Finite Group Databasebeta

← #10849342

Subgroups — #10849342

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2C2×#4890C211yesmaximal, characteristic, complemented
3C2×#4794C211yesmaximal, characteristic, complemented
4#76071C211yesmaximal, characteristic, complemented
5C3×#1863C211yesmaximal, characteristic, complemented
6#71628C211yesmaximal, characteristic, complemented
7#70638C211yesmaximal, characteristic, complemented
8#70279C211yesmaximal, characteristic, complemented
9#70245C211yesmaximal, characteristic, complemented
10#70230C211yesmaximal, characteristic, complemented
11#69627C211yesmaximal, characteristic
12#68894C211yesmaximal, characteristic, complemented
13#68782C211yesmaximal, characteristic, complemented
14#68629C211yesmaximal, characteristic
15#68620C211yesmaximal, characteristic, complemented
16#25660—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×#947C2211yescharacteristic
20#5585—24no—
21#5585—24no—
22C2×#894—21no—
23C2×#894—21no—
24C2×#893C2211yescharacteristic
25C2×#893C2211yescharacteristic
26C2×#888C2211yescharacteristic
27C2×#875—21no—
28C2×#874C2211yescharacteristic
29C2×C4×D12—21no—
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×#107C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#821C2211yescharacteristic
41C2×#815C2211yescharacteristic
42C2×Dic3⋊C8C2211yescharacteristic
43#4984C2211yescharacteristic
44#4979—21no—
45#4975—21no—
46Dic3×C4⋊C4C2211yescharacteristic
47#4972C2211yescharacteristic
48C2×#809C2211yescharacteristic
49C2×#809C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#807C2211yescharacteristic
53#4928C2211yescharacteristic
54#4890—24no—
55#4794—24no—
56#4656—21no—
57#4646—21no—
58C3×#340C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#559—31no—
65C2×#489—31no—
66C2×#474—31no—
67#2567—31no—
68#2042—31no—
69#2042—31no—
70#2041—31no—
71#2041—31no—
72#1985—31no—
73#1983—31no—
74#1863—31no—
75#1863D311yescharacteristic, complemented
76#1862—31no—
77#1862—31no—
78#1762—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×D12—21no—
81C22×D12—21no—
82C22×D12C2311yescharacteristic
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×C4×D3—21no—
86C22×Dic6—21no—
87C22×Dic6D411yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99#947—22no—
100#947—22no—
101#947—22no—
102#947D414yes—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4D411yescharacteristic
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#894—28no—
124#894—28no—
125#893—22no—
126#893—22no—
127#893—22no—
128#893D414yes—
129#888—22no—
130#888—22no—
131#875—28no—
132#874—22no—
133#874—22no—
134C4×D12—28no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#107—22no—
138C3×#103—22no—
139C3×#103—22no—
140C3×#95—21no—
141#835—22no—
142#835—22no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147#821—22no—
148#821—22no—
149#821—22no—
150#821—22no—
151#815—22no—
152Dic3⋊C8—22no—
153#809—22no—
154#809—22no—
155#809—22no—
156#809—22no—
157#807—22no—
158#559—64no—
159#559—64no—
160C2×C4⋊Q8—31no—
161C2×#127—31no—
162C2×#124—31no—
163C2×#124—31no—
164C2×#124—61no—
