Finite Group Databasebeta

← #10819883

Subgroups — #10819883

Order 768.

986 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4889C211yesmaximal, characteristic
2C2×#4873C211yesmaximal, characteristic, complemented
3C2×#4715C211yesmaximal, characteristic
4#73654C211yesmaximal, characteristic
5#73650C211yesmaximal, characteristic, complemented
6C3×#1776C211yesmaximal, characteristic, complemented
7#70575C211yesmaximal, characteristic, complemented
8#70574C211yesmaximal, characteristic
9#69810C211yesmaximal, characteristic, complemented
10#69488C211yesmaximal, characteristic
11#69486C211yesmaximal, characteristic, complemented
12#68909C211yesmaximal, characteristic
13#68908C211yesmaximal, characteristic, complemented
14#68908C211yesmaximal, characteristic, complemented
15#68896C211yesmaximal, characteristic
16#23194—31nomaximal, Sylow
17C22×Dic12C2211yescharacteristic
18C22×Dic12C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×#887C2211yescharacteristic
25C2×#875C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32C2×#815C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4988C2211yescharacteristic
35#4973C2211yescharacteristic
36#4973C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46#4940—21no—
47C4×(C2×D3)⋊C4—21no—
48C4×C4⋊Dic3—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50#4928—21no—
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4889—24no—
53#4873—24no—
54#4715C418yescomplemented
55#4670—21no—
56#4668—21no—
57#4653—21no—
58#4653—21no—
59C3×#340C2211yescharacteristic
60C3×#340C2211yescharacteristic
61#4550C2211yescharacteristic
62#4550C2211yescharacteristic
63C2×#497—31no—
64C2×#440—31no—
65C2×C4×Q16—31no—
66#2367—31no—
67#2365—31no—
68#1983—31no—
69#1981—31no—
70#1971—31no—
71#1968—31no—
72#1880—31no—
73#1878—31no—
74#1878—31no—
75#1877—31no—
76#1776D311yescharacteristic, complemented
77#1771—31no—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3D411yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3C2311yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C2×Dic12—22no—
112C2×Dic12—22no—
113C2×Dic12—41no—
114C2×Dic12—41no—
115C2×Dic12C2×C414yescomplemented
116C2×Dic12C2×C414yescomplemented
117C2×#204—22no—
118C2×#204—22no—
119C2×#204—41no—
120C2×#204—41no—
121C2×#204C2×C414yes—
122C2×#204C2×C414yes—
123#893—22no—
124#893—41no—
125#887—22no—
126#887—22no—
127#887—22no—
128#887—41no—
129#887—41no—
130#887—41no—
131#875—22no—
132#875C2×C414yes—
133#869—22no—
134#869C2×C414yescomplemented
135C3×#107D414yes—
136C3×#106—22no—
137C3×#97C2×C414yescomplemented
138C3×#95—21no—
139C3×#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95D411yescharacteristic
147C3⋊#95D411yescharacteristic
148#815—22no—
149C8×Dic3—22no—
150#810—22no—
151#810—22no—
152#810—22no—
153#810—41no—
154#810—41no—
155#810—41no—
156#809—22no—
157#809—41no—
158C4⋊(C3⋊C8)—22no—
159C22×Q16—31no—
160C22×Q16—31no—
161C22×SD16—31no—
162C22×SD16—31no—
163C2×C4⋊Q8—31no—
164C2×#124—31no—
165C2×C4×Q8—31no—
166C2×C4×Q8—31no—
167C2×C4×Q8—31no—
168C2×C4×D4—31no—
169#497—38no—
170#440—64no—
171C4×Q16—64no—
172C2×#107C2×D311yescharacteristic
173C2×#106C2×D311yescharacteristic
174C2×C4⋊C8—31no—
175C2×#103—31no—
176C2×#103—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#102—31no—
182C2×#102—31no—
183C2×#97—31no—
184C2×#97C2×D311yescharacteristic
185C2×C4×C8—31no—
186#391—31no—
187#391—31no—
188#391—31no—
189#390—31no—
190#389—61no—
191#388—61no—
192#388—61no—
193#387—61no—
194#387—61no—
195#385—61no—
196C4×C4⋊C4—31no—
197C4×C4⋊C4—61no—
198C4×C4⋊C4C2×D311yescharacteristic
199C4×C4⋊C4C2×D311yescharacteristic
200C4×C22⋊C4—61no—
201#340C2×D311yescharacteristic
