Finite Group Databasebeta

← #10828498

Subgroups — #10828498

Order 768.

1072 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5578C211yesmaximal, characteristic
2C2×#4868C211yesmaximal, characteristic
3C2×#4814C211yesmaximal, characteristic
4#75963C211yesmaximal, characteristic
5C3×#1865C211yesmaximal, characteristic, complemented
6#71628C211yesmaximal, characteristic, complemented
7#70565C211yesmaximal, characteristic, complemented
8#70284C211yesmaximal, characteristic, complemented
9#70269C211yesmaximal, characteristic
10#70246C211yesmaximal, characteristic, complemented
11#69632C211yesmaximal, characteristic, complemented
12#68879C211yesmaximal, characteristic
13#68826C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68664C211yesmaximal, characteristic
16#24065—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#944C2211yescharacteristic
20#5578—24no—
21#5578—24no—
22C2×#897C2211yescharacteristic
23C2×#896—21no—
24C2×#892C2211yescharacteristic
25C2×#892C2211yescharacteristic
26C2×#889—21no—
27C2×#888C2211yescharacteristic
28C2×#872—21no—
29C2×C42×D3C2211yescharacteristic
30C2×C4×Dic6—21no—
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#106C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×(C2×D3)⋊C8C2211yescharacteristic
39C2×(C2×D3)⋊C8C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43#4989—21no—
44#4984C2211yescharacteristic
45Dic3×C4⋊C4—21no—
46#4972C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4928C2211yescharacteristic
52#4868—24no—
53#4814—24no—
54#4668—21no—
55#4649—21no—
56C3×#340C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#533—31no—
65C2×#485—31no—
66C2×#485—31no—
67#2296—31no—
68#2058—31no—
69#2057—31no—
70#2043—31no—
71#2043—31no—
72#1873—31no—
73#1873—31no—
74#1865—31no—
75#1865D311yescharacteristic, complemented
76#1863—31no—
77#1863—31no—
78#1817—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C4×D3D411yescharacteristic
81C22×C4×D3D411yescharacteristic
82C22×C4×D3C2311yescharacteristic
83C22×Dic6—21no—
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96#946—22no—
97#946D414yes—
98#944—22no—
99#944—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#897—22no—
125#897D414yes—
126#896—28no—
127#892—22no—
128#892—22no—
129#892—22no—
130#892—22no—
131#889—28no—
132#888—22no—
133#888—22no—
134#872—28no—
135C42×D3—22no—
136C42×D3—22no—
137C4×Dic6—28no—
138C4×Dic6—22no—
139C4×Dic6—22no—
140C3×#106—22no—
141C3×#103—22no—
142C3×#103—22no—
143C3×#95—21no—
144#835—22no—
145#835—22no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149(C2×D3)⋊C8—22no—
150(C2×D3)⋊C8—22no—
151#818—22no—
152#816—22no—
153#816—22no—
154#810—22no—
155#810—22no—
156#808—22no—
157#807—22no—
158#533—64no—
159#533—64no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—31no—
162C2×#125—61no—
163C2×C22⋊Q8—31no—
164C2×C22⋊Q8—31no—
165C2×C22⋊Q8—61no—
166C2×C4×Q8—31no—
167C2×C4×Q8—61no—
168C2×C4×Q8C2×D311yescharacteristic
169C2×#117—31no—
170C2×#117—31no—
171C2×#117—61no—
172C22×C42—31no—
173#485—64no—
174#485—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#106—31no—
178C2×#106C2×D311yescharacteristic
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#103—31no—
185C2×#103C2×D311yescharacteristic
186C2×#103C2×D311yescharacteristic
187C2×C22⋊C8—31no—
188C2×C22⋊C8—31no—
189#389—31no—
190#389—31no—
191#389—31no—
192#389C2×D311yescharacteristic
193#385—61no—
194#385—61no—
195C4×C4⋊C4—61no—
196C4×C4⋊C4—61no—
197#340—31no—
198#340—31no—
