Finite Group Databasebeta

← #11712653

Subgroups — #11712653

Order 768.

539 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C22×#131⋊C3C211yesmaximal, characteristic
2#85519C212yesmaximal
3#73126C212yesmaximal, complemented
4#73125C212yesmaximal, complemented
5#20678—31nomaximal, Sylow
6C22×#41⋊C3C2211yescharacteristic
7C2×#131⋊C3C414yescomplemented
8C2×#131⋊C3C2211yescharacteristic
9C2×#131⋊C3C2211yescharacteristic
10#5426C2214yes—
11#5424—22no—
12#5420—22no—
13#5106—41nomaximal
14C22×#131D311yescharacteristic
15C22×#108—31no—
16C2×#427—31no—
17#2896—32no—
18#1816—32no—
19#1816—32no—
20#1815—32no—
21#1815—32no—
22#1742—32no—
23C2×#41⋊C3C2×C412yes—
24C2×#41⋊C3D411yescharacteristic
25C2×#41⋊C3Q812yes—
26C2×#41⋊C3C2311yescharacteristic
27C22×SL(2,𝔽3)D411yescharacteristic
28C22×C3×C8—41no—
29#926—42no—
30#131⋊C3C2×C412yes—
31#131⋊C3C2×C412yes—
32#859—24no—
33#818—42no—
34#817—42no—
35C22×#41C2×D311yescharacteristic
36C2×#131—31no—
37C2×#131—31no—
38C2×#131—34no—
39C2×#131Dic314yescomplemented
40C2×#131C2×D311yescharacteristic
41C2×#131C2×D311yescharacteristic
42C22×M16—31no—
43C22×M16—32no—
44C23×C8—31no—
45C2×#117—32no—
46C2×#108—32no—
47C2×#108—32no—
48C2×#108—64no—
49#427—61no—
50#427—61no—
51#427—61no—
52#427—61no—
53C2×#107—32no—
54C2×#106—32no—
55#419—62no—
56#418—62no—
57#417—34no—
58#417—34no—
59#416—62no—
60#416—62no—
61#343—34no—
62#343—34no—
63C22×C3×C4—41no—
64#41⋊C3C4⋊C412yes—
65#41⋊C3C22×C411yescharacteristic
66#41⋊C3C2×Q811yescharacteristic
67C2×SL(2,𝔽3)C4⋊C412yes—
68C2×SL(2,𝔽3)C2×D411yescharacteristic
69C2×C3×C8—44no—
70C2×C3×C8—41no—
71C2×C3×C8—41no—
72#217—82no—
73C4⋊Dic3—44no—
74C4×Dic3—82no—
75C2×#41—32no—
76C2×#41—31no—
77C2×#41—32no—
78C2×#41—31no—
79C2×#41Dic612yes—
80C2×#41D1211yescharacteristic
81C2×#41C2×Dic312yes—
82C2×#41C22×D311yescharacteristic
83C22×Q8D1211yescharacteristic
84C22×D4—31no—
85C23×C4—31no—
86#131—34no—
87#131—34no—
88#131—32no—
89#131—32no—
90#131—62no—
91#131—62no—
92#131C2×Dic312yes—
93#131C2×Dic312yes—
94C2×M16—34no—
95C2×M16—31no—
96C2×M16—31no—
97C2×M16—34no—
98C2×M16—62no—
99C2×M16—61no—
100C2×M16—62no—
101C2×M16—64no—
102C2×M16—64no—
103C22×C8—31no—
104C22×C8—34no—
105C22×C8—34no—
106C22×C8—31no—
107C22×C8—31no—
108C22×C8—31no—
109C22×C8—31no—
110C22×C8—62no—
111C22×C8S411yescharacteristic
112#117—62no—
113#117—62no—
114#117—62no—
115#117—62no—
116C2×C4⋊C4—34no—
117C2×C22⋊C4—62no—
118C2×C42—62no—
119#108—64no—
120#108—64no—
121#107—62no—
122#107—62no—
123#106—62no—
124#106—62no—
125#103—64no—
126#103—64no—
127#102—64no—
128#102—64no—
129C23×C3—41no—
130C2×C3×C4—42no—
131C2×C3×C4—41no—
132C2×C3×C4—42no—
133C2×C3×C4—41no—
134C2×Dic3—84no—
135SL(2,𝔽3)C2×C4⋊C411yescharacteristic
136C3×C8—42no—
137C3×C8—42no—
138C24—31no—
