Finite Group Databasebeta

← #72276644

Subgroups — #72276644

Order 2688.

1156 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 149 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
1ΓL(1,𝔽8)×#518C211yesmaximal, characteristic, complemented
2C2×#6500⋊C3C211yesmaximal, characteristic
3C2×#6408⋊C3C211yesmaximal, characteristic, complemented
4#89727⋊C3C212yesmaximal, complemented
5#89321⋊C3C212yesmaximal, complemented
6#11729408C311yesmaximal, characteristic, complemented, Hall
7C24×AGL(1,𝔽7)C2211yescharacteristic
8C22×D4×ΓL(1,𝔽8)C2211yescharacteristic
9C22×(C22⋊D7)⋊C3—21no—
10C4×D4×ΓL(1,𝔽8)—24no—
11ΓL(1,𝔽8)×#117—21no—
12C2×C22⋊C4×ΓL(1,𝔽8)—21no—
13C2×C22⋊C4×ΓL(1,𝔽8)C2212yes—
14C2×(C22⋊Dic7)⋊C3C2212yes—
15C2×(C22⋊Dic7)⋊C3C2211yescharacteristic
16#6500⋊C3—22no—
17#6500⋊C3C414yescomplemented
18C2×(C2×AGL(1,𝔽7))⋊C4—21no—
19C2×(C2×AGL(1,𝔽7))⋊C4—24no—
20#6408⋊C3—22no—
21C7×#518C2×C311yescharacteristic, complemented
22C2×#6500C2×C311yescharacteristic
23C2×#6408C2×C311yescharacteristic, complemented
24#89727C2×C312yescomplemented
25#89321C2×C312yescomplemented
26C3×#1889—71nomaximal, Hall
27C24×ΓL(1,𝔽8)D411yescharacteristic
28C24×ΓL(1,𝔽8)C2311yescharacteristic
29C23×AGL(1,𝔽7)—21no—
30C23×AGL(1,𝔽7)—22no—
31C23×AGL(1,𝔽7)—21no—
32C23×AGL(1,𝔽7)—22no—
33C23×AGL(1,𝔽7)—44no—
34C23×AGL(1,𝔽7)C2×C412yescomplemented
35C2×D4×ΓL(1,𝔽8)—22no—
36C2×D4×ΓL(1,𝔽8)—22no—
37C2×D4×ΓL(1,𝔽8)C2×C412yes—
38C2×D4×ΓL(1,𝔽8)D412yes—
39C22×C4×ΓL(1,𝔽8)—22no—
40C22×C4×ΓL(1,𝔽8)D411yescharacteristic
41C22×C4×ΓL(1,𝔽8)D414yescomplemented
42C2×(C22⋊D7)⋊C3—24no—
43C2×(C22⋊D7)⋊C3—44no—
44C22×ΓL(1,𝔽8)⋊C4—21no—
45C22×ΓL(1,𝔽8)⋊C4—22no—
46C4⋊C4×ΓL(1,𝔽8)—22no—
47C22⋊C4×ΓL(1,𝔽8)—22no—
48C22⋊C4×ΓL(1,𝔽8)—22no—
49C22⋊C4×ΓL(1,𝔽8)—22no—
50C42×ΓL(1,𝔽8)—22no—
51(C22⋊Dic7)⋊C3—21no—
52(C22⋊Dic7)⋊C3—22no—
53(C22⋊Dic7)⋊C3—22no—
54(C22⋊Dic7)⋊C3C2×C412yescomplemented
55(C2×AGL(1,𝔽7))⋊C4—24no—
56(C2×AGL(1,𝔽7))⋊C4—48no—
57C24×D7C22×C311yescharacteristic
58C22×C7×D4C22×C311yescharacteristic
59C22×C22⋊D7—21no—
60C4×C7×D4—24no—
61C7×#117—21no—
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4C22×C312yes—
64C2×C22⋊Dic7C22×C312yes—
65C2×C22⋊Dic7C22×C311yescharacteristic
66#6500—22no—
67#6500C3×C414yescomplemented
68C2×(C2×D7)⋊C4—21no—
69C2×(C2×D7)⋊C4—24no—
70#6408—22no—
71C2×C3×#120—71no—
72C3×#518—71no—
73C2×C3×#99—71no—
