Finite Group Databasebeta

← #89723⋊C3 #13051849

Subgroups — #89723⋊C3 #13051849

Order 1344.

380 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 120 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×C22⋊C4×ΓL(1,𝔽8)C211yesmaximal, characteristic
2C2×(C22⋊Dic7)⋊C3C211yesmaximal, characteristic
3C2×(C22⋊Dic7)⋊C3C211yesmaximal, characteristic
4C2×C4⋊(ΓL(1,𝔽8)⋊C4)C211yesmaximal, characteristic, complemented
5C2×C4×ΓL(1,𝔽8)⋊C4C211yesmaximal, characteristic, complemented
6ΓL(1,𝔽8)⋊#95C211yesmaximal, characteristic, complemented
7ΓL(1,𝔽8)⋊#95C211yesmaximal, characteristic, complemented
8#89723C311yesmaximal, characteristic, complemented, Hall
9C24×ΓL(1,𝔽8)C2211yescharacteristic
10C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
11C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
12C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
13C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
14C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
15C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
16C22⋊C4×ΓL(1,𝔽8)C414yescomplemented
17(C22⋊Dic7)⋊C3—22no—
18(C22⋊Dic7)⋊C3—22no—
19C4⋊(ΓL(1,𝔽8)⋊C4)—22no—
20C4×ΓL(1,𝔽8)⋊C4—22no—
21C2×C7×C22⋊C4C2×C311yescharacteristic
22C2×C22⋊Dic7C2×C311yescharacteristic
23C2×C22⋊Dic7C2×C311yescharacteristic
24C2×C4⋊Dic7C2×C311yescharacteristic, complemented
25C2×C4×Dic7C2×C311yescharacteristic, complemented
26C7⋊#95C2×C311yescharacteristic, complemented
27C7⋊#95C2×C311yescharacteristic, complemented
28C3×#388—71nomaximal, Hall
29C23×ΓL(1,𝔽8)—21no—
30C23×ΓL(1,𝔽8)—21no—
31C23×ΓL(1,𝔽8)—21no—
32C23×ΓL(1,𝔽8)—21no—
33C23×ΓL(1,𝔽8)—21no—
34C23×ΓL(1,𝔽8)—21no—
35C23×ΓL(1,𝔽8)C2×C412yes—
36C23×ΓL(1,𝔽8)C2311yescharacteristic
37C2×C4×ΓL(1,𝔽8)—21no—
38C2×C4×ΓL(1,𝔽8)—21no—
39C2×C4×ΓL(1,𝔽8)—21no—
40C2×C4×ΓL(1,𝔽8)—21no—
41C2×C4×ΓL(1,𝔽8)C2×C412yes—
42C2×C4×ΓL(1,𝔽8)C2×C412yes—
43C2×ΓL(1,𝔽8)⋊C4—21no—
44C2×ΓL(1,𝔽8)⋊C4—21no—
45C2×ΓL(1,𝔽8)⋊C4—21no—
46C2×ΓL(1,𝔽8)⋊C4—21no—
47C2×ΓL(1,𝔽8)⋊C4—21no—
48C2×ΓL(1,𝔽8)⋊C4—21no—
49C2×ΓL(1,𝔽8)⋊C4—21no—
50C2×ΓL(1,𝔽8)⋊C4—21no—
51C2×ΓL(1,𝔽8)⋊C4—21no—
52C2×ΓL(1,𝔽8)⋊C4—21no—
53C2×ΓL(1,𝔽8)⋊C4D412yes—
54C2×ΓL(1,𝔽8)⋊C4D412yes—
55C24×C7C22×C311yescharacteristic
56C22×C4×C7C22×C311yescharacteristic
57C22×C4×C7C22×C311yescharacteristic
58C22×Dic7C22×C311yescharacteristic
59C22×Dic7C22×C311yescharacteristic
60C22×Dic7C22×C311yescharacteristic
61C22×Dic7C22×C311yescharacteristic
62C7×C22⋊C4C3×C414yescomplemented
63C22⋊Dic7—22no—
64C22⋊Dic7—22no—
65C4⋊Dic7—22no—
66C4×Dic7—22no—
67C2×C3×C4⋊C4—71no—
68C2×C3×C22⋊C4—71no—
69C2×C3×C22⋊C4—71no—
70C2×C3×C22⋊C4—71no—
71C2×C3×C42—71no—
72C3×#95—71no—
73C3×#95—71no—
74C22×ΓL(1,𝔽8)—22no—
75C22×ΓL(1,𝔽8)—22no—
76C22×ΓL(1,𝔽8)—22no—
77C22×ΓL(1,𝔽8)—41no—
78C22×ΓL(1,𝔽8)—41no—
