Finite Group Databasebeta

← #10819882

Subgroups — #10819882

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×#5190C211yesmaximal, characteristic
2#82465C211yesmaximal, characteristic, complemented
3C2×#5026C211yesmaximal, characteristic
4C2×#4790C211yesmaximal, characteristic, complemented
5#76493C211yesmaximal, characteristic
6C3×#1878C211yesmaximal, characteristic, complemented
7#72065C211yesmaximal, characteristic, complemented
8#70257C211yesmaximal, characteristic
9#70230C211yesmaximal, characteristic
10#70229C211yesmaximal, characteristic, complemented
11#69678C211yesmaximal, characteristic, complemented
12#68909C211yesmaximal, characteristic
13#68786C211yesmaximal, characteristic, complemented
14#68672C211yesmaximal, characteristic
15#68654C211yesmaximal, characteristic, complemented
16#23194—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×#944C2211yescharacteristic
22C22×#216C2211yescharacteristic
23C22×#215C2211yescharacteristic
24C22×#204C2211yescharacteristic
25C2×#887C2211yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C3×#391C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5190C418yescomplemented
34C2×#821C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#815C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#5026—24no—
40#4988C2211yescharacteristic
41#4978—21no—
42#4976—21no—
43Dic3×C4⋊C4C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4933—21no—
50C4×Dic3⋊C4—21no—
51C4×Dic3⋊C4C2211yescharacteristic
52#4929C2211yescharacteristic
53C2×#803C2211yescharacteristic
54#4790—24no—
55#4670—21no—
56#4669—21no—
57#4668—21no—
58#4666—21no—
59C3×#340C2211yescharacteristic
60C3×#336C2211yescharacteristic
61#4551C2211yescharacteristic
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#1878D311yescharacteristic, complemented
75#1877—31no—
76#1776—31no—
77#1771—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×Q8C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6C2311yescharacteristic
83C2×C3×Q16—41no—
84C2×C3×Q16—41no—
85C2×C3×Q16C2×C414yescomplemented
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×Q8—22no—
89C3×C4×Q8—41no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95#947—22no—
96#947C2×C414yes—
97Q8×Dic3—22no—
98Q8×Dic3—41no—
99#944—22no—
100#944C2×C414yescomplemented
101C2×#216—22no—
102C2×#216—22no—
103C2×#216C2×C414yescomplemented
104C2×#215—22no—
105C2×#215—22no—
106C2×#215C2×C414yes—
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×C4⋊Dic3—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4D411yescharacteristic
119C2×Dic3⋊C4D411yescharacteristic
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3C2311yescharacteristic
125C2×C4×Dic3C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C2×#204—41no—
129C2×#204—41no—
130C2×#204C2×C414yes—
131#887—22no—
132#887—41no—
133C4×D12—22no—
134C4×D12—41no—
135C3×#103—22no—
136C3×#103—41no—
137C3×#95—21no—
138C3×#95—21no—
139#835—22no—
140#835—41no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95D411yescharacteristic
148C3⋊#95D411yescharacteristic
149#821—41no—
150#816—22no—
151#815C2×C414yescomplemented
152Dic3⋊C8—22no—
153C8×Dic3—22no—
154#810—41no—
155#809—22no—
156#808—22no—
157#807D414yes—
158#803—22no—
159C22×Q16—31no—
160C22×Q16C2×D311yescharacteristic
161C22×SD16—31no—
162C22×SD16—31no—
163C2×C4⋊Q8—31no—
164C2×#124—31no—
165C2×C4×Q8—31no—
166C2×C4×Q8—31no—
167C2×C4×Q8C2×D311yescharacteristic
168C2×C4×D4—31no—
169#497—38no—
170#440—64no—
171C4×Q16—64no—
172C2×#107—31no—
173C2×#106—31no—
174C2×C4⋊C8—31no—
175C2×#103—31no—
176C2×#103—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103C2×D311yescharacteristic
180C2×#103C2×D311yescharacteristic
181C2×#102—31no—
182C2×#102—31no—
183C2×#97—31no—
184C2×#97—31no—
185C2×C4×C8—31no—
186#391—31no—
187#391—31no—
188#391C2×D311yescharacteristic
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—31no—
198C4×C4⋊C4—31no—
199C4×C4⋊C4—61no—
200C4×C22⋊C4—61no—
201#340—31no—
202#340C2×D311yescharacteristic
203#336—31no—
204#336C2×D311yescharacteristic
205C23×D3—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8C22×C412yes—
217C2×C3×Q8C22×C412yes—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12C22×C412yes—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×C4×D3—81no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6C22×C412yes—
247C3×Q16—44no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8C22×C412yes—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—42no—
258C3×C4⋊C4C2×D412yes—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—42no—
262#216—28no—
263#215—28no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—81no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—42no—
273C4⋊Dic3C2×D412yes—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—22no—
277Dic3⋊C4—41no—
278Dic3⋊C4—42no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3—42no—
282C4×Dic3—42no—
283C4×Dic3—42no—
284C4×Dic3—81no—
285C4×Dic3C22×C412yes—
286C4×Dic3#4112yes—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8C22×C412yes—
291C2×C3⋊C8C2×D412yes—
292C2×C3⋊C8#4112yes—
293#204—44no—
294C22×Q8—31no—
295C22×Q8C22×D311yescharacteristic
296C22×Q8C22×D311yescharacteristic
297C22×D4—31no—
298C23×C4—61no—
