Finite Group Databasebeta

← #11714072

Subgroups — #11714072

Order 768.

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

normal subgroups only

This group has 389 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#85337C211yesmaximal, characteristic
2#85336C211yesmaximal, characteristic
3C4×#131⋊C3C211yesmaximal, characteristic
4#73124C214yesmaximal
5#33061—31nomaximal, Sylow
6C2×#984C2212yes—
7C2×#131⋊C3C2212yes—
8C4×#41⋊C3C2211yescharacteristic
9#5426C2212yes—
10#5391—21no—
11C4⋊#226—21no—
12C4⋊#226—21no—
13#5386—21no—
14#4681—41nomaximal
15C2×#487—31no—
16#3079—31no—
17#3039—31no—
18C4×#131D311yescharacteristic
19#1997—31no—
20#1966—31no—
21#1930—34no—
22#1908—34no—
23#1837—31no—
24C2×#41⋊C3C2311yescharacteristic
25#984—24no—
26C2×#226—22no—
27#869—41no—
28#869—41no—
29#131⋊C3—22no—
30C4×SL(2,𝔽3)D411yescharacteristic
31C4×SL(2,𝔽3)D411yescharacteristic
32#859—22no—
33C3×C4×C8—41no—
34#816—44no—
35C2×#137—32no—
36C2×#131C2×D312yes—
37C2×C4⋊Q8—31no—
38C2×C4⋊Q8—31no—
39C4×#41C2×D311yescharacteristic
40#487—62no—
41#487—61no—
42#487—61no—
43#477—61no—
44#476—61no—
45#475—61no—
46#474—61no—
47#462—61no—
48#462—61no—
49#461—61no—
50#461—61no—
51#428—32no—
52#423—31no—
53#423—31no—
54#419—32no—
55#417—32no—
56C2×#103—34no—
57C2×#101—34no—
58#405—32no—
59C4×M16—31no—
60C2×C4×C8—31no—
61#343—34no—
62#341—34no—
63C2×Dic6—42no—
64C2×Dic6—81no—
65#41⋊C3#4112yes—
66C2×SL(2,𝔽3)C2×D411yescharacteristic
67#226—42no—
68C2×C3×C8—42no—
69C3×C42—41no—
70C4⋊Dic3—42no—
71C4⋊Dic3—81no—
72C4⋊Dic3—81no—
73C4⋊Dic3—81no—
74C2×#41C22×D311yescharacteristic
75C22×Q8—32no—
76C22×Q8—61no—
77#137—64no—
78#137—64no—
79C2×Q16—62no—
80C2×SD16—62no—
81#131—22no—
82#131—62no—
83C2×M16—32no—
84C2×M16—32no—
85C2×M16—32no—
86C22×C8—32no—
87C4⋊Q8—61no—
88C4⋊Q8—61no—
89C4⋊Q8—61no—
90C4⋊Q8—61no—
91C4⋊Q8—61no—
92C4⋊Q8—61no—
93C4⋊Q8—61no—
94C4⋊Q8—61no—
95C4×Q8D1211yescharacteristic
96C4×Q8D1211yescharacteristic
97C4×D4—31no—
98C4×D4—31no—
99#117—31no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—61no—
102C2×C4⋊C4—61no—
103C2×C4⋊C4—61no—
104C2×C42—31no—
105#107—62no—
106#106—62no—
107C4⋊C8—61no—
108C4⋊C8—61no—
109C4⋊C8—61no—
110C4⋊C8—61no—
111#103—62no—
112#103—62no—
113#103—64no—
114#103—62no—
115#103—64no—
116#102—62no—
117#101—68no—
118#97—61no—
119C4×C8—32no—
120C4×C8—31no—
121C4×C8S411yescharacteristic
122C2×C3×C4—41no—
123C2×C3×C4—41no—
124C2×C3×C4—41no—
125C2×Dic3—82no—
126C2×Dic3—82no—
127Dic6—84no—
128Dic6—162no—
129SL(2,𝔽3)#12411yescharacteristic
130C3×C8—82no—
131#41—32no—
132#41#23512yes—
133C2×Q8—64no—
134C2×Q8—62no—
135C2×Q8—64no—
136C2×Q8—62no—
137C2×Q8—121no—
138C2×Q8—121no—
139C2×Q8—242no—
140C2×Q8C2×D1211yescharacteristic
141C2×D4—31no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—62no—
146C22×C4—62no—
147Q16—122no—
148Q16—122no—
149SD16—124no—
150M16—34no—
151M16—62no—
152M16—64no—
153M16—64no—
154C2×C8—32no—
155C2×C8—62no—
156C2×C8—62no—
157C2×C8—62no—
158C2×C8—62no—
159C2×C8C2×S412yes—
160C4⋊C4—31no—
161C4⋊C4—31no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—121no—
165C4⋊C4—121no—
166C4⋊C4—121no—
167C4⋊C4—121no—
168C4⋊C4—121no—
169C4⋊C4—121no—
170C22⋊C4—61no—
171C42—31no—
172C42—31no—
173C42—31no—
174C42C2×S411yescharacteristic
175C22×C3—41no—
176C3×C4—42no—
177C3×C4—42no—
178C3×C4—42no—
179Dic3—162no—
180Dic3—162no—
181C23—31no—
182Q8—31no—
183Q8—122no—
184Q8—122no—
185Q8—124no—
186Q8—242no—
187Q8—242no—
188Q8#87511yescharacteristic
189D4—32no—
190D4—61no—
191C2×C4—31no—
192C2×C4—32no—
193C2×C4—32no—
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—124no—
199C2×C4—122no—
200C2×C4—122no—
201C2×C4—242no—
202C2×C4C4⋊S411yescharacteristic
203C2×C4C4⋊S411yescharacteristic
204C2×C4C22×S411yescharacteristic
205C8—22no—
206C8—62no—
207C8—122no—
208C8—122no—
209C2×C3—41no—
210C2×C3—41no—
211C2×C3—41no—
212C22—32no—
213C22—61no—
214C22—61no—
215C22C2×C4⋊S411yescharacteristic
216C4—31no—
217C4—31no—
218C4—61no—
219C4—61no—
220C4—242no—
221C4—242no—
222C4#540312yes—
223C4#540612yes—
224C4#591212yes—
225C3—41noSylow
226C2—62no—
227C2#8529011yescharacteristic
228C2C2×#540311yescharacteristic
229C2C2×#540611yescharacteristic
subgroupG/H
center, socleC22C2×C4⋊S4
commutator, prime radical, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC4×#131D3
hypercenterC4×C8S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Maximal subgroups

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

subgroupindex
#853372
#853362
C4×#131⋊C32
#731242
#330613
#46814