Finite Group Databasebeta

← ΓL(1,𝔽8)×#398 #13054929

Subgroups — ΓL(1,𝔽8)×#398 #13054929

Order 1344.

300 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 117 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×C4⋊C4×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C2×C4⋊C4×ΓL(1,𝔽8)C211yesmaximal, characteristic
3C2×C4⋊C4×ΓL(1,𝔽8)C212yesmaximal
4ΓL(1,𝔽8)×#95C212yesmaximal
5ΓL(1,𝔽8)×#95C211yesmaximal, characteristic
6C7×#398C311yesmaximal, characteristic, complemented, Hall
7C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
8C22×C4×ΓL(1,𝔽8)C2212yes—
9C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
10C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
11C22×C4×ΓL(1,𝔽8)C2212yes—
12C4⋊C4×ΓL(1,𝔽8)—22no—
13C4⋊C4×ΓL(1,𝔽8)—24no—
14C4⋊C4×ΓL(1,𝔽8)—22no—
15C2×C7×C4⋊C4C2×C311yescharacteristic
16C2×C7×C4⋊C4C2×C311yescharacteristic
17C2×C7×C4⋊C4C2×C312yes—
18C7×#95C2×C312yes—
19C7×#95C2×C311yescharacteristic
20C3×#398—71nomaximal, Hall
21C23×ΓL(1,𝔽8)C2311yescharacteristic
22C2×C4×ΓL(1,𝔽8)—21no—
23C2×C4×ΓL(1,𝔽8)—22no—
24C2×C4×ΓL(1,𝔽8)—21no—
25C2×C4×ΓL(1,𝔽8)—22no—
26C2×C4×ΓL(1,𝔽8)—22no—
27C2×C4×ΓL(1,𝔽8)—22no—
28C2×C4×ΓL(1,𝔽8)—22no—
29C2×C4×ΓL(1,𝔽8)—21no—
30C2×C4×ΓL(1,𝔽8)—22no—
31C2×C4×ΓL(1,𝔽8)—22no—
32C2×C4×ΓL(1,𝔽8)D412yes—
33C2×C4×ΓL(1,𝔽8)D412yes—
34C2×C4×ΓL(1,𝔽8)Q814yes—
35C22×C4×C7C22×C311yescharacteristic
36C22×C4×C7C22×C312yes—
37C22×C4×C7C22×C311yescharacteristic
38C22×C4×C7C22×C311yescharacteristic
39C22×C4×C7C22×C312yes—
40C7×C4⋊C4—22no—
41C7×C4⋊C4—24no—
42C7×C4⋊C4—22no—
43C2×C3×C4⋊C4—72no—
44C2×C3×C4⋊C4—71no—
45C2×C3×C4⋊C4—71no—
46C3×#95—71no—
47C3×#95—72no—
48C22×ΓL(1,𝔽8)C2×D411yescharacteristic
49C22×ΓL(1,𝔽8)C2×D411yescharacteristic
50C22×ΓL(1,𝔽8)C2×Q812yes—
51C22×ΓL(1,𝔽8)#4111yescharacteristic
52C22×ΓL(1,𝔽8)#4112yes—
53C4×ΓL(1,𝔽8)—22no—
54C4×ΓL(1,𝔽8)—24no—
55C4×ΓL(1,𝔽8)—22no—
56C4×ΓL(1,𝔽8)—42no—
57C4×ΓL(1,𝔽8)—41no—
58#398ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
59C23×C7C23×C311yescharacteristic
60C2×C4×C7—21no—
61C2×C4×C7—22no—
62C2×C4×C7—21no—
63C2×C4×C7—22no—
64C2×C4×C7—22no—
65C2×C4×C7—22no—
66C2×C4×C7—22no—
67C2×C4×C7—21no—
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7C3×D412yes—
71C2×C4×C7C3×D412yes—
72C2×C4×C7C3×Q814yes—
73C22×C3×C4—72no—
74C22×C3×C4—71no—
75C22×C3×C4—71no—
76C22×C3×C4—71no—
77C22×C3×C4—72no—
78C3×C4⋊C4—142no—
79C3×C4⋊C4—144no—
80C3×C4⋊C4—142no—
81C2×ΓL(1,𝔽8)#12111yescharacteristic
82C2×ΓL(1,𝔽8)C22⋊Q812yes—
83C2×ΓL(1,𝔽8)#12311yescharacteristic