165C2×#123—61no—
166C2×#123—61no—
167C2×C4×Q8—31no—
168C2×C4×Q8C2×D311yescharacteristic
169C2×C4×D4—31no—
170C2×C4×D4—31no—
171C2×C4×D4—61no—
172#489—64no—
173#474—64no—
174C2×#107—31no—
175C2×#107C2×D311yescharacteristic
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103C2×D311yescharacteristic
180C2×#103C2×D311yescharacteristic
181C2×#102—31no—
182C2×#102—31no—
183C2×#102—31no—
184C2×#102—31no—
185C2×#102—31no—
186C2×#102—31no—
187C2×#102—31no—
188C2×#102—31no—
189C2×#97—31no—
190#395—61no—
191#395—61no—
192#391—61no—
193#389—31no—
194#389—31no—
195#389—31no—
196#389C2×D311yescharacteristic
197#384—61no—
198C4×C4⋊C4—31no—
199#340—31no—
200#340C2×D311yescharacteristic
201#336—31no—
202#336—31no—
203#336—31no—
204#336C2×D311yescharacteristic
205C23×D3—21no—
206C23×D3—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D412yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—24no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—24no—
238C2×D12—42no—
239C2×D12—42no—
240C2×D12—81no—
241C2×D12—81no—
242C2×D12C2×D412yes—
243C2×D12C2×D412yes—
244C2×C4×D3—24no—
245C2×C4×D3—24no—
246C2×C4×D3—81no—
247C2×C4×D3—81no—
248C2×C4×D3—81no—
249C2×C4×D3—81no—
250C2×Dic6—22no—
251C2×Dic6—22no—
252C2×Dic6—41no—
253C2×Dic6—42no—
254C2×Dic6—81no—
255C2×Dic6—81no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C3×C4⋊C4—24no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—81no—
268C3×C4⋊C4C2×D412yes—
269C3×C4⋊C4#4112yes—
270C3×C42—24no—
271C3×C42—21no—
272C3×C42—21no—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—24no—
275(C2×D3)⋊C4—24no—
276(C2×D3)⋊C4—24no—
277(C2×D3)⋊C4—24no—
278(C2×D3)⋊C4—24no—
279C4⋊Dic3—21no—
280C4⋊Dic3—22no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—81no—
284C4⋊Dic3#4112yes—
285Dic3⋊C4—22no—
286Dic3⋊C4—22no—
287Dic3⋊C4—42no—
288Dic3⋊C4—81no—
289Dic3⋊C4—81no—
290C4×Dic3—21no—
291C4×Dic3—21no—
292C4×Dic3—42no—
293C4×Dic3C2×D412yes—
294C4×Dic3C2×D412yes—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C22×Q8—31no—
302C22×Q8—61no—
303C22×Q8C22×D311yescharacteristic
304C22×D4—31no—
305C22×D4—31no—
306C22×D4—61no—
307C22×D4—61no—
308C23×C4—61no—
309C23×C4—61no—
310C22×C8—31no—
311C22×C8—31no—
312C22×C8C22×D311yescharacteristic
313C22×C8C22×D311yescharacteristic
314C4⋊Q8—62no—
315C4⋊Q8—62no—
316#127—62no—
317#127—62no—
318#124—34no—
319#124—68no—
320#124—62no—
321#124—62no—
322#124—62no—
323#123—68no—
324#123—68no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—62no—
328C4×Q8—62no—
329C4×D4—34no—
330C4×D4—68no—
331C4×D4—62no—
332C4×D4—62no—
333C4×D4—62no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—31no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C4⋊C4C22×D311yescharacteristic
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C42—21no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—61no—
358C2×C42C22×D311yescharacteristic
359#107—22no—
360#107—62no—
361C4⋊C8—62no—
362#103—22no—
363#103—22no—
364#103—62no—
365#103—62no—
366#102—62no—
367#102—62no—
368#102—62no—
369#102—62no—
370#102—62no—
371#102—62no—
372#102—62no—
373#102—62no—
374#97—62no—
