Finite Group Databasebeta

← #10849253

Subgroups — #10849253

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2#82483C211yesmaximal, characteristic, complemented
3C2×#4847C211yesmaximal, characteristic, complemented
4C2×#4697C211yesmaximal, characteristic, complemented
5C3×#1984C211yesmaximal, characteristic, complemented
6#70458C211yesmaximal, characteristic, complemented
7C4×#821C211yesmaximal, characteristic, complemented
8#68684C211yesmaximal, characteristic, complemented
9#68683C211yesmaximal, characteristic, complemented
10#68671C211yesmaximal, characteristic, complemented
11#68670C211yesmaximal, characteristic, complemented
12#68630C211yesmaximal, characteristic, complemented
13#68630C211yesmaximal, characteristic, complemented
14#68629C211yesmaximal, characteristic
15#68629C211yesmaximal, characteristic
16#25653—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947—21no—
19#5585—24no—
20#5585—24no—
21C2×#894—21no—
22C2×#894—21no—
23C2×#893—21no—
24C2×#875C2211yescharacteristic
25C2×#875C2211yescharacteristic
26C2×#874C2211yescharacteristic
27C2×C4×D12C2211yescharacteristic
28C2×C4×D12C2211yescharacteristic
29C2×#870C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4×C8C2211yescharacteristic
33#5230—21no—
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#821C2211yescharacteristic
41C2×#821C2211yescharacteristic
42C2×#818C2211yescharacteristic
43C2×#817C2211yescharacteristic
44C2×#816C2211yescharacteristic
45C2×#816C2211yescharacteristic
46C2×#816C2211yescharacteristic
47C2×#816C2211yescharacteristic
48#4983—21no—
49#4976—21no—
50#4976—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52#4928C2211yescharacteristic
53#4928C2211yescharacteristic
54#4928C2211yescharacteristic
55#4928C2211yescharacteristic
56#4847—24no—
57#4697—24no—
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60C3×#336C2211yescharacteristic
61C3×#336C2211yescharacteristic
62#4551C2211yescharacteristic
63#4551C2211yescharacteristic
64C2×#559—31no—
65C2×#488—31no—
66C2×#478—31no—
67#2584—31no—
68#2042—31no—
69#2042—31no—
70#2041—31no—
71#2041—31no—
72#1984D311yescharacteristic, complemented
73#1983—31no—
74#1865—31no—
75#1864—31no—
76#1863—31no—
77#1862—31no—
78C4×#102—31no—
79C22×C3×Q8D411yescharacteristic
80C22×D12—21no—
81C22×D12—21no—
82C22×D12C2311yescharacteristic
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C22×C4×D3—21no—
86C22×Dic6C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99#947—28no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120#894—28no—
121#894—28no—
122#893—28no—
123#875—22no—
124#875—22no—
125#875—22no—
126#875D414yes—
127#874—22no—
128#874—22no—
129C4×D12—22no—
130C4×D12—22no—
131C4×D12—22no—
132C4×D12D414yes—
133#870—22no—
134#870—22no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×C4⋊C8—22no—
138C3×C4×C8—22no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#821—22no—
145#821—22no—
146#821—22no—
147#821—22no—
148#821—22no—
149#821—22no—
150#821—22no—
151#821—22no—
152#818—22no—
153#817—22no—
154#816—22no—
155#816—22no—
156#816—22no—
157#816—22no—
158#559—64no—
159#559—64no—
160C2×#127—31no—
161C2×#125—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#488—64no—
173#478—64no—
174C2×#107—31no—
175C2×#106—31no—
176C2×C4⋊C8C2×D311yescharacteristic
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#103—31no—
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×C4×C8C2×D311yescharacteristic
190#398—61no—
191#395—61no—
192#389—31no—
193#389—31no—
194#389—31no—
195#389—31no—
196#389—61no—
197#389—61no—
198C4×C4⋊C4—31no—
199#340—31no—
200#340—31no—
201#336C2×D311yescharacteristic
202#336C2×D311yescharacteristic
203#336C2×D311yescharacteristic
204#336C2×D311yescharacteristic
205C23×D3—21no—
206C23×D3—21no—
207C2×C3×Q8—22no—
208C2×C3×Q8—22no—
209C2×C3×Q8—41no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—24no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—24no—
