Finite Group Databasebeta

← #11713027

Subgroups — #11713027

Order 768.

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

normal subgroups only

This group has 457 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
1C2×#5404C211yesmaximal, characteristic
2#85365C212yesmaximal
3#2912⋊C3C211yesmaximal, characteristic, complemented
4#73125C211yesmaximal, characteristic, complemented
5#73075C212yesmaximal, complemented
6#22690—31nomaximal, Sylow
7C2×#985C2211yescharacteristic
8C2×#131⋊C3C2211yescharacteristic
9#520⋊C3C2212yes—
10#5426C2212yes—
11C2×GU(2,𝔽3)C2211yescharacteristic
12#5404C414yescomplemented
13#5394—22no—
14C4×#226—22no—
15#4860—41nomaximal
16C2×#443—31no—
17#2951—32no—
18#2912—31no—
19#2912D311yescharacteristic, complemented
20#1952—32no—
21#1860—32no—
22#1860—32no—
23#1846—31no—
24#1815—31no—
25#1815—31no—
26#1772—31no—
27C2×#41⋊C3C2311yescharacteristic
28#985C2×C412yes—
29#985Q812yes—
30C2×GL(2,𝔽3)D411yescharacteristic
31C2×#226D411yescharacteristic
32C2×D3×C8—41no—
33#892—42no—
34#131⋊C3C2×C412yescomplemented
35C4×SL(2,𝔽3)—22no—
36GU(2,𝔽3)C2×C412yescomplemented
37#859—22no—
38C3×#106—41no—
39#817—41no—
40#807—42no—
41C2×#135—31no—
42C2×#131—31no—
43C2×#131C2×D311yescharacteristic
44#520—32no—
45#520C2×D312yes—
46#443—34no—
47#443—34no—
48#440—62no—
49#440—62no—
50C4×Q16—62no—
51C4×D8—62no—
52C2×#108—31no—
53C2×#108—31no—
54#428—32no—
55#428—32no—
56#428—62no—
57#427—31no—
58#427—61no—
59C2×#107—32no—
60C2×#106—31no—
61C2×#106—31no—
62C2×#106—61no—
63C2×#104—31no—
64C2×#104—31no—
65#417—32no—
66#417—32no—
67#405—32no—
68C2×C4×C8—31no—
69#382—32no—
70#343—32no—
71#343—32no—
72#339—32no—
73C2×C4×D3—41no—
74#41⋊C3C22×C411yescharacteristic
75#41⋊C3C2×Q811yescharacteristic
76C2×SL(2,𝔽3)C2×D411yescharacteristic
77GL(2,𝔽3)C4⋊C412yes—
78#226C4⋊C412yes—
79C2×C3×C8—41no—
80C3×C4⋊C4—42no—
81(C2×D3)⋊C4—82no—
82C4⋊Dic3—42no—
83C4×Dic3—82no—
84C2×C3⋊C8—41no—
85D3×C8—44no—
86C2×#41—31no—
87C2×#41C22×D311yescharacteristic
88#135—32no—
89#135—32no—
90#135—32no—
91#135—32no—
92C2×Q16—31no—
93C2×SD16—31no—
94C2×SD16—31no—
95C2×D8—31no—
96#131—32no—
97#131—34no—
98#131—62no—
99#131C4×D312yes—
100C2×M16—31no—
101C2×M16—31no—
102C2×M16—32no—
103C2×M16—61no—
104C22×C8—31no—
105C22×C8—31no—
106C22×C8—61no—
107C4×Q8—22no—
108C4×Q8—62no—
109C4×D4—62no—
110C4×D4—62no—
111C4×D4—122no—
112#117—32no—
113#117—62no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—62no—
117C2×C42—31no—
118#108—34no—
119#108—62no—
120#107—32no—
121#107—62no—
122#107—62no—
123#107—62no—
124#106—31no—
125#106—61no—
126#106—62no—
127#106—61no—
128#106—61no—
129#106S411yescharacteristic
130#104—34no—
131#104—32no—
132#104—32no—
133#103—62no—