375#95—21no—
376#95—61no—
377#95—61no—
378#95—61no—
379#95—61no—
380C23×C3C22×D411yescharacteristic
381C22×D3—22no—
382C22×D3—22no—
383C22×D3—41no—
384C22×D3—41no—
385C22×D3—41no—
386C22×D3—41no—
387C22×D3—41no—
388C22×D3—41no—
389C22×D3—41no—
390C22×D3—41no—
391C22×D3—41no—
392C22×D3—41no—
393C22×D3—41no—
394C22×D3—41no—
395C3×Q8—22no—
396C3×Q8—22no—
397C3×Q8—41no—
398C3×Q8—41no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4#13712yes—
417C2×C3×C4C22×D411yescharacteristic
418C2×C3×C4C22×D411yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C4C2×#4111yescharacteristic
421C2×C3×C42−1+411yescharacteristic
422C2×C3×C42−1+411yescharacteristic
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—22no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3#12012yes—
440C2×Dic3#12012yes—
441D12—22no—
442D12—22no—
443D12—22no—
444D12—22no—
445D12—48no—
446D12—41no—
447D12—41no—
448D12—41no—
449D12—41no—
450D12—84no—
451C4×D3—84no—
452C4×D3—84no—
453Dic6—44no—
454Dic6—84no—
455C3×C8—41no—
456C3×C8—41no—
457C3⋊C8—41no—
458C3⋊C8—41no—
459C24—61no—
460C24—61no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—62no—
467C2×Q8—62no—
468C2×Q8—121no—
469C2×Q8—122no—
470C2×Q8—241no—
471C2×Q8—241no—
472C2×Q8D3×D412yes—
473C2×D4—32no—
474C2×D4—32no—
475C2×D4—64no—
476C2×D4—64no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—61no—
481C2×D4—61no—
482C2×D4—61no—
483C2×D4—61no—
484C2×D4—61no—
485C2×D4—61no—
486C2×D4—61no—
487C2×D4—122no—
488C2×D4—122no—
489C2×D4—241no—
490C2×D4—241no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—31no—
496C22×C4—31no—
497C22×C4—31no—
498C22×C4—31no—
499C22×C4—64no—
500C22×C4—64no—
501C22×C4—61no—
502C22×C4—61no—
503C22×C4—241no—
504C22×C4—241no—
505C22×C4—241no—
506C22×C4—241no—
507C22×C4D3×D411yescharacteristic
508C22×C4D3×D411yescharacteristic
509C22×C4C23×D311yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—61no—
517C2×C8—61no—
518C2×C8—61no—
519C2×C8—61no—
520C2×C8—61no—
521C2×C8—61no—
522C4⋊C4—24no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—32no—
528C4⋊C4—62no—
529C4⋊C4—62no—
530C4⋊C4—62no—
531C4⋊C4—61no—
532C4⋊C4—61no—
533C4⋊C4—61no—
534C4⋊C4—81no—
535C4⋊C4—122no—
536C4⋊C4—241no—
537C4⋊C4—241no—
538C4⋊C4—241no—
539C4⋊C4#23512yes—
540C4⋊C4D3×D412yes—
541C22⋊C4—64no—
542C22⋊C4—64no—
543C22⋊C4—64no—
544C22⋊C4—64no—
545C22⋊C4—64no—
546C22⋊C4—64no—
547C42—24no—
548C42—21no—
549C42—21no—
550C42—32no—
551C42—32no—
552C42—61no—
553C42—61no—
554C42—122no—
555C22×C3C2×#12011yescharacteristic
556C22×C3#54811yescharacteristic
557C22×C3#54811yescharacteristic
558C22×C3C2×#13711yescharacteristic
559C22×C3#57511yescharacteristic
560C22×C3#57611yescharacteristic
561C22×C3#57611yescharacteristic
562C2×D3—42no—
563C2×D3—42no—
564C2×D3—42no—
565C2×D3—42no—
566C2×D3—42no—
567C2×D3—42no—
568C2×D3—81no—
569C2×D3—81no—
570C2×D3—81no—
571C2×D3—81no—
572C2×D3—81no—
573C2×D3—81no—
574C2×D3—81no—
575C2×D3—81no—
576C3×C4—22no—
577C3×C4—42no—
578C3×C4—42no—
579C3×C4—42no—
580C3×C4—41no—
581C3×C4—81no—
582C3×C4#53211yescharacteristic
583C3×C4#53611yescharacteristic
