Finite Group Databasebeta

← GO+(4,𝔽2)×ΓL(1,𝔽8)⋊C4 #72217776

Subgroups — GO+(4,𝔽2)×ΓL(1,𝔽8)⋊C4 #72217776

Order 6048.

656 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 91 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
1Dic7×GO+(4,𝔽2)C311yesmaximal, characteristic, complemented
2#72217594C211yesmaximal, characteristic, complemented
3#72217560C212yesmaximal, complemented
4C2×ΓL(1,𝔽8)×GO+(4,𝔽2)C211yesmaximal, characteristic
5D32×ΓL(1,𝔽8)⋊C4C212yesmaximal, complemented
6#157×ΓL(1,𝔽8)⋊C4C211yesmaximal, characteristic, complemented
7C2×D32×ΓL(1,𝔽8)C2212yes—
8C2×ΓL(1,𝔽8)⋊#157C2211yescharacteristic
9C2×ΓL(1,𝔽8)×#157C2211yescharacteristic
10ΓL(1,𝔽8)×GO+(4,𝔽2)C414yescomplemented
11#92693⋊C3C2212yes—
12D3×C3⋊(ΓL(1,𝔽8)⋊C4)—22no—
13C3⋊D3×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
14C3×D3×ΓL(1,𝔽8)⋊C4—22no—
15C2×C7×GO+(4,𝔽2)C2×C311yescharacteristic
16D32×Dic7C2×C312yescomplemented
17#11759408C2×C311yescharacteristic, complemented
18D32⋊Dic7C2×C312yescomplemented
19Dic7×#157C2×C311yescharacteristic, complemented
20C2×D3×ΓL(1,𝔽8)⋊C4—62no—
21C3×C4×GO+(4,𝔽2)—71nomaximal, Hall
22C2×C3⋊D3×ΓL(1,𝔽8)C2311yescharacteristic
23C2×C3×D3×ΓL(1,𝔽8)—22no—
24D32×ΓL(1,𝔽8)—22no—
25D32×ΓL(1,𝔽8)C2×C414yescomplemented
26ΓL(1,𝔽8)⋊#157—21no—
27ΓL(1,𝔽8)×#157C2×C412yescomplemented
28C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4))D411yescharacteristic, complemented
29C3×C3⋊(ΓL(1,𝔽8)⋊C4)—22no—
30C32×ΓL(1,𝔽8)⋊C4D411yescharacteristic, complemented
31D4×ΓL(1,𝔽8)⋊C4—91nomaximal
32C2×D32×C7C22×C312yes—
33C2×C7⋊#157C22×C311yescharacteristic
34C2×C7×#157C22×C311yescharacteristic
35C22×D3×ΓL(1,𝔽8)—62no—
36C7×GO+(4,𝔽2)C3×C414yescomplemented
37#92693C22×C312yes—
38D3×Dic21—22no—
39C3⋊D3×Dic7C22×C311yescharacteristic
40C3×D3×Dic7—22no—
41C2×C3⋊(ΓL(1,𝔽8)⋊C4)—62no—
42C2×C3×ΓL(1,𝔽8)⋊C4—62no—
43D3×ΓL(1,𝔽8)⋊C4—22no—
44D3×ΓL(1,𝔽8)⋊C4—42no—
45D3×ΓL(1,𝔽8)⋊C4—62no—
46D3×ΓL(1,𝔽8)⋊C4—62no—
47D3×ΓL(1,𝔽8)⋊C4—62no—
48C2×C3×GO+(4,𝔽2)—71no—
49C3×C4×D32—72no—
50C3×C4×#157—71no—
51C3×#3773—71no—
52C3×D32⋊C4—72no—
53C2×C32×ΓL(1,𝔽8)C2×D411yescharacteristic
54C3⋊D3×ΓL(1,𝔽8)C22×C411yescharacteristic
55C3⋊D3×ΓL(1,𝔽8)#4111yescharacteristic
56C3×D3×ΓL(1,𝔽8)—24no—
57C2×D4×ΓL(1,𝔽8)—91no—
58C2×D3×Dic7—62no—
59C22×ΓL(1,𝔽8)⋊C4—92no—
60(C22⋊Dic7)⋊C3—92no—
61C4⋊(ΓL(1,𝔽8)⋊C4)—91no—
62C4×ΓL(1,𝔽8)⋊C4—91no—
63C4×GO+(4,𝔽2)—71no—
64C2×C7×C3⋊D3C23×C311yescharacteristic