134#103—62no—
135#102—62no—
136#102—62no—
137#97—62no—
138C4×C8—34no—
139C4×C8—62no—
140#95—122no—
141C22×D3—41no—
142C2×C3×C4—41no—
143C2×C3×C4—82no—
144C2×Dic3—41no—
145C2×Dic3—82no—
146C4×D3—42no—
147C4×D3—42no—
148SL(2,𝔽3)C2×C4⋊C411yescharacteristic
149C3×C8—42no—
150C3⋊C8—42no—
151#41—31no—
152#41—31no—
153#41—32no—
154#41—32no—
155#41—61no—
156#41—61no—
157#41C2×C4×D311yescharacteristic
158#41D3×Q811yescharacteristic
159C2×Q8—31no—
160C2×Q8D3×D411yescharacteristic
161C2×D4—31no—
162C2×D4—31no—
163C2×D4—61no—
164C22×C4—31no—
165C22×C4—61no—
166C22×C4—61no—
167C22×C4—62no—
168C22×C4—62no—
169C22×C4—122no—
170Q16—32no—
171Q16—32no—
172SD16—32no—
173SD16—32no—
174SD16—32no—
175SD16—32no—
176D8—32no—
177D8—32no—
178M16—32no—
179M16—32no—
180M16—61no—
181M16—64no—
182M16—61no—
183M16—62no—
184M16—62no—
185C2×C8—32no—
186C2×C8—31no—
187C2×C8—31no—
188C2×C8—32no—
189C2×C8—31no—
190C2×C8—32no—
191C2×C8—32no—
192C2×C8—61no—
193C2×C8—64no—
194C2×C8—62no—
195C2×C8C2×S411yescharacteristic
196C4⋊C4—32no—
197C4⋊C4—62no—
198C4⋊C4—62no—
199C4⋊C4—62no—
200C4⋊C4—122no—
201C4⋊C4C2×S412yes—
202C22⋊C4—62no—
203C22⋊C4—122no—
204C42—32no—
205C42—62no—
206C42—61no—
207C42—122no—
208C22×C3—41no—
209C2×D3—42no—
210C2×D3—82no—
211C3×C4—41no—
212C3×C4—41no—
213C3×C4—162no—
214Dic3—42no—
215Dic3—162no—
216C23—31no—
217C23—61no—
218Q8—31no—
219Q8—32no—
220Q8—61no—
221Q8D3×C4⋊C411yescharacteristic
222D4—31no—
223D4—31no—
224D4—32no—
225D4—61no—
226D4—61no—
227D4—62no—
228D4—62no—
229C2×C4—22no—
230C2×C4—31no—
231C2×C4—31no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—61no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4—242no—
243C2×C4—242no—
244C2×C4—482no—
245C2×C4C22×S411yescharacteristic
246C8—32no—
247C8—32no—
248C8—32no—
249C8—62no—
250C8—122no—
251C8C4×S412yes—
252C2×C3—41no—
253C2×C3—42no—
254D3—82no—
255C22—31no—
256C22—31no—
257C22—62no—
258C22—62no—
259C22—122no—
260C22D4×S411yescharacteristic
261C4—31no—
262C4—31no—
263C4—42no—
264C4—62no—
265C4—122no—
266C4—122no—
267C4—242no—
268C4C2×C4×S411yescharacteristic
269C4Q8×S411yescharacteristic
270C3—41noSylow
271C2—61no—
272C2—61no—
273C2—122no—
274C2C4⋊C4×S411yescharacteristic
275C2#8547212yes—
subgroupG/H
center, socleC22D4×S4
commutator#41⋊C3C22×C4
FrattiniC2×C4C22×S4
Fitting#2912D3
prime radicalC2×#985C22
cosocleC2×#41⋊C3C23
hypercenter#106S4
hypocenterSL(2,𝔽3)C2×C4⋊C4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
3. #2912⋊C3C2semidirect
4. #73125C2semidirect
5. #73075C2semidirect
12. #5404C4semidirect
19. #2912D3semidirect
34. #131⋊C3C2×C4semidirect
36. GU(2,𝔽3)C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#54042
#853652
#2912⋊C32
#731252
#730752
#226903
#48604