Finite Group Databasebeta

← #72276714

Subgroups — #72276714

Order 2688.

1156 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 153 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×ΓL(1,𝔽8)×#120C211yesmaximal, characteristic
2#90489⋊C3C211yesmaximal, characteristic, complemented
3#13052147C212yesmaximal, complemented
4C2×#6408⋊C3C211yesmaximal, characteristic, complemented
5#89322⋊C3C212yesmaximal, complemented
6#11729423C311yesmaximal, characteristic, complemented, Hall
7C25×ΓL(1,𝔽8)C2211yescharacteristic
8C22×D4×ΓL(1,𝔽8)—21no—
9C22×D4×ΓL(1,𝔽8)C2211yescharacteristic
10ΓL(1,𝔽8)×#120—22no—
11ΓL(1,𝔽8)×#120C414yescomplemented
12C2×C22⋊C4×ΓL(1,𝔽8)—21no—
13C2×C22⋊C4×ΓL(1,𝔽8)C2211yescharacteristic
14C2×(C22⋊Dic7)⋊C3—21no—
15C2×(C22⋊Dic7)⋊C3—24no—
16C2×(C22⋊Dic7)⋊C3C2212yes—
17C2×(C22⋊Dic7)⋊C3C2212yes—
18D4×ΓL(1,𝔽8)⋊C4—24no—
19#6489⋊C3—21no—
20#6408⋊C3—22no—
21C2×C7×#120C2×C311yescharacteristic
22#90489C2×C311yescharacteristic, complemented
23#90019C2×C312yescomplemented
24C2×#6408C2×C311yescharacteristic, complemented
25#89322C2×C312yescomplemented
26C3×#1889—71nomaximal, Hall
27C24×ΓL(1,𝔽8)—22no—
28C24×ΓL(1,𝔽8)—21no—
29C24×ΓL(1,𝔽8)—21no—
30C24×ΓL(1,𝔽8)—22no—
31C24×ΓL(1,𝔽8)—44no—
32C24×ΓL(1,𝔽8)C2×C412yescomplemented
33C24×ΓL(1,𝔽8)D411yescharacteristic
34C24×ΓL(1,𝔽8)C2311yescharacteristic
35C2×D4×ΓL(1,𝔽8)—22no—
36C2×D4×ΓL(1,𝔽8)—22no—
37C2×D4×ΓL(1,𝔽8)—24no—
38C2×D4×ΓL(1,𝔽8)—44no—
39C2×D4×ΓL(1,𝔽8)C2×C412yes—
40C2×D4×ΓL(1,𝔽8)D412yes—
41C22×C4×ΓL(1,𝔽8)—21no—
42C22×C4×ΓL(1,𝔽8)D411yescharacteristic
43C22×ΓL(1,𝔽8)⋊C4—22no—
44C22×ΓL(1,𝔽8)⋊C4—22no—
45C22×ΓL(1,𝔽8)⋊C4D414yescomplemented
46C22⋊C4×ΓL(1,𝔽8)—21no—
47C22⋊C4×ΓL(1,𝔽8)—24no—
48C22⋊C4×ΓL(1,𝔽8)C2×C412yescomplemented
49(C22⋊Dic7)⋊C3—22no—
50(C22⋊Dic7)⋊C3—22no—
51(C22⋊Dic7)⋊C3—22no—
52(C22⋊Dic7)⋊C3—22no—
53(C22⋊Dic7)⋊C3—22no—
54(C22⋊Dic7)⋊C3—48no—
55C4⋊(ΓL(1,𝔽8)⋊C4)—22no—
56C4×ΓL(1,𝔽8)⋊C4—22no—
57C25×C7C22×C311yescharacteristic
58C22×C7×D4—21no—
59C22×C7×D4C22×C311yescharacteristic
60C7×#120—22no—
61C7×#120C3×C414yescomplemented
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4C22×C311yescharacteristic
64C2×C22⋊Dic7—21no—
65C2×C22⋊Dic7—24no—
66C2×C22⋊Dic7C22×C312yes—
67C2×C22⋊Dic7C22×C312yes—
68D4×Dic7—24no—
69#6489—21no—
70#6408—22no—
