Finite Group Databasebeta

← #72246293

Subgroups — #72246293

Order 9408.

840 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 118 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
1#72246276C212yesmaximal, complemented
2#72246271C211yesmaximal, characteristic, complemented
3#72246214C311yesmaximal, characteristic, complemented, Hall
4C7×ΓL(1,𝔽8)×H(5,𝔽2)C211yesmaximal, characteristic, complemented
5#72245607C211yesmaximal, characteristic, complemented
6C4×ΓL(1,𝔽8)⋊Dic7C2211yescharacteristic, complemented
7#72245208C212yesmaximal, complemented
8#72245198—22no—
9#72245191—22no—
10C7×#41×ΓL(1,𝔽8)—22no—
11C7×#41×ΓL(1,𝔽8)C2212yes—
12#72245172C2212yescomplemented
13C2×#11924202—22no—
14C2×C4⋊(ΓL(1,𝔽8)⋊D7)C2211yescharacteristic, complemented
15C2×C7×D4×ΓL(1,𝔽8)—22no—
16C2×C7×D4×ΓL(1,𝔽8)—22no—
17C2×C7×D4×ΓL(1,𝔽8)C2211yescharacteristic
18C72×H(5,𝔽2)C2×C311yescharacteristic, complemented
19#13096589C2×C312yescomplemented
20#13096573C2×C311yescharacteristic, complemented
21#13096155C2×C312yescomplemented
22#13096152C2×C311yescharacteristic, complemented
23C3×#90013—71nomaximal
24#90013⋊C3—71nomaximal
25C23×C7×ΓL(1,𝔽8)—22no—
26C23×C7×ΓL(1,𝔽8)D412yes—
27C22×ΓL(1,𝔽8)⋊D7—21no—
28C7×Q8×ΓL(1,𝔽8)D412yescomplemented
29C7×D4×ΓL(1,𝔽8)—24no—
30C7×D4×ΓL(1,𝔽8)—22no—
31C7×D4×ΓL(1,𝔽8)—22no—
32C7×D4×ΓL(1,𝔽8)D412yescomplemented
33C2×C4×C7×ΓL(1,𝔽8)—22no—
34C2×C4×C7×ΓL(1,𝔽8)—22no—
35C2×C4×C7×ΓL(1,𝔽8)C2311yescharacteristic
36#11924202—44no—
37C2×ΓL(1,𝔽8)⋊Dic7—21no—
38C4⋊(ΓL(1,𝔽8)⋊D7)—22no—
39ΓL(1,𝔽8)⋊(C7⋊C8)—22no—
40C72×#41—22no—
41C72×#41C22×C312yes—
42C2×C72×D4—22no—
43C2×C72×D4—22no—
44C2×C72×D4C22×C311yescharacteristic
45C2×C22⋊(C7⋊D7)—22no—
46C2×C4⋊(C7⋊D7)C22×C311yescharacteristic, complemented
47#11716073—22no—
48#11716071—22no—
49C4×C7⋊Dic7C22×C311yescharacteristic, complemented
50#11716066C22×C312yescomplemented
51C3×C7×H(5,𝔽2)—141no—
52ΓL(1,𝔽8)×H(5,𝔽2)D711yescharacteristic, complemented
53C3×#6512—72no—
54C3×#6503—71no—
55#6512⋊C3—72no—
56#6503⋊C3—71no—
57C3×#6411—72no—
58C3×#6407—71no—
59#6411⋊C3—72no—
60#6407⋊C3—71no—
61C22×C7×ΓL(1,𝔽8)—22no—
62C22×C7×ΓL(1,𝔽8)—22no—
63C22×C7×ΓL(1,𝔽8)—42no—
64C22×C7×ΓL(1,𝔽8)—82no—
65C22×C7×ΓL(1,𝔽8)C2×D411yescharacteristic
66C2×ΓL(1,𝔽8)⋊D7—42no—
67C2×ΓL(1,𝔽8)⋊D7—41no—
