Finite Group Databasebeta

← #10865182

Subgroups — #10865182

Order 768.

720 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3097C211yesmaximal, characteristic, complemented
2#83401C211yesmaximal, characteristic, complemented
3#82564C211yesmaximal, characteristic
4#79197C211yesmaximal, characteristic, complemented
5#71867C211yesmaximal, characteristic
6#71665C211yesmaximal, characteristic, complemented
7#70673C211yesmaximal, characteristic, complemented
8#70667C211yesmaximal, characteristic, complemented
9#70659C211yesmaximal, characteristic, complemented
10#70343C211yesmaximal, characteristic, complemented
11#70338C211yesmaximal, characteristic, complemented
12#70265C211yesmaximal, characteristic, complemented
13#70226C211yesmaximal, characteristic, complemented
14#70003C211yesmaximal, characteristic
15#69997C211yesmaximal, characteristic, complemented
16#26851—31nomaximal, Sylow
17C3×#536C2211yescharacteristic
18C3×#520C2211yescharacteristic
19C2×#949C2211yescharacteristic
20C2×#947C2211yescharacteristic
21#5521C2211yescharacteristic
22#5519C2211yescharacteristic
23C3×#485—21no—
24C3×#485—21no—
25C3×#484—21no—
26C3×#483—21no—
27C3×#428C2211yescharacteristic
28C3×#427C2211yescharacteristic
29C3×#419C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C3×#408C2211yescharacteristic
32C2×#837C2211yescharacteristic
33#5233C2211yescharacteristic
34#5232C2211yescharacteristic
35C2×#836C2211yescharacteristic
36#5225C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#823C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#819C2211yescharacteristic
41#5102C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#5082—21no—
44#5079—21no—
45#5056—21no—
46#5055—21no—
47#5026—21no—
48C4×#215—21no—
49#5014—21no—
50C4×#213—21no—
51#5001C2211yescharacteristic
52#4997C2211yescharacteristic
53#4997C2211yescharacteristic
54#4995C2211yescharacteristic
55#4977C2211yescharacteristic
56C2×#809C2211yescharacteristic
57#4964C2211yescharacteristic
58#4891—21no—
59#4890—21no—
60#4871—21no—
61#4870—21no—
62#4558C2211yescharacteristic
63#4555C2211yescharacteristic
64#4554C2211yescharacteristic
65#4532C2211yescharacteristic
66#4532C2211yescharacteristic
67#4530C2211yescharacteristic
68#3137—31no—
69#3097D311yescharacteristic, complemented
70#3081—31no—
71#2946—31no—
72#2064—31no—
73#2061—31no—
74#2051—31no—
75#2003—31no—
76#1958—31no—
77#1937—31no—
78#1931—31no—
79#1930—31no—
80#1885—31no—
81#1860—31no—
82#1772—31no—
83C2×C3×#41C2311yescharacteristic
84C22×C3×Q8C2311yescharacteristic
85C22×D12C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4⋊Q8—21no—
89C3×#124—21no—
90C3×#123—41no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×D4—21no—
95C3×C4×D4—41no—
96C3×#117—21no—
97C3×#117C2311yescharacteristic
98C3×#117C2311yescharacteristic
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4C2311yescharacteristic
101#949—22no—
102#949—22no—
103#947—22no—
104#947—22no—
105C2×#215—21no—
106C2×#213—21no—
