Finite Group Databasebeta

← #10796320

Subgroups — #10796320

Order 768.

1898 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5828C211yesmaximal, characteristic
2C23×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
3C2×C3×#408C211yesmaximal, characteristic, complemented
4C2×#5234C211yesmaximal, characteristic, complemented
5C2×#5126C211yesmaximal, characteristic, complemented
6#71980C211yesmaximal, characteristic, complemented
7#71965C211yesmaximal, characteristic, complemented
8#69958C218yesmaximal, complemented
9#20896—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C23×C4×D3C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C23×C3×C8C2211yescharacteristic
14#5828—24no—
15#5828—24no—
16C2×#946C412yes—
17C2×#938C412yes—
18C2×#937C412yes—
19C2×#935C412yes—
20C2×C4×#67C2214yes—
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C22×ΓL(1,𝔽25)—24no—
25C22×ΓL(1,𝔽25)—24no—
26C22×ΓL(1,𝔽25)—24no—
27C22×ΓL(1,𝔽25)C2214yes—
28C3×#408—24no—
29C2×C3×C22⋊C8C2214yescomplemented
30#5234—24no—
31C2×#830C2214yescomplemented
32#5126—24no—
33C2×(C2×D3)⋊C8C2214yes—
34#4550C2214yescomplemented
35#4550C2214yescomplemented
36C23×M16—31no—
37C2×#525—31no—
38C2×#408—31no—
39C2×#408D311yescharacteristic, complemented
40C2×#407—31no—
41#1870—38no—
42#1729—31no—
43#1729—31no—
44C24×D3C2×C411yescharacteristic
45C2×C3×#41—22no—
46C2×C3×#41—22no—
47C2×C3×#41—22no—
48C2×C3×#41—22no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C22×C3×Q8C2×C411yescharacteristic
52C22×C3×D4C2×C412yes—
53C22×C3×D4C2×C411yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C22×#67C2×C414yes—
57C23×Dic3C2×C411yescharacteristic
58C22×C4×D3—22no—
59C22×C4×D3—22no—
60C22×C4×D3—22no—
61C22×C4×D3—22no—
62C22×C4×D3C2×C412yes—
63C22×C4×D3D414yes—
64C22×C4×D3D414yes—
65C22×C4×D3C2312yes—
66C2×C3×M16—22no—
67C2×C3×M16—22no—
68C2×C3×M16—22no—
69C22×C3×C8—22no—
70C22×C3×C8—22no—
71C22×C3×C8D414yescomplemented
72C22×C3×C8C2312yes—
73C22×C3×C8C2312yes—
74C2×#217C2×C412yes—
75C2×#217C2×C414yes—
76#946—28no—
77#938—28no—
78#937—28no—
79#935—28no—
80C4×#67—28no—
81C4×#67—28no—
82C2×(C2×D3)⋊C4C2×C414yes—
83#926—22no—
84#926—22no—
85C2×C4⋊Dic3C2×C412yes—
86C2×Dic3⋊C4C2×C414yes—
87C2×C4×Dic3C2312yes—
88C2×#208—22no—
89C2×#208—22no—
90C2×#208—22no—
91C22×C3⋊C8—22no—
92C22×C3⋊C8—22no—
93C22×C3⋊C8D414yescomplemented
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2312yes—
96C2×ΓL(1,𝔽25)—232no—
97C2×ΓL(1,𝔽25)—28no—
98C2×ΓL(1,𝔽25)—28no—
99C2×ΓL(1,𝔽25)—28no—
100C3×C22⋊C8—28no—
101#830—28no—
102(C2×D3)⋊C8—28no—
103C22×#41C2×D311yescharacteristic
104C24×C4—31no—
105C22×M16—31no—
106C22×M16—34no—
107C22×M16—64no—
108C22×M16—64no—
109C22×M16—64no—
110C22×M16C2×D311yescharacteristic
111C23×C8—31no—
112C23×C8C2×D311yescharacteristic
113#525—64no—
114#525—64no—
115C2×#123—32no—
116C2×C22⋊Q8—32no—
117C2×#121—32no—
118C2×#120—32no—
119C2×C4×D4—34no—
120C2×#117—31no—
121#408—24no—
122#408—64no—
123#407—64no—
124C2×C22⋊C8—34no—
125C2×C22⋊C8—34no—
126C2×C22⋊C8C2×D314yescomplemented
127#336—34no—
128#336—34no—
129C24×C3C22×C412yes—
130C24×C3C22×C411yescharacteristic
131C23×D3—22no—
132C23×D3—22no—
133C23×D3—44no—
134C23×D3C22⋊C414yes—
135C23×D3C22×C412yes—
136C3×#41—44no—
137C3×#41—44no—
138C3×#41—44no—
139C3×#41—44no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×D4—24no—
144C2×C3×D4—24no—
145C2×C3×D4—24no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4C22×C412yes—
156C22×C3×C4C22×C411yescharacteristic
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4#4112yes—