68C4×C7×ΓL(1,𝔽8)—22no—
69C4×C7×ΓL(1,𝔽8)C2×D412yes—
70ΓL(1,𝔽8)⋊Dic7—22no—
71#90013—71no—
72#90013—71no—
73#90013—212no—
74C23×C72—22no—
75C23×C72C3×D412yes—
76C22×C7⋊D7—21no—
77C72×Q8C3×D412yescomplemented
78C72×D4—24no—
79C72×D4—22no—
80C72×D4—22no—
81C72×D4C3×D412yescomplemented
82C2×C4×C72—22no—
83C2×C4×C72—22no—
84C2×C4×C72C23×C311yescharacteristic
85C22⋊(C7⋊D7)—44no—
86C2×C7⋊Dic7—21no—
87C4⋊(C7⋊D7)—22no—
88C7⋊(C7⋊C8)—22no—
89C3×C7×#41—142no—
90C3×C7×#41—142no—
91C2×C3×C7×D4—142no—
92C2×C3×C7×D4—142no—
93C2×C3×C7×D4—141no—
94C2×C3×C22⋊D7—142no—
95C2×C3×D28—71no—
96#41×ΓL(1,𝔽8)—22no—
97#41×ΓL(1,𝔽8)C2×D712yes—
98C2×D4×ΓL(1,𝔽8)—22no—
99C2×D4×ΓL(1,𝔽8)—22no—
100C2×D4×ΓL(1,𝔽8)C2×D711yescharacteristic
101C2×(C22⋊D7)⋊C3—142no—
102C2×C4⋊AGL(1,𝔽7)—71no—
103C3×#1143—142no—
104C3×#1141—142no—
105C3×C4×Dic7—71no—
106C3×#1136—72no—
107#1143⋊C3—142no—
108#1141⋊C3—142no—
109C4×ΓL(1,𝔽8)⋊C4—71no—
110#1136⋊C3—72no—
111C2×C7×ΓL(1,𝔽8)—21no—
112C2×C7×ΓL(1,𝔽8)—42no—
113C2×C7×ΓL(1,𝔽8)—42no—
114C2×C7×ΓL(1,𝔽8)#12011yescharacteristic
115ΓL(1,𝔽8)⋊D7—81no—
116C7×H(5,𝔽2)—32no—
117C7×H(5,𝔽2)AGL(1,𝔽7)11yescharacteristic, complemented
118C7×H(5,𝔽2)C3×D711yescharacteristic, complemented
119#6512—72no—
120#6512—72no—
121#6512—214no—
122#6503—71no—
123#6503—71no—
124#6503—212no—
125#6411—72no—
126#6411—72no—
127#6411—214no—
128#6407—71no—
129#6407—71no—
130#6407—212no—
131C22×C72—22no—
132C22×C72—22no—
133C22×C72—42no—
134C22×C72—82no—
135C22×C72C2×C3×D411yescharacteristic
136C2×C7⋊D7—42no—
137C2×C7⋊D7—41no—
138C4×C72—22no—
139C4×C72C2×C3×D412yes—
140C7⋊Dic7—22no—
141C3×#453—491noHall
142C23×C3×C7—142no—
143C23×C3×C7—142no—
144C22×C3×D7—141no—
145C23×ΓL(1,𝔽8)—22no—
146C23×ΓL(1,𝔽8)C22⋊D712yes—
147C22×AGL(1,𝔽7)—141no—
148C3×C7×Q8—142no—
149C3×C7×D4—144no—
150C3×C7×D4—142no—
151C3×C7×D4—142no—
152C3×C7×D4—142no—
153C2×C3×C4×C7—142no—
154C2×C3×C4×C7—142no—
155C2×C3×C4×C7—141no—
156C3×C22⋊D7—284no—
157C2×C3×Dic7—141no—
158C3×D28—142no—
159Q8×ΓL(1,𝔽8)C22⋊D712yescomplemented
160D4×ΓL(1,𝔽8)—24no—
161D4×ΓL(1,𝔽8)—22no—
162D4×ΓL(1,𝔽8)—22no—
163D4×ΓL(1,𝔽8)C22⋊D712yescomplemented
164C2×C4×ΓL(1,𝔽8)—22no—
165C2×C4×ΓL(1,𝔽8)—22no—
166C2×C4×ΓL(1,𝔽8)C22×D711yescharacteristic
167(C22⋊D7)⋊C3—284no—
168C2×ΓL(1,𝔽8)⋊C4—141no—
169C4⋊AGL(1,𝔽7)—142no—
170C3×C7⋊C8—142no—
