Finite Group Databasebeta

← #10800519

Subgroups — #10800519

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85837C211yesmaximal, characteristic
2#78518C211yesmaximal, characteristic, complemented
3#73382C211yesmaximal, characteristic, complemented
4C3×#1759C211yesmaximal, characteristic, complemented
5C4×#837C211yesmaximal, characteristic, complemented
6#70383C211yesmaximal, characteristic, complemented
7#70382C211yesmaximal, characteristic, complemented
8#70357C211yesmaximal, characteristic, complemented
9#70356C211yesmaximal, characteristic, complemented
10C4×#825C211yesmaximal, characteristic, complemented
11#69430C211yesmaximal, characteristic, complemented
12#68757C211yesmaximal, characteristic, complemented
13#68756C211yesmaximal, characteristic, complemented
14#68732C211yesmaximal, characteristic, complemented
15#68731C211yesmaximal, characteristic, complemented
16#21419—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18#5832C2211yescharacteristic
19#5828C2211yescharacteristic
20#5583C412yescomplemented
21#5582C412yescomplemented
22#5564—22no—
23#5562C412yescomplemented
24#5561C412yescomplemented
25#5560—22no—
26#5528—22no—
27#5524C2211yescharacteristic
28#5517C2211yescharacteristic
29#5479—22no—
30C4×#235C2211yescharacteristic
31C2×#872C2211yescharacteristic
32C2×#872C2211yescharacteristic
33C2×C42×D3C2211yescharacteristic
34C2×C3×#104C2211yescharacteristic
35C2×C3×#104C2211yescharacteristic
36C3×C4×M16C2211yescharacteristic
37C3×#376C2211yescharacteristic
38C2×#837C2211yescharacteristic
39C2×#837C2211yescharacteristic
40C2×#825C2211yescharacteristic
41C2×#825C2211yescharacteristic
42#5126C2211yescharacteristic
43#5126C2211yescharacteristic
44#5118C2211yescharacteristic
45#5118C2211yescharacteristic
46#4932C2211yescharacteristic
47C42×Dic3C2211yescharacteristic
48C42×Dic3C2211yescharacteristic
49C2×#805C2211yescharacteristic
50C2×#805C2211yescharacteristic
51C4×#208C2211yescharacteristic
52#4709—22no—
53#4690—22no—
54#4666—22no—
55#4649—22no—
56C3×#339C2211yescharacteristic
57C3×#339C2211yescharacteristic
58#4555C2211yescharacteristic
59#4555C2211yescharacteristic
60#4532C2211yescharacteristic
61#4532C2211yescharacteristic
62#4523C2211yescharacteristic
63#4523C2211yescharacteristic
64#3455—31no—
65#2921—31no—
66C4×#117—31no—
67#1995—31no—
68#1994—31no—
69#1963—31no—
70#1962—31no—
71#1897—31no—
72#1897—31no—
73#1896—31no—
74#1896—31no—
75#1759—31no—
76#1759D311yescharacteristic, complemented
77C4×#104—31no—
78C4×#104—31no—
79C2×C3×#41C2311yescharacteristic
80C2×#235C2311yescharacteristic
81C22×C4×D3—21no—
82C22×C4×D3C2311yescharacteristic
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C3×C4×Q8C2×C412yescomplemented
86C3×C4×D4—22no—
87C3×C4×D4C2×C412yescomplemented
88C3×#117—21no—
89C3×#117C2311yescharacteristic
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94#947C2×C411yescharacteristic, complemented
95#946C2×C412yes—
96#944C2×C411yescharacteristic, complemented
97#940C2×C411yescharacteristic, complemented
98#939—22no—
99#938C2×C412yes—
100#937—21no—
101#936C2×C411yescharacteristic, complemented
