Finite Group Databasebeta

← #72276585

Subgroups — #72276585

Order 2688.

1988 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
2#90482⋊C3C211yesmaximal, characteristic, complemented
3C2×#6471⋊C3C212yesmaximal, complemented
4ΓL(1,𝔽8)×#407C211yesmaximal, characteristic, complemented
5#89903⋊C3C212yesmaximal, complemented
6#89504⋊C3C214yesmaximal, complemented
7#89503⋊C3C214yesmaximal, complemented
8#11736082C311yesmaximal, characteristic, complemented, Hall
9C24×AGL(1,𝔽7)—21no—
10C23×C4×ΓL(1,𝔽8)C2211yescharacteristic
11C2×#1157⋊C3C2211yescharacteristic, complemented
12C22×C4⋊AGL(1,𝔽7)—22no—
13C22×C4⋊AGL(1,𝔽7)—21no—
14C22×C4⋊AGL(1,𝔽7)—28no—
15C22×C4⋊AGL(1,𝔽7)C2214yes—
16C22×C4⋊AGL(1,𝔽7)C2212yes—
17C2×M16×ΓL(1,𝔽8)C2212yescomplemented
18#10620789—21no—
19#6493⋊C3C2212yescomplemented
20#6492⋊C3—21no—
21C4×(C22⋊D7)⋊C3—22no—
22#6489⋊C3C2211yescharacteristic, complemented
23#6487⋊C3C2212yescomplemented
24#6471⋊C3—28no—
25C2×D56⋊C3C2214yescomplemented
26C2×#1132⋊C3C2214yescomplemented
27ΓL(1,𝔽8)×C22⋊C8C2214yescomplemented
28#6395⋊C3C2218yescomplemented
29C23×D28C2×C311yescharacteristic, complemented
30#90482C2×C311yescharacteristic, complemented
31C2×#6471C2×C312yescomplemented
32C7×#407C2×C311yescharacteristic, complemented
33#89903C2×C312yescomplemented
34#89504C2×C314yescomplemented
35#89503C2×C314yescomplemented
36C3×#3003—71nomaximal, Hall
37C24×ΓL(1,𝔽8)D411yescharacteristic
38C23×AGL(1,𝔽7)—28no—
39C23×AGL(1,𝔽7)—24no—
40C23×AGL(1,𝔽7)—22no—
41C23×AGL(1,𝔽7)—48no—
42C22×C4×ΓL(1,𝔽8)—24no—
43C22×C4×ΓL(1,𝔽8)D412yes—
44C22×C4×ΓL(1,𝔽8)D411yescharacteristic
45C22×C4×ΓL(1,𝔽8)C2312yes—
46C22×C4×ΓL(1,𝔽8)C2311yescharacteristic
47C2×(C22⋊D7)⋊C3—21no—
48C2×(C22⋊D7)⋊C3—22no—
49#1157⋊C3—24no—
50C2×C4⋊AGL(1,𝔽7)—28no—
51C2×C4⋊AGL(1,𝔽7)—216no—
52C2×C4⋊AGL(1,𝔽7)—24no—
53C2×C4⋊AGL(1,𝔽7)—22no—
54C2×C4⋊AGL(1,𝔽7)—24no—
55C2×C4⋊AGL(1,𝔽7)—216no—
56C2×C4⋊AGL(1,𝔽7)D418yescomplemented
57C2×C4⋊AGL(1,𝔽7)C2314yes—
58C2×C4⋊AGL(1,𝔽7)C2311yescharacteristic, complemented
59C2×C4×AGL(1,𝔽7)—21no—
60C2×ΓL(1,𝔽8)⋊Q8C2311yescharacteristic, complemented
61M16×ΓL(1,𝔽8)—24no—
62C2×C8×ΓL(1,𝔽8)C2314yescomplemented
63(C22⋊Dic7)⋊C3—22no—
64(C22⋊Dic7)⋊C3—21no—
65(C2×AGL(1,𝔽7))⋊C4—22no—
66C4⋊(ΓL(1,𝔽8)⋊C4)C2312yescomplemented
67(ΓL(1,𝔽8)⋊C4)⋊C4—22no—
68C4×ΓL(1,𝔽8)⋊C4—21no—
69D56⋊C3—28no—
70#1132⋊C3—28no—
71C24×D7—21no—
72C23×C4×C7C22×C311yescharacteristic
73C2×#1157C22×C311yescharacteristic, complemented
74C22×D28—22no—
75C22×D28—21no—
76C22×D28—28no—
77C22×D28C22×C314yes—
