Finite Group Databasebeta

← #10818123

Subgroups — #10818123

Order 768.

866 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4898C211yesmaximal, characteristic
2#79202C211yesmaximal, characteristic, complemented
3#79201C211yesmaximal, characteristic
4#79146C211yesmaximal, characteristic, complemented
5#78572C211yesmaximal, characteristic, complemented
6C3×#1952C211yesmaximal, characteristic, complemented
7#70703C211yesmaximal, characteristic, complemented
8#70677C211yesmaximal, characteristic, complemented
9#70674C211yesmaximal, characteristic, complemented
10#70629C211yesmaximal, characteristic, complemented
11#69888C211yesmaximal, characteristic, complemented
12#69005C211yesmaximal, characteristic
13#69004C211yesmaximal, characteristic, complemented
14#68983C211yesmaximal, characteristic, complemented
15#68982C211yesmaximal, characteristic, complemented
16#23049—31nomaximal, Sylow
17C2×#909C2211yescharacteristic
18C2×#908C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#901C2211yescharacteristic
21#5521C2211yescharacteristic
22#5516C2211yescharacteristic
23#5504C2211yescharacteristic
24#5504C2211yescharacteristic
25C2×C3×#108C2211yescharacteristic
26C3×#428C2211yescharacteristic
27C3×#428C2211yescharacteristic
28C2×C3×#107C2211yescharacteristic
29C3×#405C2211yescharacteristic
30C2×#825C2211yescharacteristic
31C2×#824C2211yescharacteristic
32#5131C2211yescharacteristic
33C2×#823C2211yescharacteristic
34#5122C2211yescharacteristic
35#5122C2211yescharacteristic
36#5120C2211yescharacteristic
37#5106C2211yescharacteristic
38C2×#815C2211yescharacteristic
39#5002C2211yescharacteristic
40#4988C2211yescharacteristic
41#4973C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#809C2211yescharacteristic
44#4964C2211yescharacteristic
45#4964C2211yescharacteristic
46#4898C418yescomplemented
47#4889—21no—
48#4889—21no—
49#4881—21no—
50#4881—21no—
51#4880—21no—
52D3×#107—21no—
53#4873—21no—
54#4872—21no—
55#4869—21no—
56#4853—21no—
57#4852—21no—
58#4852—21no—
59#4710—21no—
60#4710—21no—
61C4×#204—21no—
62C4×#204—21no—
63C3×#343C2211yescharacteristic
64C3×#343C2211yescharacteristic
65#4558C2211yescharacteristic
66#4555C2211yescharacteristic
67#4555C2211yescharacteristic
68#4532C2211yescharacteristic
69C2×#471—31no—
70#2949—31no—
71#2947—31no—
72#2946—31no—
73#2913—31no—
74#1973—31no—
75#1952D311yescharacteristic, complemented
76#1931—31no—
77#1930—31no—
78#1883—31no—
79#1882—31no—
80#1857—31no—
81#1856—31no—
82#1773—31no—
83#1765—31no—
84C2×#235C2311yescharacteristic
85C22×D12C2311yescharacteristic
86C22×Dic6C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#117C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C4×#67—41no—
95C4×#67—41no—
96C2×(C2×D3)⋊C4—41no—
97#926C2311yescharacteristic
98#926C2311yescharacteristic
99C2×C4×Dic3C2311yescharacteristic
100C2×#208C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102#909—22no—
103#909C2×C414yescomplemented
104#908—22no—
105#908C2×C414yes—
106C2×Dic12—21no—
107C2×D24—21no—
108C2×#204—21no—
109C2×#204—21no—
110C2×#204—21no—
111C2×#204—21no—
112C2×#204—22no—
113C2×#204C2×C414yescomplemented
114#901—22no—
115#901C2×C414yescomplemented
116C2×ΓL(1,𝔽25)—21no—
117C2×D3×C8—21no—
118#893—21no—
119#893—21no—
120#893—21no—
121#893—21no—
122#892—21no—
123#892—21no—
124D3×C4⋊C4—21no—
125D3×C4⋊C4—21no—
126#887—21no—
127#887—21no—
128#887—21no—
129#887—21no—
130D3×C22⋊C4—41no—
131#877—41no—
132C4×D12—21no—
133C4×D12—21no—
134C4×Dic6—21no—
135C4×Dic6—21no—
