Finite Group Databasebeta

← #11711859

Subgroups — #11711859

Order 768.

336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 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#85322C211yesmaximal, characteristic
2C2×C8×SL(2,𝔽3)C211yesmaximal, characteristic
3#73055C211yesmaximal, characteristic
4#73009C212yesmaximal, complemented
5#226⋊C8C212yesmaximal, complemented
6#17906—31nomaximal, Sylow
7C2×C4×SL(2,𝔽3)C2211yescharacteristic
8#5419C412yes—
9#5391C411yescharacteristic
10C4⋊#226C411yescharacteristic
11C4×#226C2212yes—
12#5102—41nomaximal
13C8×SL(2,𝔽3)C2211yescharacteristic
14C8×SL(2,𝔽3)C2211yescharacteristic
15SL(2,𝔽3)⋊C8C2212yes—
16#3104—31no—
17#2960—31no—
18C2×C8×Q8D311yescharacteristic
19#1587—32no—
20#1587—32no—
21#1579—32no—
22#1579—32no—
23#1570—31no—
24#1487—31no—
25#1481—31no—
26#1452—31no—
27C22×SL(2,𝔽3)C2×C411yescharacteristic
28C2×#226C2×C412yes—
29C22×C3×C8—41no—
30#926—41no—
31C2×#208—41no—
32C4×SL(2,𝔽3)C2×C411yescharacteristic
33C4×SL(2,𝔽3)C2311yescharacteristic
34#859C2×C412yes—
35Dic3⋊C8—42no—
36Dic3⋊C8—42no—
37#533—31no—
38C2×C4×Q8C2×D311yescharacteristic
39#484—32no—
40#477—31no—
41#477—31no—
42#462—31no—
43#462—31no—
44#451—32no—
45#446—32no—
46#446—61no—
47C8×Q8—31no—
48C8×Q8—31no—
49C8×Q8—32no—
50C8×Q8—61no—
51C8×Q8C2×D311yescharacteristic
52C8×Q8C2×D311yescharacteristic
53C4×Q16—32no—
54C4×Q16—32no—
55#431—31no—
56#431—31no—
57#424—61no—
58#423—31no—
59C2×C4⋊C8—31no—
60#405—61no—
61C2×C4×C8—31no—
62#334—32no—
63#334—32no—
64Q8⋊C8—32no—
65Q8⋊C8—32no—
66Q8⋊C8—32no—
67Q8⋊C8—32no—
68#322—32no—
69#322—32no—
70C22×C3×C4—41no—
71C2×SL(2,𝔽3)—21no—
72C2×SL(2,𝔽3)C22×C411yescharacteristic
73#226—22no—
74C2×C3×C8—41no—
75C2×C3×C8—41no—
76C2×C3×C8—82no—
77#217—42no—
78C4⋊Dic3—41no—
79C4⋊Dic3—41no—
80C4×Dic3—42no—
81#208—82no—
82C2×C3⋊C8—42no—
83C22×Q8C4×D311yescharacteristic
84C2×Q16—32no—
85C2×Q16—32no—
86C2×M16—61no—
87C22×C8—31no—
88C22×C8S411yescharacteristic
89C4⋊Q8—31no—
90C4⋊Q8—31no—
91#125—61no—
92C22⋊Q8—32no—
93C22⋊Q8—61no—
94C4×Q8—31no—
95C4×Q8—31no—
96C4×Q8—32no—
97C4×Q8—61no—
98C4×Q8C4×D311yescharacteristic
99C4×Q8C22×D311yescharacteristic
100#117—61no—
101C2×C4⋊C4—31no—
102C2×C42—31no—
103#107—32no—
104#107—32no—
105C4⋊C8—31no—
106C4⋊C8—31no—
107C4⋊C8—31no—
108C4⋊C8—31no—
109C4⋊C8—32no—
110C4⋊C8—61no—
111C4⋊C8—61no—
112C4⋊C8—61no—
113C4⋊C8—62no—
114C4⋊C8—62no—
115#103—32no—
116#103—32no—
117#103—32no—
118#103—32no—
119C22⋊C8—32no—
120C22⋊C8—61no—
121#97—62no—
122C4×C8—32no—
123C4×C8—31no—
124C4×C8—31no—
125C4×C8—31no—
126C4×C8—31no—
127C4×C8—32no—
128C4×C8—62no—
129C23×C3—41no—
130C2×C3×C4—41no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—41no—
134C2×Dic3—42no—
135C2×Dic3—42no—
136SL(2,𝔽3)#13111yescharacteristic
137C3×C8—81no—
138C3×C8—81no—
139C3⋊C8—82no—
140C2×Q8—21no—
141C2×Q8—31no—
142C2×Q8—32no—
143C2×Q8—61no—
144C2×Q8—61no—
145C2×Q8C2×C4×D311yescharacteristic
146C22×C4—31no—
147C22×C4—31no—
148C22×C4C2×S411yescharacteristic
149Q16—62no—
150Q16—64no—
151Q16—62no—
152M16—122no—
153C2×C8—22no—
154C2×C8—32no—
155C2×C8—32no—
156C2×C8—31no—
157C2×C8—31no—
158C2×C8—62no—
159C2×C8—62no—
160C2×C8—62no—
161C2×C8C2×S411yescharacteristic
162C2×C8C2×S411yescharacteristic
163C4⋊C4—31no—
164C4⋊C4—31no—
165C4⋊C4—32no—
166C4⋊C4—61no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—61no—
170C4⋊C4—61no—
171C4⋊C4—61no—
172C22⋊C4—62no—
173C42—31no—
174C42—31no—
175C42—31no—
176C42—31no—
177C42—62no—
178C22×C3—42no—
179C22×C3—41no—
180C22×C3—81no—
181C22×C3—81no—
182C3×C4—42no—
183C3×C4—41no—
184C3×C4—41no—
185Dic3—82no—
186Dic3—82no—
187C23C4×S411yescharacteristic
188Q8—31no—
189Q8—32no—
190Q8—61no—
191Q8—62no—
192Q8—62no—
193Q8—122no—
194Q8#90111yescharacteristic
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—121no—
204C2×C4C4×S411yescharacteristic
205C2×C4#98312yes—
206C2×C4#98412yes—
207C2×C4C22×S411yescharacteristic
208C8—21no—
209C8—21no—
210C8—62no—
211C8—62no—
212C8—61no—
213C8—61no—
214C8—122no—
215C8—122no—
216C2×C3—41no—
217C2×C3—41no—
218C2×C3—41no—
219C2×C3—82no—
220C22—21no—
221C22—21no—
222C22#538812yes—
223C22C2×C4×S411yescharacteristic
224C4—31no—
225C4—31no—
226C4—61no—
227C4—61no—
228C4—61no—
229C4—122no—
230C4—122no—
231C4#538812yes—
232C4C2×#98311yescharacteristic
233C4C2×#98411yescharacteristic
234C3—41noSylow
235C2—22no—
236C2C2×#538811yescharacteristic
237C2#8543711yescharacteristic
238C2#8544811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingC2×C8×Q8D3
socleC22C2×C4×S4
prime radicalC22×SL(2,𝔽3)C2×C4
cosocleC4×SL(2,𝔽3)C23
hypercenterC22×C8S4
hypocenterSL(2,𝔽3)#131
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
4. #73009C2semidirect
5. #226⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#853222
C2×C8×SL(2,𝔽3)2
#730552
#730092
#226⋊C82
#179063
#51024