Finite Group Databasebeta

← #10875985

Subgroups — #10875985

Order 768.

782 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2511C211yesmaximal, characteristic, complemented
2C4⋊C4×D12C211yesmaximal, characteristic
3#69687C211yesmaximal, characteristic
4#69687C211yesmaximal, characteristic
5#69687C211yesmaximal, characteristic
6#69682C211yesmaximal, characteristic, complemented
7#69675C211yesmaximal, characteristic, complemented
8#69670C211yesmaximal, characteristic, complemented
9#69626C211yesmaximal, characteristic
10#69612C211yesmaximal, characteristic, complemented
11#69584C211yesmaximal, characteristic
12#69571C211yesmaximal, characteristic, complemented
13#69483C211yesmaximal, characteristic, complemented
14#69474C211yesmaximal, characteristic
15#69460C211yesmaximal, characteristic, complemented
16#27573—31nomaximal, Sylow
17C22×#215C2211yescharacteristic
18C2×D3×C4⋊C4—21no—
19C2×C4×D12C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C3×#400—21no—
23C3×#400—21no—
24C3×#391C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#387—21no—
29C3×#387—21no—
30C3×#384C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36#4987—21no—
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41#4935C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4668—21no—
47#4668—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2511D311yescharacteristic, complemented
61D4×C4⋊C4—31no—
62#2100—31no—
63#2099—31no—
64#2072—31no—
65#2066—31no—
66#1935—31no—
67#1920—31no—
68#1917—31no—
69#1877—31no—
70#1877—31no—
71#1877—31no—
72#1864—31no—
73#1802—31no—
74#1762—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×#215—22no—
93C2×#215—22no—
94C2×#215—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109D3×C4⋊C4—44no—
110C4×D12—22no—
111C4×D12—22no—
112C4×D12—22no—
113C4×D12—22no—
114C4×D12—41no—
115C4×D12—41no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95D411yescharacteristic
125C3×#95D411yescharacteristic
126#835—22no—
127#835—41no—
128#835—41no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#809—22no—
132#809—22no—
133#809—22no—
134#808—22no—
135C4⋊(C3⋊C8)—41no—
136#803—41no—
137C22×SD16—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×C4×D4—31no—
141C22×C4⋊C4—61no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#97—31no—
151#400—21no—
152#400—21no—
153#391C2×D311yescharacteristic
154#391C2×D311yescharacteristic
155#391C2×D311yescharacteristic
156#389—31no—
157#389—31no—
158#389C2×D311yescharacteristic
159#387—21no—
160#387—21no—
161#385—61no—
162#385—61no—
163#384—31no—
164#384C2×D311yescharacteristic
165#380—61no—
166C4×C4⋊C4C2×D311yescharacteristic
167C4×C4⋊C4C2×D311yescharacteristic
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179#336—31no—
180C23×D3—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12#4112yes—
205C2×D12#4112yes—
206C2×D12#4112yes—
207C2×C4×D3—24no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3—81no—
215C2×C4×D3—81no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4—81no—
226C3×C4⋊C4—81no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—81no—
234#215—44no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—42no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3—81no—
245C4⋊Dic3#4112yes—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C22×Q8C22×D311yescharacteristic
261C22×D4—31no—
262C23×C4—61no—
263C23×C4—61no—
264C23×C4—61no—
265C2×SD16—62no—
266C2×SD16—62no—
267C2×SD16—62no—
268C22×C8—31no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—31no—
272C4×D4—62no—
273C4×D4—62no—
274C4×D4—62no—
275C4×D4—62no—
276C4×D4—121no—
277C4×D4—121no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—124no—
289C2×C4⋊C4#6711yescharacteristic
290C2×C4⋊C4#6711yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—21no—
298C2×C42C22×D311yescharacteristic
299C2×C42C22×D311yescharacteristic
300C2×C42C22×D311yescharacteristic
301#106—62no—
302C4⋊C8—121no—
303#103—62no—
304#103—121no—
305#103—121no—
306#102—62no—
307#102—62no—
308#102—62no—
309#97—121no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—61no—
319#95—61no—
320#95#6711yescharacteristic
321#95#6711yescharacteristic
322C23×C3C22×D411yescharacteristic
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C3×Q8—41no—
331C3×Q8—41no—