136C3×#108C2×C414yescomplemented
137C3×#107—21no—
138C3×#107—21no—
139C3×#107—21no—
140C3×#107—21no—
141C3×#106—21no—
142C3×#106—21no—
143C3×#97—21no—
144C3×C4×C8—21no—
145C3⋊#95—41no—
146#825—22no—
147#824C2×C414yescomplemented
148#823C2×C414yescomplemented
149#821—21no—
150#821—21no—
151#818D411yescharacteristic
152#818D411yescharacteristic
153#817D411yescharacteristic
154#817D411yescharacteristic
155#816—21no—
156#816—21no—
157#815—22no—
158#815—21no—
159#815—21no—
160C8×Dic3—21no—
161C8×Dic3—21no—
162#810—21no—
163#810—21no—
164#810D411yescharacteristic
165#810D411yescharacteristic
166#809—21no—
167#809—21no—
168#809D411yescharacteristic
169#809D411yescharacteristic
170#808—21no—
171#808—21no—
172#807—21no—
173#807—21no—
174C2×#137—31no—
175C2×AGL(1,ℤ/8)—31no—
176C22×SD16—31no—
177C2×#131—31no—
178#520—31no—
179#520—31no—
180#519—31no—
181#518—31no—
182#471—38no—
183#442—61no—
184#441—61no—
185#440—61no—
186#440—61no—
187#440—61no—
188#440—61no—
189C4×Q16—61no—
190C4×SD16—61no—
191C4×SD16—61no—
192C4×SD16—61no—
193C4×SD16—61no—
194C4×D8—61no—
195C2×#108C2×D311yescharacteristic
196#428—31no—
197#428—31no—
198#428—61no—
199#428—61no—
200#428C2×D311yescharacteristic
201#428C2×D311yescharacteristic
202#427—31no—
203#427—61no—
204C2×#107—61no—
205C2×#107C2×D311yescharacteristic
206C2×#104—31no—
207#419—31no—
208#418—31no—
209#416—31no—
210C2×#103—31no—
211C2×#102—31no—
212C2×#101—31no—
213C2×#100—31no—
214#405—31no—
215#405—31no—
216#405C2×D311yescharacteristic
217C2×#97—31no—
218#391—31no—
219#390—31no—
220#343C2×D311yescharacteristic
221#343C2×D311yescharacteristic
222#341—31no—
223#339—31no—
224#339—31no—
225#339—31no—
226C23×D3—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4C2411yescharacteristic
230C2×#67—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233#235—21no—
234#235—21no—
235#235C22×C412yes—
236#235#4112yes—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—21no—
241C2×D12—21no—
242C2×D12C22×C412yes—
243C2×D12C2×D411yescharacteristic
244C2×C4×D3—21no—
245C2×C4×D3—41no—
246C2×C4×D3—41no—
247C2×C4×D3—41no—
248C2×C4×D3—41no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6C22×C412yes—
255C2×Dic6C2×D411yescharacteristic
256C3×M16—21no—
257C3×M16—21no—
258C3×M16—21no—
259C3×M16—21no—
260C3×M16C22×C412yes—
261C3×M16C22×C412yes—
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C2×C3×C8—21no—
265C2×C3×C8C22×C412yes—
266C2×C3×C8C2×D411yescharacteristic
267C2×C3×C8C2×D411yescharacteristic
268C2×C3×C8#4112yes—
269C3×C4⋊C4—21no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—21no—
272C3×C4⋊C4—21no—
273C3×C4⋊C4C2×D411yescharacteristic
274C3×C4⋊C4C2×D411yescharacteristic
275C3×C22⋊C4—21no—
276C3×C42—21no—
277#217—21no—
278#217—21no—
279(C2×D3)⋊C4—41no—
280(C2×D3)⋊C4—41no—
281(C2×D3)⋊C4—41no—
282(C2×D3)⋊C4—41no—
283(C2×D3)⋊C4—82no—
284C4⋊Dic3C2×D411yescharacteristic
285C4⋊Dic3C2×D411yescharacteristic
286C4⋊Dic3C2×D411yescharacteristic
287C4⋊Dic3C2×D411yescharacteristic
288Dic3⋊C4—41no—
289Dic3⋊C4—41no—
290Dic3⋊C4—41no—
291Dic3⋊C4—41no—
292C4×Dic3—21no—
293C4×Dic3—21no—
294C4×Dic3—21no—
295C4×Dic3—21no—
296C4×Dic3—41no—
297C4×Dic3—41no—
298C4×Dic3—41no—
299C4×Dic3—41no—
300#208—21no—
301#208C22×C412yes—
302C2×C3⋊C8—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8—21no—
305C2×C3⋊C8C2×D411yescharacteristic
306C2×C3⋊C8C2×D411yescharacteristic
307Dic12—24no—
308D24—24no—
309#204—24no—
310#204—24no—
311#204—24no—
312#204—24no—
313ΓL(1,𝔽25)—24no—
314D3×C8—24no—
315C2×#41—31no—