74C3×#379—72no—
75C3×#342—72no—
76C23×ΓL(1,𝔽8)—21no—
77C23×ΓL(1,𝔽8)—21no—
78C23×ΓL(1,𝔽8)—42no—
79C23×ΓL(1,𝔽8)—42no—
80C23×ΓL(1,𝔽8)C22⋊C412yescomplemented
81C23×ΓL(1,𝔽8)C22×C411yescharacteristic
82C23×ΓL(1,𝔽8)C2×D411yescharacteristic
83C23×ΓL(1,𝔽8)C2×D412yes—
84C23×ΓL(1,𝔽8)C2×D411yescharacteristic
85C23×ΓL(1,𝔽8)#4112yes—
86C22×AGL(1,𝔽7)—24no—
87C22×AGL(1,𝔽7)—28no—
88C22×AGL(1,𝔽7)—22no—
89C22×AGL(1,𝔽7)—24no—
90C22×AGL(1,𝔽7)—44no—
91C22×AGL(1,𝔽7)—44no—
92C22×AGL(1,𝔽7)—48no—
93C22×AGL(1,𝔽7)—44no—
94C22×AGL(1,𝔽7)—42no—
95C22×AGL(1,𝔽7)—41no—
96C22×AGL(1,𝔽7)—42no—
97C22×AGL(1,𝔽7)—41no—
98D4×ΓL(1,𝔽8)—24no—
99D4×ΓL(1,𝔽8)—24no—
100C2×C4×ΓL(1,𝔽8)—21no—
101C2×C4×ΓL(1,𝔽8)—21no—
102C2×C4×ΓL(1,𝔽8)—24no—
103C2×C4×ΓL(1,𝔽8)—44no—
104C2×C4×ΓL(1,𝔽8)—82no—
105C2×C4×ΓL(1,𝔽8)C22⋊C412yescomplemented
106(C22⋊D7)⋊C3—48no—
107C2×ΓL(1,𝔽8)⋊C4—22no—
108C2×ΓL(1,𝔽8)⋊C4—42no—
109C2×ΓL(1,𝔽8)⋊C4—44no—
110C2×ΓL(1,𝔽8)⋊C4—42no—
111#1889—71noSylow
112C24×C7C3×D411yescharacteristic
113C24×C7C23×C311yescharacteristic
114C23×D7—22no—
115C23×D7—21no—
116C23×D7—21no—
117C23×D7—22no—
118C23×D7—44no—
119C23×D7C2×C3×C412yescomplemented
120C2×C7×D4—22no—
121C2×C7×D4—22no—
122C2×C7×D4C2×C3×C412yes—
123C2×C7×D4C3×D412yes—
124C22×C4×C7—22no—
125C22×C4×C7C3×D411yescharacteristic
126C22×C4×C7C3×D414yescomplemented
127C2×C22⋊D7—24no—
128C2×C22⋊D7—44no—
129C22×Dic7—21no—
130C22×Dic7—22no—
131C7×C4⋊C4—22no—
132C7×C22⋊C4—22no—
133C7×C22⋊C4—22no—
134C7×C22⋊C4—22no—
135C42×C7—22no—
136C22⋊Dic7—21no—
137C22⋊Dic7—22no—
138C22⋊Dic7—22no—
139C22⋊Dic7C2×C3×C412yescomplemented
140(C2×D7)⋊C4—24no—
141(C2×D7)⋊C4—48no—
142C25×C3—71no—
143C22×C3×D4—71no—
144C22×C3×D4—141no—
145C3×#120—74no—
146C3×#120—142no—
147C3×C4×D4—144no—
148C3×#117—141no—
149C2×C3×C22⋊C4—71no—
150C2×C3×C22⋊C4—72no—
151C2×C3×C22⋊C4—72no—
152C2×C3×C22⋊C4—141no—
153C2×C3×C22⋊C4—141no—
154C2×C3×C22⋊C4—144no—
155C3×#99—142no—
156C22×ΓL(1,𝔽8)—24no—
157C22×ΓL(1,𝔽8)—22no—
158C22×ΓL(1,𝔽8)—22no—
159C22×ΓL(1,𝔽8)—41no—
160C22×ΓL(1,𝔽8)—41no—
161C22×ΓL(1,𝔽8)—42no—
162C22×ΓL(1,𝔽8)—42no—
163C22×ΓL(1,𝔽8)—42no—
164C22×ΓL(1,𝔽8)—81no—
165C22×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
166C22×ΓL(1,𝔽8)C4×D412yes—
167C22×ΓL(1,𝔽8)#12011yescharacteristic
168C22×ΓL(1,𝔽8)#12112yes—
169C22×ΓL(1,𝔽8)#12311yescharacteristic
170C2×AGL(1,𝔽7)—24no—
171C2×AGL(1,𝔽7)—28no—
