Finite Group Databasebeta

← #10845500

Subgroups — #10845500

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#435C211yesmaximal, characteristic, complemented
2#82415C211yesmaximal, characteristic
3#82382C211yesmaximal, characteristic, complemented
4C2×#4842C211yesmaximal, characteristic, complemented
5#71990C211yesmaximal, characteristic, complemented
6#71988C211yesmaximal, characteristic, complemented
7#71973C211yesmaximal, characteristic, complemented
8#71970C211yesmaximal, characteristic, complemented
9#70455C211yesmaximal, characteristic, complemented
10#69940C211yesmaximal, characteristic, complemented
11#69940C211yesmaximal, characteristic, complemented
12#69935C211yesmaximal, characteristic, complemented
13#69934C211yesmaximal, characteristic, complemented
14#69828C211yesmaximal, characteristic, complemented
15#69812C211yesmaximal, characteristic, complemented
16#25354—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C23×C3×C8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#936—21no—
21C2×#930C2211yescharacteristic
22C2×#930C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C3×#435—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30#5223—21no—
31#5212C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34#5212C2211yescharacteristic
35#5212C2211yescharacteristic
36#5212C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#821C2211yescharacteristic
41C2×#818C2211yescharacteristic
42C2×#817C2211yescharacteristic
43C2×#817C2211yescharacteristic
44C2×#817C2211yescharacteristic
45#4992—21no—
46#4973—21no—
47#4956—21no—
48#4956—21no—
49#4953—21no—
50#4953—21no—
51#4936C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53#4842—24no—
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#476—31no—
63C2×#435D311yescharacteristic, complemented
64#2547—31no—
65#2535—31no—
66#2077—31no—
67#2077—31no—
68#2034—31no—
69#2033—31no—
70#1993—31no—
71#1949—31no—
72#1895—31no—
73#1893—31no—
74#1809—31no—
75#1807—31no—
76#1762—31no—
77C22×C3×D4D411yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×#67—21no—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C22×Dic6—21no—
85C2×C3×M16—22no—
86C2×C3×M16—22no—
87C2×C3×M16—41no—
88C22×C3×C8—22no—
89C22×C3×C8—22no—
90C22×C3×C8—41no—
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2311yescharacteristic
94C22×C3×C8C2311yescharacteristic
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106C2×#217—21no—
107#936—44no—
108#930—22no—
109#930—22no—
110#930—41no—
111#930—41no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C2×C4×Dic3—21no—
124C4×D12—22no—
125C4×D12—22no—
126#869—22no—
127#869—22no—
128C3×C4⋊C8D414yescomplemented
129C3×C22⋊C8—22no—
130C3×C22⋊C8—22no—
131C3×#97—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#821—22no—
142#821—22no—
143#821—22no—
144#821—22no—
145#818—41no—
146#817—22no—
147#817—22no—
148#817—41no—
149C22×M16C2×D311yescharacteristic
150C23×C8C2×D311yescharacteristic
151C2×C4⋊Q8—31no—
152C2×#124—61no—
153C2×#121—31no—
154C2×#121—31no—
155C2×C4×D4—31no—
156C2×C4×D4C2×D311yescharacteristic
157#476—64no—
158#435—24no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106—31no—
163C2×C4⋊C8C2×D311yescharacteristic
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C22⋊C8C2×D311yescharacteristic
169C2×C22⋊C8C2×D311yescharacteristic
170C2×#97C2×D311yescharacteristic
171#397—61no—
172#397—61no—
173#397—61no—
174#392—61no—
175#391—61no—
176#386—61no—
177#386—61no—
178#382—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184#380—31no—
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3C2×D411yescharacteristic
195C24×C3C2×D411yescharacteristic
196C23×D3—21no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—41no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C2×#67—42no—
218C2×#67—42no—
219C2×#67—42no—
220C2×#67—42no—
221C2×#67—81no—
222C2×#67—81no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12#4112yes—
236C2×C4×D3—42no—
237C2×C4×D3—81no—
238C2×C4×D3—81no—
239C2×Dic6—42no—
240C2×Dic6—42no—
241C2×Dic6—42no—
242C3×M16—44no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—44no—
253C2×C3×C8C2×D412yes—
254C2×C3×C8C2×D412yes—
255C2×C3×C8#4112yes—
256C3×C4⋊C4—22no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C42C2×D412yes—
260C3×C42#4112yes—
261#217—42no—
262#217—42no—
263#217—42no—
264#217—42no—
265#217—81no—
266#217—81no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—42no—
269(C2×D3)⋊C4—42no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—21no—
284C4⋊Dic3—42no—
285C4×Dic3—42no—
286C22×Q8—61no—
287C22×D4—31no—
288C22×D4—61no—
289C22×D4—61no—
290C22×D4D1211yescharacteristic
291C23×C4—61no—
292C23×C4C22×D311yescharacteristic
293C23×C4C22×D311yescharacteristic
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—41no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—41no—
300C22×C8C22×D311yescharacteristic
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C22×C8C22×D311yescharacteristic
304C4⋊Q8—62no—
305C4⋊Q8—62no—
306#124—124no—
307#121—62no—
308#121—62no—
309#121—121no—
310#121—121no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—62no—
314C4×D4—62no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4D1211yescharacteristic