316C22×Q8—31no—
317C22×D4—31no—
318#137—34no—
319#137—62no—
320AGL(1,ℤ/8)—34no—
321AGL(1,ℤ/8)—62no—
322C2×Q16—61no—
323C2×SD16—34no—
324C2×SD16—61no—
325C2×SD16—61no—
326C2×SD16—61no—
327C2×SD16—61no—
328C2×SD16—62no—
329C2×D8—61no—
330#131—34no—
331#131—62no—
332C2×M16—31no—
333C2×M16—61no—
334C2×M16C22×D311yescharacteristic
335C2×M16C22×D311yescharacteristic
336C2×M16C22×D311yescharacteristic
337C22×C8—31no—
338C22×C8—61no—
339C22×C8C22×D311yescharacteristic
340C4×Q8—61no—
341C4×Q8—61no—
342C4×Q8—61no—
343C4×Q8—61no—
344C4×Q8—61no—
345C4×Q8—61no—
346C4×D4—61no—
347C4×D4—61no—
348C4×D4—61no—
349C4×D4—61no—
350C4×D4—61no—
351C4×D4—61no—
352C4×D4—121no—
353C4×D4—121no—
354#117—31no—
355#117—31no—
356#117—61no—
357#117—61no—
358#117—121no—
359#117C22×D311yescharacteristic
360C2×C4⋊C4—61no—
361C2×C4⋊C4—61no—
362C2×C4⋊C4C22×D311yescharacteristic
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C22⋊C4—121no—
365C2×C22⋊C4—121no—
366C2×C42—31no—
367#108C4×D314yescomplemented
368#107—21no—
369#107—21no—
370#107—21no—
371#107—21no—
372#107—31no—
373#107—31no—
374#107—61no—
375#107—61no—
376#106—21no—
377#106—21no—
378#106—31no—
379#106—31no—
380#106—61no—
381#106—61no—
382#104—62no—
383#103—31no—
384#103—31no—
385#103—61no—
386#103—61no—
387#103—61no—
388#103—61no—
389#102—31no—
390#102—31no—
391#102—61no—
392#102—61no—
393#102—61no—
394#102—61no—
395#101—34no—
396#100—34no—
397#97—21no—
398#97—61no—
399#97—61no—
400#97—62no—
401C4×C8—21no—
402C4×C8—61no—
403C4×C8—61no—
404#95—121no—
405C23×C3H(5,𝔽2)11yescharacteristic
406C22×D3—21no—
407C22×D3—22no—
408C22×D3—41no—
409C22×D3—41no—
410C22×D3—81no—
411C22×D3—81no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—81no—
417C2×C3×C4—81no—
418C2×C3×C4C23×C411yescharacteristic
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C2×#4111yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423C2×C3×C42−1+411yescharacteristic
424#67—22no—
425#67—22no—
426C2×Dic3—21no—
427C2×Dic3—22no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437D12—22no—
438D12—22no—
439D12—21no—
440D12—41no—
441D12—41no—
442D12C4×D412yes—
443C4×D3—22no—
444C4×D3—22no—
445C4×D3—161no—
446C4×D3—161no—
447C4×D3—161no—
448C4×D3—161no—
449Dic6—21no—
450Dic6—22no—
451Dic6—22no—
452Dic6—41no—
453Dic6—41no—
454Dic6C4×D412yes—
455C3×C8—22no—
456C3×C8—22no—
457C3×C8—41no—
458C3×C8C4×D412yes—
459C3×C8C4×D412yes—
460C3⋊C8—22no—
461C3⋊C8—21no—
462C3⋊C8—21no—
463C24—61no—
464#41—32no—
465#41—32no—
466#41—61no—
467#41—61no—
468C2×Q8—32no—
469C2×Q8—31no—
470C2×Q8—61no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8—61no—
475C2×D4—32no—
476C2×D4—31no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—61no—
481C2×D4—61no—
482C2×D4—61no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—121no—
489C22×C4—121no—
490C22×C4—121no—
491C22×C4—121no—
492C22×C4C23×D311yescharacteristic
493Q16—64no—
494SD16—64no—
495SD16—64no—
496SD16—64no—
497SD16—64no—
498D8—64no—
499M16—21no—
500M16—21no—
501M16—21no—
502M16—21no—
503M16—32no—
504M16—61no—
505M16—64no—
506M16C2×C4×D312yes—
507M16C2×C4×D312yes—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—31no—
512C2×C8—31no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—64no—
517C2×C8C2×C4×D312yes—
518C2×C8D3×D411yescharacteristic
519C2×C8D3×D411yescharacteristic
520C2×C8#23912yes—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—31no—
526C4⋊C4—31no—
527C4⋊C4—31no—
528C4⋊C4—31no—
529C4⋊C4—121no—
530C4⋊C4—121no—
531C4⋊C4—121no—
532C4⋊C4—121no—
