Finite Group Databasebeta

← #10868097

Subgroups — #10868097

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73512C211yesmaximal, characteristic, complemented
2#73467C211yesmaximal, characteristic
3C3×#1924C211yesmaximal, characteristic, complemented
4#70483C211yesmaximal, characteristic, complemented
5#70453C211yesmaximal, characteristic, complemented
6#69934C211yesmaximal, characteristic
7#69934C211yesmaximal, characteristic
8#69928C211yesmaximal, characteristic, complemented
9#69925C211yesmaximal, characteristic, complemented
10#69819C211yesmaximal, characteristic, complemented
11#69812C211yesmaximal, characteristic, complemented
12#68671C211yesmaximal, characteristic, complemented
13#68635C211yesmaximal, characteristic, complemented
14#68624C211yesmaximal, characteristic, complemented
15#68623C211yesmaximal, characteristic, complemented
16#27047—31nomaximal, Sylow
17C2×#930C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C2×#67⋊C4—21no—
20C2×#879—21no—
21C2×C4×D12C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C22⋊C8C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×C4×C22⋊C4C2211yescharacteristic
27#5212C2211yescharacteristic
28#5212C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#817C2211yescharacteristic
37#4958—21no—
38#4953—21no—
39#4939C2211yescharacteristic
40#4935C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4670—21no—
47#4668—21no—
48#4659—21no—
49#4653—21no—
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2496—31no—
61#2361—31no—
62#2071—31no—
63#2065—31no—
64#2033—31no—
65#2033—31no—
66#1947—31no—
67#1943—31no—
68#1924D311yescharacteristic, complemented
69#1921—31no—
70#1920—31no—
71#1864—31no—
72#1853—31no—
73#1824—31no—
74#1762—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3—21no—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C22⋊C4D411yescharacteristic
85C2×C3×C22⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90#930—22no—
91#930—22no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—41no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113#67⋊C4—44no—
114#879—44no—
115C4×D12—22no—
116C4×D12—41no—
117C3×C4⋊C8—22no—
118C3×C22⋊C8—22no—
119C3×C22⋊C8—41no—
120C3×#97—41no—
121C3×#95D411yescharacteristic
122C3×#95D411yescharacteristic
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#821—22no—
131#821—22no—
132#821—22no—
133#821—41no—
134#818—22no—
135#818—22no—
136#817—22no—
137#817—41no—
138C2×#126—61no—
139C2×#121—31no—
140C2×C4×D4—31no—
141C2×C4×D4—61no—
142C22×C4⋊C4—31no—
143C2×#107—31no—
144C2×#107—31no—
145C2×#106—31no—
146C2×#106—31no—
147C2×C4⋊C8C2×D311yescharacteristic
148C2×#102—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×C22⋊C8C2×D311yescharacteristic
153C2×C22⋊C8C2×D311yescharacteristic
154C2×#97C2×D311yescharacteristic
155#400—61no—
156#397—61no—
157#390—31no—
158#389—31no—
159#389—31no—
160#389—31no—
161#388—61no—
162#387—61no—
163#386—61no—
164#385—61no—
165#384—31no—
166#380—31no—
167#380—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170C4×C22⋊C4C2×D311yescharacteristic
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#336C2×D311yescharacteristic
180#336C2×D311yescharacteristic
181C24×C3C2×D411yescharacteristic
182C23×D3—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C2×#67—24no—
196C2×#67—42no—
197C2×#67—42no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C22×Dic3—81no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12#4112yes—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C4×D3—81no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8#4112yes—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235#217—42no—
236#217—42no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—81no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—42no—
254C4⋊Dic3—42no—
255C4⋊Dic3—42no—
256C4⋊Dic3#4112yes—
257C4⋊Dic3#4112yes—
258Dic3⋊C4—42no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261C4×Dic3—42no—
262C4×Dic3—42no—
263C22×D4—31no—
264C22×D4—61no—
265C23×C4—61no—
266C23×C4—61no—
267C23×C4C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272#126—124no—
273#121—62no—
274#121—62no—
275C4×D4—62no—
276C4×D4—121no—
277C4×D4—124no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—121no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4D1211yescharacteristic
300C2×C22⋊C4D1211yescharacteristic
301C2×C42—61no—
302C2×C42—61no—
303C2×C42C22×D311yescharacteristic
304C2×C42C22×D311yescharacteristic
305#107—62no—
306#107—62no—
307#106—62no—
308#106—121no—
309C4⋊C8—22no—
310#102—62no—
311#102—62no—
312#102—62no—
313#102—121no—
314C22⋊C8—22no—
315C22⋊C8—41no—
316#97—41no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95D1211yescharacteristic
325#95D1211yescharacteristic
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3AGL(1,ℤ/8)12yes—
