Finite Group Databasebeta

← #10824471

Subgroups — #10824471

Order 768.

956 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
1C3×#2319C211yesmaximal, characteristic, complemented
2C2×#5014C211yesmaximal, characteristic, complemented
3C2×C4×#213C211yesmaximal, characteristic, complemented
4#73389C211yesmaximal, characteristic, complemented
5#69658C211yesmaximal, characteristic
6#69650C211yesmaximal, characteristic, complemented
7#69650C211yesmaximal, characteristic, complemented
8#69641C211yesmaximal, characteristic, complemented
9#69630C211yesmaximal, characteristic, complemented
10#69629C211yesmaximal, characteristic, complemented
11#69627C211yesmaximal, characteristic, complemented
12#69609C211yesmaximal, characteristic, complemented
13#69474C211yesmaximal, characteristic, complemented
14C4×#809C211yesmaximal, characteristic, complemented
15#69436C211yesmaximal, characteristic, complemented
16#23688—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C22×#213C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C2×#874C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C3×#390C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#388—21no—
28C3×#388—21no—
29C3×#381—21no—
30C3×#380—21no—
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×#376C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35#5014—24no—
36C4×#213—24no—
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4939C2211yescharacteristic
46#4939C2211yescharacteristic
47#4936C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50C2×#803C2211yescharacteristic
51C2×C4×C3⋊C8C2211yescharacteristic
52#4665—21no—
53#4665—21no—
54#4665—21no—
55#4665—21no—
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#442—31no—
63C2×C4×D8—31no—
64#2328—31no—
65#2319D311yescharacteristic, complemented
66#2085—31no—
67#2073—31no—
68#2067—31no—
69#2065—31no—
70#2054—31no—
71#2054—31no—
72#2041—31no—
73#2011—31no—
74#1768—31no—
75#1762—31no—
76C4×#102—31no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×D12—21no—
80C22×D12—21no—
81C22×D12C2311yescharacteristic
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C22×C4×D3—21no—
85C3×C4×D4—22no—
86C3×C4×D4D414yescomplemented
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#213—22no—
102C2×#213—22no—
103C2×#213—22no—
104C2×#213—22no—
105C2×#213—22no—
106C2×#213—22no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#874—22no—
120#874—22no—
121C4×D12—22no—
122C4×D12—22no—
123C4×D12—22no—
124C4×D12—22no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95D411yescharacteristic
129C3×#95D411yescharacteristic
130C3×#95D411yescharacteristic
131C3×#95D411yescharacteristic
132#832—22no—
133#832—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#809—22no—
137#809—22no—
138#809—22no—
139#809—22no—
140#809—22no—
141#809—22no—
142#808—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—22no—
146#803—22no—
147C4×C3⋊C8—22no—
148C22×D8—31no—
149C22×D8—31no—
150C2×#127—31no—
151C2×C4×D4—31no—
152C2×C4×D4—31no—
153C2×C4×D4C2×D311yescharacteristic
154#442—64no—
155C4×D8—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#102—31no—
161C2×#102—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×C4×C8—31no—
170#390—31no—
171#390—31no—
172#390C2×D311yescharacteristic
173#390C2×D311yescharacteristic
174#389C2×D311yescharacteristic
175#389C2×D311yescharacteristic
176#388—21no—
177#388—21no—
178#386—61no—
179#386—61no—
180#386—61no—
181#386—61no—
182#382—31no—
183#381—21no—
184#380—21no—
185C4×C4⋊C4C2×D311yescharacteristic
186#376C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C23×D3—21no—
195C23×D3—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4C2×D412yes—
202C22×C3×C4—21no—
203C22×C3×C4—24no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—81no—
207C22×C3×C4—81no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C22×Dic3—21no—
216C22×Dic3—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—24no—
228C2×D12—24no—
229C2×D12—81no—
230C2×D12—81no—
231C2×D12—81no—
232C2×D12—81no—
233C2×D12#4112yes—
234C2×D12#4112yes—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3—81no—
238C2×C4×D3—81no—
239C2×C4×D3—81no—
240C2×C4×D3—81no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—81no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4C2×D412yes—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—81no—
251C3×C22⋊C4—81no—
252C3×C22⋊C4—81no—
253C3×C22⋊C4—81no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—81no—
259C3×C42C2×D412yes—
260C3×C42#4112yes—
261#213—28no—
262#213—28no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—81no—
266(C2×D3)⋊C4—81no—
267(C2×D3)⋊C4—81no—
268(C2×D3)⋊C4—81no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8#4112yes—
284C2×C3⋊C8#4112yes—
285C22×D4—31no—
286C22×D4—31no—
287C22×D4—61no—
288C22×D4—61no—
289C22×D4C22×D311yescharacteristic
290C23×C4—21no—
291C23×C4—61no—
292C23×C4—61no—
293C2×D8—62no—
294C2×D8—62no—
295C2×D8—62no—
296C2×D8—62no—
297C2×D8—62no—
298C2×D8—62no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8—31no—
303#127—62no—
304#127—62no—
305C4×D4—22no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—62no—
310C4×D4D1214yescomplemented
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
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×C42—21no—
330C2×C42C22×D311yescharacteristic
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#106—62no—
335C4⋊C8—62no—