171ΓL(1,𝔽8)⋊C8—142no—
172C7×ΓL(1,𝔽8)#45311yescharacteristic, complemented, Hall
173C7×#41—22no—
174C7×#41—22no—
175C7×#41—34no—
176C7×#41—64no—
177C7×#41C2×AGL(1,𝔽7)12yes—
178C7×#41C2×C3×D712yes—
179C2×C7×D4—22no—
180C2×C7×D4—22no—
181C2×C7×D4—22no—
182C2×C7×D4—22no—
183C2×C7×D4—32no—
184C2×C7×D4—64no—
185C2×C7×D4—64no—
186C2×C7×D4C2×AGL(1,𝔽7)11yescharacteristic
187C2×C7×D4C2×C3×D711yescharacteristic
188C2×C22⋊D7—142no—
189C2×C22⋊D7—142no—
190C2×C22⋊D7—424no—
191C2×D28—71no—
192C2×D28—71no—
193C2×D28—212no—
194#1143—142no—
195#1143—142no—
196#1143—424no—
197#1141—142no—
198#1141—142no—
199#1141—424no—
200C4×Dic7—71no—
201C4×Dic7—71no—
202C4×Dic7—212no—
203#1136—72no—
204#1136—72no—
205#1136—214no—
206C2×C72—21no—
207C2×C72—42no—
208C2×C72—42no—
209C2×C72C3×#12011yescharacteristic
210C7⋊D7—81no—
211C3×H(5,𝔽2)—141no—
212C3×AGL(1,ℤ/8)—492no—
213C3×#127—491no—
214C3×#104—492no—
215C3×#100—491no—
216C22×C3×C7—142no—
217C22×C3×C7—142no—
218C22×C3×C7—141no—
219C22×C3×C7—562no—
220C22×C3×C7—562no—
221C2×C3×D7—281no—
222C2×C3×D7—282no—
223C22×ΓL(1,𝔽8)—22no—
224C22×ΓL(1,𝔽8)—22no—
225C22×ΓL(1,𝔽8)—42no—
226C22×ΓL(1,𝔽8)—82no—
227C22×ΓL(1,𝔽8)C2×C22⋊D711yescharacteristic
228C2×AGL(1,𝔽7)—282no—
229C2×AGL(1,𝔽7)—281no—
230C3×C4×C7—142no—
231C3×C4×C7—142no—
232C3×Dic7—142no—
233C4×ΓL(1,𝔽8)—22no—
234C4×ΓL(1,𝔽8)C2×C22⋊D712yes—
235ΓL(1,𝔽8)⋊C4—142no—
236#453—491noSylow
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—34no—
240C23×C7—64no—
241C23×C7(C22⋊D7)⋊C312yes—
242C23×C7C3×C22⋊D712yes—
243C22×D7—141no—
244C22×D7—141no—
245C22×D7—422no—
246C7×Q8—34no—
247C7×Q8(C22⋊D7)⋊C312yescomplemented
248C7×Q8C3×C22⋊D712yescomplemented
249C7×D4—24no—
250C7×D4—24no—
251C7×D4—22no—
252C7×D4—22no—
253C7×D4—22no—
254C7×D4—22no—
255C7×D4—34no—
256C7×D4—64no—
257C7×D4—64no—
258C7×D4—68no—
259C7×D4(C22⋊D7)⋊C312yescomplemented
260C7×D4C3×C22⋊D712yescomplemented
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—32no—
266C2×C4×C7—64no—
267C2×C4×C7—64no—
268C2×C4×C7C22×AGL(1,𝔽7)11yescharacteristic
269C2×C4×C7C22×C3×D711yescharacteristic
270C22⋊D7—284no—
271C22⋊D7—284no—
272C22⋊D7—848no—
273C2×Dic7—141no—
274C2×Dic7—141no—
275C2×Dic7—422no—
276D28—142no—
277D28—142no—
278D28—424no—
279C7⋊C8—142no—
280C7⋊C8—142no—
281C7⋊C8—424no—