79C22×ΓL(1,𝔽8)—41no—
80C22×ΓL(1,𝔽8)—41no—
81C22×ΓL(1,𝔽8)C22×C411yescharacteristic
82C22×ΓL(1,𝔽8)C2×D411yescharacteristic
83C22×ΓL(1,𝔽8)C2×D411yescharacteristic
84C22×ΓL(1,𝔽8)#4111yescharacteristic
85C22×ΓL(1,𝔽8)#4111yescharacteristic
86C22×ΓL(1,𝔽8)#4111yescharacteristic
87C22×ΓL(1,𝔽8)#4111yescharacteristic
88C4×ΓL(1,𝔽8)—22no—
89C4×ΓL(1,𝔽8)—22no—
90ΓL(1,𝔽8)⋊C4—22no—
91ΓL(1,𝔽8)⋊C4—22no—
92ΓL(1,𝔽8)⋊C4—41no—
93ΓL(1,𝔽8)⋊C4—41no—
94#388—71noSylow
95C23×C7—21no—
96C23×C7—21no—
97C23×C7—21no—
98C23×C7—21no—
99C23×C7—21no—
100C23×C7—21no—
101C23×C7C2×C3×C412yes—
102C23×C7C23×C311yescharacteristic
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7C2×C3×C412yes—
108C2×C4×C7C2×C3×C412yes—
109C2×Dic7—21no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—21no—
116C2×Dic7—21no—
117C2×Dic7—21no—
118C2×Dic7—21no—
119C2×Dic7C3×D412yes—
120C2×Dic7C3×D412yes—
121C24×C3—71no—
122C22×C3×C4—71no—
123C22×C3×C4—71no—
124C22×C3×C4—71no—
125C22×C3×C4—71no—
126C22×C3×C4—71no—
127C22×C3×C4—71no—
128C3×C4⋊C4—142no—
129C3×C22⋊C4—74no—
130C3×C22⋊C4—142no—
131C3×C22⋊C4—142no—
132C3×C42—142no—
133C2×ΓL(1,𝔽8)—42no—
134C2×ΓL(1,𝔽8)#11711yescharacteristic
135C2×ΓL(1,𝔽8)C4×D411yescharacteristic
136C2×ΓL(1,𝔽8)C4×D411yescharacteristic
137C2×ΓL(1,𝔽8)#12111yescharacteristic
138C2×ΓL(1,𝔽8)#12311yescharacteristic
139C2×ΓL(1,𝔽8)#12411yescharacteristic
140C2×ΓL(1,𝔽8)#12611yescharacteristic
141C2×C4⋊C4—71no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—71no—
144C2×C22⋊C4AGL(1,𝔽7)11yescharacteristic
145C2×C42—71no—
146#95—71no—
147#95—71no—
148C22×C7—22no—
149C22×C7—22no—
150C22×C7—22no—
151C22×C7—41no—
152C22×C7—41no—
153C22×C7—41no—
154C22×C7—41no—
155C22×C7C22×C3×C411yescharacteristic
156C22×C7C2×C3×D411yescharacteristic
157C22×C7C2×C3×D411yescharacteristic
158C22×C7C3×#4111yescharacteristic
159C22×C7C3×#4111yescharacteristic
160C22×C7C3×#4111yescharacteristic
161C22×C7C3×#4111yescharacteristic
162C4×C7—22no—
163C4×C7—22no—
164Dic7—22no—
165Dic7—22no—
166Dic7—41no—
167Dic7—41no—
168C23×C3—72no—
169C23×C3—71no—
170C23×C3—141no—
171C23×C3—141no—
172C23×C3—141no—
173C23×C3—141no—
174C23×C3—141no—
175C23×C3—141no—
176C2×C3×C4—72no—
177C2×C3×C4—72no—
178C2×C3×C4—72no—
179C2×C3×C4—72no—
180C2×C3×C4—141no—
181C2×C3×C4—141no—
182C2×C3×C4—141no—
183C2×C3×C4—141no—
184C2×C3×C4—141no—
185C2×C3×C4—141no—
186C2×C3×C4—141no—
187C2×C3×C4—141no—
188C2×C3×C4—141no—
189C2×C3×C4—141no—
190C2×C3×C4—141no—
191C2×C3×C4—141no—
192C2×C3×C4—141no—
193C2×C3×C4—141no—
194ΓL(1,𝔽8)#38811yescharacteristic, complemented, Hall
195C24C2×AGL(1,𝔽7)11yescharacteristic
196C22×C4—71no—
197C22×C4—71no—
198C22×C4—71no—
199C22×C4—71no—
200C22×C4C2×AGL(1,𝔽7)11yescharacteristic
201C22×C4C2×AGL(1,𝔽7)11yescharacteristic
202C4⋊C4—142no—