78C22×D28C22×C312yes—
79C2×C7×M16C22×C312yescomplemented
80#6516—21no—
81#6493C22×C312yescomplemented
82#6492—21no—
83C4×C22⋊D7—22no—
84#6489C22×C311yescharacteristic, complemented
85#6487C22×C312yescomplemented
86#6471—28no—
87C2×D56C22×C314yescomplemented
88C2×#1132C22×C314yescomplemented
89C7×C22⋊C8C22×C314yescomplemented
90#6395C22×C318yescomplemented
91C23×C3×D4—71no—
92C2×C3×AGL(1,ℤ/8)—72no—
93C3×#525—71no—
94C3×#450—74no—
95C3×#447—74no—
96C3×#418—72no—
97C3×#407—71no—
98C23×ΓL(1,𝔽8)—22no—
99C23×ΓL(1,𝔽8)—24no—
100C23×ΓL(1,𝔽8)C2×D411yescharacteristic
101C23×ΓL(1,𝔽8)C2×D412yes—
102C22×AGL(1,𝔽7)—216no—
103C22×AGL(1,𝔽7)—28no—
104C22×AGL(1,𝔽7)—24no—
105C22×AGL(1,𝔽7)—21no—
106C22×AGL(1,𝔽7)—432no—
107C22×AGL(1,𝔽7)—48no—
108C22×AGL(1,𝔽7)—48no—
109C22×AGL(1,𝔽7)—48no—
110C2×C4×ΓL(1,𝔽8)—22no—
111C2×C4×ΓL(1,𝔽8)—24no—
112C2×C4×ΓL(1,𝔽8)—24no—
113C2×C4×ΓL(1,𝔽8)C2×D412yes—
114C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
115C2×C4×ΓL(1,𝔽8)C2×D414yes—
116C2×C4×ΓL(1,𝔽8)C2411yescharacteristic
117(C22⋊D7)⋊C3—42no—
118(C22⋊D7)⋊C3—44no—
119C2×ΓL(1,𝔽8)⋊C4—21no—
120C2×ΓL(1,𝔽8)⋊C4—22no—
121C4⋊AGL(1,𝔽7)—216no—
122C4⋊AGL(1,𝔽7)—28no—
123C4⋊AGL(1,𝔽7)—24no—
124C4⋊AGL(1,𝔽7)—22no—
125C4⋊AGL(1,𝔽7)C2×D418yes—
126C4×AGL(1,𝔽7)—42no—
127ΓL(1,𝔽8)⋊Q8—22no—
128C8×ΓL(1,𝔽8)—24no—
129#3003—71noSylow
130C24×C7C3×D411yescharacteristic
131C23×D7—28no—
132C23×D7—24no—
133C23×D7—22no—
134C23×D7—48no—
135C22×C4×C7—24no—
136C22×C4×C7C3×D412yes—
137C22×C4×C7C3×D411yescharacteristic
138C22×C4×C7C23×C312yes—
139C22×C4×C7C23×C311yescharacteristic
140C2×C22⋊D7—22no—
141C2×C22⋊D7—21no—
142#1157—24no—
143C2×D28—28no—
144C2×D28—216no—
145C2×D28—24no—
146C2×D28—22no—
147C2×D28—24no—
148C2×D28—216no—
149C2×D28C3×D418yescomplemented
150C2×D28C23×C314yes—
151C2×D28C23×C311yescharacteristic, complemented
152C2×C4×D7—21no—
153C2×Dic14C23×C311yescharacteristic, complemented
154C7×M16—24no—
155C2×C7×C8C23×C314yescomplemented
156C22⋊Dic7—22no—
157C22⋊Dic7—21no—
158(C2×D7)⋊C4—22no—
159C4⋊Dic7C23×C312yescomplemented
160Dic7⋊C4—22no—
161C4×Dic7—21no—
162D56—28no—
163#1132—28no—
164C25×C3—141no—
165C2×C3×#41—71no—
166C22×C3×D4—72no—
167C22×C3×D4—74no—
168C22×C3×D4—142no—
169C22×C3×D4—141no—
170C22×C3×D4—148no—
171C23×C3×C4—71no—
172C3×AGL(1,ℤ/8)—148no—
173C2×C3×SD16—74no—
174C2×C3×D8—74no—
175C2×C3×M16—72no—
176C3×#123—141no—
177C3×C22⋊Q8—72no—
178C3×#121—72no—
179C3×#120—141no—
180C3×C4×D4—142no—
181C3×#117—71no—
182C3×#102—78no—
183C3×C22⋊C8—74no—
184C22×ΓL(1,𝔽8)—28no—
