Finite Group Databasebeta

← #10845293

Subgroups — #10845293

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4890C211yesmaximal, characteristic, complemented
2C2×#67⋊C8C211yesmaximal, characteristic, complemented
3#73937C211yesmaximal, characteristic, complemented
4#73935C211yesmaximal, characteristic
5C3×#2075C211yesmaximal, characteristic, complemented
6#70630C211yesmaximal, characteristic, complemented
7#69934C211yesmaximal, characteristic, complemented
8#69515C211yesmaximal, characteristic, complemented
9#69512C211yesmaximal, characteristic, complemented
10#68889C211yesmaximal, characteristic, complemented
11#68883C211yesmaximal, characteristic, complemented
12#68822C211yesmaximal, characteristic, complemented
13#68803C211yesmaximal, characteristic, complemented
14#68795C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#25338—31nomaximal, Sylow
17C2×#930C2211yescharacteristic
18C2×C4×#67C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21C2×#895C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×#885—21no—
25C2×#884—21no—
26C2×C3×#107C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×#380C2211yescharacteristic
29C3×#380C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4975C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4959—21no—
47#4952—21no—
48#4890—24no—
49#67⋊C8—24no—
50#4675—21no—
51#4672—21no—
52#4667—21no—
53#4646—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#474—31no—
63C2×C8×D4—31no—
64#2547—31no—
65#2537—31no—
66#2075—31no—
67#2075D311yescharacteristic, complemented
68#2033—31no—
69#2033—31no—
70#1990—31no—
71#1947—31no—
72#1893—31no—
73#1893—31no—
74#1808—31no—
75#1808—31no—
76C4×#102—31no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C22×#67D411yescharacteristic
80C22×D12—21no—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3C2311yescharacteristic
84C22×Dic6—21no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×#217—21no—
94C2×#217D411yescharacteristic
95#930—22no—
96#930—41no—
97C4×#67—22no—
98C4×#67—22no—
99C2×(C2×D3)⋊C4—21no—
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)⋊C4D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4D411yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8—22no—
116C22×C3⋊C8—22no—
117C22×C3⋊C8—41no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×D3×C8—22no—
121C2×D3×C8—22no—
122C2×D3×C8—41no—
123#895—22no—
124#895—41no—
125#893—22no—
126#893—22no—
127#888—22no—
128#888—22no—
129#885—44no—
130#884—44no—
131C3×#107—22no—
132C3×C22⋊C8—22no—
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(C2×D3)⋊C8—22no—
143#818—41no—
144Dic3⋊C8D414yescomplemented
145C8×Dic3—22no—
146#809—22no—
147#809—22no—
148#807—22no—
149#807—41no—
150C23×C8—31no—
151C23×C8—31no—
152C2×C4⋊Q8—31no—
153C2×#124—61no—
154C2×#121—31no—
155C2×#121—31no—
156C2×#121—61no—
157C2×C4×D4—31no—
158C2×C4×D4—31no—
159#474—64no—
160C8×D4—64no—
161C2×#107—31no—
162C2×#107—31no—
163C2×#107—31no—
164C2×#107C2×D311yescharacteristic
165C2×C4⋊C8—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×C22⋊C8—31no—
171C2×C22⋊C8C2×D311yescharacteristic
172C2×C4×C8—31no—
173#397—61no—
174#397—61no—
175#394—61no—
176#394—61no—
177#391—61no—
178#384—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—61no—
184#380C2×D311yescharacteristic
185#380C2×D311yescharacteristic
186C4×C4⋊C4—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194#340C2×D311yescharacteristic
195C24×C3C2×D411yescharacteristic
196C23×D3—21no—
197C23×D3C2×D411yescharacteristic
198C22×C3×C4—21no—
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×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C2×#67—22no—
211C2×#67—22no—
212C2×#67—42no—
213C2×#67—42no—
214C2×#67—81no—
215C2×#67D814yes—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—42no—