102#935—21no—
103#930—22no—
104#929—22no—
105#929—22no—
106C4×#67—22no—
107C4×#67—22no—
108C4×#67—22no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4D412yes—
111#926C2311yescharacteristic
112#924—22no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4D412yes—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—22no—
117C2×C4×Dic3—22no—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C2×#208C2311yescharacteristic
122C2×#208C2311yescharacteristic
123#898—22no—
124#897C2×C412yes—
125#896—22no—
126#895C2×C412yes—
127#894—22no—
128#890C2×C412yescomplemented
129#889—22no—
130#888C2×C412yescomplemented
131#886—21no—
132#885—22no—
133#884—22no—
134#883—22no—
135#882—21no—
136#879—22no—
137#878—22no—
138#876—22no—
139#876—22no—
140#875—22no—
141C4×D12—22no—
142C4×D12C2×C412yescomplemented
143#872—21no—
144#872—22no—
145#872—22no—
146#872C2×C412yes—
147#872C2×C412yes—
148#872D412yes—
149#872D412yes—
150C42×D3—21no—
151C42×D3—22no—
152C42×D3—22no—
153#870—22no—
154C4×Dic6—22no—
155C4×Dic6C2×C412yescomplemented
156C3×#104—22no—
157C3×#104—22no—
158C3×#97—21no—
159C3×C4×C8—21no—
160C3×#95—22no—
161#837—22no—
162#837—22no—
163C3⋊#95—22no—
164#825—22no—
165#825—22no—
166(C2×D3)⋊C8—22no—
167(C2×D3)⋊C8—22no—
168Dic3⋊C8—22no—
169Dic3⋊C8—22no—
170#805—22no—
171#805—22no—
172#803—21no—
173C4×C3⋊C8—21no—
174#542—32no—
175#542—32no—
176#541—32no—
177#540—32no—
178#539—62no—
179#538—62no—
180#533—31no—
181#532—31no—
182#529—62no—
183#529—62no—
184#525—31no—
185#525—31no—
186C4×#41—31no—
187C4×#41C2×D311yescharacteristic
188C2×#117—31no—
189C2×#117—31no—
190C22×C42—31no—
191#433—62no—
192#433—62no—
193#423—31no—
194#423—31no—
195C2×#104—31no—
196C2×#104—31no—
197C2×#104—31no—
198C2×#104—31no—
199C2×#104—31no—
200C2×#104—31no—
201C2×#104C2×D311yescharacteristic
202C2×#104C2×D311yescharacteristic
203#407—31no—
204#407—31no—
205C4×M16—31no—
206C4×M16C2×D311yescharacteristic
207C4×C4⋊C4—62no—
208C4×C22⋊C4—62no—
209#376—31no—
210#376C2×D311yescharacteristic
211C43—31no—
212C43—31no—
213#339—31no—
214#339—31no—
215#339—31no—
216#339—31no—
217#339—31no—
218#339—31no—
219#339C2×D311yescharacteristic
220#339C2×D311yescharacteristic
221C23×D3C2×D411yescharacteristic
222C3×#41—21no—
223C3×#41—21no—
224C3×#41—21no—
225C3×#41—21no—
226C2×C3×Q8C22×C411yescharacteristic
227C2×C3×D4—21no—
228C2×C3×D4C22×C411yescharacteristic
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—22no—
233C22×C3×C4C2×D412yes—
234C22×C3×C4C2411yescharacteristic
235C2×#67—21no—
236C2×#67—22no—
237C2×#67—22no—
238C22×Dic3—21no—
239C22×Dic3—21no—
240C22×Dic3C2×D411yescharacteristic
241C22×Dic3C2×D412yes—
242#235—21no—
243#235—21no—
244#235—21no—
245#235—21no—
246C2×D12—22no—
247C2×D12C22×C411yescharacteristic
248C2×C4×D3—21no—
249C2×C4×D3—21no—
250C2×C4×D3—21no—
251C2×C4×D3—21no—
252C2×C4×D3—24no—
253C2×C4×D3—21no—
254C2×C4×D3—22no—
255C2×C4×D3—44no—
256C2×C4×D3C22×C412yes—
257C2×C4×D3C2×D412yes—
258C2×Dic6—22no—
259C2×Dic6C22×C411yescharacteristic
260C3×M16—21no—
261C3×M16—21no—
262C3×M16—21no—
263C3×M16—21no—
264C2×C3×C8—21no—
265C2×C3×C8—21no—