172C2×AGL(1,𝔽7)—42no—
173C2×AGL(1,𝔽7)—48no—
174C2×AGL(1,𝔽7)—44no—
175C2×AGL(1,𝔽7)—44no—
176C2×AGL(1,𝔽7)—42no—
177C2×AGL(1,𝔽7)—44no—
178C4×ΓL(1,𝔽8)—22no—
179C4×ΓL(1,𝔽8)—44no—
180ΓL(1,𝔽8)⋊C4—42no—
181ΓL(1,𝔽8)⋊C4—82no—
182C2×#120—71no—
183#518AGL(1,𝔽7)11yescharacteristic, complemented
184C2×#99—71no—
185#379—72no—
186#342—72no—
187C23×C7—21no—
188C23×C7—21no—
189C23×C7—42no—
190C23×C7—42no—
191C23×C7C3×C22⋊C412yescomplemented
192C23×C7C22×C3×C411yescharacteristic
193C23×C7C2×C3×D411yescharacteristic
194C23×C7C2×C3×D411yescharacteristic
195C23×C7C2×C3×D412yes—
196C23×C7C3×#4112yes—
197C22×D7—28no—
198C22×D7—24no—
199C22×D7—22no—
200C22×D7—24no—
201C22×D7—44no—
202C22×D7—44no—
203C22×D7—48no—
204C22×D7—44no—
205C22×D7—42no—
206C22×D7—41no—
207C22×D7—42no—
208C22×D7—41no—
209C7×D4—24no—
210C7×D4—24no—
211C2×C4×C7—21no—
212C2×C4×C7—21no—
213C2×C4×C7—24no—
214C2×C4×C7—44no—
215C2×C4×C7—82no—
216C2×C4×C7C3×C22⋊C412yescomplemented
217C22⋊D7—48no—
218C2×Dic7—22no—
219C2×Dic7—42no—
220C2×Dic7—44no—
221C2×Dic7—42no—
222C24×C3—72no—
223C24×C3—71no—
224C24×C3—71no—
225C24×C3—142no—
226C24×C3—141no—
227C24×C3—142no—
228C24×C3—141no—
229C24×C3—284no—
230C2×C3×D4—72no—
231C2×C3×D4—72no—
232C2×C3×D4—142no—
233C2×C3×D4—142no—
234C2×C3×D4—144no—
235C2×C3×D4—284no—
236C22×C3×C4—71no—
237C22×C3×C4—74no—
238C22×C3×C4—142no—
239C22×C3×C4—141no—
240C22×C3×C4—142no—
241C3×C4⋊C4—142no—
242C3×C22⋊C4—72no—
243C3×C22⋊C4—142no—
244C3×C22⋊C4—142no—
245C3×C22⋊C4—142no—
246C3×C22⋊C4—144no—
247C3×C22⋊C4—141no—
248C3×C22⋊C4—142no—
249C3×C22⋊C4—142no—
250C3×C22⋊C4—288no—
251C3×C42—142no—
252C2×ΓL(1,𝔽8)—21no—
253C2×ΓL(1,𝔽8)—22no—
254C2×ΓL(1,𝔽8)—22no—
255C2×ΓL(1,𝔽8)—21no—
256C2×ΓL(1,𝔽8)—42no—
257C2×ΓL(1,𝔽8)#38611yescharacteristic
258C2×ΓL(1,𝔽8)B(4,𝔽2)12yes—
259AGL(1,𝔽7)—44no—
260C25—71no—
261C22×D4—141no—
262C22×D4C2×AGL(1,𝔽7)11yescharacteristic
263#120—74no—
264#120—142no—
265C4×D4—24no—
266#117—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—71no—
269C2×C22⋊C4—72no—
270C2×C22⋊C4—141no—
271C2×C22⋊C4—144no—
272C2×C22⋊C4C2×AGL(1,𝔽7)12yes—
273#99—142no—
274C22×C7—24no—
275C22×C7—22no—
276C22×C7—22no—
277C22×C7—41no—
278C22×C7—41no—
279C22×C7—42no—
280C22×C7—42no—
281C22×C7—42no—
282C22×C7—81no—
283C22×C7C2×C3×C22⋊C411yescharacteristic
284C22×C7C3×C4×D412yes—
285C22×C7C3×#12011yescharacteristic