333C23×C3C22×D411yescharacteristic
334C22×D3—22no—
335C22×D3—41no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—41no—
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—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4AGL(1,ℤ/8)12yes—
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4H(5,𝔽2)11yescharacteristic
365C2×C3×C42−1+411yescharacteristic
366#67—48no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—84no—
389D12—22no—
390D12—22no—
391D12—41no—
392D12—41no—
393C4×D3—84no—
394C3×C8—22no—
395C3×C8—41no—
396C3×C8—41no—
397C3×C8—41no—
398C24—61no—
399C24C2×D1211yescharacteristic
400C2×D4—32no—
401C2×D4—64no—
402C2×D4—61no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—122no—
408C2×D4—122no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—122no—
423C22×C4—122no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4—241no—
427C22×C4—241no—
428C22×C4C2×D1211yescharacteristic
429C22×C4C2×D1211yescharacteristic
430C22×C4C2×D1211yescharacteristic
431C22×C4C2×D1211yescharacteristic
432C22×C4C2×D1211yescharacteristic
433C22×C4C23×D311yescharacteristic
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8#23712yes—
446C4⋊C4—32no—
447C4⋊C4—32no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4—122no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—64no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—122no—
472C22⋊C4—122no—
473C22⋊C4—241no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—122no—
479C42—122no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3#52511yescharacteristic
488C22×C3#54611yescharacteristic
489C22×C3#54711yescharacteristic
490C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
491C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
492C22×C3#57711yescharacteristic
493C22×C3#57811yescharacteristic
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—42no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—81no—
500C2×D3—81no—
501C3×C4—22no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4#52911yescharacteristic
507C3×C4#53911yescharacteristic
508C3×C4#55311yescharacteristic
509C3×C4#55611yescharacteristic
510Dic3—42no—
511Dic3—42no—
512Dic3—81no—
513Dic3—81no—
514Dic3—81no—
515Dic3—81no—
516Dic3—81no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—62no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23#90812yes—
531C23C22×D1211yescharacteristic
532D4—62no—
533D4—62no—
534D4—128no—
535D4—121no—
536D4—121no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—244no—
577C2×C4—244no—
578C2×C4#90812yes—
579C2×C4C2×#23711yescharacteristic
580C2×C4#100411yescharacteristic
581C2×C4D3×#4111yescharacteristic
582C2×C4D3×#4111yescharacteristic
583C2×C4D3×#4111yescharacteristic
584C2×C4#101011yescharacteristic
585C8—22no—
586C8—41no—
587C8—41no—
588C8—41no—
589C2×C3—42no—
590C2×C3#246611yescharacteristic
591C2×C3#310911yescharacteristic
592C2×C3#312011yescharacteristic
593C2×C3#330811yescharacteristic
594C2×C3#332411yescharacteristic
595C2×C3#334311yescharacteristic
596C2×C3#334511yescharacteristic
597D3—82no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—122no—
606C22—122no—
607C22—122no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22#552411yescharacteristic
613C22#555411yescharacteristic
614C22#555511yescharacteristic
615C22C2×#90811yescharacteristic
616C22C2×#90811yescharacteristic
617C22#575111yescharacteristic
618C22#575311yescharacteristic
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—122no—
625C4—122no—
626C4—241no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4—241no—
631C4#560111yescharacteristic
632C4#566011yescharacteristic
633C4#568011yescharacteristic
634C4#570211yescharacteristic
635C3#2704711yescharacteristic, complemented, Sylow
636C2—42no—
637C2—242no—
638C2#7551611yescharacteristic
639C2#7934311yescharacteristic
640C2#7938511yescharacteristic
641C2#7946911yescharacteristic
642C2#7947111yescharacteristic
643C2#7950911yescharacteristic
644C2#7955511yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1924C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1924D3
hypocenterC3#27047
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1924

Complements

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

NH ≅ G/Ntype
1. #73512C2semidirect
3. C3×#1924C2semidirect
4. #70483C2semidirect
5. #70453C2semidirect
8. #69928C2semidirect
9. #69925C2semidirect
10. #69819C2semidirect
11. #69812C2semidirect
12. #68671C2semidirect
13. #68635C2semidirect
14. #68624C2semidirect
15. #68623C2semidirect
68. #1924D3semidirect
635. C3#27047semidirect

Maximal subgroups

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

subgroupindex
#735122
#734672
C3×#19242
#704832
#704532
#699342
#699282
#699252
#698192
#698122
#686712
#686352
#686242
#686232
#270473