Finite Group Databasebeta

← #10845440

Subgroups — #10845440

Order 768.

904 conjugacy classes of subgroups.

The subgroup lattice has 720 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#82384C211yesmaximal, characteristic, complemented
4C2×#4846C211yesmaximal, characteristic, complemented
5#71990C211yesmaximal, characteristic, complemented
6#71988C211yesmaximal, characteristic, complemented
7#71973C211yesmaximal, characteristic, complemented
8#71971C211yesmaximal, characteristic, complemented
9#70455C211yesmaximal, characteristic, complemented
10#69939C211yesmaximal, characteristic, complemented
11#69938C211yesmaximal, characteristic, complemented
12#69935C211yesmaximal, characteristic, complemented
13#69934C211yesmaximal, characteristic, complemented
14#69828C211yesmaximal, characteristic, complemented
15#69812C211yesmaximal, characteristic, complemented
16#25350—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C23×C3×C8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#930C2211yescharacteristic
21C2×#930C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C3×#435—24no—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29#5223—21no—
30#5221—21no—
31#5212C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34#5212C2211yescharacteristic
35#5212C2211yescharacteristic
36#5212C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#821C2211yescharacteristic
41C2×#818C2211yescharacteristic
42C2×#818C2211yescharacteristic
43C2×#818C2211yescharacteristic
44C2×#817C2211yescharacteristic
45#4992—21no—
46#4977—21no—
47#4956—21no—
48#4956—21no—
49#4947—21no—
50#4947—21no—
51#4936C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53#4846—24no—
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#474—31no—
63C2×#435D311yescharacteristic, complemented
64#2563—31no—
65#2535—31no—
66#2076—31no—
67#2075—31no—
68#2034—31no—
69#2033—31no—
70#1993—31no—
71#1949—31no—
72#1895—31no—
73#1893—31no—
74#1809—31no—
75#1808—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#930—22no—
108#930—22no—
109#930—41no—
110#930—41no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4×Dic3—21no—
123C4×D12—22no—
124C4×D12—22no—
125#869—22no—
126#869—22no—
127C3×C4⋊C8D414yescomplemented
128C3×C22⋊C8—22no—
129C3×C22⋊C8—22no—
130C3×#97—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#821—22no—
141#821—22no—
142#821—22no—
143#821—22no—
144#818—22no—
145#818—22no—
146#818—41no—
147#817—41no—
148C22×M16C2×D311yescharacteristic
149C23×C8C2×D311yescharacteristic
150C2×C4⋊Q8—31no—
151C2×#121—31no—
152C2×#121—31no—
153C2×C4×D4—31no—
154C2×C4×D4C2×D311yescharacteristic
155#474—64no—
156#435—24no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×C22⋊C8C2×D311yescharacteristic
167C2×C22⋊C8C2×D311yescharacteristic
168C2×#97C2×D311yescharacteristic
169#397—61no—
170#393—61no—
171#393—61no—
172#392—61no—
173#390—61no—
174#386—61no—
175#386—61no—
176#382—31no—
177#382—61no—
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184C4×C4⋊C4—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3C2×D411yescharacteristic
194C24×C3C2×D411yescharacteristic
195C23×D3—21no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—41no—
199C22×C3×C4—21no—
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×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C2×#67—42no—
217C2×#67—42no—
218C2×#67—42no—
219C2×#67—42no—
220C2×#67—81no—
221C2×#67—81no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12#4112yes—
235C2×C4×D3—42no—
236C2×C4×D3—81no—
237C2×C4×D3—81no—
238C2×Dic6—42no—
239C2×Dic6—81no—
240C2×Dic6—81no—
241C3×M16—44no—
242C2×C3×C8—21no—
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—44no—
252C2×C3×C8C2×D412yes—
253C2×C3×C8C2×D412yes—
254C2×C3×C8#4112yes—
255C3×C4⋊C4—22no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—22no—
258C3×C42C2×D412yes—
259C3×C42#4112yes—
260#217—81no—
261#217—81no—
262#217—81no—
263#217—81no—
264#217—81no—
265#217—81no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—42no—
269C4⋊Dic3—21no—
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—42no—
284C4×Dic3—42no—
285C22×Q8—61no—
286C22×D4—31no—
287C22×D4—61no—
288C22×D4—61no—
289C22×D4D1211yescharacteristic
290C23×C4—61no—
291C23×C4C22×D311yescharacteristic
292C23×C4C22×D311yescharacteristic
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—41no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—41no—
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C4⋊Q8—62no—
304C4⋊Q8—62no—
305#121—62no—
306#121—62no—
307#121—121no—
308#121—121no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—62no—
312C4×D4—62no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4D1211yescharacteristic
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
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⋊C4D1211yescharacteristic
332C2×C22⋊C4D1211yescharacteristic
333C2×C42—61no—
334C2×C42C22×D311yescharacteristic
335#107—62no—
336#107—62no—
337#107—121no—
338#106—121no—
339C4⋊C8#6714yescomplemented
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344C22⋊C8—22no—
345C22⋊C8—22no—
346#97—22no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—21no—
357C23×C3—21no—
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#13712yes—