584C3×C4#54711yescharacteristic
585C3×C4#54711yescharacteristic
586Dic3—22no—
587Dic3—22no—
588Dic3—42no—
589Dic3—41no—
590Dic3—41no—
591Dic3—81no—
592C23—62no—
593C23—62no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23—121no—
602C23—121no—
603C23—121no—
604C23—121no—
605C23—121no—
606C23C2×D3×D411yescharacteristic
607Q8—22no—
608Q8—22no—
609Q8—41no—
610Q8—41no—
611Q8—124no—
612Q8—244no—
613D4—62no—
614D4—62no—
615D4—62no—
616D4—62no—
617D4—128no—
618D4—121no—
619D4—121no—
620D4—121no—
621D4—121no—
622D4—244no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—32no—
632C2×C4—32no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—61no—
643C2×C4—61no—
644C2×C4—61no—
645C2×C4—61no—
646C2×C4—62no—
647C2×C4—61no—
648C2×C4—61no—
649C2×C4—61no—
650C2×C4—61no—
651C2×C4—61no—
652C2×C4—61no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—121no—
658C2×C4—244no—
659C2×C4—244no—
660C2×C4#91812yes—
661C2×C4C2×#23511yescharacteristic
662C2×C4C2×D3×D411yescharacteristic
663C2×C4C2×D3×D411yescharacteristic
664C2×C4#100711yescharacteristic
665C2×C4D3×#4111yescharacteristic
666C2×C4#101011yescharacteristic
667C8—41no—
668C8—41no—
669C8—121no—
670C8—121no—
671C2×C3#243211yescharacteristic
672C2×C3#300711yescharacteristic
673C2×C3#300811yescharacteristic
674C2×C3#337011yescharacteristic
675C2×C3#337011yescharacteristic
676C2×C3#337511yescharacteristic
677C2×C3#337511yescharacteristic
678D3—82no—
679D3—82no—
680C22—122no—
681C22—122no—
682C22—122no—
683C22—122no—
684C22—122no—
685C22—122no—
686C22—241no—
687C22—241no—
688C22—241no—
689C22—241no—
690C22—241no—
691C22—241no—
692C22—241no—
693C22—241no—
694C22#555211yescharacteristic
695C22D3×#12011yescharacteristic
696C22#563311yescharacteristic
697C22C2×#91811yescharacteristic
698C22#576911yescharacteristic
699C22#577411yescharacteristic
700C22#577711yescharacteristic
701C4—22no—
702C4—42no—
703C4—42no—
704C4—41no—
705C4—42no—
706C4—62no—
707C4—62no—
708C4—81no—
709C4—121no—
710C4—122no—
711C4—121no—
712C4—241no—
713C4#550711yescharacteristic
714C4#554811yescharacteristic
715C4#562811yescharacteristic
716C4#565511yescharacteristic
717C3#2566011yescharacteristic, complemented, Sylow
718C2—242no—
719C2—242no—
720C2#7441111yescharacteristic
721C2#7970311yescharacteristic
722C2#7975311yescharacteristic
723C2#8008511yescharacteristic
724C2#8018411yescharacteristic
725C2#8113911yescharacteristic
726C2#8121711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1863C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1863D3
hypocenterC3#25660
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
2. C2×#4890C2semidirect
3. C2×#4794C2semidirect
4. #76071C2semidirect
5. C3×#1863C2semidirect
6. #71628C2semidirect
7. #70638C2semidirect
8. #70279C2semidirect
9. #70245C2semidirect
10. #70230C2semidirect
12. #68894C2semidirect
13. #68782C2semidirect
15. #68620C2semidirect
75. #1863D3semidirect
717. C3#25660semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C2×#48902
C2×#47942
#760712
C3×#18632
#716282
#706382
#702792
#702452
#702302
#696272
#688942
#687822
#686292
#686202
#256603