235C2×D12—42no—
236C2×D12—42no—
237C2×D12—81no—
238C2×D12—81no—
239C2×D12C2×D412yes—
240C2×D12C2×D412yes—
241C2×C4×D3—24no—
242C2×C4×D3—24no—
243C2×C4×D3—81no—
244C2×C4×D3—81no—
245C2×C4×D3—81no—
246C2×C4×D3—81no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6C2×D412yes—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8—21no—
265C3×C4⋊C4—22no—
266C3×C4⋊C4—22no—
267C3×C4⋊C4—22no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—21no—
272C3×C42C2×D412yes—
273C3×C42C2×D412yes—
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—
279(C2×D3)⋊C4—24no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—21no—
284C4⋊Dic3—21no—
285C4⋊Dic3—21no—
286C4⋊Dic3—21no—
287C4⋊Dic3C2×D412yes—
288C4⋊Dic3#4112yes—
289C4⋊Dic3#4112yes—
290Dic3⋊C4—24no—
291Dic3⋊C4—42no—
292Dic3⋊C4—42no—
293Dic3⋊C4—42no—
294Dic3⋊C4—42no—
295Dic3⋊C4—81no—
296Dic3⋊C4—81no—
297C4×Dic3—24no—
298C4×Dic3—81no—
299C22×Q8—31no—
300C22×Q8D1211yescharacteristic
301C22×D4—31no—
302C22×D4—31no—
303C22×D4—61no—
304C22×D4—61no—
305C23×C4—61no—
306C23×C4—61no—
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309C22×C8C22×D311yescharacteristic
310C22×C8C22×D311yescharacteristic
311#127—62no—
312#127—62no—
313#125—62no—
314#125—62no—
315#124—34no—
316#124—62no—
317#124—62no—
318#124—68no—
319#124—62no—
320#123—68no—
321#123—68no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8—62no—
325C4×Q8—62no—
326C4×D4—34no—
327C4×D4—62no—
328C4×D4—62no—
329C4×D4—68no—
330C4×D4—62no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4D1211yescharacteristic
344C2×C4⋊C4D1211yescharacteristic
345C2×C4⋊C4D1211yescharacteristic
346C2×C22⋊C4—61no—
347C2×C22⋊C4—61no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C42—61no—
353C2×C42—61no—
354C2×C42C22×D311yescharacteristic
355C2×C42C22×D311yescharacteristic
356C2×C42C22×D311yescharacteristic
357#107—62no—
358#106—62no—
359C4⋊C8—22no—
360#103—62no—
361#103—62no—
362#103—62no—
363#103—62no—
364#102—62no—
365#102—62no—
366#102—62no—
367#102—62no—
368#102—62no—
369#102—62no—
370#102—62no—
371#102—62no—
372C4×C8—22no—
373#95—61no—
374#95—61no—
375#95—61no—
376#95—61no—
377#95—61no—
378C23×C3C22×D411yescharacteristic
379C22×D3—22no—
380C22×D3—22no—
381C22×D3—41no—
382C22×D3—41no—
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—
393C3×Q8—44no—
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—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4#12012yes—
410C2×C3×C4#12012yes—
411C2×C3×C4#13512yes—
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416C2×C3×C42−1+411yescharacteristic
417C2×C3×C42−1+411yescharacteristic
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436D12—22no—
437D12—22no—
438D12—22no—
439D12—22no—
440D12—48no—
441D12—41no—
442D12—41no—
443D12—41no—
444D12—41no—
445D12—84no—
446C4×D3—84no—
447C4×D3—84no—
448Dic6—22no—
449Dic6—22no—
450Dic6—41no—
451Dic6—41no—
452C3×C8—41no—
453C3×C8—41no—
454C3×C8—41no—
455C3×C8—41no—
456C24—61no—
457C24—61no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—32no—
461C2×Q8—41no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8—61no—
467C2×D4—32no—
468C2×D4—32no—
469C2×D4—64no—
470C2×D4—64no—
471C2×D4—61no—
472C2×D4—61no—
473C2×D4—61no—
474C2×D4—61no—
475C2×D4—61no—
476C2×D4—61no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—61no—
481C2×D4—122no—
482C2×D4—122no—
483C2×D4—241no—
484C2×D4—241no—
485C22×C4—64no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—64no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—241no—
494C22×C4—241no—
495C22×C4—241no—
496C22×C4—241no—
497C22×C4C2×D1211yescharacteristic
498C22×C4C2×D1211yescharacteristic
499C22×C4C2×D1211yescharacteristic
500C22×C4C2×D1211yescharacteristic
501C22×C4C2×D1211yescharacteristic
502C22×C4C2×D1211yescharacteristic
503C22×C4C23×D311yescharacteristic
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—32no—
