Finite Group Databasebeta

← #11712885

Subgroups — #11712885

Order 768.

368 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 63 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#85334C211yesmaximal, characteristic
2C4×#131⋊C3C211yesmaximal, characteristic
3#73123C212yesmaximal
4#73122C212yesmaximal
5C4⋊GU(2,𝔽3)C211yesmaximal, characteristic
6#21628—31nomaximal, Sylow
7C2×#131⋊C3C2211yescharacteristic
8C2×#131⋊C3C2211yescharacteristic
9C4×#41⋊C3C2211yescharacteristic
10#5424C2212yes—
11C2×GU(2,𝔽3)C2212yes—
12#5391C411yescharacteristic, complemented
13#5389C412yescomplemented
14#5386C411yescharacteristic, complemented
15#4679—41nomaximal
16#3076—31no—
17#2960—31no—
18C4×#131D311yescharacteristic
19#1986—31no—
20#1950—31no—
21#1922—32no—
22#1866—32no—
23#1806—31no—
24#1773—32no—
25#1770—31no—
26#1765—32no—
27C2×#41⋊C3C2311yescharacteristic
28C4×Dic6—41no—
29#131⋊C3—21no—
30#131⋊C3Q812yescomplemented
31C4×SL(2,𝔽3)C2×C411yescharacteristic
32C4×SL(2,𝔽3)C2×C411yescharacteristic
33GU(2,𝔽3)—22no—
34#859C2×C412yes—
35#859C2×C412yes—
36C3×C4×C8—41no—
37#815—42no—
38Dic3⋊C8—42no—
39C4⋊(C3⋊C8)—41no—
40C2×#131C2×D311yescharacteristic
41C2×#131C2×D311yescharacteristic
42#533—31no—
43C4×#41C2×D311yescharacteristic
44#479—32no—
45#478—32no—
46#477—31no—
47#476—31no—
48#475—31no—
49#474—31no—
50#446—61no—
51#446—61no—
52C8×Q8—61no—
53C8×Q8—61no—
54C2×#108—32no—
55#427—32no—
56#424—32no—
57#423—31no—
58C2×C4⋊C8—31no—
59C2×#104—32no—
60#416—32no—
61#405—31no—
62#405—31no—
63#405—32no—
64C4×M16—31no—
65C2×#97—32no—
66C2×C4×C8—31no—
67C4×C4⋊C4—31no—
68#341—32no—
69#339—32no—
70#339—32no—
71#339—32no—
72#336—32no—
73C2×Dic6—41no—
74#41⋊C3C2×Q811yescharacteristic
75#41⋊C3#4111yescharacteristic
76C2×SL(2,𝔽3)C22×C411yescharacteristic
77C2×C3×C8—41no—
78C2×C3×C8—41no—
79C3×C42—41no—
80C4⋊Dic3—41no—
81Dic3⋊C4—42no—
82C4×Dic3—42no—
83C2×C3⋊C8—42no—
84C2×#41C22×D311yescharacteristic
85#131—21no—
86#131—32no—
87#131—61no—
88#131Dic612yescomplemented
89C2×M16—31no—
90C2×M16—31no—
91C2×M16—32no—
92C22×C8—31no—
93C22×C8—32no—
94C22×C8—31no—
95C4⋊Q8—31no—
96C4⋊Q8—31no—
97#125—32no—
98C22⋊Q8—61no—
99C22⋊Q8—61no—
100C4×Q8—61no—
101C4×Q8—61no—
102C4×Q8C4×D311yescharacteristic
103C4×Q8C4×D311yescharacteristic
104C4×D4—31no—
105C4×D4—31no—
106#117—31no—
107#117—32no—
108C2×C4⋊C4—61no—
109C2×C42—31no—
110C2×C42—32no—
111#108—64no—
112#107—32no—
113#107—32no—
114#106—32no—
115#106—32no—
116C4⋊C8—31no—
117C4⋊C8—31no—
118C4⋊C8—32no—
119C4⋊C8—61no—
120C4⋊C8—62no—
121C4⋊C8—62no—
122C4⋊C8—61no—
123#104—62no—
124#104—62no—
125#103—32no—
126#103—32no—
127#102—32no—
128#102—32no—
129#97—61no—
130#97—64no—
131#97—62no—