332C3×Q8—41no—
333C3×Q8—41no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4AGL(1,ℤ/8)12yes—
363C2×C3×C4#13712yes—
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C42−1+411yescharacteristic
370C2×Dic3—22no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379D12—22no—
380D12—22no—
381D12—22no—
382D12#12614yes—
383C4×D3—48no—
384C4×D3—84no—
385C4×D3—84no—
386C3⋊C8—41no—
387C3⋊C8—41no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C24—61no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×D4—32no—
398C2×D4—32no—
399C2×D4—32no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—64no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—122no—
412C22×C4—122no—
413C22×C4—122no—
414C22×C4—122no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4—241no—
418C22×C4—241no—
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C2×#6711yescharacteristic
423C22×C4C2×#6711yescharacteristic
424C22×C4C2×#6711yescharacteristic
425C22×C4C23×D311yescharacteristic
426SD16—124no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—32no—
448C4⋊C4—42no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—241no—
460C22⋊C4—64no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—81no—
470C22×C3#52511yescharacteristic
471C22×C3#54611yescharacteristic
472C22×C3#54811yescharacteristic
473C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
474C22×C3C2×#13711yescharacteristic
475C22×C3#57711yescharacteristic
476C22×C3#57711yescharacteristic
477C2×D3—24no—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—42no—
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—42no—
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4C2×#12611yescharacteristic
495C3×C4#53911yescharacteristic
496C3×C4#55211yescharacteristic
497C3×C4#55311yescharacteristic
498Dic3—42no—
499Dic3—81no—
500Dic3—81no—
501C23—62no—
502C23—62no—
503C23—62no—
504C23—121no—
505C23—121no—
506C23—121no—
507C23—121no—
508C23C22×#6711yescharacteristic
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513D4—34no—
514D4—62no—
515D4—62no—
516D4—62no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—62no—
546C2×C4—128no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—244no—
556C2×C4—244no—
557C2×C4#94212yes—
558C2×C4#94912yes—
559C2×C4D3×#4111yescharacteristic
560C2×C4D3×#4111yescharacteristic
561C2×C4D3×#4111yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C2×C4#100911yescharacteristic
564C2×C4#101011yescharacteristic
565C8—121no—
566C8—121no—
567C8—121no—
568C8—121no—
569C2×C3#248111yescharacteristic
570C2×C3#311011yescharacteristic
571C2×C3#312511yescharacteristic
572C2×C3#331011yescharacteristic
573C2×C3#333111yescharacteristic
574C2×C3#337211yescharacteristic
575C2×C3#337311yescharacteristic
576D3—44no—
577C22—64no—
578C22—122no—
579C22—122no—
580C22—122no—
581C22—122no—
582C22—122no—
583C22—122no—
584C22C2×#94211yescharacteristic
585C22C2×#94911yescharacteristic
586C22#582811yescharacteristic
587C22#582911yescharacteristic
588C22#583111yescharacteristic
589C22#583611yescharacteristic
590C22#583711yescharacteristic
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—122no—
602C4—241no—
603C4—241no—
604C4D3×#12611yescharacteristic
605C4#570711yescharacteristic
606C4#570811yescharacteristic
607C4#570911yescharacteristic
608C3#2757311yescharacteristic, complemented, Sylow
609C2—124no—
610C2#8308211yescharacteristic
611C2#8335311yescharacteristic
612C2#8346211yescharacteristic
613C2#8359911yescharacteristic
614C2#8363611yescharacteristic
615C2#8369211yescharacteristic
616C2#8371511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2511C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2511D3
hypocenterC3#27573
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2511

Complements

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

NH ≅ G/Ntype
1. C3×#2511C2semidirect
6. #69682C2semidirect
7. #69675C2semidirect
8. #69670C2semidirect
10. #69612C2semidirect
12. #69571C2semidirect
13. #69483C2semidirect
15. #69460C2semidirect
60. #2511D3semidirect
608. C3#27573semidirect

Maximal subgroups

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

subgroupindex
C3×#25112
C4⋊C4×D122
#696872
#696822
#696752
#696702
#696262
#696122
#695842
#695712
#694832
#694742
#694602
#275733