286C22×C7C3×#12112yes—
287C22×C7C3×#12311yescharacteristic
288C2×D7—24no—
289C2×D7—28no—
290C2×D7—42no—
291C2×D7—48no—
292C2×D7—44no—
293C2×D7—44no—
294C2×D7—42no—
295C2×D7—44no—
296C4×C7—22no—
297C4×C7—44no—
298Dic7—42no—
299Dic7—82no—
300C23×C3—71no—
301C23×C3—71no—
302C23×C3—72no—
303C23×C3—71no—
304C23×C3—72no—
305C23×C3—72no—
306C23×C3—141no—
307C23×C3—141no—
308C23×C3—142no—
309C23×C3—144no—
310C23×C3—148no—
311C23×C3—144no—
312C23×C3—282no—
313C23×C3—282no—
314C23×C3—281no—
315C23×C3—282no—
316C23×C3—281no—
317C23×C3—284no—
318C23×C3—284no—
319C23×C3—282no—
320C23×C3—284no—
321C23×C3—288no—
322C3×D4—144no—
323C3×D4—144no—
324C3×D4—288no—
325C2×C3×C4—72no—
326C2×C3×C4—141no—
327C2×C3×C4—141no—
328C2×C3×C4—144no—
329C2×C3×C4—142no—
330C2×C3×C4—284no—
331C2×C3×C4—282no—
332C2×C3×C4—282no—
333C2×C3×C4—284no—
334C2×C3×C4—562no—
335ΓL(1,𝔽8)#188911yescharacteristic, complemented, Hall
336C24—72no—
337C24—142no—
338C24—141no—
339C24—141no—
340C24—142no—
341C24—284no—
342C24(C22⋊D7)⋊C311yescharacteristic
343C24C22×AGL(1,𝔽7)11yescharacteristic
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—144no—
347C2×D4—284no—
348C2×D4C4×AGL(1,𝔽7)12yes—
349C2×D4C4⋊AGL(1,𝔽7)12yes—
350C22×C4—22no—
351C22×C4—141no—
352C22×C4—142no—
353C22×C4(C22⋊D7)⋊C311yescharacteristic
354C22×C4(C22⋊D7)⋊C314yescomplemented
355C4⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—72no—
360C22⋊C4—141no—
361C22⋊C4—144no—
362C22⋊C4—142no—
363C22⋊C4—142no—
364C22⋊C4—288no—
365C42—22no—
366C2×C7—21no—
367C2×C7—22no—
368C2×C7—22no—
369C2×C7—21no—
370C2×C7—42no—
371C2×C7C3×#38611yescharacteristic
372C2×C7C3×B(4,𝔽2)12yes—
373D7—44no—
374C22×C3—71no—
375C22×C3—72no—
376C22×C3—71no—
377C22×C3—72no—
378C22×C3—71no—
379C22×C3—144no—
380C22×C3—142no—
381C22×C3—142no—
382C22×C3—144no—
383C22×C3—148no—
384C22×C3—282no—
385C22×C3—281no—
386C22×C3—281no—
387C22×C3—282no—
388C22×C3—282no—
389C22×C3—282no—
390C22×C3—284no—
391C22×C3—282no—
392C22×C3—288no—
393C22×C3—284no—
394C22×C3—284no—
395C22×C3—561no—
396C3×C4—142no—
397C3×C4—284no—
398C3×C4—282no—
399C3×C4—562no—
400C23—21no—
401C23—21no—
402C23—42no—
403C23—42no—
404C23—148no—
405C23—144no—
406C23—142no—
407C23—144no—
408C23—284no—
409C23—282no—
410C23—284no—
411C23—284no—
412C23—288no—
413C23—281no—
414C23—282no—
415C23—281no—
416C23(C2×AGL(1,𝔽7))⋊C412yescomplemented
