Finite Group Databasebeta

← #72254331

Subgroups — #72254331

Order 2016.

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

normal subgroups only

This group has 206 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#66728⋊C3C211yesmaximal, characteristic
2ΓL(1,𝔽8)×#208C211yesmaximal, characteristic, complemented
3C3×C4×ΓL(1,𝔽8)⋊C4C211yesmaximal, characteristic, complemented
4#10620516C311yesmaximal, characteristic, complemented
5#6399⋊C3—31nomaximal
6C2×C3×C4×ΓL(1,𝔽8)C2211yescharacteristic
7#4268⋊C3—21no—
8C4⋊(C3⋊AGL(1,𝔽7))C411yescharacteristic, complemented
9C4×C3⋊AGL(1,𝔽7)—21no—
10Dic42⋊C3C411yescharacteristic, complemented
11C2×C3×ΓL(1,𝔽8)⋊C4—21no—
12ΓL(1,𝔽8)×C3⋊C8—21no—
13#66728C2×C311yescharacteristic
14#1157⋊C3—31no—
15C7×#208C2×C311yescharacteristic, complemented
16C3×C4×Dic7C2×C311yescharacteristic, complemented
17M16×ΓL(1,𝔽8)—31no—
18C4×ΓL(1,𝔽8)⋊C4—21no—
19C4×ΓL(1,𝔽8)⋊C4D311yescharacteristic, complemented
20C3×#805—71nomaximal, Hall
21C22×C3×ΓL(1,𝔽8)D411yescharacteristic
22C2×C3⋊AGL(1,𝔽7)—21no—
23C3×C4×ΓL(1,𝔽8)C2×C411yescharacteristic
24C3×C4×ΓL(1,𝔽8)D411yescharacteristic
25C3⋊(ΓL(1,𝔽8)⋊C4)—21no—
26C3×ΓL(1,𝔽8)⋊C4—22no—
27#6399—31noHall
28C2×C3×C4×C7C22×C311yescharacteristic
29#4268—21no—
30D84C3×C411yescharacteristic, complemented
31C4×D21—21no—
32Dic42C3×C411yescharacteristic, complemented
33C2×C3×Dic7—21no—
34C2×C4×ΓL(1,𝔽8)—21no—
35C2×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
36(C22⋊D7)⋊C3—61no—
37C2×ΓL(1,𝔽8)⋊C4—22no—
38C2×ΓL(1,𝔽8)⋊C4—21no—
39C4⋊AGL(1,𝔽7)—31no—
40C4×AGL(1,𝔽7)—61no—
41ΓL(1,𝔽8)⋊Q8—31no—
42C7×C3⋊C8—21no—
43C8×ΓL(1,𝔽8)—61no—
44C3×#235—71no—
45C32×C42—71no—
46C3×#208—141no—
47C2×C3×ΓL(1,𝔽8)—21no—
48C2×C3×ΓL(1,𝔽8)C22⋊C411yescharacteristic
49C3⋊AGL(1,𝔽7)—41no—
50#1157—31no—
51C7×M16—31no—
52C4×Dic7C3×D311yescharacteristic, complemented
53C3×#104—211no—
54#805—71no—
55C22×C3×C7C3×D411yescharacteristic
56C2×D21—21no—
57C22×ΓL(1,𝔽8)—21no—
58C22×ΓL(1,𝔽8)#6711yescharacteristic
59C2×AGL(1,𝔽7)—61no—
60C3×C4×C7C2×C3×C411yescharacteristic
61C3×C4×C7C3×D411yescharacteristic
62Dic21—21no—
63C3×Dic7—22no—
64C4×ΓL(1,𝔽8)—21no—
65C4×ΓL(1,𝔽8)—21no—
66C4×ΓL(1,𝔽8)C4×D311yescharacteristic
67C4×ΓL(1,𝔽8)D1211yescharacteristic
68ΓL(1,𝔽8)⋊C4—24no—
69ΓL(1,𝔽8)⋊C4—22no—
70ΓL(1,𝔽8)⋊C4—61no—
71C2×C32×C4—141no—
72C2×C32×C4—141no—
73C3×#67—141no—
74C3×D12—71no—
75C3×C4×D3—141no—
76C3×Dic6—71no—
77C3×C3⋊C8—141no—
78C3×ΓL(1,𝔽8)#10411yescharacteristic, complemented, Hall
79C2×C4×C7C2×C3×D311yescharacteristic
80C22⋊D7—61no—
81C2×Dic7—21no—
82D28—31no—
83C4×D7—61no—
84Dic14—31no—
85C7×C8—61no—
86C3×#41—211no—
87#235—71no—
88C3×M16—421no—
89C3×C42—71no—
90C3×C42—71no—
91C3×C42—141no—
92#208AGL(1,𝔽7)11yescharacteristic, complemented
93C2×C3×C7—21no—
94C2×C3×C7C3×C22⋊C411yescharacteristic
95D21—41no—
96C2×ΓL(1,𝔽8)—22no—
