Finite Group Databasebeta

← #10845342

Subgroups — #10845342

Order 768.

914 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82416C211yesmaximal, characteristic
2C2×#5009C211yesmaximal, characteristic, complemented
3C2×C8×D12C211yesmaximal, characteristic, complemented
4#73806C211yesmaximal, characteristic, complemented
5C3×C4×#102C211yesmaximal, characteristic, complemented
6#70207C211yesmaximal, characteristic, complemented
7#70206C211yesmaximal, characteristic, complemented
8#70202C211yesmaximal, characteristic, complemented
9#70202C211yesmaximal, characteristic, complemented
10#69778C211yesmaximal, characteristic, complemented
11#69468C211yesmaximal, characteristic, complemented
12#68890C211yesmaximal, characteristic, complemented
13#68869C211yesmaximal, characteristic, complemented
14#68868C211yesmaximal, characteristic, complemented
15#68833C211yesmaximal, characteristic, complemented
16#25342—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#938C2211yescharacteristic
19C2×#938C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21C22×D3×C8C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×#870C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5213—21no—
31#5213—21no—
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34#5009—24no—
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4987C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4960—21no—
46#4953—21no—
47#4940—21no—
48#4940—21no—
49#4936C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C8×D12—24no—
52#4672—21no—
53#4659—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478—31no—
63C2×C8×D4—31no—
64#2562—31no—
65#2535—31no—
66#2077—31no—
67#2076—31no—
68#2034—31no—
69#2033—31no—
70#1991—31no—
71#1949—31no—
72#1893—31no—
73#1893—31no—
74#1808—31no—
75#1807—31no—
76C4×#102D311yescharacteristic, complemented
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×#67—21no—
80C22×#67—21no—
81C22×D12D411yescharacteristic
82C22×C4×D3C2311yescharacteristic
83C22×C4×D3C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96#938—22no—
97#938—22no—
98#938—41no—
99#938—41no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4D411yescharacteristic
107C2×(C2×D3)⋊C4D411yescharacteristic
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C2×D3×C8—22no—
118C2×D3×C8—22no—
119C2×D3×C8—22no—
120C2×D3×C8—22no—
121C2×D3×C8—41no—
122C2×D3×C8—41no—
123C4×D12—22no—
124C4×D12—22no—
125#870—22no—
126#870—22no—
127C3×#102—22no—
128C3×#102—22no—
129C3×C4×C8—22no—
130C3×#95—21no—
131#832—22no—
132#832—22no—
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(C2×D3)⋊C8—22no—
142(C2×D3)⋊C8—22no—
143#808—22no—
144#808—41no—
145#807—22no—
146#807—41no—
147C4⋊(C3⋊C8)D414yescomplemented
148C23×C8—31no—
149C23×C8—31no—
150C2×#125—31no—
151C2×#121—31no—
152C2×#121—31no—
153C2×C4×D4—31no—
154C2×C4×D4C2×D311yescharacteristic
155#478—64no—
156C8×D4—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102C2×D311yescharacteristic
165C2×#102C2×D311yescharacteristic
166C2×C22⋊C8—31no—
167C2×C22⋊C8—31no—
168C2×C4×C8C2×D311yescharacteristic
169#400—61no—
170#397—61no—
171#394—61no—
172#392—61no—
173#388—61no—
174#388—61no—
175#386—61no—
176#386—61no—
177#382—31no—
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184C4×C4⋊C4C2×D311yescharacteristic
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C24×C3—21no—
194C23×D3C2×D411yescharacteristic
195C23×D3C2×D411yescharacteristic
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4#4112yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—42no—
206C22×C3×C4—81no—
207C22×C3×C4—81no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C2×#67—42no—
212C2×#67—42no—
213C2×#67—42no—
214C2×#67—42no—
215C2×#67—81no—
216C2×#67—81no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C2×D12—22no—
224C2×D12—22no—
225C2×D12—22no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8#4112yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C22⋊C4—42no—
251C3×C42—81no—
252C3×C42C2×D412yes—
253C3×C42#4112yes—
254#217—42no—
255#217—42no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264C4⋊Dic3—22no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273Dic3⋊C4—42no—
274Dic3⋊C4—42no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8C2×D412yes—
280C2×C3⋊C8C2×D412yes—
281D3×C8—44no—
282D3×C8—44no—
283C22×D4—31no—
284C22×D4—61no—
285C22×D4—61no—
286C22×D4C22×D311yescharacteristic
287C23×C4—21no—
288C23×C4—31no—
289C23×C4—31no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8—62no—
293C22×C8—62no—
294C22×C8—62no—
295C22×C8—62no—
296C22×C8—121no—
297C22×C8—121no—
298C22×C8C22×D311yescharacteristic
299C22×C8C22×D311yescharacteristic
300#125—62no—
301#125—62no—
302#121—62no—
303#121—62no—
304#121—121no—
305#121—121no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—62no—
309C4×D4—62no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—31no—
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×C42—21no—
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#107—121no—
335#106—62no—
336#106—121no—
337C4⋊C8—34no—
338#102—22no—
339#102—22no—
340#102—62no—
341#102—62no—
342C22⋊C8—62no—
343C22⋊C8—62no—
344C4×C8—22no—
345#95—21no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3—22no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3C22×D411yescharacteristic
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3#13512yes—