162C22×C3×C4#4112yes—
163C22×C3×C4C2411yescharacteristic
164C2×#67—28no—
165C2×#67—48no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3C22⋊C414yes—
169C22×Dic3C22×C414yes—
170C22×Dic3C22×C412yes—
171C2×C4×D3—24no—
172C2×C4×D3—24no—
173C2×C4×D3—28no—
174C2×C4×D3—28no—
175C2×C4×D3—28no—
176C2×C4×D3—28no—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3C22⋊C418yes—
182C2×C4×D3C2×D414yes—
183C2×C4×D3C2×D414yes—
184C3×M16—44no—
185C2×C3×C8—28no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8C2×D414yes—
192#217—24no—
193#217—28no—
194(C2×D3)⋊C4—28no—
195C4⋊Dic3—24no—
196Dic3⋊C4—28no—
197C4×Dic3—22no—
198C4×Dic3—22no—
199#208—44no—
200C2×C3⋊C8—28no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8C2×D414yes—
207ΓL(1,𝔽25)—232no—
208C25—31no—
209C2×#41—22no—
210C2×#41—22no—
211C2×#41—22no—
212C2×#41—22no—
213C2×#41—22no—
214C2×#41—22no—
215C22×Q8C4×D311yescharacteristic
216C22×D4—34no—
217C22×D4C4×D312yes—
218C22×D4C4×D311yescharacteristic
219C23×C4—31no—
220C23×C4—32no—
221C23×C4—34no—
222C23×C4—34no—
223C23×C4—32no—
224C23×C4—62no—
225C23×C4—62no—
226C23×C4—62no—
227C23×C4—62no—
228C23×C4C22×D312yes—
229C23×C4C22×D311yescharacteristic
230C2×M16—22no—
231C2×M16—22no—
232C2×M16—22no—
233C2×M16—68no—
234C2×M16—68no—
235C2×M16—632no—
236C2×M16—62no—
237C2×M16—62no—
238C2×M16—62no—
239C2×M16—68no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—34no—
243C22×C8—32no—
244C22×C8—32no—
245C22×C8—62no—
246C22×C8—62no—
247C22×C8#6714yescomplemented
248C22×C8C22×D312yes—
249C22×C8C22×D312yes—
250#123—68no—
251C22⋊Q8—68no—
252#121—68no—
253#120—68no—
254C4×D4—68no—
255C4×D4—68no—
256#117—62no—
257#117—62no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—34no—
260C2×C22⋊C4—32no—
261C2×C22⋊C4—34no—
262C2×C22⋊C4—34no—
263C2×C42—32no—
264C22⋊C8—28no—
265C22⋊C8—68no—
266C22⋊C8—68no—
267C23×C3—24no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—28no—
272C23×C3C2×C22⋊C412yes—
273C23×C3C4×D412yes—
274C23×C3C4×D412yes—
275C23×C3C23×C411yescharacteristic
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—24no—
279C22×D3—24no—
280C22×D3—28no—
281C22×D3—48no—
282C22×D3—48no—
283C22×D3—48no—
284C22×D3C2×C22⋊C414yes—
285C3×Q8—44no—
286C3×D4—48no—
287C3×D4—44no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4C2×C22⋊C412yes—
296C2×C3×C4C4×D412yes—
297C2×C3×C4C4×D412yes—
298C2×C3×C4#12014yes—
299C2×C3×C4#12114yes—
300C2×C3×C4#12114yes—
301C2×C3×C4#12314yes—
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308#67—416no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—28no—
312C2×Dic3—24no—
313C2×Dic3—28no—
314C2×Dic3C2×C22⋊C414yes—
315C4×D3—24no—
316C4×D3—24no—
317C4×D3—24no—
318C4×D3—24no—
319C4×D3—216no—
320C3×C8—24no—
321C3×C8—42no—
322C3⋊C8—24no—
323C3⋊C8—42no—
324C24—32no—
325C24—34no—
326C24—62no—
327C24—62no—
328C24—124no—
329C24C2×C4×D312yes—
330C24C2×C4×D311yescharacteristic
331#41—44no—
332#41—44no—
333#41—44no—
334#41—44no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—68no—
345C2×D4—128no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—34no—
353C22×C4—32no—
354C22×C4—38no—
355C22×C4—34no—
356C22×C4—34no—
357C22×C4—34no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—64no—
361C22×C4—68no—
362C22×C4—68no—
363C22×C4—64no—
364C22×C4—68no—
365C22×C4—68no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4C2×C4×D312yes—
371C22×C4C2×C4×D311yescharacteristic
372C22×C4#23512yes—
373C22×C4D3×D412yes—
374C22×C4D3×D412yes—
375C22×C4D3×D412yes—
376C22×C4#23712yes—