282C72C3×#45311yescharacteristic, complemented, Sylow
283C3×#41—142no—
284C3×#41—142no—
285C2×C3×D4—142no—
286C2×C3×D4—142no—
287C2×C3×D4—141no—
288C2×C3×D4—491no—
289C2×C3×D4—982no—
290C3×SD16—982no—
291C3×D8—982no—
292C3×M16—492no—
293C3×C42—491no—
294C2×C3×C7—141no—
295C2×C3×C7—281no—
296C2×C3×C7—282no—
297C2×C3×C7—282no—
298C3×D7—561no—
299C2×ΓL(1,𝔽8)—21no—
300C2×ΓL(1,𝔽8)—42no—
301C2×ΓL(1,𝔽8)—42no—
302C2×ΓL(1,𝔽8)#651611yescharacteristic
303AGL(1,𝔽7)—561no—
304H(5,𝔽2)ΓL(1,𝔽8)⋊D711yescharacteristic, complemented
305AGL(1,ℤ/8)—492no—
306#127—491no—
307#104—492no—
308#100—491no—
309C22×C7—22no—
310C22×C7—22no—
311C22×C7—22no—
312C22×C7—22no—
313C22×C7—32no—
314C22×C7—42no—
315C22×C7—42no—
316C22×C7—64no—
317C22×C7—64no—
318C22×C7—82no—
319C22×C7—82no—
320C22×C7—124no—
321C22×C7—244no—
322C22×C7C2×(C22⋊D7)⋊C311yescharacteristic
323C22×C7C2×C3×C22⋊D711yescharacteristic
324C2×D7—282no—
325C2×D7—282no—
326C2×D7—281no—
327C2×D7—281no—
328C2×D7—842no—
329C2×D7—844no—
330C4×C7—22no—
331C4×C7—22no—
332C4×C7—34no—
333C4×C7—64no—
334C4×C7C2×(C22⋊D7)⋊C312yes—
335C4×C7C2×C3×C22⋊D712yes—
336Dic7—142no—
337Dic7—142no—
338Dic7—424no—
339C23×C3—142no—
340C23×C3—142no—
341C23×C3—981no—
342C3×Q8—142no—
343C3×D4—144no—
344C3×D4—142no—
345C3×D4—142no—
346C3×D4—142no—
347C3×D4—982no—
348C3×D4—1964no—
349C2×C3×C4—142no—
350C2×C3×C4—142no—
351C2×C3×C4—141no—
352C2×C3×C4—981no—
353C3×C8—982no—
354C3×C7—141no—
355ΓL(1,𝔽8)#9001311yescharacteristic, complemented
356#41—22no—
357#41C2×ΓL(1,𝔽8)⋊D712yes—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—491no—
361C2×D4—982no—
362C2×D4C2×ΓL(1,𝔽8)⋊D711yescharacteristic
363SD16—982no—
364D8—982no—
365M16—492no—
366C42—491no—
367C2×C7—21no—
368C2×C7—21no—
369C2×C7—32no—
370C2×C7—42no—
371C2×C7—42no—
372C2×C7—42no—
373C2×C7—42no—
374C2×C7—62no—
375C2×C7—124no—
376C2×C7—124no—
377C2×C7#1062078911yescharacteristic
378C2×C7C3×#651611yescharacteristic
379D7—561no—
380D7—561no—
381D7—1682no—
382C22×C3—142no—
383C22×C3—142no—
384C22×C3—141no—
385C22×C3—562no—
386C22×C3—562no—
387C22×C3—1961no—
388C22×C3—1962no—
389C3×C4—142no—
390C3×C4—142no—
391C3×C4—982no—
392C23—22no—
393C23—981no—
394C23#1192420212yes—
395Q8#1192420212yescomplemented
396D4—24no—
397D4—22no—
398D4—22no—