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4D1211yescharacteristic
334C2×C22⋊C4D1211yescharacteristic
335C2×C42—61no—
336C2×C42C22×D311yescharacteristic
337#107—121no—
338#106—62no—
339#106—62no—
340#106—121no—
341C4⋊C8#6714yescomplemented
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346C22⋊C8—22no—
347C22⋊C8—22no—
348#97—22no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3AGL(1,ℤ/8)12yes—
372C23×C3C22×D411yescharacteristic
373C22×D3—22no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C3×D4—44no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4#12112yes—
400C2×C3×C4#12112yes—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407#67—84no—
408#67—84no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430D12—22no—
431D12—22no—
432D12—41no—
433D12—41no—
434C4×D3—84no—
435Dic6—84no—
436C3×C8—22no—
437C3×C8—22no—
438C3×C8—22no—
439C3×C8—41no—
440C24—61no—
441C24C2×D1211yescharacteristic
442C24C2×D1211yescharacteristic
443C2×Q8—122no—
444C2×Q8—122no—
445C2×Q8—122no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—32no—
449C2×D4—41no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—122no—
458C2×D4—122no—
459C2×D4—241no—
460C2×D4—241no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—122no—
481C22×C4—241no—
482C22×C4—241no—
483C22×C4C2×D1211yescharacteristic
484C22×C4C2×D1211yescharacteristic
485C22×C4C2×D1211yescharacteristic
486C22×C4C2×D1211yescharacteristic
487C22×C4C23×D311yescharacteristic
488M16—44no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—44no—
499C2×C8#23712yes—
500C2×C8C2×#6712yes—
501C2×C8C2×#6712yes—
502C4⋊C4—22no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—122no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—122no—
527C22⋊C4—241no—
528C22⋊C4—241no—
529C42—122no—
530C42#23512yes—
531C42C2×#6712yes—
532C22×C3—22no—
533C22×C3—22no—
534C22×C3—22no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3—41no—
546C22×C3C2×#12111yescharacteristic
547C22×C3#54611yescharacteristic
548C22×C3#54711yescharacteristic
549C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
550C22×C3#57511yescharacteristic
551C22×C3#57711yescharacteristic
552C22×C3#57711yescharacteristic
553C2×D3—42no—
554C2×D3—42no—
555C2×D3—42no—
556C2×D3—81no—
557C2×D3—81no—
558C2×D3—81no—
559C2×D3—81no—
560C3×C4—22no—
561C3×C4—22no—
562C3×C4—41no—
563C3×C4—41no—
564C3×C4#52911yescharacteristic
565C3×C4#54211yescharacteristic
566C3×C4#54611yescharacteristic
567C3×C4#54811yescharacteristic
568Dic3—81no—
569Dic3—81no—
570Dic3—81no—
571Dic3—81no—
572Dic3—81no—
573Dic3—81no—
574Dic3—81no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—62no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23#90812yes—
596C23C22×D1211yescharacteristic
597Q8—244no—
598D4—44no—
599D4—62no—
600D4—62no—
601D4—121no—
602D4—121no—
603D4—244no—
604D4—244no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—244no—
645C2×C4#93012yes—
646C2×C4#93012yes—
647C2×C4C2×#23511yescharacteristic
648C2×C4C2×#23711yescharacteristic
649C2×C4#100411yescharacteristic
650C2×C4D3×#4111yescharacteristic
651C2×C4#101011yescharacteristic
652C2×C4C22×#6711yescharacteristic
653C8—22no—
654C8—22no—
655C8—22no—
656C8—41no—
657C2×C3—42no—
658C2×C3—81no—
659C2×C3#242711yescharacteristic
660C2×C3#305511yescharacteristic
661C2×C3#305711yescharacteristic
662C2×C3#330611yescharacteristic
663C2×C3#331111yescharacteristic
664C2×C3#334511yescharacteristic
665C2×C3#335011yescharacteristic
666D3—82no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—122no—
682C22—122no—
683C22—122no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22#555411yescharacteristic
689C22#555511yescharacteristic
690C22C2×#90811yescharacteristic
691C22#574811yescharacteristic
692C22#575111yescharacteristic
693C22#575111yescharacteristic
694C22C2×#93011yescharacteristic
695C4—22no—
696C4—22no—
697C4—41no—
698C4—41no—
699C4—241no—
700C4—241no—
701C4—241no—
702C4—241no—
703C4—241no—
704C4—241no—
705C4—241no—
706C4#555711yescharacteristic
707C4#556111yescharacteristic
708C4#580611yescharacteristic
709C4#583111yescharacteristic
710C3#2535411yescharacteristic, complemented, Sylow
711C2—42no—
712C2—81no—
713C2—242no—
714C2#7947111yescharacteristic
715C2#7947611yescharacteristic
716C2#7949911yescharacteristic
717C2#7950211yescharacteristic
718C2#8240811yescharacteristic
719C2#8377211yescharacteristic
720C2#8379811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#435C2
socleC23×C3C22×D4
prime radical#82415C2
hypercenterC2×#435D3
hypocenterC3#25354
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#435

Complements

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

NH ≅ G/Ntype
1. C2×C3×#435C2semidirect
3. #82382C2semidirect
4. C2×#4842C2semidirect
5. #71990C2semidirect
6. #71988C2semidirect
7. #71973C2semidirect
8. #71970C2semidirect
9. #70455C2semidirect
10. #69940C2semidirect
11. #69940C2semidirect
12. #69935C2semidirect
13. #69934C2semidirect
14. #69828C2semidirect
15. #69812C2semidirect
63. C2×#435D3semidirect
128. C3×C4⋊C8D4semidirect
341. C4⋊C8#67semidirect
710. C3#25354semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4352
#824152
#823822
C2×#48422
#719902
#719882
#719732
#719702
#704552
#699402
#699352
#699342
#698282
#698122
#253543