228C2×D12—42no—
229C2×D12—81no—
230C2×D12#4112yes—
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—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×Dic6—42no—
241C2×Dic6—42no—
242C2×Dic6—42no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8C2×D412yes—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C22⋊C4—42no—
258C3×C22⋊C4—42no—
259#217—22no—
260#217—42no—
261(C2×D3)⋊C4—22no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—42no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274Dic3⋊C4—22no—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—81no—
278C4×Dic3—42no—
279C4×Dic3C2×D412yes—
280C4×Dic3#4112yes—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—44no—
286C2×C3⋊C8C2×D412yes—
287C2×C3⋊C8#4112yes—
288D3×C8—44no—
289C22×Q8—61no—
290C22×D4—31no—
291C22×D4—31no—
292C22×D4—61no—
293C22×D4—61no—
294C23×C4—31no—
295C23×C4—61no—
296C23×C4C22×D311yescharacteristic
297C22×C8—31no—
298C22×C8—31no—
299C22×C8—62no—
300C22×C8—62no—
301C22×C8—62no—
302C22×C8—62no—
303C22×C8—121no—
304C22×C8—121no—
305C22×C8C22×D311yescharacteristic
306C22×C8C22×D311yescharacteristic
307C4⋊Q8—62no—
308C4⋊Q8—62no—
309#124—124no—
310#121—62no—
311#121—62no—
312#121—124no—
313#121—121no—
314#121—121no—
315C4×D4—62no—
316C4×D4—62no—
317C4×D4—62no—
318C4×D4—62no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—31no—
333C2×C22⋊C4—31no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C42—31no—
342C2×C42—61no—
343#107—22no—
344#107—62no—
345#107—121no—
346#107—121no—
347C4⋊C8—34no—
348#102—62no—
349#102—62no—
350#102—62no—
351#102—62no—
352C22⋊C8—22no—
353C22⋊C8—62no—
354C4×C8—62no—
355#95—21no—
356#95—21no—
357#95—61no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3C2×D812yes—
369C23×C3C22×D411yescharacteristic
370C22×D3—21no—
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—22no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3C2×D812yes—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
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—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4H(5,𝔽2)11yescharacteristic
408C2×C3×C42−1+411yescharacteristic
409#67—28no—
410#67—84no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
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#12112yes—
426C2×Dic3#12112yes—
427C2×Dic3C2×D812yes—
428C2×Dic3#13512yes—
429D12—22no—
430D12—22no—
431D12—41no—
432D12—41no—
433D12—84no—
434C4×D3—41no—
435C4×D3—41no—
436C4×D3—41no—
437C4×D3—41no—
438C4×D3—84no—
439Dic6—84no—
440C3×C8—22no—
441C3×C8—41no—
442C3⋊C8—22no—
443C3⋊C8—22no—
444C24—31no—
445C24—61no—
446C24D3×D411yescharacteristic
447C2×Q8—122no—
448C2×Q8—122no—
449C2×Q8—122no—
450C2×D4—34no—
451C2×D4—32no—
452C2×D4—62no—
453C2×D4—62no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—122no—
460C2×D4—122no—
461C2×D4—122no—
462C2×D4—122no—
463C2×D4—241no—
464C2×D4—241no—
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—21no—
474C22×C4—21no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—122no—
488C22×C4—122no—
489C22×C4—122no—
490C22×C4D3×D411yescharacteristic
491C22×C4C23×D311yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—32no—
498C2×C8—32no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8—124no—
504C2×C8—124no—
505C2×C8D3×D412yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—62no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—61no—
520C4⋊C4—61no—
521C4⋊C4—122no—
522C4⋊C4—122no—
523C4⋊C4—241no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—62no—
527C22⋊C4—62no—
528C22⋊C4—122no—
529C22⋊C4—122no—
530C22⋊C4—122no—
531C22⋊C4—122no—
532C22⋊C4—122no—
533C22⋊C4—122no—
534C22⋊C4—122no—
535C42—32no—
536C42—32no—
537C42—122no—
538C22×C3—22no—
539C22×C3—22no—
540C22×C3—22no—
