Finite Group Databasebeta

← #11713635

Subgroups — #11713635

Order 768.

681 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 681 conjugacy classes of subgroups — too many to draw. Only its 67 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×#5425C214yesmaximal, complemented
3C2×C4⋊#226C211yesmaximal, characteristic, complemented
4C4×#859C211yesmaximal, characteristic, complemented
5#25755—31nomaximal, Sylow
6C22×#41⋊C3C2212yes—
7C22×#226C2212yes—
8C4⋊(#41⋊C3)—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10#5425—28no—
11C2×#859C2212yes—
12C4⋊#226—22no—
13#4975—41nomaximal
14C2×#550D311yescharacteristic
15C2×#492—31no—
16C2×#462—31no—
17C2×#452—34no—
18#2542—31no—
19#2000—34no—
20#1982—31no—
21#1954—31no—
22C4×#103—31no—
23C2×#41⋊C3—24no—
24C2×#41⋊C3—22no—
25C22×SL(2,𝔽3)C2311yescharacteristic
26C2×#226—22no—
27C2×#226—24no—
28C2×C3×C4⋊C4—41no—
29C2×C4⋊Dic3—41no—
30C2×Dic3⋊C4—44no—
31C2×C4×Dic3—41no—
32C4×SL(2,𝔽3)D412yes—
33#859D414yescomplemented
34C22×#41C2×D312yes—
35C22×Q16—32no—
36C22×SD16—34no—
37#550—22no—
38#550—62no—
39C2×C4⋊Q8—31no—
40C2×#127—31no—
41C2×C22⋊Q8—34no—
42C2×#121—32no—
43C2×C4×Q8C2×D311yescharacteristic
44C2×C4×D4—31no—
45#492—64no—
46#462—62no—
47#462—62no—
48#452—68no—
49#452—68no—
50C2×C4⋊C8—31no—
51C2×#103—32no—
52C2×#103—32no—
53C2×#102—34no—
54C2×C22⋊C8—34no—
55C2×C4×C8—31no—
56#399—62no—
57#390—61no—
58#384—61no—
59C4×C4⋊C4—31no—
60#340—32no—
61C22×C3×C4—41no—
62C22×C3×C4—42no—
63C22×Dic3—42no—
64C22×Dic3—42no—
65#41⋊C3—42no—
66C2×SL(2,𝔽3)C2×D412yes—
67C2×SL(2,𝔽3)C2×D411yescharacteristic
68#226—42no—
69C3×C4⋊C4—82no—
70C4⋊Dic3—82no—
71Dic3⋊C4—88no—
72C4×Dic3—44no—
73C2×#41—24no—
74C2×#41—22no—
75C2×#41—62no—
76C2×#41—64no—
77C22×Q8—32no—
78C22×Q8C22×D311yescharacteristic
79C22×D4—32no—
80C22×D4—31no—
81C22×D4—61no—
82C23×C4—32no—
83C2×Q16—64no—
84C2×Q16—62no—
85C2×Q16—64no—
86C2×Q16—62no—
87C2×SD16—68no—
88C2×SD16—68no—
89C2×SD16—124no—
90C22×C8—32no—
91C22×C8—32no—
92C4⋊Q8—62no—
93C4⋊Q8—62no—
94#127—34no—
95#127—62no—
96C22⋊Q8—68no—
97C22⋊Q8—124no—
98#121—64no—
99#121—64no—
100C4×Q8—32no—
101C4×Q8#6712yes—
102C4×D4—62no—
103C4×D4—62no—
104C2×C4⋊C4—31no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—61no—
107C2×C4⋊C4—64no—
108C2×C4⋊C4—61no—
109C2×C4⋊C4S411yescharacteristic
110C2×C22⋊C4—32no—
111C2×C22⋊C4—64no—
112C2×C42—31no—
113C2×C42—61no—
114C4⋊C8—62no—
115#103—34no—
116#103—34no—
117#103—64no—
118#102—68no—
119C22⋊C8—68no—
120C4×C8—62no—
121#95—61no—
122C23×C3—41no—
123C2×C3×C4—42no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4—84no—
127C2×C3×C4—82no—
128C2×Dic3—44no—
129C2×Dic3—44no—
130C2×Dic3—44no—
131C2×Dic3—84no—
132C2×Dic3—82no—