203C22⋊C4—142no—
204C22⋊C4—142no—
205C22⋊C4ΓL(1,𝔽8)⋊C414yescomplemented
206C42—142no—
207C2×C7—42no—
208C2×C7C3×#11711yescharacteristic
209C2×C7C3×C4×D411yescharacteristic
210C2×C7C3×C4×D411yescharacteristic
211C2×C7C3×#12111yescharacteristic
212C2×C7C3×#12311yescharacteristic
213C2×C7C3×#12411yescharacteristic
214C2×C7C3×#12611yescharacteristic
215C22×C3—71no—
216C22×C3—71no—
217C22×C3—71no—
218C22×C3—71no—
219C22×C3—71no—
220C22×C3—71no—
221C22×C3—71no—
222C22×C3—142no—
223C22×C3—142no—
224C22×C3—142no—
225C22×C3—281no—
226C22×C3—281no—
227C22×C3—281no—
228C22×C3—281no—
229C3×C4—142no—
230C3×C4—142no—
231C3×C4—142no—
232C3×C4—142no—
233C3×C4—281no—
234C3×C4—281no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23C2×ΓL(1,𝔽8)⋊C412yes—
242C23C22×AGL(1,𝔽7)11yescharacteristic
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—72no—
248C2×C4—72no—
249C2×C4—141no—
250C2×C4—141no—
251C2×C4—141no—
252C2×C4—141no—
253C2×C4—141no—
254C2×C4—141no—
255C2×C4—141no—
256C2×C4—141no—
257C2×C4—141no—
258C2×C4—141no—
259C2×C4C2×ΓL(1,𝔽8)⋊C412yes—
260C2×C4C2×ΓL(1,𝔽8)⋊C412yes—
261C7C3×#38811yescharacteristic, complemented, Sylow
262C2×C3—71no—
263C2×C3—71no—
264C2×C3—71no—
265C2×C3—71no—
266C2×C3—71no—
267C2×C3—71no—
268C2×C3—71no—
269C2×C3—282no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22#1157⋊C311yescharacteristic
278C22#1157⋊C311yescharacteristic
279C22D4×AGL(1,𝔽7)11yescharacteristic
280C22D4×AGL(1,𝔽7)11yescharacteristic
281C22#1159⋊C311yescharacteristic
282C22#1159⋊C311yescharacteristic
283C22C22×ΓL(1,𝔽8)⋊C411yescharacteristic
284C4—22no—
285C4—22no—
286C4—142no—
287C4—142no—
288C4—281no—
289C4—281no—
290C3—71noSylow
291C2—42no—
292C2#6442⋊C311yescharacteristic
293C2#6446⋊C311yescharacteristic
294C2#6447⋊C311yescharacteristic
295C2#6448⋊C311yescharacteristic
296C2#6489⋊C311yescharacteristic
297C2D4×ΓL(1,𝔽8)⋊C411yescharacteristic
298C2D4×ΓL(1,𝔽8)⋊C411yescharacteristic
subgroupG/H
center, FrattiniC23C22×AGL(1,𝔽7)
commutatorC22×C7C22×C3×C4
FittingC2×C7×C22⋊C4C2×C3
socle, cosocleC23×C7C23×C3
prime radicalC24×ΓL(1,𝔽8)C22
hypercenterC2×C22⋊C4AGL(1,𝔽7)
hypocenterC7C3×#388
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊(ΓL(1,𝔽8)⋊C4)C2semidirect
5. C2×C4×ΓL(1,𝔽8)⋊C4C2semidirect
6. ΓL(1,𝔽8)⋊#95C2semidirect
7. ΓL(1,𝔽8)⋊#95C2semidirect
8. #89723C3semidirect
16. C22⋊C4×ΓL(1,𝔽8)C4semidirect
24. C2×C4⋊Dic7C2×C3semidirect
25. C2×C4×Dic7C2×C3semidirect
26. C7⋊#95C2×C3semidirect
27. C7⋊#95C2×C3semidirect
62. C7×C22⋊C4C3×C4semidirect
194. ΓL(1,𝔽8)#388semidirect
205. C22⋊C4ΓL(1,𝔽8)⋊C4semidirect
261. C7C3×#388semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×ΓL(1,𝔽8)2
C2×(C22⋊Dic7)⋊C32
C2×C4⋊(ΓL(1,𝔽8)⋊C4)2
C2×C4×ΓL(1,𝔽8)⋊C42
ΓL(1,𝔽8)⋊#952
#897233
C3×#3887