Finite Group Databasebeta

← #10824803

Subgroups — #10824803

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4711C211yesmaximal, characteristic, complemented
2C2×#4709C211yesmaximal, characteristic, complemented
3#73763C211yesmaximal, characteristic, complemented
4#73763C211yesmaximal, characteristic, complemented
5C3×#1776C211yesmaximal, characteristic, complemented
6#70639C211yesmaximal, characteristic, complemented
7#70634C211yesmaximal, characteristic, complemented
8#70585C211yesmaximal, characteristic, complemented
9#70581C211yesmaximal, characteristic, complemented
10C4×#809C211yesmaximal, characteristic, complemented
11C4×#809C211yesmaximal, characteristic, complemented
12#68886C211yesmaximal, characteristic
13#68862C211yesmaximal, characteristic, complemented
14#68840C211yesmaximal, characteristic, complemented
15#68832C211yesmaximal, characteristic, complemented
16#23712—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C2×#895C2211yescharacteristic
19C2×#895C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×#872C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4988C2211yescharacteristic
36#4988C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4984C2211yescharacteristic
40#4984C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45#4940—21no—
46#4940—21no—
47#4939—21no—
48#4939—21no—
49C2×C4×C3⋊C8C2211yescharacteristic
50#4711—24no—
51#4709—24no—
52#4668—21no—
53#4668—21no—
54#4665—21no—
55#4665—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#442—31no—
63C2×#433—31no—
64#2324—31no—
65#2324—31no—
66#2088—31no—
67#2077—31no—
68#2067—31no—
69#2065—31no—
70#2054—31no—
71#2034—31no—
72#2033—31no—
73#1999—31no—
74#1776D311yescharacteristic, complemented
75C4×#102—31no—
76C4×#102—31no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×C4×D3C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42—21no—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4D411yescharacteristic
104C2×(C2×D3)⋊C4D411yescharacteristic
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4D411yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C2×D24—22no—
115C2×D24—22no—
116C2×D24—22no—
117#895—22no—
118#895—22no—
119#895—22no—
120#895—22no—
121C4×D12—22no—
122C4×D12—22no—
123C4×D12—22no—
124C4×D12—22no—
125#872—22no—
126#872—22no—
127C3×#107—22no—
128C3×#106—22no—
129C3×#97D414yescomplemented
130C3×#95—21no—
131C3×#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#821—22no—
137#821—22no—
138(C2×D3)⋊C8—22no—
139(C2×D3)⋊C8—22no—
140#817—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143#809—22no—
144#809—22no—
145#809—22no—
146#809—22no—
147C4×C3⋊C8—22no—
148C22×D8—31no—
149C2×#121—31no—
150C2×#121—31no—
151C2×C4×D4—31no—
152C2×C4×D4—31no—
153C2×#117—31no—
154#442—64no—
155#433—64no—
156C2×#107C2×D311yescharacteristic
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#102—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×#97C2×D311yescharacteristic
170C2×C4×C8—31no—
171#390—31no—
172#390—31no—
173#390—61no—
174#390—61no—
175#389—31no—
176#389—31no—
177#388—61no—
178#388—61no—
179#386—61no—
180#386—61no—
181#385—61no—
182#385—61no—
183#380—31no—
184#380—31no—
185C4×C4⋊C4C2×D311yescharacteristic
186C4×C4⋊C4C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C23×D3—21no—
194C23×D3—21no—
195C23×D3C2×D411yescharacteristic
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—24no—
219C2×D12—24no—
220C2×D12—81no—
221C2×D12—81no—
222C2×D12—81no—
223C2×D12—81no—
224C2×D12#4112yes—
225C2×D12#4112yes—
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—42no—
233C2×C4×D3—42no—
234C2×C4×D3—81no—
235C2×C4×D3—81no—
236C2×C4×D3—81no—
237C2×C4×D3—81no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8C2×D412yes—
243C2×C3×C8C2×D412yes—
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×C4⋊C4#4112yes—
251C3×C4⋊C4#4112yes—
252C3×C42—81no—
253C3×C42—81no—
254C3×C42C2×D412yes—
255C3×C42#4112yes—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264(C2×D3)⋊C4—81no—
265(C2×D3)⋊C4—81no—
266(C2×D3)⋊C4—81no—
267(C2×D3)⋊C4—81no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—81no—
273C4⋊Dic3—81no—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284D24—28no—
285C22×D4—31no—
286C22×D4—31no—
287C22×D4—61no—
288C22×D4—61no—
289C23×C4—31no—
290C23×C4—61no—
291C23×C4—61no—
292C2×D8—62no—
293C2×D8—62no—
294C2×D8—62no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299#121—62no—
300#121—62no—
301#121—62no—
302#121—62no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—62no—
307#117—62no—
308#117—62no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C22⋊C4—31no—
320C2×C22⋊C4—31no—
321C2×C22⋊C4—61no—
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×C42—21no—
332C2×C42—21no—
333C2×C42—31no—
334C2×C42C22×D311yescharacteristic
335#107—22no—
336#106—22no—
337#106—62no—
338C4⋊C8—62no—
339C4⋊C8—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346C22⋊C8—62no—
347C22⋊C8—62no—
348#97D1214yescomplemented
349C4×C8—62no—
350#95—21no—
351#95—21no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3C22×D411yescharacteristic
357C22×D3—21no—
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
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—41no—