202#340C2×D311yescharacteristic
203#336—31no—
204#336—31no—
205C23×D3—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12C22×C412yes—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—81no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6C22×C412yes—
245C2×Dic6C22×C412yes—
246C2×Dic6C22×C412yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8C22×C412yes—
250C2×C3×C8C22×C412yes—
251C2×C3×C8C2×D412yes—
252C2×C3×C8#4112yes—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4C2×D412yes—
260C3×C4⋊C4C2×D412yes—
261C3×C42—42no—
262C3×C42—81no—
263C3×C42C22×C412yes—
264C3×C42#4112yes—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—24no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—42no—
269(C2×D3)⋊C4—81no—
270C4⋊Dic3—24no—
271C4⋊Dic3—24no—
272C4⋊Dic3—22no—
273C4⋊Dic3—41no—
274Dic3⋊C4—42no—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3—42no—
282C4×Dic3—42no—
283C4×Dic3—42no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290Dic12—28no—
291Dic12—44no—
292#204—28no—
293#204—44no—
294C22×Q8—31no—
295C22×Q8—31no—
296C22×Q8—31no—
297C22×D4—31no—
298C23×C4—61no—
299C2×Q16—34no—
300C2×Q16—34no—
301C2×Q16—62no—
302C2×Q16—62no—
303C2×Q16—121no—
304C2×Q16—121no—
305C2×SD16—34no—
306C2×SD16—34no—
307C2×SD16—62no—
308C2×SD16—62no—
309C2×SD16—121no—
310C2×SD16—121no—
311C22×C8—31no—
312C22×C8—31no—
313C22×C8C22×D311yescharacteristic
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—34no—
316C4⋊Q8—62no—
317#124—34no—
318#124—62no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×Q8—62no—
322C4×Q8—121no—
323C4×Q8—121no—
324C4×Q8—121no—
325C4×D4—62no—
326C4×D4—121no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—31no—
346C2×C42—31no—
347C2×C42—61no—
348C2×C42—61no—
349C2×C42—61no—
350C2×C42C22×D311yescharacteristic
351#107D1214yes—
352#106—22no—
353C4⋊C8—62no—
354#103—62no—
355#103—62no—
356#103—62no—
357#103—121no—
358#103—121no—
359#103—121no—
360#102—62no—
361#102—121no—
362#97—62no—
363#97C4×D314yescomplemented
364C4×C8—62no—
365#95—21no—
366#95—21no—
367#95—31no—
368#95—31no—
369#95—61no—
370#95—61no—
371#95—61no—
372#95—61no—
373#95—61no—
374#95—61no—
375C23×C3C22×D411yescharacteristic
376C22×D3—22no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
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×C4C4×D412yes—
400C2×C3×C4C4×D412yes—
401C2×C3×C4C23×C411yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C4H(5,𝔽2)11yescharacteristic
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3#13712yes—
428D12—22no—
429D12—22no—
430D12—41no—
431D12—41no—
432C4×D3—84no—
433Dic6—22no—
434Dic6—22no—
435Dic6—22no—
436Dic6—22no—
437Dic6—22no—
438Dic6—22no—
439Dic6—41no—
440Dic6—41no—
441Dic6—41no—
442Dic6—41no—
443Dic6—41no—
444Dic6—41no—
445C3×C8—22no—
446C3×C8C4×D414yes—
447C3⋊C8—41no—
448C3⋊C8—41no—
449C24—61no—
450C2×Q8—32no—
451C2×Q8—32no—
452C2×Q8—32no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8—61no—
467C2×Q8—61no—
468C2×D4—32no—
469C2×D4—61no—
470C2×D4—61no—
471C2×D4—61no—
472C2×D4—61no—
473C2×D4—61no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—31no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4—241no—
488C22×C4D3×D411yescharacteristic
489C22×C4D3×D411yescharacteristic
490C22×C4C23×D311yescharacteristic
491Q16—68no—
492Q16—124no—
493SD16—68no—
494SD16—124no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8C2×C4×D312yes—
504C2×C8C2×C4×D312yes—
505C2×C8C2×D1212yes—
506C2×C8#23512yes—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—62no—
514C4⋊C4—64no—
515C4⋊C4—64no—
516C4⋊C4—121no—
517C4⋊C4—122no—
518C4⋊C4—122no—
519C4⋊C4—122no—
520C4⋊C4C2×D1212yes—
521C4⋊C4C2×D1212yes—