370C23×C3C22×D411yescharacteristic
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C3×D4—44no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
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—41no—
390C2×C3×C4—41no—
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#12112yes—
398C2×C3×C4#12112yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4H(5,𝔽2)11yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405#67—84no—
406#67—84no—
407C2×Dic3—41no—
408C2×Dic3—41no—
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—
428D12—22no—
429D12—22no—
430D12—41no—
431D12—41no—
432C4×D3—84no—
433Dic6—84no—
434C3×C8—22no—
435C3×C8—22no—
436C3×C8—22no—
437C3×C8—41no—
438C24—61no—
439C24C2×D1211yescharacteristic
440C24C2×D1211yescharacteristic
441C2×Q8—122no—
442C2×Q8—241no—
443C2×Q8—241no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—32no—
447C2×D4—41no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—241no—
458C2×D4—241no—
459C22×C4—21no—
460C22×C4—21no—
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—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—122no—
479C22×C4—241no—
480C22×C4—241no—
481C22×C4C2×D1211yescharacteristic
482C22×C4C2×D1211yescharacteristic
483C22×C4C2×D1211yescharacteristic
484C22×C4C2×D1211yescharacteristic
485C22×C4C23×D311yescharacteristic
486M16—44no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—44no—
497C2×C8#23712yes—
498C2×C8C2×#6712yes—
499C2×C8C2×#6712yes—
500C4⋊C4—22no—
501C4⋊C4—61no—
502C4⋊C4—61no—
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—122no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C22⋊C4—241no—
524C22⋊C4—241no—
525C22⋊C4—241no—
526C22⋊C4—241no—
527C42—122no—
528C42#23512yes—
529C42C2×#6712yes—
530C22×C3—22no—
531C22×C3—22no—
532C22×C3—22no—
533C22×C3—41no—
534C22×C3—41no—
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×C3C2×#12111yescharacteristic
545C22×C3#54611yescharacteristic
546C22×C3#54711yescharacteristic
547C22×C3C2×#13711yescharacteristic
548C22×C3#57511yescharacteristic
549C22×C3#57611yescharacteristic
550C22×C3#57611yescharacteristic
551C2×D3—42no—
552C2×D3—42no—
553C2×D3—42no—
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C2×D3—81no—
558C3×C4—22no—
559C3×C4—22no—
560C3×C4—41no—
561C3×C4—41no—
562C3×C4#52911yescharacteristic
563C3×C4#54211yescharacteristic
564C3×C4#54611yescharacteristic
565C3×C4#54811yescharacteristic
566Dic3—81no—
567Dic3—81no—
568Dic3—81no—
569Dic3—81no—
570Dic3—81no—
571Dic3—81no—
572Dic3—81no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—62no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23#90912yes—
594C23C22×D1211yescharacteristic
595Q8—244no—
596D4—44no—
597D4—62no—
598D4—62no—
599D4—121no—
600D4—121no—
601D4—244no—
602D4—244no—
603C2×C4—21no—
604C2×C4—21no—
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—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—121no—
622C2×C4—121no—
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—244no—
643C2×C4#93012yes—
644C2×C4#93012yes—
645C2×C4C2×#23511yescharacteristic
646C2×C4C2×#23711yescharacteristic
647C2×C4#100411yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C2×C4#101011yescharacteristic
650C2×C4C22×#6711yescharacteristic
651C8—22no—
652C8—22no—
653C8—22no—
654C8—41no—
655C2×C3—42no—
656C2×C3—81no—
657C2×C3#242711yescharacteristic
658C2×C3#305611yescharacteristic
659C2×C3#305811yescharacteristic
660C2×C3#331411yescharacteristic
661C2×C3#331911yescharacteristic
662C2×C3#334611yescharacteristic
663C2×C3#334711yescharacteristic
664D3—82no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—122no—
680C22—122no—
681C22—122no—
682C22—241no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22#555411yescharacteristic
687C22#555511yescharacteristic
688C22C2×#90911yescharacteristic
689C22#574811yescharacteristic
690C22#575211yescharacteristic
691C22#575211yescharacteristic
692C22C2×#93011yescharacteristic
693C4—22no—
694C4—22no—
695C4—41no—
696C4—41no—
697C4—241no—
698C4—241no—
699C4—241no—
700C4—241no—
701C4—241no—
702C4—241no—
703C4—241no—
704C4#555711yescharacteristic
705C4#556111yescharacteristic
706C4#580611yescharacteristic
707C4#583111yescharacteristic
708C3#2535011yescharacteristic, complemented, Sylow
709C2—42no—
710C2—81no—
711C2—242no—
712C2#7947211yescharacteristic
713C2#7947311yescharacteristic
714C2#7950311yescharacteristic
715C2#7950611yescharacteristic
716C2#8240811yescharacteristic
717C2#8377611yescharacteristic
718C2#8379911yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#435C2
socleC23×C3C22×D4
prime radical#82415C2
hypercenterC2×#435D3
hypocenterC3#25350
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. #82384C2semidirect
4. C2×#4846C2semidirect
5. #71990C2semidirect
6. #71988C2semidirect
7. #71973C2semidirect
8. #71971C2semidirect
9. #70455C2semidirect
10. #69939C2semidirect
11. #69938C2semidirect
12. #69935C2semidirect
13. #69934C2semidirect
14. #69828C2semidirect
15. #69812C2semidirect
63. C2×#435D3semidirect
127. C3×C4⋊C8D4semidirect
339. C4⋊C8#67semidirect
708. C3#25350semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4352
#824152
#823842
C2×#48462
#719902
#719882
#719732
#719712
#704552
#699392
#699382
#699352
#699342
#698282
#698122
#253503