Finite Group Databasebeta

← #72276697

Subgroups — #72276697

Order 4032.

292 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 25 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
1ΓL(1,𝔽8)×#997C211yesmaximal, characteristic, complemented
2#13052183C312yesmaximal, complemented
3#13052163C311yesmaximal, characteristic, complemented
4#13052144C311yesmaximal, characteristic, complemented
5C2×A4×AGL(1,𝔽7)—41nomaximal
6C7⋊#997C2×C312yescomplemented
7C7×#997C2×C311yescharacteristic, complemented
8ΓL(1,𝔽8)×H(5,𝔽2)C2×C311yescharacteristic, complemented
9#6500⋊C3—31no—
10#6408⋊C3—31no—
11C3×B(4,𝔽2)⋊C3—71nomaximal, Hall
12C2×A4×ΓL(1,𝔽8)—41no—
13C2×A4×ΓL(1,𝔽8)—81no—
14A4×AGL(1,𝔽7)—42no—
15ΓL(1,𝔽8)×SL(2,𝔽3)—81no—
16#90016C3211yescharacteristic, complemented, Hall
17C2×D7⋊A4—42no—
18C2×A4×D7—41no—
19C23×AGL(1,𝔽7)A411yescharacteristic, complemented
20#41×ΓL(1,𝔽8)—61no—
21C2×D4×ΓL(1,𝔽8)—31no—
22C2×D4×ΓL(1,𝔽8)—61no—
23C2×(C22⋊D7)⋊C3—61no—
24(C22⋊Dic7)⋊C3—31no—
25(C2×AGL(1,𝔽7))⋊C4—61no—
26C3×#997—71no—
27C2×C3×AGL(1,𝔽7)—161no—
28A4×ΓL(1,𝔽8)—41no—
29A4×ΓL(1,𝔽8)—81no—
30C7×H(5,𝔽2)C2×C3211yescharacteristic, complemented
31#6500—31no—
32#6408—31no—
33C3×B(4,𝔽2)—71no—
34B(4,𝔽2)⋊C3—72no—
35B(4,𝔽2)⋊C3—71no—
36C2×C22⋊ΓL(1,𝔽8)—42no—
37C2×C22⋊ΓL(1,𝔽8)—82no—
38C2×C7×A4—41no—
39C2×C7×A4—81no—
40C23×ΓL(1,𝔽8)—21no—
41C23×ΓL(1,𝔽8)—31no—
42C23×ΓL(1,𝔽8)C2×A411yescharacteristic
43D7⋊A4—44no—
44A4×D7—42no—
45C22×AGL(1,𝔽7)—42no—
46C22×AGL(1,𝔽7)—61no—
47C7⋊SL(2,𝔽3)—82no—
48C7×SL(2,𝔽3)—81no—
49Q8×ΓL(1,𝔽8)—21no—
50D4×ΓL(1,𝔽8)—62no—
51D4×ΓL(1,𝔽8)—61no—
52C2×C4×ΓL(1,𝔽8)—31no—
53C2×C4×ΓL(1,𝔽8)—61no—
54(C22⋊D7)⋊C3—122no—
55C2×ΓL(1,𝔽8)⋊C4—61no—
56C22×C3×A4—281no—
57C2×C3×ΓL(1,𝔽8)—161no—
58C3×AGL(1,𝔽7)—162no—
59C23×D7C3×A411yescharacteristic, complemented
60C7×#41—61no—
61C2×C7×D4—31no—
62C2×C7×D4—61no—
63C2×C22⋊D7—61no—
64C22⋊Dic7—31no—
65(C2×D7)⋊C4—61no—
66C3×H(5,𝔽2)—71no—
67#997—72no—
68#997AGL(1,𝔽7)11yescharacteristic, complemented
69C3×#120—211no—
70C3×#99—211no—
71C2×C3×D7—161no—
72C22⋊ΓL(1,𝔽8)—42no—
73C22⋊ΓL(1,𝔽8)—82no—
74C7×A4—41no—
75C7×A4—81no—
76C22×ΓL(1,𝔽8)—31no—
77C22×ΓL(1,𝔽8)—41no—
78C22×ΓL(1,𝔽8)—61no—
79C22×ΓL(1,𝔽8)—81no—
80C22×ΓL(1,𝔽8)—121no—
81C2×AGL(1,𝔽7)—41no—
82C2×AGL(1,𝔽7)—62no—
83C2×AGL(1,𝔽7)—121no—
84C2×AGL(1,𝔽7)—162no—
85C4×ΓL(1,𝔽8)—61no—
86ΓL(1,𝔽8)⋊C4—121no—
87C2×C3×A4—281no—
88C2×C3×A4—282no—
89C2×C3×A4—561no—
90C3×SL(2,𝔽3)—561no—
91B(4,𝔽2)—71noSylow
92C3×ΓL(1,𝔽8)—161noHall
93C23×C7—21no—
94C23×C7—31no—
95C23×C7C2×C3×A411yescharacteristic
96C22×D7—42no—
97C22×D7—61no—
98C7×Q8—21no—
99C7×D4—62no—
100C7×D4—61no—
101C2×C4×C7—31no—
102C2×C4×C7—61no—
103C22⋊D7—122no—
104C2×Dic7—61no—
105C24×C3—71no—
106C22×A4—281no—
107C22×A4—282no—
108C3×#41—421no—
109C2×C3×D4—211no—
110C2×C3×D4—421no—
111C2×C3×D4—421no—
112C3×C22⋊C4—211no—
113C3×C22⋊C4—421no—
