Finite Group Databasebeta

← #72276617

Subgroups — #72276617

Order 4032.

628 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 38 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×ΓL(1,𝔽8)×C22⋊A4C211yesmaximal, characteristic
2C7⋊#5936C311yesmaximal, characteristic, complemented
3#13052147—31nomaximal
4ΓL(1,𝔽8)×C22⋊A4C411yescharacteristic, complemented
5A4⋊(ΓL(1,𝔽8)⋊C4)—43nomaximal
6C2×C22⋊(C22⋊ΓL(1,𝔽8))—21no—
7C2×C7×C22⋊A4C2×C311yescharacteristic
8C25×ΓL(1,𝔽8)D311yescharacteristic
9C2×(C22⋊Dic7)⋊C3—33no—
10C2×(C22⋊Dic7)⋊C3—33no—
11C3×#5936—71nomaximal, Hall
12C2×A4×ΓL(1,𝔽8)—43no—
13C2×A4×ΓL(1,𝔽8)—81no—
14#90019—31noHall
15C22⋊(C22⋊ΓL(1,𝔽8))—21no—
16C7×C22⋊A4C3×C411yescharacteristic, complemented
17C24×ΓL(1,𝔽8)—33no—
18C24×ΓL(1,𝔽8)—33no—
19C24×ΓL(1,𝔽8)—61no—
20C24×ΓL(1,𝔽8)—61no—
21C24×ΓL(1,𝔽8)Dic311yescharacteristic, complemented
22C22×ΓL(1,𝔽8)⋊C4—33no—
23C22×ΓL(1,𝔽8)⋊C4—31no—
24C7⋊SL(2,ℤ/4)—43no—
25(C22⋊Dic7)⋊C3—63no—
26(C22⋊Dic7)⋊C3—63no—
27(C22⋊Dic7)⋊C3—63no—
28(C22⋊Dic7)⋊C3—63no—
29C2×C3×C22⋊A4—71no—
30A4×ΓL(1,𝔽8)—43no—
31A4×ΓL(1,𝔽8)—81no—
32C3⋊(ΓL(1,𝔽8)⋊C4)—161no—
33C25×C7C3×D311yescharacteristic
34C2×C22⋊Dic7—33no—
35C2×C22⋊Dic7—33no—
36#5936—71no—
37C3×#379—211no—
38C2×C22⋊ΓL(1,𝔽8)—82no—
39C2×C22⋊ΓL(1,𝔽8)—83no—
40C2×C7×A4—43no—
41C2×C7×A4—81no—
42C23×ΓL(1,𝔽8)—21no—
43C23×ΓL(1,𝔽8)—31no—
44C23×ΓL(1,𝔽8)—33no—
45C23×ΓL(1,𝔽8)—33no—
46C23×ΓL(1,𝔽8)—33no—
47C23×ΓL(1,𝔽8)—33no—
48C23×ΓL(1,𝔽8)—33no—
49C23×ΓL(1,𝔽8)—61no—
50C23×ΓL(1,𝔽8)—61no—
51C23×ΓL(1,𝔽8)—66no—
52C23×ΓL(1,𝔽8)—63no—
53C23×ΓL(1,𝔽8)—63no—
54C23×ΓL(1,𝔽8)—63no—
55C23×ΓL(1,𝔽8)S413yes—
56C2×ΓL(1,𝔽8)⋊C4—66no—
57C2×ΓL(1,𝔽8)⋊C4—63no—
58C2×ΓL(1,𝔽8)⋊C4—63no—
59C3×C22⋊A4—71no—
60C3×SL(2,ℤ/4)—283no—
61C2×C3×ΓL(1,𝔽8)—161no—
62C24×C7—33no—
63C24×C7—33no—
64C24×C7—61no—
65C24×C7—61no—
66C24×C7C3×Dic311yescharacteristic, complemented
67C22×Dic7—33no—
68C22×Dic7—31no—
69C22⋊Dic7—63no—
70C22⋊Dic7—63no—
71C22⋊Dic7—63no—
72C22⋊Dic7—63no—
73C25×C3—71no—
74C2×C22⋊A4—141no—
75C2×C22⋊A4AGL(1,𝔽7)11yescharacteristic
76C2×C3×C22⋊C4—213no—
77C2×C3×C22⋊C4—213no—
78C22⋊ΓL(1,𝔽8)—82no—
79C22⋊ΓL(1,𝔽8)—83no—
80C7×A4—43no—
81C7×A4—81no—
