Finite Group Databasebeta

← #89994⋊C3 #13052121

Subgroups — #89994⋊C3 #13052121

Order 1344.

476 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C22×D4×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C2×(C22⋊Dic7)⋊C3C212yesmaximal, complemented
3C2×(C22⋊Dic7)⋊C3C212yesmaximal, complemented
4C2×(ΓL(1,𝔽8)⋊C4)⋊C4C211yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)⋊#95C211yesmaximal, characteristic, complemented
6#89994C311yesmaximal, characteristic, complemented, Hall
7C24×ΓL(1,𝔽8)C2212yes—
8C2×D4×ΓL(1,𝔽8)—24no—
9C2×D4×ΓL(1,𝔽8)—22no—
10C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
11C22×ΓL(1,𝔽8)⋊C4C2212yes—
12C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
13C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
14(C22⋊Dic7)⋊C3—24no—
15(C22⋊Dic7)⋊C3—24no—
16(ΓL(1,𝔽8)⋊C4)⋊C4—22no—
17C22×C7×D4C2×C311yescharacteristic
18C2×C22⋊Dic7C2×C312yescomplemented
19C2×C22⋊Dic7C2×C312yescomplemented
20C2×Dic7⋊C4C2×C311yescharacteristic, complemented
21C7⋊#95C2×C311yescharacteristic, complemented
22C3×#394—71nomaximal, Hall
23C23×ΓL(1,𝔽8)—22no—
24C23×ΓL(1,𝔽8)—22no—
25C23×ΓL(1,𝔽8)—22no—
26C23×ΓL(1,𝔽8)—22no—
27C23×ΓL(1,𝔽8)—22no—
28C23×ΓL(1,𝔽8)—22no—
29C23×ΓL(1,𝔽8)D414yes—
30C23×ΓL(1,𝔽8)C2311yescharacteristic
31D4×ΓL(1,𝔽8)—28no—
32C2×C4×ΓL(1,𝔽8)—22no—
33C2×C4×ΓL(1,𝔽8)D412yes—
34C2×ΓL(1,𝔽8)⋊C4—21no—
35C2×ΓL(1,𝔽8)⋊C4—21no—
36C2×ΓL(1,𝔽8)⋊C4—21no—
37C2×ΓL(1,𝔽8)⋊C4—22no—
38C2×ΓL(1,𝔽8)⋊C4—22no—
39C2×ΓL(1,𝔽8)⋊C4—22no—
40C2×ΓL(1,𝔽8)⋊C4—22no—
41C2×ΓL(1,𝔽8)⋊C4D412yescomplemented
42C24×C7C22×C312yes—
43C2×C7×D4—24no—
44C2×C7×D4—22no—
45C22×C4×C7C22×C311yescharacteristic
46C22×Dic7C22×C312yes—
47C22×Dic7C22×C311yescharacteristic
48C22×Dic7C22×C311yescharacteristic
49C22⋊Dic7—24no—
50C22⋊Dic7—24no—
51Dic7⋊C4—22no—
52C22×C3×D4—71no—
53C2×C3×C4⋊C4—71no—
54C2×C3×C22⋊C4—72no—
55C2×C3×C22⋊C4—72no—
56C3×#95—71no—
57C22×ΓL(1,𝔽8)—24no—
58C22×ΓL(1,𝔽8)—24no—
59C22×ΓL(1,𝔽8)—24no—
60C22×ΓL(1,𝔽8)—42no—
61C22×ΓL(1,𝔽8)—42no—
62C22×ΓL(1,𝔽8)—42no—
63C22×ΓL(1,𝔽8)—42no—
64C22×ΓL(1,𝔽8)C2×D412yes—
65C22×ΓL(1,𝔽8)C2×D411yescharacteristic
66C22×ΓL(1,𝔽8)C2×D411yescharacteristic
67C22×ΓL(1,𝔽8)#4111yescharacteristic
68C22×ΓL(1,𝔽8)#4112yes—
69C4×ΓL(1,𝔽8)—22no—
70ΓL(1,𝔽8)⋊C4—22no—
71ΓL(1,𝔽8)⋊C4—41no—
72ΓL(1,𝔽8)⋊C4—42no—
73#394—71noSylow
74C23×C7—22no—
75C23×C7—22no—
76C23×C7—22no—
77C23×C7—22no—
78C23×C7—22no—
79C23×C7—22no—
80C23×C7C3×D414yes—
81C23×C7C23×C311yescharacteristic
82C7×D4—28no—
83C2×C4×C7—22no—
84C2×C4×C7C3×D412yes—
85C2×Dic7—21no—
86C2×Dic7—21no—
87C2×Dic7—21no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7C3×D412yescomplemented
93C24×C3—72no—
94C2×C3×D4—142no—
95C2×C3×D4—144no—
96C22×C3×C4—71no—
97C22×C3×C4—72no—
98C22×C3×C4—71no—
99C22×C3×C4—71no—
100C3×C4⋊C4—142no—
101C3×C22⋊C4—144no—
102C3×C22⋊C4—144no—
103C2×ΓL(1,𝔽8)—44no—
104C2×ΓL(1,𝔽8)#12011yescharacteristic
105C2×ΓL(1,𝔽8)#12111yescharacteristic
106C2×ΓL(1,𝔽8)#12112yes—
107C2×ΓL(1,𝔽8)#12312yes—
108C2×ΓL(1,𝔽8)#12411yescharacteristic
109C22×D4AGL(1,𝔽7)11yescharacteristic
110C2×C4⋊C4—71no—
111C2×C22⋊C4—72no—