199#340—31no—
200#340C2×D311yescharacteristic
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C23×D3C2×D411yescharacteristic
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8C2×D412yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—22no—
233C2×C4×D3—22no—
234C2×C4×D3—22no—
235C2×C4×D3—41no—
236C2×C4×D3C2×D412yes—
237C2×C4×D3C2×D412yes—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—24no—
243C2×Dic6—21no—
244C2×Dic6—42no—
245C2×Dic6—42no—
246C2×Dic6C2×D412yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—42no—
259C3×C4⋊C4C2×D412yes—
260C3×C4⋊C4#4112yes—
261C3×C42—24no—
262C3×C42—21no—
263C3×C42—21no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—24no—
267(C2×D3)⋊C4—42no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—24no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—42no—
274C4⋊Dic3C2×D412yes—
275C4⋊Dic3#4112yes—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—24no—
279Dic3⋊C4—24no—
280Dic3⋊C4—24no—
281Dic3⋊C4—42no—
282Dic3⋊C4—42no—
283C4×Dic3—24no—
284C4×Dic3—24no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C4×Dic3—22no—
288C4×Dic3—22no—
289C4×Dic3—42no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C22×Q8—31no—
297C22×Q8—61no—
298C22×Q8C22×D311yescharacteristic
299C23×C4—31no—
300C23×C4—31no—
301C23×C4—31no—
302C22×C8—31no—
303C22×C8—31no—
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308C4⋊Q8—62no—
309C4⋊Q8—62no—
310#125—68no—
311C22⋊Q8—34no—
312C22⋊Q8—34no—
313C22⋊Q8—62no—
314C22⋊Q8—68no—
315C22⋊Q8—62no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—62no—
319C4×Q8—62no—
320C4×Q8—68no—
321#117—68no—
322#117—62no—
323#117—62no—
324#117—62no—
325#117—62no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C42—21no—
348C2×C42—31no—
349C2×C42—31no—
350C2×C42—31no—
351C2×C42—62no—
352C2×C42—62no—
353C2×C42—61no—
354C2×C42—61no—
355C2×C42—61no—
356C2×C42C22×D311yescharacteristic
357#107—62no—
358#107—62no—
359#106—22no—
360#106—62no—
361#103—22no—
362#103—22no—
363#103—62no—
364#103—62no—
365#103—62no—
366#103—62no—
367#103—62no—
368#103—62no—
369C22⋊C8—62no—
370C22⋊C8—62no—
371#95—21no—
372#95—61no—
373#95—61no—
374#95—61no—
375C23×C3C22×D411yescharacteristic
376C22×D3—21no—
377C22×D3—21no—
378C22×D3—21no—
379C22×D3—21no—
380C22×D3—21no—
381C22×D3—21no—
382C22×D3#12012yes—
383C3×Q8—22no—
384C3×Q8—22no—
385C3×Q8—41no—
386C3×Q8—41no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
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×C4—41no—
404C2×C3×C4#13712yes—
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C42−1+411yescharacteristic
410C2×C3×C42−1+411yescharacteristic
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3#12012yes—
430C2×Dic3#13512yes—
431C2×Dic3#13712yes—
432C4×D3—28no—
433C4×D3—22no—
434C4×D3—22no—
435C4×D3—22no—
436C4×D3—22no—
437C4×D3—44no—
438Dic6—22no—
439Dic6—22no—
440Dic6—48no—
441Dic6—41no—
442Dic6—41no—
443C3×C8—41no—
444C3×C8—41no—
445C3⋊C8—41no—
446C3⋊C8—41no—
447C24—31no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—32no—
454C2×Q8—64no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—122no—
461C2×Q8—122no—
462C2×Q8D3×D412yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—121no—
486C22×C4D3×D411yescharacteristic
487C22×C4D3×D411yescharacteristic
488C22×C4C23×D311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—32no—
507C4⋊C4—32no—
508C4⋊C4—42no—
509C4⋊C4—64no—
510C4⋊C4—64no—
511C4⋊C4—64no—
512C4⋊C4—64no—
513C4⋊C4—64no—