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×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417D12—22no—
418D12—22no—
419D12—22no—
420D12—22no—
421D12—41no—
422D12—41no—
423D12—41no—
424D12—41no—
425D12—84no—
426D12—84no—
427C4×D3—41no—
428C4×D3—41no—
429C4×D3—41no—
430C4×D3—41no—
431C4×D3—84no—
432C4×D3—84no—
433C3×C8—22no—
434C3×C8—22no—
435C3⋊C8—41no—
436C3⋊C8—41no—
437C24—31no—
438C24—61no—
439C24—61no—
440C2×D4—32no—
441C2×D4—32no—
442C2×D4—64no—
443C2×D4—64no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—241no—
455C2×D4—241no—
456C2×D4—241no—
457C2×D4—241no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
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—122no—
474C22×C4—122no—
475C22×C4—241no—
476C22×C4—241no—
477C22×C4—241no—
478C22×C4—241no—
479C22×C4D3×D411yescharacteristic
480C22×C4D3×D411yescharacteristic
481C22×C4C23×D311yescharacteristic
482D8—68no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8C2×D1212yes—
494C2×C8C2×D1212yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—62no—
502C4⋊C4—62no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—241no—
508C4⋊C4—241no—
509C4⋊C4#23512yes—
510C4⋊C4#23512yes—
511C22⋊C4—64no—
512C22⋊C4—64no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—241no—
518C22⋊C4—241no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C42—61no—
524C42—61no—
525C42—81no—
526C42—81no—
527C42C2×D1212yes—
528C42#23912yes—
529C22×C3C2×#12111yescharacteristic
530C22×C3#52511yescharacteristic
531C22×C3#52511yescharacteristic
532C22×C3#57611yescharacteristic
533C22×C3#57611yescharacteristic
534C22×C3#57711yescharacteristic
535C22×C3#57711yescharacteristic
536C2×D3—41no—
537C2×D3—42no—
538C2×D3—41no—
539C2×D3—42no—
540C2×D3—41no—
541C2×D3—42no—
542C2×D3—41no—
543C2×D3—42no—
544C2×D3—41no—
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—41no—
548C2×D3—41no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C2×D3—81no—
553C2×D3—81no—
554C2×D3—81no—
555C2×D3—81no—
556C2×D3—81no—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—42no—
560C3×C4—42no—
561C3×C4—81no—
562C3×C4—81no—
563C3×C4#52511yescharacteristic
564C3×C4#52911yescharacteristic
565C3×C4#52911yescharacteristic
566C3×C4#53211yescharacteristic
567Dic3—41no—
568Dic3—41no—
569Dic3—41no—
570Dic3—81no—
571Dic3—81no—
572C23—61no—
573C23—61no—
574C23—61no—
575C23—61no—
576C23—61no—
577C23—62no—
578C23—62no—
579C23—61no—
580C23—61no—
581C23—121no—
582C23—121no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23—121no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23C2×D3×D411yescharacteristic
594D4—62no—
595D4—62no—
596D4—62no—
597D4—62no—
598D4—121no—
599D4—121no—
600D4—121no—
601D4—121no—
602D4—244no—
603D4—244no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
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—244no—
640C2×C4—244no—
641C2×C4#89512yes—
642C2×C4#89512yes—
643C2×C4C22×D1211yescharacteristic
644C2×C4C2×#23511yescharacteristic
645C2×C4C2×#23511yescharacteristic
646C2×C4C2×#23911yescharacteristic
647C2×C4D3×#4111yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C8—22no—
650C8—22no—
651C8—121no—
652C8—121no—
653C2×C3#239211yescharacteristic
654C2×C3#306211yescharacteristic
655C2×C3#306211yescharacteristic
656C2×C3#311811yescharacteristic
657C2×C3#311911yescharacteristic
658C2×C3#312211yescharacteristic
659C2×C3#312511yescharacteristic
660D3—81no—
661D3—82no—
662D3—82no—
663C22—121no—
664C22—122no—
665C22—121no—
666C22—122no—
667C22—122no—
668C22—121no—
669C22—121no—
670C22—121no—
671C22—122no—
672C22—121no—
673C22—122no—
674C22—121no—
675C22—122no—
676C22—241no—
677C22—241no—
678C22—241no—
679C22—241no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22C2×#89511yescharacteristic
685C22#554711yescharacteristic
686C22#554711yescharacteristic
687C22#577411yescharacteristic
688C22#577411yescharacteristic
689C22#577811yescharacteristic
690C22#577811yescharacteristic
691C4—22no—
692C4—22no—
693C4—42no—
694C4—42no—
695C4—81no—
696C4—81no—
697C4—121no—
698C4—121no—
699C4—121no—
700C4—241no—
701C4—241no—
702C4#547711yescharacteristic
703C4#552411yescharacteristic
704C4#558011yescharacteristic
705C4#558011yescharacteristic
706C3#2371211yescharacteristic, complemented, Sylow
707C2—241no—
708C2—242no—
709C2—242no—
710C2#7377611yescharacteristic
711C2#7921311yescharacteristic
712C2#7921311yescharacteristic
713C2#7978511yescharacteristic
714C2#7981211yescharacteristic
715C2#7985011yescharacteristic
716C2#7987411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1776C2
socleC23×C3C22×D4
prime radical#68886C2
hypercenter#1776D3
hypocenterC3#23712
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
1. C2×#4711C2semidirect
2. C2×#4709C2semidirect
3. #73763C2semidirect
4. #73763C2semidirect
5. C3×#1776C2semidirect
6. #70639C2semidirect
7. #70634C2semidirect
8. #70585C2semidirect
9. #70581C2semidirect
10. C4×#809C2semidirect
11. C4×#809C2semidirect
13. #68862C2semidirect
14. #68840C2semidirect
15. #68832C2semidirect
74. #1776D3semidirect
129. C3×#97D4semidirect
348. #97D12semidirect
706. C3#23712semidirect

Maximal subgroups

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

subgroupindex
C2×#47112
C2×#47092
#737632
C3×#17762
#706392
#706342
#705852
#705812
C4×#8092
#688862
#688622
#688402
#688322
#237123