417C23C2×C4×AGL(1,𝔽7)11yescharacteristic
418C23C2×C4⋊AGL(1,𝔽7)11yescharacteristic
419C23#1157⋊C312yes—
420C23C2×(C22⋊D7)⋊C311yescharacteristic
421C23C2×(C22⋊D7)⋊C312yes—
422D4—24no—
423D4—24no—
424D4—288no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—44no—
429C2×C4—82no—
430C2×C4—142no—
431C2×C4—282no—
432C2×C4—284no—
433C2×C4—282no—
434C2×C4(C2×AGL(1,𝔽7))⋊C412yescomplemented
435C7C3×#188911yescharacteristic, complemented, Sylow
436C2×C3—71no—
437C2×C3—72no—
438C2×C3—141no—
439C2×C3—142no—
440C2×C3—142no—
441C2×C3—141no—
442C2×C3—282no—
443C2×C3—284no—
444C22—24no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—81no—
453C22—144no—
454C22—148no—
455C22—282no—
456C22—288no—
457C22—284no—
458C22—284no—
459C22—282no—
460C22—284no—
461C22C2×(C2×AGL(1,𝔽7))⋊C411yescharacteristic
462C22C4×(C22⋊D7)⋊C312yes—
463C22#6492⋊C311yescharacteristic
464C22#6493⋊C312yes—
465C22#1062078911yescharacteristic
466C4—22no—
467C4—44no—
468C4—282no—
469C4—562no—
470C3—71noSylow
471C2—21no—
472C2—22no—
473C2—21no—
474C2—22no—
475C2—42no—
476C2—284no—
477C2#89985⋊C311yescharacteristic
478C2#1305214412yes—
subgroupG/H
centerC22#10620789
commutatorC23×C7C22×C3×C4
FrattiniC24C22×AGL(1,𝔽7)
FittingC7×#518C2×C3
socleC22×C7C3×#120
prime radicalC2×#6500⋊C3C2
cosocleC24×C7C23×C3
hypercenter#518AGL(1,𝔽7)
hypocenterC7C3×#1889
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#518C2semidirect
3. C2×#6408⋊C3C2semidirect
4. #89727⋊C3C2semidirect
5. #89321⋊C3C2semidirect
6. #11729408C3semidirect
17. #6500⋊C3C4semidirect
21. C7×#518C2×C3semidirect
23. C2×#6408C2×C3semidirect
24. #89727C2×C3semidirect
25. #89321C2×C3semidirect
34. C23×AGL(1,𝔽7)C2×C4semidirect
41. C22×C4×ΓL(1,𝔽8)D4semidirect
54. (C22⋊Dic7)⋊C3C2×C4semidirect
67. #6500C3×C4semidirect
80. C23×ΓL(1,𝔽8)C22⋊C4semidirect
105. C2×C4×ΓL(1,𝔽8)C22⋊C4semidirect
119. C23×D7C2×C3×C4semidirect
126. C22×C4×C7C3×D4semidirect
139. C22⋊Dic7C2×C3×C4semidirect
183. #518AGL(1,𝔽7)semidirect
191. C23×C7C3×C22⋊C4semidirect
216. C2×C4×C7C3×C22⋊C4semidirect
335. ΓL(1,𝔽8)#1889semidirect
354. C22×C4(C22⋊D7)⋊C3semidirect
416. C23(C2×AGL(1,𝔽7))⋊C4semidirect
434. C2×C4(C2×AGL(1,𝔽7))⋊C4semidirect
435. C7C3×#1889semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#5182
C2×#6500⋊C32
C2×#6408⋊C32
#89727⋊C32
#89321⋊C32
#117294083
C3×#18897