112C2×C22⋊C4—72no—
113#95—71no—
114C22×C7—24no—
115C22×C7—24no—
116C22×C7—24no—
117C22×C7—42no—
118C22×C7—42no—
119C22×C7—42no—
120C22×C7—42no—
121C22×C7C2×C3×D411yescharacteristic
122C22×C7C2×C3×D412yes—
123C22×C7C2×C3×D411yescharacteristic
124C22×C7C3×#4111yescharacteristic
125C22×C7C3×#4112yes—
126C4×C7—22no—
127Dic7—22no—
128Dic7—41no—
129Dic7—42no—
130C23×C3—74no—
131C23×C3—71no—
132C23×C3—142no—
133C23×C3—142no—
134C23×C3—142no—
135C23×C3—142no—
136C23×C3—142no—
137C23×C3—142no—
138C3×D4—148no—
139C2×C3×C4—72no—
140C2×C3×C4—72no—
141C2×C3×C4—142no—
142C2×C3×C4—141no—
143C2×C3×C4—141no—
144C2×C3×C4—142no—
145C2×C3×C4—142no—
146C2×C3×C4—142no—
147C2×C3×C4—141no—
148C2×C3×C4—142no—
149ΓL(1,𝔽8)#39411yescharacteristic, complemented, Hall
150C24C2×AGL(1,𝔽7)12yes—
151C2×D4—24no—
152C2×D4—22no—
153C22×C4—71no—
154C22×C4—72no—
155C22×C4—71no—
156C22×C4C2×AGL(1,𝔽7)11yescharacteristic
157C4⋊C4—142no—
158C22⋊C4—144no—
159C22⋊C4—144no—
160C2×C7—44no—
161C2×C7C3×#12011yescharacteristic
162C2×C7C3×#12111yescharacteristic
163C2×C7C3×#12112yes—
164C2×C7C3×#12312yes—
165C2×C7C3×#12411yescharacteristic
166C22×C3—71no—
167C22×C3—71no—
168C22×C3—72no—
169C22×C3—71no—
170C22×C3—72no—
171C22×C3—144no—
172C22×C3—144no—
173C22×C3—144no—
174C22×C3—282no—
175C22×C3—282no—
176C22×C3—282no—
177C22×C3—282no—
178C3×C4—142no—
179C3×C4—142no—
180C3×C4—282no—
181C3×C4—281no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23(C22⋊D7)⋊C314yes—
189C23C22×AGL(1,𝔽7)11yescharacteristic
190D4—28no—
191C2×C4—22no—
192C2×C4—72no—
193C2×C4—141no—
194C2×C4—141no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—141no—
200C2×C4(C22⋊D7)⋊C312yes—
201C7C3×#39411yescharacteristic, complemented, Sylow
202C2×C3—72no—
203C2×C3—71no—
204C2×C3—71no—
205C2×C3—72no—
206C2×C3—71no—
207C2×C3—284no—
208C22—24no—
209C22—24no—
210C22—24no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22D4×AGL(1,𝔽7)11yescharacteristic
216C22#1159⋊C311yescharacteristic
217C22#1159⋊C312yes—
218C22C2×(C22⋊D7)⋊C312yes—
219C22C2×(C22⋊D7)⋊C311yescharacteristic
220C4—22no—
221C4—142no—
222C4—282no—
223C4—281no—
224C3—71noSylow
225C2—44no—
226C2#6498⋊C312yes—
227C2#6499⋊C311yescharacteristic
228C2#6501⋊C311yescharacteristic
229C2#6502⋊C312yes—
230C2#1062078911yescharacteristic
subgroupG/H
center, FrattiniC23C22×AGL(1,𝔽7)
commutator, socle, cosocleC23×C7C23×C3
FittingC22×C7×D4C2×C3
prime radicalC22×D4×ΓL(1,𝔽8)C2
hypercenterC22×D4AGL(1,𝔽7)
hypocenterC7C3×#394
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊Dic7)⋊C3C2semidirect
3. C2×(C22⋊Dic7)⋊C3C2semidirect
4. C2×(ΓL(1,𝔽8)⋊C4)⋊C4C2semidirect
5. ΓL(1,𝔽8)⋊#95C2semidirect
6. #89994C3semidirect
18. C2×C22⋊Dic7C2×C3semidirect
19. C2×C22⋊Dic7C2×C3semidirect
20. C2×Dic7⋊C4C2×C3semidirect
21. C7⋊#95C2×C3semidirect
41. C2×ΓL(1,𝔽8)⋊C4D4semidirect
92. C2×Dic7C3×D4semidirect
149. ΓL(1,𝔽8)#394semidirect
201. C7C3×#394semidirect

Maximal subgroups

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

subgroupindex
C22×D4×ΓL(1,𝔽8)2
C2×(C22⋊Dic7)⋊C32
C2×(ΓL(1,𝔽8)⋊C4)⋊C42
ΓL(1,𝔽8)⋊#952
#899943
C3×#3947