71C2×C3×#120—71no—
72C3×#518—71no—
73C2×C3×#99—71no—
74C3×#379—72no—
75C3×#342—72no—
76C23×ΓL(1,𝔽8)—24no—
77C23×ΓL(1,𝔽8)—28no—
78C23×ΓL(1,𝔽8)—22no—
79C23×ΓL(1,𝔽8)—24no—
80C23×ΓL(1,𝔽8)—21no—
81C23×ΓL(1,𝔽8)—21no—
82C23×ΓL(1,𝔽8)—44no—
83C23×ΓL(1,𝔽8)—42no—
84C23×ΓL(1,𝔽8)—44no—
85C23×ΓL(1,𝔽8)—42no—
86C23×ΓL(1,𝔽8)—44no—
87C23×ΓL(1,𝔽8)—48no—
88C23×ΓL(1,𝔽8)—42no—
89C23×ΓL(1,𝔽8)—41no—
90C23×ΓL(1,𝔽8)—42no—
91C23×ΓL(1,𝔽8)—41no—
92C23×ΓL(1,𝔽8)C22⋊C412yescomplemented
93C23×ΓL(1,𝔽8)C22×C411yescharacteristic
94C23×ΓL(1,𝔽8)C2×D411yescharacteristic
95C23×ΓL(1,𝔽8)C2×D412yes—
96C23×ΓL(1,𝔽8)C2×D411yescharacteristic
97C23×ΓL(1,𝔽8)#4112yes—
98D4×ΓL(1,𝔽8)—24no—
99D4×ΓL(1,𝔽8)—24no—
100D4×ΓL(1,𝔽8)—48no—
101C2×C4×ΓL(1,𝔽8)—21no—
102C2×C4×ΓL(1,𝔽8)—21no—
103C2×C4×ΓL(1,𝔽8)—22no—
104C2×C4×ΓL(1,𝔽8)—42no—
105C2×C4×ΓL(1,𝔽8)C22⋊C412yescomplemented
106C2×ΓL(1,𝔽8)⋊C4—24no—
107C2×ΓL(1,𝔽8)⋊C4—44no—
108C2×ΓL(1,𝔽8)⋊C4—44no—
109C2×ΓL(1,𝔽8)⋊C4—42no—
110C2×ΓL(1,𝔽8)⋊C4—82no—
111#1889—71noSylow
112C24×C7—22no—
113C24×C7—21no—
114C24×C7—21no—
115C24×C7—22no—
116C24×C7—44no—
117C24×C7C2×C3×C412yescomplemented
118C24×C7C3×D411yescharacteristic
119C24×C7C23×C311yescharacteristic
120C2×C7×D4—22no—
121C2×C7×D4—22no—
122C2×C7×D4—24no—
123C2×C7×D4—44no—
124C2×C7×D4C2×C3×C412yes—
125C2×C7×D4C3×D412yes—
126C22×C4×C7—21no—
127C22×C4×C7C3×D411yescharacteristic
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7C3×D414yescomplemented
131C7×C22⋊C4—21no—
132C7×C22⋊C4—24no—
133C7×C22⋊C4C2×C3×C412yescomplemented
134C22⋊Dic7—22no—
135C22⋊Dic7—22no—
136C22⋊Dic7—22no—
137C22⋊Dic7—22no—
138C22⋊Dic7—22no—
139C22⋊Dic7—48no—
140C4⋊Dic7—22no—
141C4×Dic7—22no—
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—72no—
150C2×C3×C22⋊C4—72no—
151C2×C3×C22⋊C4—71no—
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)—24no—
159C22×ΓL(1,𝔽8)—28no—
160C22×ΓL(1,𝔽8)—22no—
161C22×ΓL(1,𝔽8)—41no—
162C22×ΓL(1,𝔽8)—41no—
163C22×ΓL(1,𝔽8)—42no—
164C22×ΓL(1,𝔽8)—42no—
165C22×ΓL(1,𝔽8)—48no—
166C22×ΓL(1,𝔽8)—42no—
167C22×ΓL(1,𝔽8)—44no—
168C22×ΓL(1,𝔽8)—44no—
169C22×ΓL(1,𝔽8)—42no—
170C22×ΓL(1,𝔽8)—42no—
171C22×ΓL(1,𝔽8)—44no—
172C22×ΓL(1,𝔽8)—81no—
173C22×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
174C22×ΓL(1,𝔽8)C4×D412yes—
175C22×ΓL(1,𝔽8)#12011yescharacteristic
176C22×ΓL(1,𝔽8)#12112yes—