266C3×C4⋊C4—22no—
267C3×C4⋊C4C22×C412yes—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—22no—
270C3×C42—22no—
271C3×C42—21no—
272C3×C42—21no—
273C3×C42—21no—
274C3×C42C22×C412yes—
275C3×C42C2×D411yescharacteristic
276C3×C42C2×D411yescharacteristic
277C3×C42#4112yes—
278C3×C42#4112yes—
279#217—21no—
280#217—22no—
281#217—22no—
282(C2×D3)⋊C4—24no—
283(C2×D3)⋊C4—22no—
284(C2×D3)⋊C4—22no—
285(C2×D3)⋊C4—22no—
286(C2×D3)⋊C4—22no—
287(C2×D3)⋊C4—22no—
288(C2×D3)⋊C4C22⋊C412yes—
289(C2×D3)⋊C4C22⋊C412yes—
290(C2×D3)⋊C4C22⋊C412yes—
291(C2×D3)⋊C4C22⋊C412yes—
292C4⋊Dic3—22no—
293C4⋊Dic3—22no—
294C4⋊Dic3C22×C412yes—
295Dic3⋊C4—24no—
296Dic3⋊C4—22no—
297Dic3⋊C4—22no—
298Dic3⋊C4—22no—
299Dic3⋊C4—22no—
300Dic3⋊C4—22no—
301Dic3⋊C4C22⋊C412yes—
302Dic3⋊C4C22⋊C412yes—
303Dic3⋊C4C22⋊C412yes—
304Dic3⋊C4C22⋊C412yes—
305C4×Dic3—24no—
306C4×Dic3—22no—
307C4×Dic3—21no—
308C4×Dic3—21no—
309C4×Dic3—21no—
310C4×Dic3—44no—
311C4×Dic3—44no—
312C4×Dic3C22×C411yescharacteristic
313C4×Dic3C22×C411yescharacteristic
314C4×Dic3C2×D412yes—
315C4×Dic3#4112yes—
316C4×Dic3#4112yes—
317#208—21no—
318#208—21no—
319#208—21no—
320#208—21no—
321C2×C3⋊C8—21no—
322C2×C3⋊C8—21no—
323C2×#41—31no—
324C2×#41C22×D311yescharacteristic
325C23×C4—31no—
326C23×C4—61no—
327C2×M16—31no—
328C2×M16—31no—
329C2×M16C22×D311yescharacteristic
330C2×M16C22×D311yescharacteristic
331C4⋊Q8—32no—
332C4⋊Q8—31no—
333#127—31no—
334#126—62no—
335#126—62no—
336#126—62no—
337#126—62no—
338#125—32no—
339#125—62no—
340#125—62no—
341#124—31no—
342#124—31no—
343#124—62no—
344#124—62no—
345#123—62no—
346#123—61no—
347#123—62no—
348#123—62no—
349#123—62no—
350#123—61no—
351C22⋊Q8—32no—
352C22⋊Q8—32no—
353C22⋊Q8—62no—
354C22⋊Q8—62no—
355C22⋊Q8—62no—
356#121—32no—
357#121—32no—
358#121—62no—
359#121—62no—
360#121—62no—
361#120—61no—
362#120—61no—
363C4×Q8—32no—
364C4×Q8—62no—
365C4×Q8C4×D312yescomplemented
366C4×D4—22no—
367C4×D4—32no—
368C4×D4—62no—
369C4×D4—62no—
370C4×D4—62no—
371C4×D4—62no—
372C4×D4C4×D312yescomplemented
373#117—21no—
374#117—32no—
375#117—32no—
376#117—32no—
377#117—32no—
378#117—31no—
379#117—62no—
380#117—62no—
381#117—61no—
382#117C22×D311yescharacteristic
383C2×C4⋊C4—32no—
384C2×C4⋊C4—61no—
385C2×C22⋊C4—32no—
386C2×C22⋊C4—61no—
387C2×C42—21no—
388C2×C42—31no—
389C2×C42—31no—
390C2×C42—31no—
391C2×C42—62no—
392C2×C42—61no—
393C2×C42—62no—
394C2×C42—61no—
395C2×C42—62no—
396C2×C42—62no—
397C2×C42C22×D311yescharacteristic
398C2×C42C22×D311yescharacteristic
399C2×C42C22×D311yescharacteristic
400C4⋊C8—62no—
401C4⋊C8—62no—
402#104—22no—
403#104—22no—
404#104—62no—
405#104—62no—
406#104—62no—
407#104—62no—
408#104—62no—
409#104—62no—
410C22⋊C8—62no—
411C22⋊C8—62no—
412#97—21no—
413#97—61no—
414C4×C8—21no—
415C4×C8—61no—
416#95—22no—
417#95—62no—
418C23×C3—21no—
419C23×C3C22×D411yescharacteristic
420C22×D3—21no—
421C22×D3—21no—
422C22×D3—21no—
423C22×D3—42no—
424C22×D3C2×C22⋊C412yes—
425C3×Q8—21no—
426C3×Q8—21no—
427C3×D4—21no—
428C3×D4—21no—
429C3×D4—41no—
430C3×D4—41no—
431C2×C3×C4—21no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4—22no—
435C2×C3×C4—42no—