185C22×ΓL(1,𝔽8)—22no—
186C22×ΓL(1,𝔽8)—22no—
187C22×ΓL(1,𝔽8)—22no—
188C22×ΓL(1,𝔽8)#12012yes—
189C22×ΓL(1,𝔽8)AGL(1,ℤ/8)14yescomplemented
190C22×ΓL(1,𝔽8)C22×D411yescharacteristic
191C2×AGL(1,𝔽7)—28no—
192C2×AGL(1,𝔽7)—416no—
193C2×AGL(1,𝔽7)—432no—
194C2×AGL(1,𝔽7)—48no—
195C2×AGL(1,𝔽7)—41no—
196C2×AGL(1,𝔽7)—42no—
197C4×ΓL(1,𝔽8)—22no—
198C4×ΓL(1,𝔽8)#12012yes—
199C4×ΓL(1,𝔽8)C22×D412yes—
200ΓL(1,𝔽8)⋊C4—41no—
201ΓL(1,𝔽8)⋊C4—42no—
202C23×D4—71no—
203C2×AGL(1,ℤ/8)—72no—
204#525—71no—
205#450—74no—
206#447—74no—
207#418—72no—
208#407AGL(1,𝔽7)11yescharacteristic, complemented
209C23×C7—22no—
210C23×C7—24no—
211C23×C7C2×C3×D411yescharacteristic
212C23×C7C2×C3×D412yes—
213C22×D7—216no—
214C22×D7—28no—
215C22×D7—24no—
216C22×D7—21no—
217C22×D7—432no—
218C22×D7—48no—
219C22×D7—48no—
220C22×D7—48no—
221C2×C4×C7—22no—
222C2×C4×C7—24no—
223C2×C4×C7—24no—
224C2×C4×C7C2×C3×D412yes—
225C2×C4×C7C2×C3×D411yescharacteristic
226C2×C4×C7C2×C3×D414yes—
227C2×C4×C7C24×C311yescharacteristic
228C22⋊D7—42no—
229C22⋊D7—44no—
230C2×Dic7—22no—
231C2×Dic7—21no—
232D28—216no—
233D28—28no—
234D28—24no—
235D28—22no—
236D28C2×C3×D418yes—
237C4×D7—42no—
238Dic14—22no—
239C7×C8—24no—
240C24×C3—71no—
241C24×C3—142no—
242C24×C3—144no—
243C24×C3—148no—
244C24×C3—288no—
245C3×#41—144no—
246C2×C3×Q8—71no—
247C2×C3×D4—74no—
248C2×C3×D4—78no—
249C2×C3×D4—71no—
250C2×C3×D4—142no—
251C2×C3×D4—144no—
252C2×C3×D4—144no—
253C2×C3×D4—1416no—
254C2×C3×D4—1416no—
255C2×C3×D4—148no—
256C2×C3×D4—142no—
257C2×C3×D4—141no—
258C22×C3×C4—72no—
259C22×C3×C4—71no—
260C22×C3×C4—71no—
261C22×C3×C4—72no—
262C22×C3×C4—144no—
263C22×C3×C4—141no—
264C3×SD16—148no—
265C3×D8—148no—
266C3×M16—144no—
267C2×C3×C8—74no—
268C3×C4⋊C4—72no—
269C3×C4⋊C4—142no—
270C3×C22⋊C4—142no—
271C3×C22⋊C4—141no—
272C3×C22⋊C4—142no—
273C3×C42—141no—
274C2×ΓL(1,𝔽8)—22no—
275C2×ΓL(1,𝔽8)—24no—
276C2×ΓL(1,𝔽8)C2×#12011yescharacteristic
277C2×ΓL(1,𝔽8)C2×AGL(1,ℤ/8)12yes—
278AGL(1,𝔽7)—48no—
279AGL(1,𝔽7)—81no—
280C25—141no—
281C2×#41—71no—
282C22×D4—72no—
283C22×D4—74no—
284C22×D4—148no—
285C22×D4—142no—
286C22×D4—141no—
287C23×C4C2×AGL(1,𝔽7)11yescharacteristic
288AGL(1,ℤ/8)—148no—
289C2×SD16—74no—
290C2×D8—74no—
291C2×M16C2×AGL(1,𝔽7)12yescomplemented
292#123—141no—
293C22⋊Q8—72no—
294#121—72no—
295#120—141no—
296C4×D4—142no—
297#117—71no—
298#102—78no—
299C22⋊C8C2×AGL(1,𝔽7)14yescomplemented
300C22×C7—28no—
301C22×C7—22no—
302C22×C7—22no—
303C22×C7—22no—
304C22×C7C3×#12012yes—
305C22×C7C3×AGL(1,ℤ/8)14yescomplemented