375C22×D3#13512yes—
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—41no—
379C3×D4—41no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—84no—
394C2×C3×C4#12112yes—
395C2×C3×C4#12112yes—
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4H(5,𝔽2)11yescharacteristic
401C2×C3×C42−1+411yescharacteristic
402#67—84no—
403#67—84no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
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—
422D12—28no—
423C4×D3—41no—
424C4×D3—41no—
425C4×D3—41no—
426C4×D3—41no—
427C4×D3—41no—
428C4×D3—41no—
429C4×D3—41no—
430C4×D3—41no—
431C3×C8—22no—
432C3×C8—41no—
433C3⋊C8—22no—
434C3⋊C8—22no—
435C24—21no—
436C24—31no—
437C24—31no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—62no—
444C2×D4—62no—
445C2×D4—62no—
446C2×D4—122no—
447C2×D4—122no—
448C2×D4—122no—
449C2×D4—122no—
450C2×D4—241no—
451C2×D4—241no—
452C2×D4#23512yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—42no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4D3×D411yescharacteristic
478C22×C4D3×D411yescharacteristic
479C22×C4C23×D311yescharacteristic
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—32no—
486C2×C8—32no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—124no—
492C2×C8—124no—
493C2×C8#23512yes—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—62no—
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—122no—
510C4⋊C4—122no—
511C22⋊C4—42no—
512C22⋊C4—62no—
513C22⋊C4—62no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—241no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C42—81no—
523C42#23512yes—
524C42D3×D412yes—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3C2×#12111yescharacteristic
533C22×C3#54611yescharacteristic
534C22×C3#54711yescharacteristic
535C22×C3C2×#13511yescharacteristic
536C22×C3C2×#13511yescharacteristic
537C22×C3#57611yescharacteristic
538C22×C3#57711yescharacteristic
539C2×D3—22no—
540C2×D3—22no—
541C2×D3—22no—
542C2×D3—22no—
543C2×D3—22no—
544C2×D3—22no—
545C2×D3—41no—
546C2×D3—41no—
547C2×D3—41no—
548C2×D3—41no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—41no—
553C3×C4—22no—
554C3×C4—22no—
555C3×C4—81no—
556C3×C4—81no—
557C3×C4—81no—
558C3×C4#52911yescharacteristic
559C3×C4#54211yescharacteristic
560C3×C4#54611yescharacteristic
561C3×C4#54811yescharacteristic
562Dic3—41no—
563Dic3—41no—
564Dic3—81no—
565Dic3—81no—
566Dic3—81no—
567Dic3—81no—
568C23—22no—
569C23—32no—
570C23—32no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—61no—
578C23—61no—
579C23—61no—
580C23—61no—
581C23—61no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—61no—
589C23C2×D3×D411yescharacteristic
590D4—22no—
591D4—22no—
592D4—41no—
593D4—41no—
594D4—68no—
595D4—244no—
596D4—244no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—84no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
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#88412yes—
638C2×C4#88412yes—
639C2×C4C2×#23511yescharacteristic
640C2×C4C2×#23511yescharacteristic
641C2×C4C2×#23511yescharacteristic
642C2×C4C2×D3×D411yescharacteristic
643C2×C4#100911yescharacteristic
644C2×C4#101011yescharacteristic
645C8—22no—
646C8—41no—
647C8—62no—
648C8—62no—
649C2×C3—82no—
650C2×C3#242711yescharacteristic
651C2×C3#305011yescharacteristic
652C2×C3#306211yescharacteristic
653C2×C3#329611yescharacteristic
654C2×C3#329711yescharacteristic
655C2×C3#333711yescharacteristic
656C2×C3#333911yescharacteristic
657D3—42no—
658D3—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—62no—
663C22—62no—
664C22—62no—
665C22—62no—
666C22—62no—
667C22—62no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—121no—
673C22—121no—
674C22—121no—
675C22—121no—
676C22—121no—
677C22—121no—
678C22—121no—
679C22—121no—
680C22C2×#88411yescharacteristic
681C22#555011yescharacteristic
682C22#555111yescharacteristic
683C22#575711yescharacteristic
684C22#576211yescharacteristic
685C22D3×#13511yescharacteristic
686C22D3×#13511yescharacteristic
687C4—22no—
688C4—22no—
689C4—81no—
690C4—81no—
691C4—81no—
692C4—121no—
693C4—121no—
694C4—241no—
695C4—241no—
696C4—241no—
697C4—241no—
698C4#547911yescharacteristic
699C4#553411yescharacteristic
700C4#555011yescharacteristic
701C4#555211yescharacteristic
702C3#2534211yescharacteristic, complemented, Sylow
703C2—82no—
704C2—122no—
705C2—122no—
706C2#7348611yescharacteristic
707C2#7875211yescharacteristic
708C2#7883811yescharacteristic
709C2#7960911yescharacteristic
710C2#7961511yescharacteristic
711C2#7967711yescharacteristic
712C2#7968911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#102C2
socleC23×C3C22×D4
prime radical#82416C2
hypercenterC4×#102D3
hypocenterC3#25342
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×#5009C2semidirect
3. C2×C8×D12C2semidirect
4. #73806C2semidirect
5. C3×C4×#102C2semidirect
6. #70207C2semidirect
7. #70206C2semidirect
8. #70202C2semidirect
9. #70202C2semidirect
10. #69778C2semidirect
11. #69468C2semidirect
12. #68890C2semidirect
13. #68869C2semidirect
14. #68868C2semidirect
15. #68833C2semidirect
76. C4×#102D3semidirect
147. C4⋊(C3⋊C8)D4semidirect
702. C3#25342semidirect

Maximal subgroups

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

subgroupindex
#824162
C2×#50092
C2×C8×D122
#738062
C3×C4×#1022
#702072
#702062
#702022
#697782
#694682
#688902
#688692
#688682
#688332
#253423