436C2×C3×C4—42no—
437C2×C3×C4—42no—
438C2×C3×C4—41no—
439C2×C3×C4—41no—
440C2×C3×C4—82no—
441C2×C3×C4C2×C22⋊C412yes—
442C2×C3×C4C2×C22⋊C412yes—
443C2×C3×C4C23×C411yescharacteristic
444C2×C3×C4C22×D411yescharacteristic
445C2×C3×C4C2×#4111yescharacteristic
446C2×C3×C4C2×#4111yescharacteristic
447C2×C3×C4C2×#4111yescharacteristic
448C2×C3×C4C2×#4111yescharacteristic
449#67—41no—
450#67—41no—
451#67—82no—
452#67—82no—
453C2×Dic3—24no—
454C2×Dic3—21no—
455C2×Dic3—22no—
456C2×Dic3—21no—
457C2×Dic3—21no—
458C2×Dic3—22no—
459C2×Dic3—22no—
460C2×Dic3—42no—
461C2×Dic3—42no—
462C2×Dic3C2×C22⋊C412yes—
463C2×Dic3C2×C22⋊C412yes—
464C2×Dic3C2×C22⋊C412yes—
465D12—21no—
466D12—21no—
467D12—82no—
468C4×D3—22no—
469C4×D3—22no—
470C4×D3—22no—
471C4×D3—22no—
472C4×D3—48no—
473C4×D3—41no—
474C4×D3—41no—
475C4×D3—82no—
476Dic6—21no—
477Dic6—21no—
478Dic6—82no—
479C3×C8—41no—
480C3×C8—41no—
481C3⋊C8—41no—
482C3⋊C8—41no—
483C24—31no—
484#41—21no—
485#41—21no—
486#41—21no—
487#41—21no—
488#41—61no—
489#41—61no—
490#41—61no—
491#41—61no—
492C2×Q8—31no—
493C2×Q8—62no—
494C2×Q8C2×C4×D311yescharacteristic
495C2×D4—21no—
496C2×D4—31no—
497C2×D4—62no—
498C2×D4—62no—
499C2×D4—61no—
500C2×D4—62no—
501C2×D4C2×C4×D311yescharacteristic
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—31no—
507C22×C4—32no—
508C22×C4—32no—
509C22×C4—32no—
510C22×C4—64no—
511C22×C4—61no—
512C22×C4—61no—
513C22×C4—61no—
514C22×C4—61no—
515C22×C4—61no—
516C22×C4—61no—
517C22×C4—61no—
518C22×C4—62no—
519C22×C4—124no—
520C22×C4D3×D412yes—
521C22×C4C23×D311yescharacteristic
522M16—21no—
523M16—21no—
524M16—21no—
525M16—21no—
526M16—61no—
527M16—61no—
528M16—61no—
529M16—61no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—61no—
533C2×C8—61no—
534C4⋊C4—22no—
535C4⋊C4—32no—
536C4⋊C4—32no—
537C4⋊C4—32no—
538C4⋊C4—32no—
539C4⋊C4—32no—
540C4⋊C4—62no—
541C4⋊C4—62no—
542C4⋊C4—62no—
543C4⋊C4—62no—
544C4⋊C4—62no—
545C4⋊C4—62no—
546C4⋊C4—64no—
547C4⋊C4—62no—
548C4⋊C4C2×C4×D312yes—
549C22⋊C4—21no—
550C22⋊C4—22no—
551C22⋊C4—32no—
552C22⋊C4—32no—
553C22⋊C4—32no—
554C22⋊C4—32no—
555C22⋊C4—62no—
556C22⋊C4—62no—
557C22⋊C4—62no—
558C22⋊C4—62no—
559C22⋊C4—64no—
560C22⋊C4—62no—
561C22⋊C4—62no—
562C22⋊C4—61no—
563C22⋊C4—62no—
564C42—22no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—31no—
569C42—31no—
570C42—32no—
571C42—32no—
572C42—32no—
573C42—64no—
574C42—62no—
575C42—61no—
576C42—61no—
577C42—61no—
578C42—124no—
579C42—124no—
580C42C2×C4×D312yes—
581C42#23512yes—
582C42#23512yes—
583C42D3×D411yescharacteristic
584C42D3×D411yescharacteristic
585C22×C3—22no—
586C22×C3—42no—
587C22×C3—41no—
588C22×C3C22×C22⋊C411yescharacteristic
589C22×C3#52511yescharacteristic
590C22×C3#52511yescharacteristic
591C2×D3—22no—
592C2×D3—22no—
593C2×D3—22no—
594C2×D3—42no—
595C2×D3—42no—
596C2×D3—42no—
597C2×D3—41no—
598C3×C4—22no—
599C3×C4—22no—
600C3×C4—42no—
601C3×C4—42no—
602C3×C4—42no—
603C3×C4—41no—
604C3×C4C4×#4111yescharacteristic
605C3×C4C4×#4111yescharacteristic
606C3×C4#53211yescharacteristic
607C3×C4#53211yescharacteristic
608Dic3—24no—
609Dic3—22no—
610Dic3—22no—
611Dic3—22no—
612Dic3—42no—