82C22×ΓL(1,𝔽8)—21no—
83C22×ΓL(1,𝔽8)—33no—
84C22×ΓL(1,𝔽8)—33no—
85C22×ΓL(1,𝔽8)—31no—
86C22×ΓL(1,𝔽8)—33no—
87C22×ΓL(1,𝔽8)—33no—
88C22×ΓL(1,𝔽8)—33no—
89C22×ΓL(1,𝔽8)—63no—
90C22×ΓL(1,𝔽8)—66no—
91C22×ΓL(1,𝔽8)—63no—
92C22×ΓL(1,𝔽8)—63no—
93C22×ΓL(1,𝔽8)—61no—
94C22×ΓL(1,𝔽8)—61no—
95C22×ΓL(1,𝔽8)SL(2,ℤ/4)13yescomplemented
96Dic21—161no—
97ΓL(1,𝔽8)⋊C4—123no—
98ΓL(1,𝔽8)⋊C4—121no—
99C2×C3×A4—283no—
100C2×C3×A4—561no—
101#379—211noSylow
102C3×ΓL(1,𝔽8)—161noHall
103C23×C7—21no—
104C23×C7—31no—
105C23×C7—33no—
106C23×C7—33no—
107C23×C7—33no—
108C23×C7—33no—
109C23×C7—33no—
110C23×C7—61no—
111C23×C7—61no—
112C23×C7—66no—
113C23×C7—63no—
114C23×C7—63no—
115C23×C7—63no—
116C23×C7C3×S413yes—
117C2×Dic7—66no—
118C2×Dic7—63no—
119C2×Dic7—63no—
120C24×C3—71no—
121C24×C3—213no—
122C24×C3—213no—
123C24×C3—421no—
124C24×C3—421no—
125C22⋊A4—141no—
126C22⋊A4ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
127C22×C3×C4—211no—
128C22×C3×C4—213no—
129SL(2,ℤ/4)—283no—
130C3×C22⋊C4—423no—
131C3×C22⋊C4—423no—
132C3×C22⋊C4—423no—
133C3×C22⋊C4—423no—
134C2×C3×C7—161no—
135C2×ΓL(1,𝔽8)—33no—
136C2×ΓL(1,𝔽8)—33no—
137C2×ΓL(1,𝔽8)—61no—
138C2×ΓL(1,𝔽8)—61no—
139C2×ΓL(1,𝔽8)—321no—
140C2×ΓL(1,𝔽8)#102011yescharacteristic
141C3×A4—283no—
142C3×A4—561no—
143C3×Dic3—1121no—
144C25C3⋊AGL(1,𝔽7)11yescharacteristic
145C2×C22⋊C4—213no—
146C2×C22⋊C4—213no—
147C22×C7—21no—
148C22×C7—33no—
149C22×C7—33no—
150C22×C7—31no—
151C22×C7—33no—
152C22×C7—33no—
153C22×C7—33no—
154C22×C7—63no—
155C22×C7—66no—
156C22×C7—63no—
157C22×C7—63no—
158C22×C7—61no—
159C22×C7—61no—
160C22×C7C3×SL(2,ℤ/4)13yescomplemented
161Dic7—123no—
162Dic7—121no—
163C23×C3—73no—
164C23×C3—141no—
165C23×C3—213no—
166C23×C3—213no—
167C23×C3—211no—
168C23×C3—213no—
169C23×C3—213no—
170C23×C3—213no—
171C23×C3—421no—
172C23×C3—421no—
173C23×C3—423no—
174C23×C3—426no—
175C23×C3—423no—
176C23×C3—423no—
177C2×A4—43no—
178C2×A4—81no—
179C2×A4—562no—
180C2×A4—563no—
181C2×C3×C4—423no—
182C2×C3×C4—423no—
183C2×C3×C4—426no—
184C3×C7—161no—
185ΓL(1,𝔽8)—321no—
186ΓL(1,𝔽8)#593611yescharacteristic, complemented
187C2×C32—1121no—
188C24—33no—
189C24—33no—
190C24—61no—
191C24—61no—