107C2×(C2×D3)⋊C4—41no—
108#926C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#893—21no—
116#893—21no—
117#890—21no—
118#888—21no—
119D3×C22⋊C4—41no—
120#878—41no—
121C4×D12—21no—
122C4×D12—21no—
123#870—21no—
124#869—21no—
125C3×#107—21no—
126C3×#107D411yescharacteristic
127C3×#107D411yescharacteristic
128C3×#106—21no—
129C3×#106D411yescharacteristic
130C3×#106D411yescharacteristic
131C3×#103—21no—
132C3×#103—21no—
133C3×#103—21no—
134C3×#103—21no—
135C3×C22⋊C8—21no—
136C3×C22⋊C8—21no—
137#837—22no—
138#836—22no—
139#835—21no—
140#835—21no—
141#835—21no—
142#835—21no—
143#835—21no—
144#835D411yescharacteristic
145#835D411yescharacteristic
146#832—21no—
147#832D411yescharacteristic
148#832D411yescharacteristic
149C3⋊#95—41no—
150#823—22no—
151#821—21no—
152#821—21no—
153#819—22no—
154Dic3⋊C8—21no—
155Dic3⋊C8—21no—
156C8×Dic3—22no—
157#809—21no—
158#809—21no—
159#809—21no—
160#809—21no—
161#808—21no—
162#808—21no—
163#807—21no—
164#807—21no—
165#803—21no—
166#803—21no—
167C4×C3⋊C8—21no—
168C4×C3⋊C8—21no—
169C2×AGL(1,ℤ/8)—31no—
170#544—31no—
171#536C2×D311yescharacteristic
172C2×#124—31no—
173#520C2×D311yescharacteristic
174#518—31no—
175#491—61no—
176#489—61no—
177#488—61no—
178#487—61no—
179#485—21no—
180#485—21no—
181#484—21no—
182#483—21no—
183#478—61no—
184#478—61no—
185#476—61no—
186#474—61no—
187#442—61no—
188#440—61no—
189C4×SD16—61no—
190C4×D8—61no—
191C2×#108—31no—
192#428—31no—
193#428—31no—
194#428C2×D311yescharacteristic
195#427C2×D311yescharacteristic
196#424—31no—
197C2×#104—31no—
198#419—31no—
199#419C2×D311yescharacteristic
200#418—31no—
201#417—31no—
202#416—31no—
203C2×#103—31no—
204C2×#103C2×D311yescharacteristic
205C2×#102—31no—
206C2×#102—31no—
207C2×#101—31no—
208C2×#100—31no—
209#408C2×D311yescharacteristic
210#405—31no—
211#405—31no—
212C2×C4×C8—31no—
213#382—31no—
214#343—31no—
215#343—31no—
216#341—31no—
217#339—31no—
218#339—31no—
219#339—31no—
220C23×D3—21no—
221C3×#41—21no—
222C3×#41—21no—
223C3×#41—21no—
224C3×#41—21no—
225C2×C3×Q8—21no—
226C2×C3×Q8—21no—
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8—21no—
231C2×C3×Q8C2×D411yescharacteristic
232C2×C3×D4—21no—
233C2×C3×D4C2×D411yescharacteristic
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4—41no—
238C22×C3×C4C2411yescharacteristic
239C22×Dic3—21no—
240C22×Dic3—21no—
241C2×D12—21no—
242C2×D12—21no—
243C2×D12—21no—
244C2×D12—21no—
245C2×D12—21no—
246C2×D12#4112yes—
247C2×C4×D3—41no—
248C2×C4×D3—41no—
249C2×Dic6—41no—
250C3×M16—21no—
251C3×M16—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8C2×D411yescharacteristic
256C2×C3×C8C2×D411yescharacteristic
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—41no—
260C3×C4⋊C4—41no—
261C3×C4⋊C4—41no—
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C4⋊C4C2×D411yescharacteristic
264C3×C4⋊C4C2×D411yescharacteristic
265C3×C4⋊C4C2×D411yescharacteristic
266C3×C22⋊C4—21no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—41no—
269C3×C22⋊C4—41no—