177C22×ΓL(1,𝔽8)#12311yescharacteristic
178C4×ΓL(1,𝔽8)—22no—
179C4×ΓL(1,𝔽8)—42no—
180ΓL(1,𝔽8)⋊C4—44no—
181ΓL(1,𝔽8)⋊C4—82no—
182C2×#120AGL(1,𝔽7)11yescharacteristic
183#518—71no—
184C2×#99—71no—
185#379—72no—
186#342—72no—
187C23×C7—28no—
188C23×C7—24no—
189C23×C7—22no—
190C23×C7—24no—
191C23×C7—21no—
192C23×C7—21no—
193C23×C7—44no—
194C23×C7—42no—
195C23×C7—44no—
196C23×C7—42no—
197C23×C7—44no—
198C23×C7—48no—
199C23×C7—42no—
200C23×C7—41no—
201C23×C7—42no—
202C23×C7—41no—
203C23×C7C3×C22⋊C412yescomplemented
204C23×C7C22×C3×C411yescharacteristic
205C23×C7C2×C3×D412yes—
206C23×C7C2×C3×D411yescharacteristic
207C23×C7C2×C3×D411yescharacteristic
208C23×C7C3×#4112yes—
209C7×D4—24no—
210C7×D4—24no—
211C7×D4—48no—
212C2×C4×C7—21no—
213C2×C4×C7—21no—
214C2×C4×C7—22no—
215C2×C4×C7—42no—
216C2×C4×C7C3×C22⋊C412yescomplemented
217C2×Dic7—24no—
218C2×Dic7—44no—
219C2×Dic7—44no—
220C2×Dic7—42no—
221C2×Dic7—82no—
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—74no—
237C22×C3×C4—71no—
238C22×C3×C4—141no—
239C22×C3×C4—142no—
240C22×C3×C4—142no—
241C3×C4⋊C4—142no—
242C3×C22⋊C4—72no—
243C3×C22⋊C4—144no—
244C3×C22⋊C4—141no—
245C3×C22⋊C4—142no—
246C3×C22⋊C4—142no—
247C3×C22⋊C4—142no—
248C3×C22⋊C4—142no—
249C3×C22⋊C4—142no—
250C3×C22⋊C4—288no—
251C3×C42—142no—
252C2×ΓL(1,𝔽8)—22no—
253C2×ΓL(1,𝔽8)—21no—
254C2×ΓL(1,𝔽8)—22no—
255C2×ΓL(1,𝔽8)—21no—
256C2×ΓL(1,𝔽8)—44no—
257C2×ΓL(1,𝔽8)—42no—
258C2×ΓL(1,𝔽8)#38611yescharacteristic
259C2×ΓL(1,𝔽8)B(4,𝔽2)12yes—
260C25C2×AGL(1,𝔽7)11yescharacteristic
261C22×D4—21no—
262C22×D4C2×AGL(1,𝔽7)11yescharacteristic
263#120—22no—
264#120ΓL(1,𝔽8)⋊C414yescomplemented
265C4×D4—144no—
266#117—141no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—72no—
269C2×C22⋊C4—72no—
270C2×C22⋊C4—141no—
271C2×C22⋊C4—144no—
272C2×C22⋊C4C2×AGL(1,𝔽7)11yescharacteristic
273#99—142no—
274C22×C7—24no—
275C22×C7—22no—
276C22×C7—24no—
277C22×C7—28no—
278C22×C7—22no—
279C22×C7—41no—
280C22×C7—41no—
281C22×C7—42no—
282C22×C7—42no—
283C22×C7—48no—
284C22×C7—42no—
285C22×C7—44no—
286C22×C7—44no—
287C22×C7—42no—
288C22×C7—42no—
289C22×C7—44no—
290C22×C7—81no—
291C22×C7C2×C3×C22⋊C411yescharacteristic
292C22×C7C3×C4×D412yes—
293C22×C7C3×#12011yescharacteristic
294C22×C7C3×#12112yes—
295C22×C7C3×#12311yescharacteristic