306C22×C7C22×C3×D411yescharacteristic
307C2×D7—28no—
308C2×D7—416no—
309C2×D7—432no—
310C2×D7—48no—
311C2×D7—42no—
312C2×D7—41no—
313C4×C7—22no—
314C4×C7C3×#12012yes—
315C4×C7C22×C3×D412yes—
316Dic7—42no—
317Dic7—41no—
318C23×C3—71no—
319C23×C3—72no—
320C23×C3—142no—
321C23×C3—144no—
322C23×C3—148no—
323C23×C3—144no—
324C23×C3—141no—
325C23×C3—1416no—
326C23×C3—288no—
327C23×C3—288no—
328C23×C3—288no—
329C23×C3—2832no—
330C3×Q8—142no—
331C3×D4—78no—
332C3×D4—148no—
333C3×D4—144no—
334C3×D4—142no—
335C3×D4—1416no—
336C3×D4—284no—
337C3×D4—282no—
338C2×C3×C4—71no—
339C2×C3×C4—72no—
340C2×C3×C4—74no—
341C2×C3×C4—71no—
342C2×C3×C4—144no—
343C2×C3×C4—144no—
344C2×C3×C4—142no—
345C2×C3×C4—142no—
346C2×C3×C4—141no—
347C2×C3×C4—282no—
348C3×C8—144no—
349ΓL(1,𝔽8)#300311yescharacteristic, complemented, Hall
350C24—148no—
351C24—144no—
352C24—142no—
353C24—288no—
354C24C4⋊AGL(1,𝔽7)11yescharacteristic
355#41—144no—
356C2×Q8—71no—
357C2×D4—78no—
358C2×D4—74no—
359C2×D4—71no—
360C2×D4—1416no—
361C2×D4—148no—
362C2×D4—1416no—
363C2×D4—142no—
364C2×D4—141no—
365C2×D4—144no—
366C2×D4—142no—
367C2×D4—144no—
368C22×C4—24no—
369C22×C4—141no—
370C22×C4C4⋊AGL(1,𝔽7)12yes—
371C22×C4C4⋊AGL(1,𝔽7)11yescharacteristic
372C22×C4C22×AGL(1,𝔽7)12yes—
373C22×C4C22×AGL(1,𝔽7)11yescharacteristic
374SD16—148no—
375D8—148no—
376M16—24no—
377C2×C8C22×AGL(1,𝔽7)14yescomplemented
378C4⋊C4—72no—
379C4⋊C4—142no—
380C22⋊C4—141no—
381C22⋊C4—142no—
382C22⋊C4—142no—
383C42—141no—
384C2×C7—22no—
385C2×C7—24no—
386C2×C7C2×C3×#12011yescharacteristic
387C2×C7C2×C3×AGL(1,ℤ/8)12yes—
388D7—48no—
389D7—81no—
390C22×C3—72no—
391C22×C3—71no—
392C22×C3—74no—
393C22×C3—142no—
394C22×C3—142no—
395C22×C3—142no—
396C22×C3—148no—
397C22×C3—148no—
398C22×C3—288no—
399C22×C3—281no—
400C22×C3—282no—
401C22×C3—2816no—
402C22×C3—2832no—
403C3×C4—72no—
404C3×C4—72no—
405C3×C4—142no—
406C3×C4—281no—
407C3×C4—282no—
408C23—22no—
409C23—24no—
410C23—1416no—
411C23—148no—
412C23—144no—
413C23—141no—
414C23—2832no—
415C23—288no—
416C23—288no—
417C23—288no—
418C23C2×C4⋊AGL(1,𝔽7)11yescharacteristic
419C23C2×C4⋊AGL(1,𝔽7)12yes—
420Q8—142no—
421D4—78no—
422D4—1416no—
423D4—148no—
424D4—144no—
425D4—142no—
426D4—282no—
427D4—284no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—142no—
432C2×C4—141no—
433C2×C4—282no—
434C2×C4C2×C4⋊AGL(1,𝔽7)12yes—
435C2×C4C2×C4⋊AGL(1,𝔽7)11yescharacteristic
436C2×C4D4×AGL(1,𝔽7)14yes—
437C2×C4C23×AGL(1,𝔽7)11yescharacteristic
438C8—24no—
439C7C3×#300311yescharacteristic, complemented, Sylow