522C22⋊C4—64no—
523C22⋊C4—64no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—241no—
527C42—42no—
528C42—61no—
529C42—61no—
530C42—61no—
531C42—61no—
532C42—81no—
533C42—122no—
534C42—122no—
535C42—122no—
536C42C2×C4×D312yes—
537C42#23912yes—
538C22×C3C2×C4×D411yescharacteristic
539C22×C3D4211yescharacteristic
540C22×C3#55011yescharacteristic
541C22×C3C2×#13711yescharacteristic
542C22×C3#57511yescharacteristic
543C22×C3#57711yescharacteristic
544C22×C3#57811yescharacteristic
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—81no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C3×C4—22no—
553C3×C4—22no—
554C3×C4—42no—
555C3×C4—42no—
556C3×C4—81no—
557C3×C4—81no—
558C3×C4C2×C4×D411yescharacteristic
559C3×C4#51811yescharacteristic
560C3×C4#55011yescharacteristic
561C3×C4#55911yescharacteristic
562Dic3—22no—
563Dic3—42no—
564Dic3—42no—
565Dic3—42no—
566Dic3—41no—
567Dic3—41no—
568Dic3—41no—
569C23—62no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23C2×D3×D411yescharacteristic
577Q8—62no—
578Q8—62no—
579Q8—62no—
580Q8—62no—
581Q8—62no—
582Q8—62no—
583Q8—121no—
584Q8—121no—
585Q8—121no—
586Q8—121no—
587Q8—121no—
588Q8—121no—
589D4—62no—
590D4—62no—
591D4—121no—
592D4—121no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—32no—
600C2×C4—41no—
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—62no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—244no—
631C2×C4#89312yes—
632C2×C4#89312yes—
633C2×C4C22×C4×D311yescharacteristic
634C2×C4C22×D1211yescharacteristic
635C2×C4C2×#23511yescharacteristic
636C2×C4C2×#23911yescharacteristic
637C2×C4#100911yescharacteristic
638C2×C4#100911yescharacteristic
639C8—22no—
640C8—121no—
641C8—121no—
642C8C4×D1214yes—
643C2×C3#236611yescharacteristic
644C2×C3C4×#13711yescharacteristic
645C2×C3#294711yescharacteristic
646C2×C3#326711yescharacteristic
647C2×C3#327011yescharacteristic
648C2×C3#340011yescharacteristic
649C2×C3#340211yescharacteristic
650D3—82no—
651C22—122no—
652C22—122no—
653C22—122no—
654C22—241no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22C2×#89311yescharacteristic
659C22D4×D1211yescharacteristic
660C22#555311yescharacteristic
661C22#577311yescharacteristic
662C22#577511yescharacteristic
663C22D3×#13711yescharacteristic
664C22#577911yescharacteristic
665C4—22no—
666C4—22no—
667C4—42no—
668C4—42no—
669C4—62no—
670C4—81no—
671C4—81no—
672C4—121no—
673C4—121no—
674C4—122no—
675C4—122no—
676C4—122no—
677C4—121no—
678C4C2×C4×D1211yescharacteristic
679C4#552111yescharacteristic
680C4#557711yescharacteristic
681C4#558511yescharacteristic
682C3#2319411yescharacteristic, complemented, Sylow
683C2—242no—
684C2#7365311yescharacteristic
685C2#7920011yescharacteristic
686C2#7920211yescharacteristic
687C2#7976411yescharacteristic
688C2#7976511yescharacteristic
689C2#7983911yescharacteristic
690C2#7987911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1776C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1776D3
hypocenterC3#23194
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
2. C2×#4873C2semidirect
5. #73650C2semidirect
6. C3×#1776C2semidirect
7. #70575C2semidirect
9. #69810C2semidirect
11. #69486C2semidirect
13. #68908C2semidirect
14. #68908C2semidirect
54. #4715C4semidirect
76. #1776D3semidirect
115. C2×Dic12C2×C4semidirect
116. C2×Dic12C2×C4semidirect
134. #869C2×C4semidirect
137. C3×#97C2×C4semidirect
363. #97C4×D3semidirect
682. C3#23194semidirect

Maximal subgroups

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

subgroupindex
C2×#48892
C2×#48732
C2×#47152
#736542
#736502
C3×#17762
#705752
#705742
#698102
#694882
#694862
#689092
#689082
#688962
#231943