139#41—32no—
140#41—31no—
141#41—32no—
142#41—32no—
143#41—32no—
144#41—32no—
145#41—31no—
146#41—61no—
147#41—61no—
148#41—62no—
149#41C4⋊Dic312yes—
150#41C2×Dic611yescharacteristic
151#41C22×Dic311yescharacteristic
152C2×Q8—31no—
153C2×Q8—32no—
154C2×Q8C4⋊Dic312yes—
155C2×Q8C2×D1211yescharacteristic
156C2×D4—31no—
157C2×D4—31no—
158C2×D4—32no—
159C2×D4—32no—
160C2×D4—61no—
161C2×D4—62no—
162C22×C4—31no—
163C22×C4—32no—
164C22×C4—32no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—32no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—64no—
172C22×C4C2×S411yescharacteristic
173M16—32no—
174M16—32no—
175M16—61no—
176M16—61no—
177M16—64no—
178M16—64no—
179M16—64no—
180M16—124no—
181C2×C8—34no—
182C2×C8—31no—
183C2×C8—31no—
184C2×C8—32no—
185C2×C8—32no—
186C2×C8—32no—
187C2×C8—32no—
188C2×C8—34no—
189C2×C8—34no—
190C2×C8—64no—
191C2×C8—124no—
192C2×C8SL(2,ℤ/4)14yes—
193C2×C8C2×S411yescharacteristic
194C2×C8C2×S411yescharacteristic
195C4⋊C4—64no—
196C4⋊C4—64no—
197C22⋊C4—122no—
198C22⋊C4—122no—
199C42—122no—
200C42—122no—
201C22×C3—41no—
202C22×C3—42no—
203C22×C3—82no—
204C3×C4—41no—
205C3×C4—42no—
206C3×C4—41no—
207Dic3—164no—
208C23—31no—
209C23—31no—
210C23—32no—
211C23—32no—
212C23—122no—
213C23C4⋊S411yescharacteristic
214Q8—32no—
215Q8—31no—
216Q8—61no—
217Q8C2×C4⋊Dic311yescharacteristic
218D4—32no—
219D4—31no—
220D4—31no—
221D4—61no—
222D4—61no—
223D4—62no—
224D4—62no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—32no—
228C2×C4—32no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—32no—
239C2×C4—124no—
240C2×C4—244no—
241C2×C4A4⋊Q812yes—
242C2×C4C4⋊S411yescharacteristic
243C2×C4C2×SL(2,ℤ/4)12yes—
244C2×C4C22×S411yescharacteristic
245C8—32no—
246C8—32no—
247C8—124no—
248C8C2×SL(2,ℤ/4)12yes—
249C8C2×SL(2,ℤ/4)12yes—
250C2×C3—42no—
251C2×C3—41no—
252C2×C3—82no—
253C22—22no—
254C22—31no—
255C22—31no—
256C22—32no—
257C22—62no—
258C22—62no—
259C22—122no—
260C22—122no—
261C22C4⋊SL(2,ℤ/4)12yes—
262C22C2×C4⋊S411yescharacteristic
263C4—31no—
264C4—31no—
265C4—32no—
266C4—244no—
267C4C4⋊SL(2,ℤ/4)12yes—
268C4C2×A4⋊Q811yescharacteristic
269C4C22×SL(2,ℤ/4)11yescharacteristic
270C3—41noSylow
271C2—22no—
272C2—61no—
273C2—61no—
274C2—62no—
275C2#8545112yes—
276C2C2×C4⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C4C4⋊S4
commutator#41⋊C3C22×C4
FrattiniC2×C4C22×S4
FittingC22×#131D3
socleC22C2×C4⋊S4
prime radicalC22×#41⋊C3C22
cosocleC2×#41⋊C3C23
hypercenterC22×C8S4
hypocenterSL(2,𝔽3)C2×C4⋊C4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. #73126C2semidirect
4. #73125C2semidirect
7. C2×#131⋊C3C4semidirect
39. C2×#131Dic3semidirect

Maximal subgroups

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

subgroupindex
C22×#131⋊C32
#855192
#731262
#731252
#206783
#51064