C4—82no—
133C2×C3×C4—82no—
134C2×C3×C4—82no—
135C2×Dic3—42no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C2×Dic3—42no—
139C2×Dic3—82no—
140C2×Dic3—82no—
141C2×Dic3—82no—
142C2×Dic3—82no—
143SL(2,𝔽3)C2×C4⋊C411yescharacteristic
144C2×Q8—31no—
145C2×Q8—31no—
146C2×Q8—31no—
147C2×Q8—32no—
148C2×Q8—32no—
149C2×Q8—32no—
150C2×Q8—61no—
151C2×Q8—61no—
152C2×Q8—61no—
153C2×Q8C2×C4×D311yescharacteristic
154C2×Q8D3×D411yescharacteristic
155C2×Q8D3×Q811yescharacteristic
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—32no—
159C22×C4—61no—
160C22×C4—61no—
161C22×C4—62no—
162C22×C4C2×S412yes—
163C22×C4C2×S411yescharacteristic
164Q16—34no—
165Q16—34no—
166Q16—34no—
167Q16—34no—
168C2×C8—32no—
169C2×C8—32no—
170C2×C8—32no—
171C2×C8—32no—
172C2×C8—61no—
173C2×C8—61no—
174C2×C8—62no—
175C2×C8—62no—
176C2×C8—62no—
177C4⋊C4—32no—
178C4⋊C4—32no—
179C4⋊C4—61no—
180C4⋊C4—62no—
181C4⋊C4—62no—
182C4⋊C4—61no—
183C4⋊C4—64no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C4⋊C4—122no—
187C4⋊C4—124no—
188C4⋊C4—124no—
189C42—32no—
190C42—32no—
191C42—62no—
192C42—62no—
193C42—64no—
194C42—124no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—42no—
199C22×C3—42no—
200C3×C4—82no—
201C3×C4—162no—
202Dic3—44no—
203Dic3—82no—
204Dic3—162no—
205C23C22×S411yescharacteristic
206Q8—31no—
207Q8—31no—
208Q8—31no—
209Q8—34no—
210Q8—61no—
211Q8—62no—
212Q8—62no—
213Q8—62no—
214Q8D3×C4⋊C411yescharacteristic
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—31no—
220C2×C4—31no—
221C2×C4—31no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—31no—
225C2×C4—32no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—122no—
235C2×C4—122no—
236C2×C4—122no—
237C2×C4—122no—
238C2×C4C4×S412yes—
239C8—34no—
240C8—62no—
241C8—122no—
242C2×C3—41no—
243C2×C3—41no—
244C2×C3—41no—
245C2×C3—42no—
246C2×C3—42no—
247C22C2×C4×S411yescharacteristic
248C22D4×S411yescharacteristic
249C22Q8×S411yescharacteristic
250C22#592312yes—
251C22#592712yes—
252C4—22no—
253C4—31no—
254C4—31no—
255C4—31no—
256C4—31no—
257C4—42no—
258C4—64no—
259C4—62no—
260C4—122no—
261C4—122no—
262C4—242no—
263C3—41noSylow
264C2C4⋊C4×S411yescharacteristic
265C2#8534111yescharacteristic
266C2#8535011yescharacteristic
267C2#8538612yes—
268C2#8541612yes—
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2351D3
prime radical, cosocleC22×SL(2,𝔽3)C23
hypercenter#95S4
hypocenterSL(2,𝔽3)C2×C4⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
13. C4×#226C4semidirect
77. #226C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#53882
C2×C4×#2262
#2351⋊C32
#730642
#730622
#226753
#46504