65C2×C3×D3×C7—22no—
66C22×C3×ΓL(1,𝔽8)—62no—
67D32×C7—22no—
68D32×C7C2×C3×C414yescomplemented
69C7⋊#157—21no—
70C7×#157C2×C3×C412yescomplemented
71C2×D3×ΓL(1,𝔽8)—22no—
72C2×D3×ΓL(1,𝔽8)—42no—
73C2×D3×ΓL(1,𝔽8)—64no—
74C2×D3×ΓL(1,𝔽8)—64no—
75C2×D3×ΓL(1,𝔽8)—62no—
76C3⋊Dic21C3×D411yescharacteristic, complemented
77C3×Dic21—22no—
78C32×Dic7C3×D411yescharacteristic, complemented
79C3⋊(ΓL(1,𝔽8)⋊C4)—22no—
80C3⋊(ΓL(1,𝔽8)⋊C4)—42no—
81C3⋊(ΓL(1,𝔽8)⋊C4)—62no—
82C3×ΓL(1,𝔽8)⋊C4—22no—
83C3×ΓL(1,𝔽8)⋊C4—42no—
84C3×ΓL(1,𝔽8)⋊C4—62no—
85D4×Dic7—91noHall
86C2×C3×D32—72no—
87C2×C3×#157—71no—
88C2×C3×#157—71no—
89C3×GO+(4,𝔽2)—74no—
90C3×C4×C3⋊D3—71no—
91C32×C4×D3—142no—
92C3×#624—72no—
93C3×D3×Dic3—142no—
94C32×ΓL(1,𝔽8)C4×D411yescharacteristic, complemented, Hall
95C22×D3×C7—62no—
96C23×ΓL(1,𝔽8)—92no—
97C2×Dic21—62no—
98C2×C3×Dic7—62no—
99D4×ΓL(1,𝔽8)—94no—
100C2×C4×ΓL(1,𝔽8)—91no—
101D3×Dic7—22no—
102D3×Dic7—62no—
103D3×Dic7—62no—
104D3×Dic7—62no—
105C2×ΓL(1,𝔽8)⋊C4—62no—
106C2×ΓL(1,𝔽8)⋊C4—92no—
107C2×ΓL(1,𝔽8)⋊C4—91no—
108C2×ΓL(1,𝔽8)⋊C4—91no—
109C2×ΓL(1,𝔽8)⋊C4—122no—
110C2×ΓL(1,𝔽8)⋊C4—182no—
111C2×GO+(4,𝔽2)AGL(1,𝔽7)11yescharacteristic
112C2×C3×C4×D3—422no—
113C4×D32—72no—
114C4×#157—71no—
115#3773—71no—
116D32⋊C4—72no—
117C2×C32×C7C2×C3×D411yescharacteristic
118C7×C3⋊D3C22×C3×C411yescharacteristic
119C7×C3⋊D3C3×#4111yescharacteristic
120C3×D3×C7—24no—
121C2×C3×ΓL(1,𝔽8)—22no—
122C2×C3×ΓL(1,𝔽8)—42no—
123C2×C3×ΓL(1,𝔽8)—64no—
124D3×ΓL(1,𝔽8)—24no—
125D3×ΓL(1,𝔽8)—44no—
126D3×ΓL(1,𝔽8)—62no—
127D3×ΓL(1,𝔽8)—62no—
128C2×C7×D4—91no—
129C22×Dic7—92no—
130C22⋊Dic7—92no—
131C4⋊Dic7—91no—
132C4×Dic7—91no—
133C2×C3×C3⋊D3—71no—
134C2×C32×D3—142no—
135C3×D32—74no—
136C3×D32—142no—
137C3×#157—72no—
138C3×#157—141no—
139C33×C4—71no—
140C3×C3⋊Dic3—71no—
141C32×Dic3—142no—
142C3×C4×D4—631no—
143C22×C3×C7—62no—
144C2×D3×C7—22no—
145C2×D3×C7—64no—
146C2×D3×C7—62no—
147C2×D3×C7—64no—
148C22×ΓL(1,𝔽8)—62no—
149C22×ΓL(1,𝔽8)—94no—
150C22×ΓL(1,𝔽8)—91no—
151C22×ΓL(1,𝔽8)—122no—
152C22×ΓL(1,𝔽8)—182no—
153Dic21—22no—
154Dic21—62no—
155C3×Dic7—22no—
156C3×Dic7—62no—
157C4×ΓL(1,𝔽8)—92no—
158ΓL(1,𝔽8)⋊C4—42no—
159ΓL(1,𝔽8)⋊C4—62no—
160ΓL(1,𝔽8)⋊C4—91no—
161ΓL(1,𝔽8)⋊C4—122no—
162ΓL(1,𝔽8)⋊C4—181no—
163ΓL(1,𝔽8)⋊C4GO+(4,𝔽2)11yescharacteristic, direct product factor
164C22×C3×D3—422no—
165C2×D32C2×AGL(1,𝔽7)12yes—
166C2×#157—71no—
167C2×#157C2×AGL(1,𝔽7)11yescharacteristic