541C22×C3—41no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3C2×#12111yescharacteristic
546C22×C3#54611yescharacteristic
547C22×C3#54711yescharacteristic
548C22×C3C22×D811yescharacteristic
549C22×C3C2×#13511yescharacteristic
550C22×C3#57711yescharacteristic
551C22×C3#57811yescharacteristic
552C2×D3—22no—
553C2×D3—22no—
554C2×D3—22no—
555C2×D3—41no—
556C2×D3—41no—
557C2×D3—41no—
558C2×D3—41no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3—42no—
562C2×D3—81no—
563C2×D3—81no—
564C2×D3—81no—
565C2×D3—81no—
566C3×C4—41no—
567C3×C4—81no—
568C3×C4—81no—
569C3×C4—81no—
570C3×C4—81no—
571C3×C4#52911yescharacteristic
572C3×C4#54211yescharacteristic
573C3×C4#54611yescharacteristic
574C3×C4#54811yescharacteristic
575Dic3—22no—
576Dic3—41no—
577Dic3—81no—
578Dic3—81no—
579Dic3—81no—
580Dic3#46614yes—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—32no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23—61no—
594C23—62no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23D3×D812yes—
602C23C2×D3×D411yescharacteristic
603Q8—244no—
604D4—68no—
605D4—62no—
606D4—62no—
607D4—121no—
608D4—121no—
609D4—244no—
610D4—244no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—32no—
615C2×C4—32no—
616C2×C4—32no—
617C2×C4—32no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—244no—
653C2×C4C2×D3×D411yescharacteristic
654C2×C4D3×#4111yescharacteristic
655C2×C4D3×#4111yescharacteristic
656C2×C4D3×#4111yescharacteristic
657C2×C4#100911yescharacteristic
658C2×C4#101011yescharacteristic
659C8—22no—
660C8—41no—
661C8—62no—
662C8—62no—
663C2×C3—42no—
664C2×C3#242711yescharacteristic
665C2×C3C2×#46611yescharacteristic
666C2×C3#306311yescharacteristic
667C2×C3#329411yescharacteristic
668C2×C3#330211yescharacteristic
669C2×C3#333411yescharacteristic
670C2×C3#334011yescharacteristic
671D3—42no—
672D3—82no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—62no—
681C22—62no—
682C22—62no—
683C22—121no—
684C22—121no—
685C22—121no—
686C22—121no—
687C22—122no—
688C22—122no—
689C22—122no—
690C22—241no—
691C22—241no—
692C22—241no—
693C22—241no—
694C22D3×#12111yescharacteristic
695C22#565411yescharacteristic
696C22#565511yescharacteristic
697C22C2×D3×D811yescharacteristic
698C22D3×#13511yescharacteristic
699C22#577811yescharacteristic
700C22#577911yescharacteristic
701C4—34no—
702C4—41no—
703C4—62no—
704C4—81no—
705C4—81no—
706C4—81no—
707C4—81no—
708C4—121no—
709C4—241no—
710C4—241no—
711C4—241no—
712C4#565411yescharacteristic
713C4#567711yescharacteristic
714C4#570611yescharacteristic
715C4#571011yescharacteristic
716C3#2533811yescharacteristic, complemented, Sylow
717C2—42no—
718C2—122no—
719C2—242no—
720C2#7746611yescharacteristic
721C2D3×#46611yescharacteristic
722C2#8090711yescharacteristic
723C2#8129111yescharacteristic
724C2#8130211yescharacteristic
725C2#8153211yescharacteristic
726C2#8154411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2075C2
socleC23×C3C22×D4
prime radical#73935C2
hypercenter#2075D3
hypocenterC3#25338
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2075

Complements

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

NH ≅ G/Ntype
1. C2×#4890C2semidirect
2. C2×#67⋊C8C2semidirect
3. #73937C2semidirect
5. C3×#2075C2semidirect
6. #70630C2semidirect
7. #69934C2semidirect
8. #69515C2semidirect
9. #69512C2semidirect
10. #68889C2semidirect
11. #68883C2semidirect
12. #68822C2semidirect
13. #68803C2semidirect
14. #68795C2semidirect
15. #68780C2semidirect
67. #2075D3semidirect
144. Dic3⋊C8D4semidirect
716. C3#25338semidirect

Maximal subgroups

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

subgroupindex
C2×#48902
C2×#67⋊C82
#739372
#739352
C3×#20752
#706302
#699342
#695152
#695122
#688892
#688832
#688222
#688032
#687952
#687802
#253383