336C4⋊C8—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#97—62no—
346C4×C8—62no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—61no—
351#95—61no—
352#95#6711yescharacteristic
353#95#6711yescharacteristic
354#95#6711yescharacteristic
355#95#6711yescharacteristic
356C23×C3—22no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3C22×D411yescharacteristic
364C22×D3—22no—
365C22×D3—22no—
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—
378C3×D4—22no—
379C3×D4—22no—
380C3×D4—41no—
381C3×D4—41no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—84no—
408C2×C3×C4#12112yes—
409C2×C3×C4#12112yes—
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422D12—22no—
423D12—22no—
424D12—22no—
425D12—22no—
426D12—41no—
427D12—41no—
428D12—41no—
429D12—41no—
430D12—84no—
431D12—84no—
432C4×D3—84no—
433C4×D3—84no—
434C3⋊C8—22no—
435C3⋊C8—22no—
436C3⋊C8—41no—
437C3⋊C8—41no—
438C24—21no—
439C24—61no—
440C24—61no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—32no—
447C2×D4—32no—
448C2×D4—64no—
449C2×D4—64no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—241no—
461C2×D4—241no—
462C2×D4—241no—
463C2×D4—241no—
464C2×D4C2×D1212yes—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4—241no—
476C22×C4—241no—
477C22×C4—241no—
478C22×C4—241no—
479C22×C4C2×#6711yescharacteristic
480C22×C4C2×#6711yescharacteristic
481C22×C4C2×#6711yescharacteristic
482C22×C4C2×#6711yescharacteristic
483C22×C4C2×#6711yescharacteristic
484C22×C4C2×#6711yescharacteristic
485C22×C4C23×D311yescharacteristic
486D8—68no—
487D8—68no—
488C2×C8—32no—
489C2×C8—32no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4C2×D1212yes—
512C22⋊C4—24no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—81no—
528C42C2×D1212yes—
529C42#23512yes—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3C2×#12111yescharacteristic
538C22×C3#52511yescharacteristic
539C22×C3#53211yescharacteristic
540C22×C3#57611yescharacteristic
541C22×C3#57611yescharacteristic
542C22×C3#57611yescharacteristic
543C22×C3#57711yescharacteristic
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—42no—
549C2×D3—42no—
550C2×D3—81no—
551C2×D3—81no—
552C2×D3—81no—
553C2×D3—81no—
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—41no—
563C3×C4—41no—
564C3×C4—42no—
565C3×C4—81no—
566C3×C4—81no—
567C3×C4#52511yescharacteristic
568C3×C4#52511yescharacteristic
569C3×C4#52911yescharacteristic
570C3×C4#52911yescharacteristic
571Dic3—81no—
572Dic3—81no—
573C23—22no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—62no—
581C23—62no—
582C23—121no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23—121no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23C22×#6711yescharacteristic
595D4—22no—
596D4—22no—
597D4—41no—
598D4—41no—
599D4—62no—
600D4—62no—
601D4—62no—
602D4—62no—
603D4—121no—
604D4—121no—
605D4—121no—
606D4—121no—
607D4—244no—
608D4—244no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
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—84no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—244no—
642C2×C4—244no—
643C2×C4#93012yes—
644C2×C4#93012yes—
645C2×C4C22×D1211yescharacteristic
646C2×C4C2×#23511yescharacteristic
647C2×C4D3×#4111yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C2×C4D3×#4111yescharacteristic
650C2×C4D3×#4111yescharacteristic
651C8—62no—
652C8—62no—
653C8—121no—
654C8—121no—
655C2×C3—82no—
656C2×C3#239211yescharacteristic
657C2×C3#304111yescharacteristic
658C2×C3#304511yescharacteristic
659C2×C3#311811yescharacteristic
660C2×C3#312211yescharacteristic
661C2×C3#317611yescharacteristic
662C2×C3#317911yescharacteristic
663D3—82no—
664D3—82no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—122no—
673C22—122no—
674C22—122no—
675C22—122no—
676C22—122no—
677C22—122no—
678C22—241no—
679C22—241no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22C2×#93011yescharacteristic
687C22#582811yescharacteristic
688C22#583211yescharacteristic
689C22#583511yescharacteristic
690C22#583511yescharacteristic
691C22#583511yescharacteristic
692C22#583711yescharacteristic
693C4—22no—
694C4—22no—
695C4—41no—
696C4—41no—
697C4—42no—
698C4—41no—
699C4—41no—
700C4—81no—
701C4—81no—
702C4—241no—
703C4—241no—
704C4#552411yescharacteristic
705C4#552411yescharacteristic
706C4#552811yescharacteristic
707C4#552811yescharacteristic
708C3#2368811yescharacteristic, complemented, Sylow
709C2—82no—
710C2—242no—
711C2—242no—
712C2#8227311yescharacteristic
713C2#8254011yescharacteristic
714C2#8256511yescharacteristic
715C2#8343211yescharacteristic
716C2#8343511yescharacteristic
717C2#8345911yescharacteristic
718C2#8346011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2319C2
socleC23×C3C22×D4
prime radical#69658C2
hypercenter#2319D3
hypocenterC3#23688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
1. C3×#2319C2semidirect
2. C2×#5014C2semidirect
3. C2×C4×#213C2semidirect
4. #73389C2semidirect
6. #69650C2semidirect
7. #69650C2semidirect
8. #69641C2semidirect
9. #69630C2semidirect
10. #69629C2semidirect
11. #69627C2semidirect
12. #69609C2semidirect
13. #69474C2semidirect
14. C4×#809C2semidirect
15. #69436C2semidirect
65. #2319D3semidirect
86. C3×C4×D4D4semidirect
310. C4×D4D12semidirect
708. C3#23688semidirect

Maximal subgroups

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

subgroupindex
C3×#23192
C2×#50142
C2×C4×#2132
#733892
#696582
#696502
#696412
#696302
#696292
#696272
#696092
#694742
C4×#8092
#694362
#236883