299C2×Q16—34no—
300C2×Q16—41no—
301C2×Q16—41no—
302C2×Q16—62no—
303C2×Q16—62no—
304C2×Q16C4×D314yescomplemented
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—22no—
320C4×Q8—41no—
321C4×Q8—62no—
322C4×Q8—62no—
323C4×Q8—121no—
324C4×Q8—121no—
325C4×D4—62no—
326C4×D4—121no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
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—31no—
345C2×C42—31no—
346C2×C42—61no—
347C2×C42—61no—
348C2×C42—61no—
349C2×C42—61no—
350C2×C42C22×D311yescharacteristic
351#107—34no—
352#106—62no—
353C4⋊C8—62no—
354#103—22no—
355#103—41no—
356#103—62no—
357#103—62no—
358#103—121no—
359#103—121no—
360#102—62no—
361#102—121no—
362#97—34no—
363#97—62no—
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—
383C3×Q8—22no—
384C3×Q8—22no—
385C3×Q8—22no—
386C3×Q8—22no—
387C3×Q8—41no—
388C3×Q8—41no—
389C3×Q8—41no—
390C3×Q8—41no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
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—41no—
408C2×C3×C4—41no—
409C2×C3×C4C23×C411yescharacteristic
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4H(5,𝔽2)11yescharacteristic
414C2×C3×C4H(5,𝔽2)11yescharacteristic
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—22no—
422C2×Dic3—21no—
423C2×Dic3—22no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3C4×D412yes—
434C2×Dic3C4×D412yes—
435C2×Dic3#13712yes—
436D12—22no—
437D12—22no—
438D12—41no—
439D12—41no—
440C4×D3—84no—
441Dic6—22no—
442Dic6—22no—
443Dic6—41no—
444Dic6—41no—
445C3×C8—22no—
446C3×C8—41no—
447C3⋊C8—41no—
448C3⋊C8C4×D414yes—
449C24—61no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—32no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8C2×C4×D312yes—
467C2×Q8C2×C4×D312yes—
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—44no—
492Q16—68no—
493SD16—68no—
494SD16—124no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—32no—
501C2×C8—32no—
502C2×C8—32no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8C2×C4×D312yes—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—32no—
511C4⋊C4—42no—
512C4⋊C4—62no—
513C4⋊C4—64no—
514C4⋊C4—64no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—121no—
519C4⋊C4—122no—
520C4⋊C4—122no—
521C4⋊C4D3×D412yes—
522C22⋊C4—64no—
523C22⋊C4—64no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—241no—
527C42—21no—
528C42—21no—
529C42—32no—
530C42—32no—
531C42—42no—
532C42—61no—
533C42—61no—
534C42—122no—
535C42—122no—
536C42—122no—
537C42—241no—
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—42no—
553C3×C4—42no—
554C3×C4—41no—
555C3×C4—42no—
556C3×C4—41no—
557C3×C4—81no—
558C3×C4C2×C4×D411yescharacteristic
559C3×C4#51811yescharacteristic
560C3×C4#55011yescharacteristic
561C3×C4#55911yescharacteristic
562Dic3—22no—
563Dic3—22no—
564Dic3—22no—
565Dic3—42no—
566Dic3—42no—
567Dic3—41no—
568Dic3—81no—
569C23—62no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23C2×D3×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—41no—
582Q8—41no—
583Q8—41no—
584Q8—41no—
585Q8—62no—
586Q8—62no—
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—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—32no—
603C2×C4—32no—
604C2×C4—32no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—62no—
623C2×C4—61no—
624C2×C4—62no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—244no—
633C2×C4C22×C4×D311yescharacteristic
634C2×C4C2×D3×D411yescharacteristic
635C2×C4#100411yescharacteristic
636C2×C4#100411yescharacteristic
637C2×C4D3×#4111yescharacteristic
638C2×C4D3×#4111yescharacteristic
639C8—22no—
640C8—34no—
641C8—41no—
642C8—121no—
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—
658C22C4×D3×D411yescharacteristic
659C22#559011yescharacteristic
660C22#559911yescharacteristic
661C22#576611yescharacteristic
662C22#577011yescharacteristic
663C22D3×#13711yescharacteristic
664C22#577711yescharacteristic
665C4—41no—
666C4—41no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—62no—
671C4—62no—
672C4—62no—
673C4—81no—
674C4—121no—
675C4—122no—
676C4—122no—
677C4—241no—
678C4#548611yescharacteristic
679C4C4×D3×D411yescharacteristic
680C4#559911yescharacteristic
681C4#566411yescharacteristic
682C3#2319411yescharacteristic, complemented, Sylow
683C2—242no—
684C2#7467911yescharacteristic
685C2#7971911yescharacteristic
686C2#7982911yescharacteristic
687C2#8020911yescharacteristic
688C2#8026511yescharacteristic
689C2#8047111yescharacteristic
690C2#8059111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1878C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1878D3
hypocenterC3#23194
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1878

Complements

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

NH ≅ G/Ntype
2. #82465C2semidirect
4. C2×#4790C2semidirect
6. C3×#1878C2semidirect
7. #72065C2semidirect
10. #70229C2semidirect
11. #69678C2semidirect
13. #68786C2semidirect
15. #68654C2semidirect
33. #5190C4semidirect
74. #1878D3semidirect
85. C2×C3×Q16C2×C4semidirect
100. #944C2×C4semidirect
103. C2×#216C2×C4semidirect
151. #815C2×C4semidirect
304. C2×Q16C4×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×#51902
#824652
C2×#50262
C2×#47902
#764932
C3×#18782
#720652
#702572
#702302
#702292
#696782
#689092
#687862
#686722
#686542
#231943