520C4⋊C4—32no—
521C4⋊C4—32no—
522C4⋊C4—64no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—122no—
531C4⋊C4—122no—
532C4⋊C4—122no—
533C4⋊C4—122no—
534C4⋊C4—241no—
535C4⋊C4—241no—
536C22⋊C4—64no—
537C22⋊C4—64no—
538C22⋊C4—64no—
539C22⋊C4—64no—
540C22⋊C4—64no—
541C22⋊C4—64no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—64no—
547C42—241no—
548C42D3×D412yes—
549C42D3×D412yes—
550C22×C3C2×#12011yescharacteristic
551C22×C3#54811yescharacteristic
552C22×C3#54811yescharacteristic
553C22×C3C2×#13511yescharacteristic
554C22×C3#57511yescharacteristic
555C22×C3#57611yescharacteristic
556C22×C3#57711yescharacteristic
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3—42no—
562C2×D3—42no—
563C2×D3—81no—
564C2×D3—81no—
565C2×D3—81no—
566C2×D3—81no—
567C2×D3—81no—
568C2×D3—81no—
569C2×D3—81no—
570C2×D3—81no—
571C3×C4—22no—
572C3×C4—22no—
573C3×C4—22no—
574C3×C4—41no—
575C3×C4—41no—
576C3×C4—41no—
577C3×C4#53211yescharacteristic
578C3×C4#53611yescharacteristic
579C3×C4#54711yescharacteristic
580C3×C4#54711yescharacteristic
581Dic3—42no—
582Dic3—42no—
583Dic3—42no—
584Dic3—42no—
585Dic3—81no—
586Dic3—81no—
587C23—62no—
588C23—62no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23—121no—
601C23C22×D1211yescharacteristic
602Q8—44no—
603Q8—62no—
604Q8—62no—
605Q8—121no—
606Q8—121no—
607D4—62no—
608D4—62no—
609D4—62no—
610D4—62no—
611D4—128no—
612D4—121no—
613D4—121no—
614D4—121no—
615D4—121no—
616D4—244no—
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—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—62no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—244no—
651C2×C4—244no—
652C2×C4#88212yes—
653C2×C4#88212yes—
654C2×C4#90412yes—
655C2×C4C2×D3×D411yescharacteristic
656C2×C4C2×D3×D411yescharacteristic
657C2×C4#100711yescharacteristic
658C2×C4#100711yescharacteristic
659C2×C4D3×#4111yescharacteristic
660C2×C4D3×#4111yescharacteristic
661C8—41no—
662C8—41no—
663C8—41no—
664C8—41no—
665C2×C3#243211yescharacteristic
666C2×C3#300211yescharacteristic
667C2×C3#301211yescharacteristic
668C2×C3#336111yescharacteristic
669C2×C3#336511yescharacteristic
670C2×C3#337011yescharacteristic
671C2×C3#337111yescharacteristic
672D3—82no—
673D3—82no—
674C22—122no—
675C22—122no—
676C22—122no—
677C22—122no—
678C22—122no—
679C22—122no—
680C22—241no—
681C22—241no—
682C22—241no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22—241no—
687C22—241no—
688C22C2×#88211yescharacteristic
689C22#557611yescharacteristic
690C22#557611yescharacteristic
691C22C2×#90411yescharacteristic
692C22#574811yescharacteristic
693C22#575111yescharacteristic
694C22#575211yescharacteristic
695C4—22no—
696C4—22no—
697C4—22no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—122no—
702C4—122no—
703C4—122no—
704C4—122no—
705C4—241no—
706C4—241no—
707C4#563111yescharacteristic
708C4#563111yescharacteristic
709C4#566811yescharacteristic
710C4#572411yescharacteristic
711C3#2565311yescharacteristic, complemented, Sylow
712C2—242no—
713C2—242no—
714C2#7649811yescharacteristic
715C2#7861011yescharacteristic
716C2#7867211yescharacteristic
717C2#7992711yescharacteristic
718C2#7993511yescharacteristic
719C2#7997111yescharacteristic
720C2#7998211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1984C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1984D3
hypocenterC3#25653
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
2. #82483C2semidirect
3. C2×#4847C2semidirect
4. C2×#4697C2semidirect
5. C3×#1984C2semidirect
6. #70458C2semidirect
7. C4×#821C2semidirect
8. #68684C2semidirect
9. #68683C2semidirect
10. #68671C2semidirect
11. #68670C2semidirect
12. #68630C2semidirect
13. #68630C2semidirect
72. #1984D3semidirect
711. C3#25653semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
#824832
C2×#48472
C2×#46972
C3×#19842
#704582
C4×#8212
#686842
#686832
#686712
#686702
#686302
#686292
#256533