Finite Group Databasebeta

← #10858989

Subgroups — #10858989

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2440C211yesmaximal, characteristic, complemented
2#73469C211yesmaximal, characteristic
3#71792C211yesmaximal, characteristic, complemented
4#71780C211yesmaximal, characteristic, complemented
5#71560C211yesmaximal, characteristic
6#71558C211yesmaximal, characteristic, complemented
7#69630C211yesmaximal, characteristic, complemented
8#69626C211yesmaximal, characteristic, complemented
9#69623C211yesmaximal, characteristic, complemented
10#69601C211yesmaximal, characteristic, complemented
11#69584C211yesmaximal, characteristic, complemented
12#69572C211yesmaximal, characteristic, complemented
13#69517C211yesmaximal, characteristic
14#69506C211yesmaximal, characteristic, complemented
15#69450C211yesmaximal, characteristic, complemented
16#26425—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8—21no—
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C3×#398—21no—
25C3×#395—21no—
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#385—21no—
29C3×#384C2211yescharacteristic
30C3×#380C2211yescharacteristic
31C3×C4×C22⋊C4C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#809C2211yescharacteristic
34C2×#809C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4957—21no—
40#4956—21no—
41#4939C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4669—21no—
47#4666—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2440D311yescharacteristic, complemented
61#2363—31no—
62#2100—31no—
63#2087—31no—
64#2067—31no—
65#2023—31no—
66#1988—31no—
67#1919—31no—
68#1895—31no—
69#1874—31no—
70#1864—31no—
71#1854—31no—
72#1851—31no—
73#1802—31no—
74#1736—31no—
75C22×C3×Q8—21no—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8—41no—
83C3×C22⋊Q8—44no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—41no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102#930—22no—
103#930Q814yescomplemented
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108#926—22no—
109#926—41no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×#208—22no—
117C2×#208—41no—
118C2×#208—41no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C4×D12—22no—
124C4×D12—41no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95D411yescharacteristic
129C3×#95D411yescharacteristic
130C3⋊#95—21no—
131C3⋊#95—21no—
132#830—22no—
133#809—22no—
134#809—41no—
135#809—41no—
136#809—41no—
137#808—22no—
138#807—41no—
139C4⋊(C3⋊C8)—41no—
140C4⋊(C3⋊C8)—41no—
141C22×M16—31no—
142C2×C4⋊Q8C2×D311yescharacteristic
143C2×C22⋊Q8—21no—
144C2×#121—31no—
145C2×C4×D4—31no—
146C2×#117—31no—
147C22×C4⋊C4C2×D311yescharacteristic
148C2×#107—31no—
149C2×#106—31no—
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×C22⋊C8—31no—
157#398—21no—
158#395—21no—
159#390—31no—
160#390—61no—
161#389—31no—
162#389C2×D311yescharacteristic
163#389C2×D311yescharacteristic
164#388—61no—
165#386—61no—
166#385—21no—
167#384C2×D311yescharacteristic
168#380C2×D311yescharacteristic
169C4×C4⋊C4—31no—
170C4×C22⋊C4—61no—
171C4×C22⋊C4C2×D311yescharacteristic
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336—31no—
183#336—31no—
184C24×C3C2×D411yescharacteristic
185C23×D3—21no—
186C2×C3×Q8—42no—
187C2×C3×Q8—42no—
188C2×C3×Q8—81no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—42no—
198C22×C3×C4—42no—
199C22×C3×C4—81no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×Q812yes—
206C22×C3×C4#4112yes—
207C22×C3×C4C2411yescharacteristic
208C2×#67—24no—
209C2×#67—81no—
210C2×#67—81no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12C2×Q812yes—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—81no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4—42no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—81no—
236C3×C4⋊C4#4112yes—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—42no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244#217—42no—
245(C2×D3)⋊C4—24no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—81no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3C2×Q812yes—
254C4⋊Dic3#4112yes—
255Dic3⋊C4—81no—
256C4×Dic3—42no—
257C4×Dic3—81no—
258#208—44no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C22×Q8—21no—
272C22×D4—31no—
273C22×D4—61no—
274C23×C4—21no—
275C23×C4—61no—
276C23×C4C22×D311yescharacteristic
277C2×M16—62no—
278C2×M16—121no—
279C2×M16—121no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8—31no—
283C22×C8—31no—
284C4⋊Q8—22no—
285C4⋊Q8—41no—
286C22⋊Q8—44no—
287#121—34no—
288#121—62no—
289C4×D4—62no—
290C4×D4—121no—
291#117—62no—
292#117—121no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—41no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4#6711yescharacteristic
316C2×C22⋊C4#6711yescharacteristic
317C2×C42—61no—
318C2×C42—61no—
319C2×C42C22×D311yescharacteristic
320C2×C42C22×D311yescharacteristic
321#107—121no—
322#106—62no—
323C4⋊C8—121no—
324C4⋊C8—121no—
325#102—62no—
326#102—121no—
327#102—121no—
328#102—121no—
329C22⋊C8—62no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—61no—