192C24C3⋊(ΓL(1,𝔽8)⋊C4)11yescharacteristic, complemented
193C22×C4—213no—
194C22×C4—211no—
195C22⋊C4—423no—
196C22⋊C4—423no—
197C22⋊C4—423no—
198C22⋊C4—423no—
199C2×C7—33no—
200C2×C7—33no—
201C2×C7—61no—
202C2×C7—61no—
203C2×C7C3×#102011yescharacteristic
204C22×C3—73no—
205C22×C3—141no—
206C22×C3—213no—
207C22×C3—213no—
208C22×C3—213no—
209C22×C3—211no—
210C22×C3—213no—
211C22×C3—213no—
212C22×C3—421no—
213C22×C3—423no—
214C22×C3—426no—
215C22×C3—423no—
216C22×C3—423no—
217C22×C3—421no—
218A4—43no—
219A4—81no—
220A4—562no—
221A4—563no—
222C3×C4—841no—
223C3×C4—843no—
224Dic3—1121no—
225C32—1121noSylow
226C23—21no—
227C23—31no—
228C23—33no—
229C23—33no—
230C23—33no—
231C23—33no—
232C23—33no—
233C23—61no—
234C23—61no—
235C23—66no—
236C23—63no—
237C23—63no—
238C23—63no—
239C23A4⋊AGL(1,𝔽7)13yes—
240C2×C4—426no—
241C2×C4—423no—
242C2×C4—423no—
243C7C3×#593611yescharacteristic, complemented, Sylow
244C2×C3—71no—
245C2×C3—161no—
246C2×C3—213no—
247C2×C3—213no—
248C2×C3—421no—
249C2×C3—421no—
250C2×C3—2241no—
251C22—21no—
252C22—33no—
253C22—33no—
254C22—31no—
255C22—33no—
256C22—33no—
257C22—33no—
258C22—63no—
259C22—66no—
260C22—63no—
261C22—63no—
262C22—61no—
263C22—61no—
264C22A4⋊(ΓL(1,𝔽8)⋊C4)13yescomplemented
265C4—843no—
266C4—841no—
267C3—71no—
268C3—161no—
269C3—2241no—
270C2—33no—
271C2—33no—
272C2—61no—
273C2—61no—
274C2#7221403211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2#72214032
commutator, hypocenterC7×C22⋊A4C3×C4
Fitting, socleC25×C7C3×D3
prime radicalC2×ΓL(1,𝔽8)×C22⋊A4C2
cosocleC2×C7×C22⋊A4C2×C3
LCSG ▷ C7×C22⋊A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C7⋊#5936C3semidirect
4. ΓL(1,𝔽8)×C22⋊A4C4semidirect
16. C7×C22⋊A4C3×C4semidirect
21. C24×ΓL(1,𝔽8)Dic3semidirect
66. C24×C7C3×Dic3semidirect
95. C22×ΓL(1,𝔽8)SL(2,ℤ/4)semidirect
126. C22⋊A4ΓL(1,𝔽8)⋊C4semidirect
160. C22×C7C3×SL(2,ℤ/4)semidirect
186. ΓL(1,𝔽8)#5936semidirect
192. C24C3⋊(ΓL(1,𝔽8)⋊C4)semidirect
243. C7C3×#5936semidirect
264. C22A4⋊(ΓL(1,𝔽8)⋊C4)semidirect

Maximal subgroups

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

subgroupindex
C2×ΓL(1,𝔽8)×C22⋊A42
C7⋊#59363
#130521473
A4⋊(ΓL(1,𝔽8)⋊C4)4
C3×#59367