296C4×C7—22no—
297C4×C7—42no—
298Dic7—44no—
299Dic7—82no—
300C23×C3—72no—
301C23×C3—72no—
302C23×C3—71no—
303C23×C3—71no—
304C23×C3—71no—
305C23×C3—72no—
306C23×C3—141no—
307C23×C3—142no—
308C23×C3—144no—
309C23×C3—141no—
310C23×C3—148no—
311C23×C3—144no—
312C23×C3—281no—
313C23×C3—281no—
314C23×C3—282no—
315C23×C3—284no—
316C23×C3—282no—
317C23×C3—284no—
318C23×C3—282no—
319C23×C3—284no—
320C23×C3—288no—
321C23×C3—282no—
322C3×D4—144no—
323C3×D4—144no—
324C3×D4—288no—
325C2×C3×C4—72no—
326C2×C3×C4—141no—
327C2×C3×C4—141no—
328C2×C3×C4—142no—
329C2×C3×C4—144no—
330C2×C3×C4—282no—
331C2×C3×C4—284no—
332C2×C3×C4—282no—
333C2×C3×C4—284no—
334C2×C3×C4—562no—
335ΓL(1,𝔽8)#188911yescharacteristic, complemented, Hall
336C24—22no—
337C24—21no—
338C24—22no—
339C24—21no—
340C24—44no—
341C24C2×ΓL(1,𝔽8)⋊C412yescomplemented
342C24(C22⋊D7)⋊C311yescharacteristic
343C24C22×AGL(1,𝔽7)11yescharacteristic
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—44no—
348C2×D4C2×ΓL(1,𝔽8)⋊C412yes—
349C2×D4(C22⋊D7)⋊C312yes—
350C22×C4—21no—
351C22×C4—74no—
352C22×C4—142no—
353C22×C4—142no—
354C22×C4(C22⋊D7)⋊C311yescharacteristic
355C4⋊C4—142no—
356C22⋊C4—21no—
357C22⋊C4—24no—
358C22⋊C4—142no—
359C22⋊C4—142no—
360C22⋊C4—142no—
361C22⋊C4—142no—
362C22⋊C4—142no—
363C22⋊C4—288no—
364C22⋊C4C2×ΓL(1,𝔽8)⋊C412yescomplemented
365C42—142no—
366C2×C7—22no—
367C2×C7—21no—
368C2×C7—22no—
369C2×C7—21no—
370C2×C7—44no—
371C2×C7—42no—
372C2×C7C3×#38611yescharacteristic
373C2×C7C3×B(4,𝔽2)12yes—
374C22×C3—72no—
375C22×C3—71no—
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—284no—
386C22×C3—282no—
387C22×C3—282no—
388C22×C3—281no—
389C22×C3—281no—
390C22×C3—282no—
391C22×C3—282no—
392C22×C3—288no—
393C22×C3—284no—
394C22×C3—284no—
395C22×C3—561no—
396C3×C4—142no—
397C3×C4—282no—
398C3×C4—284no—
399C3×C4—562no—
400C23—28no—
401C23—24no—
402C23—24no—
403C23—21no—
404C23—22no—
405C23—21no—
406C23—44no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—44no—
411C23—48no—
412C23—42no—
413C23—41no—
414C23—41no—
415C23—42no—
416C23(C22⋊Dic7)⋊C312yescomplemented
417C23D4×AGL(1,𝔽7)12yes—
418C23#1159⋊C312yes—
419C23C22×ΓL(1,𝔽8)⋊C411yescharacteristic
420C23C2×(C22⋊D7)⋊C311yescharacteristic
421C23C2×(C22⋊D7)⋊C311yescharacteristic