133SL(2,𝔽3)#12711yescharacteristic
134C24—32no—
135#41—42no—
136#41—124no—
137#41—124no—
138#41—122no—
139C2×Q8—32no—
140C2×Q8—31no—
141C2×Q8—62no—
142C2×Q8—64no—
143C2×Q8—64no—
144C2×Q8—62no—
145C2×Q8D3×D412yes—
146C2×Q8C2×#6711yescharacteristic
147C2×D4—34no—
148C2×D4—64no—
149C2×D4—64no—
150C2×D4—62no—
151C2×D4—64no—
152C2×D4—62no—
153C2×D4—62no—
154C2×D4—62no—
155C2×D4—242no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—64no—
159C22×C4—62no—
160C22×C4—62no—
161C22×C4—64no—
162C22×C4—62no—
163C22×C4—62no—
164C22×C4C2×S412yes—
165C22×C4C2×S411yescharacteristic
166Q16—124no—
167Q16—122no—
168Q16—122no—
169SD16—1216no—
170C2×C8—34no—
171C2×C8—62no—
172C2×C8—62no—
173C2×C8—64no—
174C2×C8—62no—
175C4⋊C4—22no—
176C4⋊C4—32no—
177C4⋊C4—34no—
178C4⋊C4—61no—
179C4⋊C4—62no—
180C4⋊C4—128no—
181C4⋊C4—122no—
182C4⋊C4—122no—
183C22⋊C4—64no—
184C22⋊C4—128no—
185C42—32no—
186C42—32no—
187C42—122no—
188C22×C3—42no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—42no—
192C22×C3—41no—
193C3×C4—44no—
194C3×C4—162no—
195Dic3—48no—
196Dic3—162no—
197C23—64no—
198C23—62no—
199C23—62no—
200C23—64no—
201C23—62no—
202C23C22×S411yescharacteristic
203Q8—32no—
204Q8—31no—
205Q8—61no—
206Q8—124no—
207Q8—122no—
208Q8—122no—
209Q8#94011yescharacteristic
210D4—68no—
211D4—122no—
212D4—122no—
213D4—124no—
214D4—244no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—32no—
218C2×C4—32no—
219C2×C4—31no—
220C2×C4—31no—
221C2×C4—32no—
222C2×C4—62no—
223C2×C4—64no—
224C2×C4—122no—
225C2×C4—124no—
226C2×C4—122no—
227C2×C4—122no—
228C2×C4—122no—
229C2×C4—124no—
230C2×C4—122no—
231C2×C4#98412yes—
232C2×C4#98512yes—
233C2×C4GL(2,ℤ/4)12yes—
234C8—64no—
235C8—122no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—42no—
239C2×C3—42no—
240C2×C3—41no—
241C22—122no—
242C22—122no—
243C22—124no—
244C22—124no—
245C22—122no—
246C22D4×S412yes—
247C22C2×#98411yescharacteristic
248C22C2×#98511yescharacteristic
249C22#592412yes—
250C22C2×GL(2,ℤ/4)11yescharacteristic
251C4—32no—
252C4—31no—
253C4—31no—
254C4—42no—
255C4—62no—
256C4—124no—
257C4—242no—
258C4#542514yescomplemented
259C3—41noSylow
260C2—242no—
261C2#8540412yes—
262C2#8540712yes—
263C2#8549611yescharacteristic
264C2C2×#542511yescharacteristic
265C2#8552711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#550D3
prime radicalC2×C4⋊(#41⋊C3)C2
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#127
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
2. C2×#5425C2semidirect
3. C2×C4⋊#226C2semidirect
4. C4×#859C2semidirect
33. #859D4semidirect
258. C4#5425semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(#41⋊C3)2
C2×#54252
C2×C4⋊#2262
C4×#8592
#257553
#49754