399D4—982no—
400D4—1964no—
401D4#1192420212yescomplemented
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—981no—
405C2×C4C22×ΓL(1,𝔽8)⋊D711yescharacteristic
406C8—982no—
407C7—32no—
408C7#90013⋊C311yescharacteristic, complemented
409C7C3×#9001311yescharacteristic, complemented
410C2×C3—141no—
411C2×C3—281no—
412C2×C3—282no—
413C2×C3—282no—
414C2×C3—3921no—
415C22—22no—
416C22—22no—
417C22—42no—
418C22—82no—
419C22—1962no—
420C22—1961no—
421C22C2×#1192420211yescharacteristic
422C4—22no—
423C4—982no—
424C4C2×#1192420212yes—
425C3—141noSylow
426C2—21no—
427C2—42no—
428C2—42no—
429C2—3921no—
430C2#7224604411yescharacteristic
subgroupG/H
centerC2#72246044
commutator, cosocleC2×C4×C72C23×C3
FrattiniC2×C4C22×ΓL(1,𝔽8)⋊D7
FittingC72×H(5,𝔽2)C2×C3
socleC2×C72C3×#120
hypercenterH(5,𝔽2)ΓL(1,𝔽8)⋊D7
hypocenterC72C3×#453
LCSG ▷ C2×C4×C72 ▷ C2×C72 ▷ C72
UCSC1 ◁ C2 ◁ C2×C4 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #72246276C2semidirect
2. #72246271C2semidirect
3. #72246214C3semidirect
4. C7×ΓL(1,𝔽8)×H(5,𝔽2)C2semidirect
5. #72245607C2semidirect
6. C4×ΓL(1,𝔽8)⋊Dic7C22semidirect
7. #72245208C2semidirect
12. #72245172C22semidirect
14. C2×C4⋊(ΓL(1,𝔽8)⋊D7)C22semidirect
18. C72×H(5,𝔽2)C2×C3semidirect
19. #13096589C2×C3semidirect
20. #13096573C2×C3semidirect
21. #13096155C2×C3semidirect
22. #13096152C2×C3semidirect
28. C7×Q8×ΓL(1,𝔽8)D4semidirect
32. C7×D4×ΓL(1,𝔽8)D4semidirect
46. C2×C4⋊(C7⋊D7)C22×C3semidirect
49. C4×C7⋊Dic7C22×C3semidirect
50. #11716066C22×C3semidirect
52. ΓL(1,𝔽8)×H(5,𝔽2)D7semidirect
77. C72×Q8C3×D4semidirect
81. C72×D4C3×D4semidirect
117. C7×H(5,𝔽2)AGL(1,𝔽7)semidirect
118. C7×H(5,𝔽2)C3×D7semidirect
159. Q8×ΓL(1,𝔽8)C22⋊D7semidirect
163. D4×ΓL(1,𝔽8)C22⋊D7semidirect
172. C7×ΓL(1,𝔽8)#453semidirect
247. C7×Q8(C22⋊D7)⋊C3semidirect
248. C7×Q8C3×C22⋊D7semidirect
259. C7×D4(C22⋊D7)⋊C3semidirect
260. C7×D4C3×C22⋊D7semidirect
282. C72C3×#453semidirect
304. H(5,𝔽2)ΓL(1,𝔽8)⋊D7semidirect
355. ΓL(1,𝔽8)#90013semidirect
395. Q8#11924202semidirect
401. D4#11924202semidirect
408. C7#90013⋊C3semidirect
409. C7C3×#90013semidirect

Maximal subgroups

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

subgroupindex
#722462762
#722462712
#722462143
C7×ΓL(1,𝔽8)×H(5,𝔽2)2
#722456072
#722452082
C3×#900137
#90013⋊C37