422D4—24no—
423D4—24no—
424D4—48no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—144no—
430C2×C4—284no—
431C2×C4—284no—
432C2×C4—282no—
433C2×C4—562no—
434C2×C4(C22⋊Dic7)⋊C312yescomplemented
435C7C3×#188911yescharacteristic, complemented, Sylow
436C2×C3—72no—
437C2×C3—71no—
438C2×C3—141no—
439C2×C3—142no—
440C2×C3—142no—
441C2×C3—141no—
442C2×C3—284no—
443C2×C3—282no—
444C22—22no—
445C22—24no—
446C22—24no—
447C22—28no—
448C22—22no—
449C22—42no—
450C22—41no—
451C22—41no—
452C22—42no—
453C22—42no—
454C22—48no—
455C22—44no—
456C22—44no—
457C22—42no—
458C22—44no—
459C22—42no—
460C22—81no—
461C22D4×ΓL(1,𝔽8)⋊C412yes—
462C22#6498⋊C311yescharacteristic
463C22#6500⋊C311yescharacteristic
464C22#6502⋊C312yes—
465C22C2×(C22⋊Dic7)⋊C311yescharacteristic
466C4—22no—
467C4—42no—
468C4—284no—
469C4—562no—
470C3—71noSylow
471C2—21no—
472C2—22no—
473C2—22no—
474C2—21no—
475C2—44no—
476C2—42no—
477C2#89804⋊C312yes—
478C2#89992⋊C311yescharacteristic
subgroupG/H
centerC22#6500⋊C3
commutatorC23×C7C22×C3×C4
FrattiniC24C22×AGL(1,𝔽7)
FittingC2×C7×#120C2×C3
socleC22×C7C3×#120
prime radicalC2×ΓL(1,𝔽8)×#120C2
cosocleC24×C7C23×C3
hypercenterC2×#120AGL(1,𝔽7)
hypocenterC7C3×#1889
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. #90489⋊C3C2semidirect
3. #13052147C2semidirect
4. C2×#6408⋊C3C2semidirect
5. #89322⋊C3C2semidirect
6. #11729423C3semidirect
11. ΓL(1,𝔽8)×#120C4semidirect
22. #90489C2×C3semidirect
23. #90019C2×C3semidirect
24. C2×#6408C2×C3semidirect
25. #89322C2×C3semidirect
32. C24×ΓL(1,𝔽8)C2×C4semidirect
45. C22×ΓL(1,𝔽8)⋊C4D4semidirect
48. C22⋊C4×ΓL(1,𝔽8)C2×C4semidirect
61. C7×#120C3×C4semidirect
92. C23×ΓL(1,𝔽8)C22⋊C4semidirect
105. C2×C4×ΓL(1,𝔽8)C22⋊C4semidirect
117. C24×C7C2×C3×C4semidirect
130. C22×Dic7C3×D4semidirect
133. C7×C22⋊C4C2×C3×C4semidirect
203. C23×C7C3×C22⋊C4semidirect
216. C2×C4×C7C3×C22⋊C4semidirect
264. #120ΓL(1,𝔽8)⋊C4semidirect
335. ΓL(1,𝔽8)#1889semidirect
341. C24C2×ΓL(1,𝔽8)⋊C4semidirect
364. C22⋊C4C2×ΓL(1,𝔽8)⋊C4semidirect
416. C23(C22⋊Dic7)⋊C3semidirect
434. C2×C4(C22⋊Dic7)⋊C3semidirect
435. C7C3×#1889semidirect

Maximal subgroups

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

subgroupindex
C2×ΓL(1,𝔽8)×#1202
#90489⋊C32
#130521472
C2×#6408⋊C32
#89322⋊C32
#117294233
C3×#18897