613Dic3—42no—
614Dic3—41no—
615C23—21no—
616C23—32no—
617C23—61no—
618C23—61no—
619C23—61no—
620C23—122no—
621C23C2×D3×D411yescharacteristic
622Q8—21no—
623Q8—21no—
624Q8—61no—
625Q8—61no—
626Q8—242no—
627D4—21no—
628D4—21no—
629D4—41no—
630D4—41no—
631D4—61no—
632D4—61no—
633D4—121no—
634D4—121no—
635D4—242no—
636D4—242no—
637D4—242no—
638C2×C4—22no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—32no—
643C2×C4—32no—
644C2×C4—32no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—64no—
651C2×C4—62no—
652C2×C4—61no—
653C2×C4—61no—
654C2×C4—62no—
655C2×C4—62no—
656C2×C4—62no—
657C2×C4—62no—
658C2×C4—62no—
659C2×C4—61no—
660C2×C4—62no—
661C2×C4—82no—
662C2×C4—128no—
663C2×C4—122no—
664C2×C4—122no—
665C2×C4—121no—
666C2×C4—121no—
667C2×C4—242no—
668C2×C4D3×C22⋊C412yes—
669C2×C4D3×C22⋊C412yes—
670C2×C4C22×C4×D311yescharacteristic
671C2×C4C2×#23511yescharacteristic
672C2×C4C2×#23511yescharacteristic
673C2×C4C2×D3×D411yescharacteristic
674C2×C4D3×#4111yescharacteristic
675C2×C4D3×#4111yescharacteristic
676C8—41no—
677C8—41no—
678C8—121no—
679C8—121no—
680C2×C3—21no—
681C2×C3—21no—
682C2×C3—81no—
683C2×C3#229711yescharacteristic
684C2×C3#290112yes—
685D3—42no—
686D3—81no—
687C22—22no—
688C22—41no—
689C22—42no—
690C22—62no—
691C22—62no—
692C22—62no—
693C22—122no—
694C22—122no—
695C22—121no—
696C22—122no—
697C22C2×D3×C22⋊C411yescharacteristic
698C22#554711yescharacteristic
699C22#554711yescharacteristic
700C4—22no—
701C4—22no—
702C4—42no—
703C4—42no—
704C4—41no—
705C4—42no—
706C4—64no—
707C4—62no—
708C4—62no—
709C4—62no—
710C4—122no—
711C4—121no—
712C4—122no—
713C4C4×#23511yescharacteristic
714C4#552211yescharacteristic
715C4#554811yescharacteristic
716C4#554811yescharacteristic
717C3#2141911yescharacteristic, complemented, Sylow
718C2—21no—
719C2—21no—
720C2—81no—
721C2—122no—
722C2—241no—
723C2#7356311yescharacteristic
724C2#7899212yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1759C2
socleC22×C3C22×C22⋊C4
prime radical#85837C2
cosocleC22×C3×C4C24
hypercenter#1759D3
hypocenterC3#21419
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
2. #78518C2semidirect
3. #73382C2semidirect
4. C3×#1759C2semidirect
5. C4×#837C2semidirect
6. #70383C2semidirect
7. #70382C2semidirect
8. #70357C2semidirect
9. #70356C2semidirect
10. C4×#825C2semidirect
11. #69430C2semidirect
12. #68757C2semidirect
13. #68756C2semidirect
14. #68732C2semidirect
15. #68731C2semidirect
20. #5583C4semidirect
21. #5582C4semidirect
23. #5562C4semidirect
24. #5561C4semidirect
76. #1759D3semidirect
85. C3×C4×Q8C2×C4semidirect
87. C3×C4×D4C2×C4semidirect
94. #947C2×C4semidirect
96. #944C2×C4semidirect
97. #940C2×C4semidirect
101. #936C2×C4semidirect
128. #890C2×C4semidirect
130. #888C2×C4semidirect
142. C4×D12C2×C4semidirect
155. C4×Dic6C2×C4semidirect
365. C4×Q8C4×D3semidirect
372. C4×D4C4×D3semidirect
717. C3#21419semidirect

Maximal subgroups

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

subgroupindex
#858372
#785182
#733822
C3×#17592
C4×#8372
#703832
#703822
#703572
#703562
C4×#8252
#694302
#687572
#687562
#687322
#687312
#214193