440C2×C3—71no—
441C2×C3—72no—
442C2×C3—142no—
443C2×C3—144no—
444C2×C3—288no—
445C2×C3—561no—
446C22—28no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—148no—
451C22—2816no—
452C22—2832no—
453C22—288no—
454C22—281no—
455C22—282no—
456C22#6445⋊C312yes—
457C22#6471⋊C314yescomplemented
458C22C22×C4⋊AGL(1,𝔽7)11yescharacteristic
459C4—22no—
460C4—281no—
461C4—282no—
462C4#6445⋊C312yes—
463C4C2×D4×AGL(1,𝔽7)12yes—
464C3—71noSylow
465C2—22no—
466C2—24no—
467C2—288no—
468C2—561no—
469C2C2×#6445⋊C311yescharacteristic
470C2C2×#6471⋊C312yes—
subgroupG/H
centerC22C22×C4⋊AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C24×C3
FrattiniC2×C4C23×AGL(1,𝔽7)
FittingC7×#407C2×C3
socleC22×C7C22×C3×D4
hypercenter#407AGL(1,𝔽7)
hypocenterC7C3×#3003
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×C4⋊AGL(1,𝔽7)C2semidirect
2. #90482⋊C3C2semidirect
3. C2×#6471⋊C3C2semidirect
4. ΓL(1,𝔽8)×#407C2semidirect
5. #89903⋊C3C2semidirect
6. #89504⋊C3C2semidirect
7. #89503⋊C3C2semidirect
8. #11736082C3semidirect
11. C2×#1157⋊C3C22semidirect
17. C2×M16×ΓL(1,𝔽8)C22semidirect
19. #6493⋊C3C22semidirect
22. #6489⋊C3C22semidirect
23. #6487⋊C3C22semidirect
25. C2×D56⋊C3C22semidirect
26. C2×#1132⋊C3C22semidirect
27. ΓL(1,𝔽8)×C22⋊C8C22semidirect
28. #6395⋊C3C22semidirect
29. C23×D28C2×C3semidirect
30. #90482C2×C3semidirect
31. C2×#6471C2×C3semidirect
32. C7×#407C2×C3semidirect
33. #89903C2×C3semidirect
34. #89504C2×C3semidirect
35. #89503C2×C3semidirect
56. C2×C4⋊AGL(1,𝔽7)D4semidirect
58. C2×C4⋊AGL(1,𝔽7)C23semidirect
60. C2×ΓL(1,𝔽8)⋊Q8C23semidirect
62. C2×C8×ΓL(1,𝔽8)C23semidirect
66. C4⋊(ΓL(1,𝔽8)⋊C4)C23semidirect
73. C2×#1157C22×C3semidirect
79. C2×C7×M16C22×C3semidirect
81. #6493C22×C3semidirect
84. #6489C22×C3semidirect
85. #6487C22×C3semidirect
87. C2×D56C22×C3semidirect
88. C2×#1132C22×C3semidirect
89. C7×C22⋊C8C22×C3semidirect
90. #6395C22×C3semidirect
149. C2×D28C3×D4semidirect
151. C2×D28C23×C3semidirect
153. C2×Dic14C23×C3semidirect
155. C2×C7×C8C23×C3semidirect
159. C4⋊Dic7C23×C3semidirect
189. C22×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
208. #407AGL(1,𝔽7)semidirect
291. C2×M16C2×AGL(1,𝔽7)semidirect
299. C22⋊C8C2×AGL(1,𝔽7)semidirect
305. C22×C7C3×AGL(1,ℤ/8)semidirect
349. ΓL(1,𝔽8)#3003semidirect
377. C2×C8C22×AGL(1,𝔽7)semidirect
439. C7C3×#3003semidirect
457. C22#6471⋊C3semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊AGL(1,𝔽7)2
#90482⋊C32
C2×#6471⋊C32
ΓL(1,𝔽8)×#4072
#89903⋊C32
#89504⋊C32
#89503⋊C32
#117360823
C3×#30037