533C4⋊C4D3×D411yescharacteristic
534C4⋊C4D3×D411yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—61no—
537C22⋊C4—61no—
538C22⋊C4—121no—
539C22⋊C4—121no—
540C22⋊C4—121no—
541C22⋊C4—121no—
542C22⋊C4—242no—
543C42—21no—
544C42—61no—
545C42—61no—
546C42—61no—
547C42—61no—
548C42—121no—
549C42—121no—
550C42—121no—
551C42—121no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3#52011yescharacteristic
555C22×C3D4211yescharacteristic
556C22×C3#55311yescharacteristic
557C2×D3—22no—
558C2×D3—41no—
559C2×D3—41no—
560C2×D3—42no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—82no—
564C2×D3—161no—
565C3×C4—41no—
566C3×C4—41no—
567C3×C4—81no—
568C3×C4—81no—
569C3×C4C2×C4×D411yescharacteristic
570C3×C4C2×C4×D411yescharacteristic
571C3×C4#54711yescharacteristic
572C3×C4#54811yescharacteristic
573Dic3—22no—
574Dic3—42no—
575Dic3—41no—
576Dic3—41no—
577Dic3—41no—
578Dic3—41no—
579Dic3—81no—
580C23—61no—
581C23—62no—
582C23—121no—
583C23—121no—
584C23—241no—
585C23—241no—
586C23#100911yescharacteristic
587Q8—32no—
588Q8—61no—
589Q8—62no—
590Q8—62no—
591Q8—121no—
592Q8—121no—
593D4—32no—
594D4—62no—
595D4—62no—
596D4—61no—
597D4—62no—
598D4—62no—
599D4—121no—
600D4—121no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—61no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—481no—
621C2×C4—481no—
622C2×C4—481no—
623C2×C4—481no—
624C2×C4C22×C4×D311yescharacteristic
625C2×C4C2×D3×D411yescharacteristic
626C2×C4C2×D3×D411yescharacteristic
627C2×C4C2×#23911yescharacteristic
628C2×C4#100711yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C8—22no—
631C8—22no—
632C8—41no—
633C8—61no—
634C8—61no—
635C8—62no—
636C8#89312yes—
637C8#89312yes—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3#236111yescharacteristic
641C2×C3#328811yescharacteristic
642C2×C3#328911yescharacteristic
643D3—42no—
644D3—82no—
645C22—21no—
646C22—21no—
647C22—62no—
648C22—121no—
649C22—121no—
650C22—122no—
651C22—242no—
652C22—241no—
653C22—241no—
654C22—481no—
655C22#557411yescharacteristic
656C22#560911yescharacteristic
657C22#569511yescharacteristic
658C4—41no—
659C4—41no—
660C4—62no—
661C4—81no—
662C4—81no—
663C4—121no—
664C4—121no—
665C4—121no—
666C4—121no—
667C4—122no—
668C4—241no—
669C4C2×#89311yescharacteristic
670C4C4×D3×D411yescharacteristic
671C4#563311yescharacteristic
672C4#572611yescharacteristic
673C3#2304911yescharacteristic, complemented, Sylow
674C2—21no—
675C2—21no—
676C2—122no—
677C2—242no—
678C2#7620611yescharacteristic
679C2#8093911yescharacteristic
680C2#8094511yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1952C2
socleC22×C3D42
prime radicalC2×#908C22
cosocleC22×C3×C4C24
hypercenter#1952D3
hypocenterC3#23049
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
2. #79202C2semidirect
4. #79146C2semidirect
5. #78572C2semidirect
6. C3×#1952C2semidirect
7. #70703C2semidirect
8. #70677C2semidirect
9. #70674C2semidirect
10. #70629C2semidirect
11. #69888C2semidirect
13. #69004C2semidirect
14. #68983C2semidirect
15. #68982C2semidirect
46. #4898C4semidirect
75. #1952D3semidirect
103. #909C2×C4semidirect
113. C2×#204C2×C4semidirect
115. #901C2×C4semidirect
136. C3×#108C2×C4semidirect
147. #824C2×C4semidirect
148. #823C2×C4semidirect
367. #108C4×D3semidirect
673. C3#23049semidirect

Maximal subgroups

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

subgroupindex
C2×#48982
#792022
#792012
#791462
#785722
C3×#19522
#707032
#706772
#706742
#706292
#698882
#690052
#690042
#689832
#689822
#230493