84C2×ΓL(1,𝔽8)#12512yes—
85C2×ΓL(1,𝔽8)C4⋊Q811yescharacteristic
86C2×C4⋊C4C2×ΓL(1,𝔽8)11yescharacteristic
87C2×C4⋊C4C2×ΓL(1,𝔽8)11yescharacteristic
88C2×C4⋊C4C2×ΓL(1,𝔽8)12yes—
89#95C2×ΓL(1,𝔽8)12yes—
90#95C2×ΓL(1,𝔽8)11yescharacteristic
91C22×C7C2×C3×D411yescharacteristic
92C22×C7C2×C3×D411yescharacteristic
93C22×C7C2×C3×Q812yes—
94C22×C7C3×#4111yescharacteristic
95C22×C7C3×#4112yes—
96C4×C7—22no—
97C4×C7—24no—
98C4×C7—22no—
99C4×C7—42no—
100C4×C7—41no—
101C23×C3—71no—
102C2×C3×C4—72no—
103C2×C3×C4—72no—
104C2×C3×C4—74no—
105C2×C3×C4—142no—
106C2×C3×C4—141no—
107C2×C3×C4—142no—
108C2×C3×C4—142no—
109C2×C3×C4—141no—
110C2×C3×C4—141no—
111C2×C3×C4—142no—
112C2×C3×C4—142no—
113C2×C3×C4—142no—
114C2×C3×C4—142no—
115ΓL(1,𝔽8)#39811yescharacteristic, direct product factor, Hall
116C22×C4C22×ΓL(1,𝔽8)11yescharacteristic
117C22×C4C22×ΓL(1,𝔽8)12yes—
118C22×C4C22×ΓL(1,𝔽8)11yescharacteristic
119C22×C4C22×ΓL(1,𝔽8)11yescharacteristic
120C22×C4C22×ΓL(1,𝔽8)12yes—
121C4⋊C4—22no—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C2×C7C3×#12111yescharacteristic
125C2×C7C3×C22⋊Q812yes—
126C2×C7C3×#12311yescharacteristic
127C2×C7C3×#12512yes—
128C2×C7C3×C4⋊Q811yescharacteristic
129C22×C3—71no—
130C22×C3—71no—
131C22×C3—72no—
132C22×C3—71no—
133C22×C3—72no—
134C3×C4—142no—
135C3×C4—144no—
136C3×C4—142no—
137C3×C4—282no—
138C3×C4—281no—
139C23C23×ΓL(1,𝔽8)11yescharacteristic
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4D4×ΓL(1,𝔽8)12yes—
151C2×C4D4×ΓL(1,𝔽8)12yes—
152C2×C4Q8×ΓL(1,𝔽8)14yes—
153C7C3×#39811yescharacteristic, complemented, Sylow
154C2×C3—71no—
155C2×C3—72no—
156C2×C3—71no—
157C2×C3—72no—
158C2×C3—71no—
159C22C2×D4×ΓL(1,𝔽8)11yescharacteristic
160C22C2×D4×ΓL(1,𝔽8)11yescharacteristic
161C22C2×Q8×ΓL(1,𝔽8)12yes—
162C22#41×ΓL(1,𝔽8)11yescharacteristic
163C22#41×ΓL(1,𝔽8)12yes—
164C4—22no—
165C4—24no—
166C4—22no—
167C4—42no—
168C4—41no—
169C3—71noSylow
170C2ΓL(1,𝔽8)×#12111yescharacteristic
171C2ΓL(1,𝔽8)×C22⋊Q812yes—
172C2ΓL(1,𝔽8)×#12311yescharacteristic
173C2ΓL(1,𝔽8)×#12512yes—
174C2ΓL(1,𝔽8)×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×ΓL(1,𝔽8)
commutator, socle, cosocleC23×C7C23×C3
FittingC7×#398C3
prime radicalC23×ΓL(1,𝔽8)C23
hypercenter#398ΓL(1,𝔽8)
hypocenterC7C3×#398
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
6. C7×#398C3semidirect
58. #398ΓL(1,𝔽8)direct (N × H)
115. ΓL(1,𝔽8)#398direct (N × H)
153. C7C3×#398semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×ΓL(1,𝔽8)2
ΓL(1,𝔽8)×#952
C7×#3983
C3×#3987