168GO+(4,𝔽2)ΓL(1,𝔽8)⋊C414yesdirect product factor
169C2×C32×C4—422no—
170C4×C3⋊D3—71no—
171C2×C3×Dic3—422no—
172C3×C4×D3—142no—
173C3×C4×D3—142no—
174C3×C4×D3—282no—
175C3×C4×D3—422no—
176C3×C4×D3—422no—
177C3×C4×D3—422no—
178#624—72no—
179D3×Dic3—142no—
180C32×C7C3×C4×D411yescharacteristic, complemented
181C3×ΓL(1,𝔽8)—22no—
182C3×ΓL(1,𝔽8)—42no—
183C23×C7—92no—
184C7×D4—94no—
185C2×C4×C7—91no—
186C2×Dic7—62no—
187C2×Dic7—92no—
188C2×Dic7—91no—
189C2×Dic7—91no—
190C2×Dic7—182no—
191C2×C33—71no—
192C3×C3⋊D3—71no—
193C3×C3⋊D3—71no—
194C32×D3—144no—
195C2×C3×D4—631no—
196C22×C3×C4—632no—
197C2×C4×D3—422no—
198C3×C4⋊C4—631no—
199C3×C22⋊C4—632no—
200C3×C42—631no—
201C2×C3×C7—22no—
202C2×C3×C7—64no—
203D3×C7—24no—
204D3×C7—62no—
205D3×C7—62no—
206C2×ΓL(1,𝔽8)—42no—
207C2×ΓL(1,𝔽8)—64no—
208C2×ΓL(1,𝔽8)—91no—
209C2×ΓL(1,𝔽8)—91no—
210C2×ΓL(1,𝔽8)—124no—
211C2×ΓL(1,𝔽8)C2×GO+(4,𝔽2)11yescharacteristic
212C22×C32—422no—
213C2×C3⋊D3C22×AGL(1,𝔽7)11yescharacteristic
214C2×C3×D3—22no—
215C2×C3×D3—142no—
216C2×C3×D3—282no—
217C2×C3×D3—424no—
218C2×C3×D3—422no—
219C2×C3×D3—424no—
220D32—22no—
221D32C2×ΓL(1,𝔽8)⋊C414yescomplemented
222#157—141no—
223#157C2×ΓL(1,𝔽8)⋊C412yescomplemented
224C32×C4—71no—
225C32×C4—142no—
226C32×C4—282no—
227C32×C4—422no—
228C3⋊Dic3—71no—
229C3×Dic3—142no—
230C3×Dic3—142no—
231C3×Dic3—282no—
232C3×Dic3—422no—
233C4×D4—631noSylow
234C22×C7—62no—
235C22×C7—94no—
236C22×C7—91no—
237C22×C7—182no—
238C4×C7—92no—
239Dic7—62no—
240Dic7—91no—
241Dic7—181no—
242Dic7C3×GO+(4,𝔽2)11yescharacteristic, complemented
243C33—71noSylow
244C23×C3—632no—
245C22×D3—62no—
246C3×D4—634no—
247C2×C3×C4—422no—
248C2×C3×C4—422no—
249C2×C3×C4—631no—
250C2×C3×C4—631no—
251C2×C3×C4—632no—
252C2×C3×C4—631no—
253C2×C3×C4—842no—
254C2×C3×C4—1262no—
255C2×Dic3—422no—
256C4×D3—142no—
257C4×D3—422no—
258C4×D3—422no—
259C4×D3—422no—
260C3×C7—22no—
261ΓL(1,𝔽8)—42no—
262ΓL(1,𝔽8)C4×GO+(4,𝔽2)11yescharacteristic, complemented
263C2×C32—142no—
264C2×C32—282no—
265C2×C32—424no—
266C2×C32D4×AGL(1,𝔽7)11yescharacteristic
267C3⋊D3#1159⋊C311yescharacteristic
268C3⋊D3C22×ΓL(1,𝔽8)⋊C411yescharacteristic
269C3×D3—24no—
270C3×D3—144no—
271C3×D3—284no—
272C3×D3—422no—
273C3×D3—422no—
274C2×D4—91no—
275C22×C4—632no—
276C4⋊C4—631no—
277C22⋊C4—632no—
278C42—631no—
279C2×C7—64no—
280C2×C7—91no—
281C2×C7—91no—
282C2×C7C2×C3×GO+(4,𝔽2)11yescharacteristic
283C22×C3—62no—
284C22×C3—422no—