514C4⋊C4—64no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—122no—
520C4⋊C4—122no—
521C4⋊C4—122no—
522C4⋊C4#23512yes—
523C4⋊C4D3×D412yes—
524C22⋊C4—64no—
525C22⋊C4—64no—
526C22⋊C4—64no—
527C22⋊C4—122no—
528C42—24no—
529C42—21no—
530C42—21no—
531C42—62no—
532C42—62no—
533C42—64no—
534C42—64no—
535C42—61no—
536C42—61no—
537C42—122no—
538C22×C3C2×#12011yescharacteristic
539C22×C3#52511yescharacteristic
540C22×C3#53611yescharacteristic
541C22×C3C2×#13511yescharacteristic
542C22×C3C2×#13711yescharacteristic
543C22×C3C2×#13711yescharacteristic
544C22×C3#57511yescharacteristic
545C2×D3—22no—
546C2×D3—22no—
547C2×D3—22no—
548C2×D3—41no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—41no—
552C3×C4—22no—
553C3×C4—42no—
554C3×C4—42no—
555C3×C4—42no—
556C3×C4—41no—
557C3×C4—81no—
558C3×C4#54711yescharacteristic
559C3×C4#54711yescharacteristic
560C3×C4#54711yescharacteristic
561C3×C4#54711yescharacteristic
562Dic3—22no—
563Dic3—22no—
564Dic3—22no—
565Dic3—42no—
566Dic3—42no—
567Dic3—42no—
568Dic3—81no—
569C23—32no—
570C23—61no—
571C23—61no—
572C23—61no—
573C23—61no—
574C23—61no—
575C23—61no—
576C23C2×D3×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581Q8—62no—
582Q8—62no—
583Q8—128no—
584Q8—121no—
585Q8—121no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—32no—
595C2×C4—32no—
596C2×C4—32no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—68no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—124no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4#91512yes—
631C2×C4C2×#23511yescharacteristic
632C2×C4C2×D3×D411yescharacteristic
633C2×C4C2×D3×D411yescharacteristic
634C2×C4#100711yescharacteristic
635C2×C4D3×#4111yescharacteristic
636C2×C4#101011yescharacteristic
637C8—41no—
638C8—41no—
639C8—121no—
640C8—121no—
641C2×C3#240611yescharacteristic
642C2×C3#300211yescharacteristic
643C2×C3#300411yescharacteristic
644C2×C3#310711yescharacteristic
645C2×C3#311611yescharacteristic
646C2×C3#323111yescharacteristic
647C2×C3#323711yescharacteristic
648D3—42no—
649C22—62no—
650C22—62no—
651C22—62no—
652C22—121no—
653C22—121no—
654C22—121no—
655C22—121no—
656C22#554711yescharacteristic
657C22D3×#12011yescharacteristic
658C22#562811yescharacteristic
659C22C2×#91511yescharacteristic
660C22D3×#13511yescharacteristic
661C22D3×#13711yescharacteristic
662C22#577711yescharacteristic
663C4—22no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—42no—
668C4—62no—
669C4—62no—
670C4—62no—
671C4—81no—
672C4—122no—
673C4—122no—
674C4—122no—
675C4—241no—
676C4#550711yescharacteristic
677C4#555111yescharacteristic
678C4#563111yescharacteristic
679C4#565511yescharacteristic
680C3#2406511yescharacteristic, complemented, Sylow
681C2—122no—
682C2#7460011yescharacteristic
683C2#7965511yescharacteristic
684C2#7984811yescharacteristic
685C2#8003911yescharacteristic
686C2#8022611yescharacteristic
687C2#8101711yescharacteristic
688C2#8123611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1865D3
hypocenterC3#24065
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
5. C3×#1865C2semidirect
6. #71628C2semidirect
7. #70565C2semidirect
8. #70284C2semidirect
10. #70246C2semidirect
11. #69632C2semidirect
13. #68826C2semidirect
14. #68683C2semidirect
75. #1865D3semidirect
680. C3#24065semidirect

Maximal subgroups

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

subgroupindex
C2×#55782
C2×#48682
C2×#48142
#759632
C3×#18652
#716282
#705652
#702842
#702692
#702462
#696322
#688792
#688262
#686832
#686642
#240653