97C2×ΓL(1,𝔽8)—21no—
98C2×ΓL(1,𝔽8)—21no—
99C2×ΓL(1,𝔽8)(C2×D3)⋊C411yescharacteristic
100AGL(1,𝔽7)—121no—
101C22×C32—141no—
102C2×C3×D3—141no—
103C32×C4—142no—
104C32×C4—141no—
105C32×C4—141no—
106C3×Dic3—141no—
107#104—211noSylow
108C22×C7C3×#6711yescharacteristic
109C2×D7—61no—
110C4×C7C3×C4×D311yescharacteristic
111C4×C7C3×D1211yescharacteristic
112Dic7—22no—
113Dic7—61no—
114C3×Q8—211no—
115C3×D4—211no—
116C3×D4—421no—
117C2×C3×C4—141no—
118C2×C3×C4—142no—
119C2×C3×C4—141no—
120C2×C3×C4—141no—
121C2×C3×C4—281no—
122C2×C3×C4—421no—
123C2×C3×C4C2×AGL(1,𝔽7)11yescharacteristic
124#67—141no—
125D12—71no—
126C4×D3—141no—
127Dic6—71no—
128C3×C8—421no—
129C3⋊C8—21no—
130C3×C7C3×#10411yescharacteristic, complemented
131ΓL(1,𝔽8)—21no—
132ΓL(1,𝔽8)#80511yescharacteristic, complemented
133C2×C32—141no—
134C2×C32—281no—
135C3×D3—281no—
136#41—211no—
137M16—31no—
138C42—71no—
139C2×C7—21no—
140C2×C7C3×(C2×D3)⋊C411yescharacteristic
141D7—121no—
142C22×C3—141no—
143C22×C3—281no—
144C22×C3—421no—
145C22×C3C4⋊AGL(1,𝔽7)11yescharacteristic
146C2×D3—141no—
147C3×C4—142no—
148C3×C4—142no—
149C3×C4—144no—
150C3×C4—141no—
151C3×C4—141no—
152C3×C4—281no—
153C3×C4—281no—
154C3×C4—421no—
155C3×C4C4×AGL(1,𝔽7)11yescharacteristic
156C3×C4(C22⋊D7)⋊C311yescharacteristic
157Dic3—141no—
158C32—141noSylow
159Q8—211no—
160D4—211no—
161D4—421no—
162C2×C4—141no—
163C2×C4—421no—
164C2×C4D3×AGL(1,𝔽7)11yescharacteristic
165C8—61no—
166C7C3×#80511yescharacteristic, complemented, Sylow
167C2×C3—21no—
168C2×C3—141no—
169C2×C3—281no—
170C2×C3—282no—
171C2×C3—281no—
172C2×C3—841no—
173C2×C3(C2×AGL(1,𝔽7))⋊C411yescharacteristic
174D3—281no—
175C22—421no—
176C22#4246⋊C311yescharacteristic
177C4—142no—
178C4—421no—
179C4(C3⋊AGL(1,𝔽7))⋊C411yescharacteristic
180C4#4247⋊C311yescharacteristic
181C3—141no—
182C3—281no—
183C3#6399⋊C311yescharacteristic, complemented
184C2—21no—
185C2—841no—
186C2#66575⋊C311yescharacteristic
subgroupG/H
centerC4#4247⋊C3
commutatorC3×C4×C7C2×C3×C4
Frattini, hypercenterC2×C4D3×AGL(1,𝔽7)
Fitting, cosocleC2×C3×C4×C7C22×C3
socleC2×C3×C7C3×C22⋊C4
prime radical#66728⋊C3C2
hypocenterC3×C7C3×#104
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. ΓL(1,𝔽8)×#208C2semidirect
3. C3×C4×ΓL(1,𝔽8)⋊C4C2semidirect
4. #10620516C3semidirect
8. C4⋊(C3⋊AGL(1,𝔽7))C4semidirect
10. Dic42⋊C3C4semidirect
15. C7×#208C2×C3semidirect
16. C3×C4×Dic7C2×C3semidirect
19. C4×ΓL(1,𝔽8)⋊C4D3semidirect
30. D84C3×C4semidirect
32. Dic42C3×C4semidirect
52. C4×Dic7C3×D3semidirect
78. C3×ΓL(1,𝔽8)#104semidirect
92. #208AGL(1,𝔽7)semidirect
130. C3×C7C3×#104semidirect
132. ΓL(1,𝔽8)#805semidirect
166. C7C3×#805semidirect
183. C3#6399⋊C3semidirect

Maximal subgroups

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

subgroupindex
#66728⋊C32
ΓL(1,𝔽8)×#2082
C3×C4×ΓL(1,𝔽8)⋊C42
#106205163
#6399⋊C33
C3×#8057