285C22×C3—631no—
286C22×C3—634no—
287C22×C3—842no—
288C22×C3—1262no—
289C2×D3—22no—
290C2×D3—64no—
291C2×D3—62no—
292C2×D3—64no—
293C3×C4—71no—
294C3×C4—142no—
295C3×C4—282no—
296C3×C4—422no—
297C3×C4—422no—
298C3×C4—632no—
299C3×C4—631no—
300C3×C4—842no—
301C3×C4—1261no—
302Dic3—142no—
303Dic3—422no—
304C32—142no—
305C32—282no—
306C32D4×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
307C23—92no—
308D4—94no—
309C2×C4—91no—
310C2×C4—422no—
311C2×C4—631no—
312C2×C4—632no—
313C2×C4—631no—
314C2×C4—1262no—
315C7C3×C4×GO+(4,𝔽2)11yescharacteristic, complemented, Sylow
316C2×C3—22no—
317C2×C3—64no—
318C2×C3—71no—
319C2×C3—282no—
320C2×C3—424no—
321C2×C3—631no—
322C2×C3—631no—
323C2×C3—844no—
324D3—24no—
325D3—62no—
326D3—62no—
327C22—62no—
328C22—94no—
329C22—91no—
330C22—182no—
331C4—71no—
332C4—92no—
333C4—422no—
334C4—631no—
335C4—1261no—
336C3—22no—
337C3—71no—
338C3—282no—
339C2—64no—
340C2—91no—
341C2—91no—
342C2AGL(1,𝔽7)×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2AGL(1,𝔽7)×GO+(4,𝔽2)
commutatorC7×C3⋊D3C22×C3×C4
Fitting, socleC2×C32×C7C2×C3×D4
prime radicalC2×ΓL(1,𝔽8)×GO+(4,𝔽2)C2
cosocleC2×C7×C3⋊D3C23×C3
hypocenterC32×C7C3×C4×D4
LCSG ▷ C7×C3⋊D3 ▷ C32×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. Dic7×GO+(4,𝔽2)C3semidirect
2. #72217594C2semidirect
3. #72217560C2semidirect
5. D32×ΓL(1,𝔽8)⋊C4C2semidirect
6. #157×ΓL(1,𝔽8)⋊C4C2semidirect
10. ΓL(1,𝔽8)×GO+(4,𝔽2)C4semidirect
16. D32×Dic7C2×C3semidirect
17. #11759408C2×C3semidirect
18. D32⋊Dic7C2×C3semidirect
19. Dic7×#157C2×C3semidirect
25. D32×ΓL(1,𝔽8)C2×C4semidirect
27. ΓL(1,𝔽8)×#157C2×C4semidirect
28. C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4))D4semidirect
30. C32×ΓL(1,𝔽8)⋊C4D4semidirect
36. C7×GO+(4,𝔽2)C3×C4semidirect
68. D32×C7C2×C3×C4semidirect
70. C7×#157C2×C3×C4semidirect
76. C3⋊Dic21C3×D4semidirect
78. C32×Dic7C3×D4semidirect
94. C32×ΓL(1,𝔽8)C4×D4semidirect
163. ΓL(1,𝔽8)⋊C4GO+(4,𝔽2)direct (N × H)
168. GO+(4,𝔽2)ΓL(1,𝔽8)⋊C4direct (N × H)
180. C32×C7C3×C4×D4semidirect
221. D32C2×ΓL(1,𝔽8)⋊C4semidirect
223. #157C2×ΓL(1,𝔽8)⋊C4semidirect
242. Dic7C3×GO+(4,𝔽2)semidirect
262. ΓL(1,𝔽8)C4×GO+(4,𝔽2)semidirect
306. C32D4×ΓL(1,𝔽8)⋊C4semidirect
315. C7C3×C4×GO+(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
Dic7×GO+(4,𝔽2)3
#722175942
#722175602
C2×ΓL(1,𝔽8)×GO+(4,𝔽2)2
D32×ΓL(1,𝔽8)⋊C42
#157×ΓL(1,𝔽8)⋊C42
C3×C4×GO+(4,𝔽2)7
D4×ΓL(1,𝔽8)⋊C49