334#95—61no—
335#95#6711yescharacteristic
336#95#6711yescharacteristic
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3C22⋊Q812yes—
344C23×C3C22×D411yescharacteristic
345C22×D3—22no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C3×Q8—84no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—84no—
382C2×C3×C4C22⋊Q812yes—
383C2×C3×C4AGL(1,ℤ/8)12yes—
384C2×C3×C4#13712yes—
385C2×C3×C4C22×Q811yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4H(5,𝔽2)11yescharacteristic
390C2×C3×C4H(5,𝔽2)11yescharacteristic
391#67—84no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401D12—22no—
402D12—22no—
403D12—41no—
404D12—41no—
405C4×D3—84no—
406C3⋊C8—41no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C3⋊C8—41no—
410C24—61no—
411C24C2×#6711yescharacteristic
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—81no—
415C2×D4—32no—
416C2×D4—64no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—61no—
422C2×D4—241no—
423C2×D4—241no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—81no—
438C22×C4—122no—
439C22×C4—122no—
440C22×C4—241no—
441C22×C4D3×Q812yes—
442C22×C4#23912yes—
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6711yescharacteristic
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C2×#6711yescharacteristic
448C22×C4C23×D311yescharacteristic
449M16—124no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—32no—
470C4⋊C4—32no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—81no—
481C4⋊C4—241no—
482C4⋊C4#23712yes—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—42no—
486C22⋊C4—64no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C22⋊C4—241no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—122no—
496C42—241no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3C2×C22⋊Q811yescharacteristic
505C22×C3#54611yescharacteristic
506C22×C3#54611yescharacteristic
507C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
508C22×C3C2×#13711yescharacteristic
509C22×C3#57611yescharacteristic
510C22×C3#57711yescharacteristic
511C2×D3—42no—
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—81no—
515C2×D3—81no—
516C2×D3—81no—
517C2×D3—81no—
518C3×C4—22no—
519C3×C4—22no—
520C3×C4—22no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—42no—
524C3×C4—81no—
525C3×C4—81no—
526C3×C4—81no—
527C3×C4#52911yescharacteristic
528C3×C4#53811yescharacteristic
529C3×C4#55411yescharacteristic
530C3×C4#55411yescharacteristic
531Dic3—42no—
532Dic3—42no—
533Dic3—81no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—62no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23#94612yes—
548C23C22×#6711yescharacteristic
549Q8—84no—
550D4—62no—
551D4—62no—
552D4—121no—
553D4—121no—
554D4—244no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—84no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—244no—
594C2×C4#93212yes—
595C2×C4#94212yes—
596C2×C4#94612yes—
597C2×C4C2×#23711yescharacteristic
598C2×C4#100411yescharacteristic
599C2×C4C2×D3×Q811yescharacteristic
600C2×C4C2×#23911yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4#100911yescharacteristic
603C8—121no—
604C8—121no—
605C8—121no—
606C8—121no—
607C2×C3—42no—
608C2×C3#245011yescharacteristic
609C2×C3#307911yescharacteristic
610C2×C3#308111yescharacteristic
611C2×C3#331411yescharacteristic
612C2×C3#331811yescharacteristic
613C2×C3#332911yescharacteristic
614C2×C3#333211yescharacteristic
615D3—82no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—122no—
624C22—122no—
625C22—122no—
626C22—241no—
627C22—241no—
628C22—241no—
629C22—241no—
630C22C2×#93211yescharacteristic
631C22#580611yescharacteristic
632C22C2×#94211yescharacteristic
633C22C2×#94611yescharacteristic
634C22#582911yescharacteristic
635C22#583511yescharacteristic
636C22#583711yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—122no—
647C4—122no—
648C4—241no—
649C4#562411yescharacteristic
650C4#563511yescharacteristic
651C4#563611yescharacteristic
652C4#564411yescharacteristic
653C3#2642511yescharacteristic, complemented, Sylow
654C2—42no—
655C2—242no—
656C2#8276511yescharacteristic
657C2#8321211yescharacteristic
658C2#8347411yescharacteristic
659C2#8348511yescharacteristic
660C2#8353311yescharacteristic
661C2#8355511yescharacteristic
662C2#8363511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2440C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2440D3
hypocenterC3#26425
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
1. C3×#2440C2semidirect
3. #71792C2semidirect
4. #71780C2semidirect
6. #71558C2semidirect
7. #69630C2semidirect
8. #69626C2semidirect
9. #69623C2semidirect
10. #69601C2semidirect
11. #69584C2semidirect
12. #69572C2semidirect
14. #69506C2semidirect
15. #69450C2semidirect
60. #2440D3semidirect
103. #930Q8semidirect
653. C3#26425semidirect

Maximal subgroups

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

subgroupindex
C3×#24402
#734692
#717922
#717802
#715602
#715582
#696302
#696262
#696232
#696012
#695842
#695722
#695172
#695062
#694502
#264253