132C4×C8—31no—
133C4×C8—31no—
134C4×C8—31no—
135C4×C8—62no—
136C4×C8S411yescharacteristic
137#95—61no—
138C2×C3×C4—41no—
139C2×C3×C4—41no—
140C2×C3×C4—41no—
141C2×Dic3—42no—
142C2×Dic3—42no—
143Dic6—82no—
144SL(2,𝔽3)C4×Q811yescharacteristic
145C3×C8—42no—
146C3×C8—81no—
147C3⋊C8—82no—
148#41—31no—
149#41—31no—
150#41C2×Dic611yescharacteristic
151#41#23511yescharacteristic
152C2×Q8—61no—
153C2×Q8—61no—
154C2×Q8C2×C4×D311yescharacteristic
155C2×D4—31no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—62no—
160M16—32no—
161M16—32no—
162M16—61no—
163M16—61no—
164M16—62no—
165M16—62no—
166C2×C8—31no—
167C2×C8—32no—
168C2×C8—32no—
169C2×C8—32no—
170C2×C8—32no—
171C2×C8—31no—
172C2×C8—61no—
173C2×C8—61no—
174C2×C8—62no—
175C2×C8—62no—
176C2×C8—62no—
177C2×C8C2×S411yescharacteristic
178C2×C8C2×S411yescharacteristic
179C4⋊C4—31no—
180C4⋊C4—31no—
181C4⋊C4—32no—
182C4⋊C4—32no—
183C4⋊C4—62no—
184C4⋊C4—61no—
185C4⋊C4—62no—
186C4⋊C4—61no—
187C4⋊C4—122no—
188C22⋊C4—61no—
189C22⋊C4—62no—
190C42—31no—
191C42—31no—
192C42—31no—
193C42—62no—
194C42—62no—
195C42—62no—
196C42C2×S411yescharacteristic
197C22×C3—41no—
198C3×C4—41no—
199C3×C4—41no—
200C3×C4—42no—
201C3×C4—81no—
202Dic3—82no—
203Dic3—82no—
204C23—31no—
205Q8—31no—
206Q8—122no—
207Q8—122no—
208Q8C4×Dic611yescharacteristic
209D4—31no—
210D4—31no—
211D4—61no—
212C2×C4—31no—
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—31no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—121no—
223C2×C4—122no—
224C2×C4—122no—
225C2×C4—122no—
226C2×C4C4×S411yescharacteristic
227C2×C4C4×S411yescharacteristic
228C2×C4C22×S411yescharacteristic
229C8—21no—
230C8—32no—
231C8—62no—
232C8—62no—
233C8—61no—
234C8—122no—
235C8A4⋊Q812yes—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—41no—
239C22—31no—
240C22—31no—
241C22—61no—
242C22—61no—
243C22C2×C4×S411yescharacteristic
244C4—21no—
245C4—31no—
246C4—31no—
247C4—61no—
248C4—61no—
249C4—122no—
250C4—122no—
251C4—242no—
252C4#540512yes—
253C4C2×A4⋊Q811yescharacteristic
254C4#591211yescharacteristic
255C3—41noSylow
256C2—61no—
257C2—61no—
258C2C4×A4⋊Q811yescharacteristic
259C2C2×#540511yescharacteristic
260C2#8544811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingC4×#131D3
socleC22C2×C4×S4
prime radical, cosocleC2×#41⋊C3C23
hypercenterC4×C8S4
hypocenterSL(2,𝔽3)C4×Q8
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
12. #5391C4semidirect
13. #5389C4semidirect
14. #5386C4semidirect
30. #131⋊C3Q8semidirect
88. #131Dic6semidirect

Maximal subgroups

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

subgroupindex
#853342
C4×#131⋊C32
#731232
#731222
C4⋊GU(2,𝔽3)2
#216283
#46794