377C22×C4C2×#6712yes—
378C22×C4C23×D311yescharacteristic
379M16—44no—
380M16—632no—
381M16—124no—
382C2×C8—28no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—34no—
389C2×C8—68no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8C2×#6714yes—
396C4⋊C4—68no—
397C4⋊C4—64no—
398C22⋊C4—68no—
399C22⋊C4—68no—
400C22⋊C4—64no—
401C42—62no—
402C42—62no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—42no—
410C22×C3#38614yes—
411C22×C3C22×C22⋊C411yescharacteristic
412C22×C3C2×C4×D411yescharacteristic
413C22×C3C2×C4×D411yescharacteristic
414C22×C3#56814yes—
415C2×D3—24no—
416C2×D3—24no—
417C2×D3—24no—
418C2×D3—416no—
419C2×D3—44no—
420C2×D3—44no—
421C3×C4—42no—
422C3×C4#38614yes—
423C3×C4C2×#12011yescharacteristic
424C3×C4C2×#12111yescharacteristic
425C3×C4C2×#12111yescharacteristic
426C3×C4C2×#12311yescharacteristic
427Dic3—24no—
428Dic3—44no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—28no—
433C23—22no—
434C23—34no—
435C23—68no—
436C23—62no—
437C23—64no—
438C23—64no—
439C23—62no—
440C23—128no—
441C23—128no—
442C23—128no—
443C23D3×C22⋊C412yes—
444C23#67⋊C412yes—
445C23C4×#6712yes—
446C23C22×C4×D311yescharacteristic
447Q8—44no—
448D4—48no—
449D4—44no—
450D4—1216no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—34no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—68no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—616no—
463C2×C4—64no—
464C2×C4—64no—
465C2×C4—64no—
466C2×C4—64no—
467C2×C4—64no—
468C2×C4—68no—
469C2×C4D3×C22⋊C412yes—
470C2×C4#67⋊C412yes—
471C2×C4#88314yes—
472C2×C4#88414yes—
473C2×C4C4×#6712yes—
474C2×C4#93714yes—
475C2×C4#93814yes—
476C2×C4C2×#23511yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C2×C4C2×D3×D411yescharacteristic
479C2×C4C2×D3×D411yescharacteristic
480C2×C4C2×#23711yescharacteristic
481C2×C4C22×#6711yescharacteristic
482C8—24no—
483C8—42no—
484C8—64no—
485C8—122no—
486C2×C3—24no—
487C2×C3—82no—
488C2×C3C2×#38611yescharacteristic
489C2×C3#287912yes—
490C2×C3#293012yes—
491C2×C3#293012yes—
492D3—44no—
493C22—24no—
494C22—24no—
495C22—24no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—42no—
500C22—64no—
501C22—64no—
502C22—64no—
503C22—1216no—
504C22—124no—
505C22—124no—
506C22#495614yes—
507C22C2×D3×C22⋊C411yescharacteristic
508C22C2×#67⋊C411yescharacteristic
509C22#575014yes—
510C22C2×C4×#6711yescharacteristic
511C4—42no—
512C4—64no—
513C4—124no—
514C4#495614yes—
515C4C2×#88311yescharacteristic
516C4C2×#88411yescharacteristic
517C4C2×#93711yescharacteristic
518C4C2×#93811yescharacteristic
519C3#2089611yescharacteristic, complemented, Sylow
520C2—24no—
521C2—82no—
522C2—124no—
523C2#7865612yes—
524C2#7866212yes—
525C2C2×#495611yescharacteristic
526C2#8379512yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC2×C3×#408C2
prime radicalC2×#5828C2
cosocleC22×C3×C4C24
hypercenterC2×#408D3
hypocenterC3#20896
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. C23×ΓL(1,𝔽25)C2semidirect
3. C2×C3×#408C2semidirect
4. C2×#5234C2semidirect
5. C2×#5126C2semidirect
6. #71980C2semidirect
7. #71965C2semidirect
8. #69958C2semidirect
29. C2×C3×C22⋊C8C22semidirect
31. C2×#830C22semidirect
34. #4550C22semidirect
35. #4550C22semidirect
39. C2×#408D3semidirect
71. C22×C3×C8D4semidirect
93. C22×C3⋊C8D4semidirect
126. C2×C22⋊C8C2×D3semidirect
247. C22×C8#67semidirect
519. C3#20896semidirect

Maximal subgroups

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

subgroupindex
C2×#58282
C23×ΓL(1,𝔽25)2
C2×C3×#4082
C2×#52342
C2×#51262
#719802
#719652
#699582
#208963