114C2×C3×C7—161no—
115C3×D7—162no—
116C2×ΓL(1,𝔽8)—61no—
117C2×ΓL(1,𝔽8)—121no—
118C2×ΓL(1,𝔽8)—162no—
119C2×ΓL(1,𝔽8)#120⋊C311yescharacteristic
120AGL(1,𝔽7)—42no—
121AGL(1,𝔽7)—164no—
122C22×C32—1121no—
123C3×A4—281no—
124C3×A4—561no—
125H(5,𝔽2)C3×AGL(1,𝔽7)11yescharacteristic, complemented
126#120—211no—
127#99—211no—
128C22×C7—31no—
129C22×C7—41no—
130C22×C7—61no—
131C22×C7—81no—
132C22×C7—121no—
133C2×D7—41no—
134C2×D7—62no—
135C2×D7—121no—
136C4×C7—61no—
137Dic7—121no—
138C23×C3—71no—
139C23×C3—141no—
140C23×C3—211no—
141C23×C3—282no—
142C23×C3—421no—
143C2×A4—41no—
144C2×A4—81no—
145C2×A4—282no—
146C2×A4—284no—
147C2×A4—282no—
148C2×A4—562no—
149C3×Q8—141no—
150C3×D4—422no—
151C3×D4—421no—
152C3×D4—842no—
153C2×C3×C4—211no—
154C2×C3×C4—421no—
155C2×C3×C4—421no—
156SL(2,𝔽3)—81no—
157SL(2,𝔽3)—562no—
158C3×C7—161no—
159ΓL(1,𝔽8)—162no—
160ΓL(1,𝔽8)B(4,𝔽2)⋊C311yescharacteristic, complemented
161C2×C32—1121no—
162C2×C32—1122no—
163C24—71no—
164#41—61no—
165C2×D4—31no—
166C2×D4—61no—
167C2×D4—421no—
168C22⋊C4—211no—
169C22⋊C4—421no—
170C2×C7—61no—
171C2×C7—121no—
172C2×C7C3×#120⋊C311yescharacteristic
173D7—42no—
174C22×C3—211no—
175C22×C3—281no—
176C22×C3—281no—
177C22×C3—421no—
178C22×C3—422no—
179C22×C3—561no—
180C22×C3—841no—
181C22×C3—841no—
182C22×C3—1121no—
183C22×C3—1122no—
184A4—41no—
185A4—81no—
186A4—282no—
187A4—562no—
188C3×C4—421no—
189C3×C4—841no—
190C32—1121noSylow
191C23—21no—
192C23—31no—
193C23—282no—
194C23—421no—
195C23A4×AGL(1,𝔽7)11yescharacteristic
196Q8—21no—
197D4—62no—
198D4—61no—
199D4—842no—
200C2×C4—31no—
201C2×C4—61no—
202C2×C4—421no—
203C7C3×B(4,𝔽2)⋊C311yescharacteristic, complemented, Sylow
204C2×C3—71no—
205C2×C3—161no—
206C2×C3—282no—
207C2×C3—421no—
208C2×C3—841no—
209C2×C3—1122no—
210C2×C3—1124no—
211C2×C3—1122no—
212C22—31no—
213C22—41no—
214C22—61no—
215C22—81no—
216C22—121no—
217C22—281no—
218C22—422no—
219C22—841no—
220C4—61no—
221C4—841no—
222C3—71no—
223C3—161no—
224C3—1122no—
225C2—61no—
226C2—121no—
227C2—282no—
228C2#7221460611yescharacteristic
subgroupG/H
center, hypercenterC2#72214606
commutator, Fitting, cosocle, hypocenterC7×H(5,𝔽2)C2×C32
FrattiniC23A4×AGL(1,𝔽7)
socleC2×C7C3×#120⋊C3
LCSG ▷ C7×H(5,𝔽2)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#997C2semidirect
2. #13052183C3semidirect
3. #13052163C3semidirect
4. #13052144C3semidirect
6. C7⋊#997C2×C3semidirect
7. C7×#997C2×C3semidirect
8. ΓL(1,𝔽8)×H(5,𝔽2)C2×C3semidirect
16. #90016C32semidirect
19. C23×AGL(1,𝔽7)A4semidirect
30. C7×H(5,𝔽2)C2×C32semidirect
59. C23×D7C3×A4semidirect
68. #997AGL(1,𝔽7)semidirect
125. H(5,𝔽2)C3×AGL(1,𝔽7)semidirect
160. ΓL(1,𝔽8)B(4,𝔽2)⋊C3semidirect
203. C7C3×B(4,𝔽2)⋊C3semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#9972
#130521833
#130521633
#130521443
C2×A4×AGL(1,𝔽7)4
C3×B(4,𝔽2)⋊C37