270C3×C22⋊C4—41no—
271C3×C42—21no—
272C3×C42—21no—
273C3×C42—41no—
274#217—21no—
275#215—24no—
276#213—24no—
277(C2×D3)⋊C4—41no—
278(C2×D3)⋊C4—41no—
279(C2×D3)⋊C4—44no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—41no—
283C4⋊Dic3C2×D411yescharacteristic
284C4⋊Dic3C2×D411yescharacteristic
285Dic3⋊C4—41no—
286Dic3⋊C4—41no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3#4112yes—
290#208—21no—
291#208—21no—
292#208—21no—
293#208—21no—
294#208—21no—
295#208#4112yes—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8C2×D411yescharacteristic
302C2×C3⋊C8C2×D411yescharacteristic
303C2×#41C22×D311yescharacteristic
304C22×Q8C22×D311yescharacteristic
305C22×D4—31no—
306AGL(1,ℤ/8)—62no—
307AGL(1,ℤ/8)—62no—
308C2×SD16—61no—
309C2×D8—61no—
310C2×M16—31no—
311C2×M16—31no—
312C2×M16—31no—
313C2×M16C22×D311yescharacteristic
314C22×C8—31no—
315C22×C8C22×D311yescharacteristic
316C4⋊Q8—21no—
317C4⋊Q8—61no—
318C4⋊Q8—61no—
319#125—61no—
320#125—61no—
321#124—21no—
322#124—62no—
323#124—62no—
324#123—41no—
325C22⋊Q8—21no—
326C22⋊Q8—21no—
327C22⋊Q8—121no—
328C4×Q8—21no—
329C4×D4—21no—
330C4×D4—41no—
331C4×D4—61no—
332C4×D4—61no—
333C4×D4—61no—
334C4×D4—61no—
335#117—21no—
336#117—31no—
337#117C22×D311yescharacteristic
338#117C22×D311yescharacteristic
339C2×C4⋊C4—21no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—121no—
343C2×C22⋊C4—121no—
344C2×C42—31no—
345#108—62no—
346#107—21no—
347#107—61no—
348#107—61no—
349#107#6711yescharacteristic
350#107#6711yescharacteristic
351#106—21no—
352#106—61no—
353#106—61no—
354#106#6711yescharacteristic
355#106#6711yescharacteristic
356C4⋊C8—61no—
357C4⋊C8—61no—
358#104—62no—
359#103—21no—
360#103—21no—
361#103—21no—
362#103—21no—
363#103—31no—
364#103—31no—
365#103—61no—
366#103—61no—
367#103—61no—
368#103—61no—
369#103—61no—
370#102—31no—
371#102—31no—
372#102—61no—
373#102—61no—
374#102—61no—
375#102—61no—
376#102—61no—
377#102—61no—
378#102—61no—
379#101—62no—
380#100—62no—
381C22⋊C8—21no—
382C22⋊C8—21no—
383#97—61no—
384#97—61no—
385C4×C8—61no—
386C4×C8—61no—
387C4×C8—62no—
388#95—121no—
389C23×C3—21no—
390C23×C3H(5,𝔽2)11yescharacteristic
391C22×D3—22no—
392C22×D3—41no—
393C22×D3—41no—
394C22×D3—81no—
395C22×D3—81no—
396C3×Q8—21no—
397C3×Q8—21no—
398C3×Q8—41no—
399C3×Q8—41no—
400C3×Q8—41no—
401C3×Q8—41no—
402C3×D4—21no—
403C3×D4—21no—
404C3×D4—41no—
405C3×D4—41no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—81no—
418C2×C3×C4—161no—
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C2×#4111yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C42−1+411yescharacteristic
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—22no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—81no—
432D12—22no—
433D12—22no—
434D12—41no—
435D12—41no—
436C4×D3—161no—
437C4×D3—161no—
438Dic6—161no—
439C3×C8—21no—
440C3×C8—21no—
441C3×C8—41no—
442C3⋊C8—22no—
443C3⋊C8—21no—
444C3⋊C8—21no—
445C3⋊C8—41no—
446C3⋊C8—41no—
447C24—61no—
448#41—21no—
449#41—21no—
450#41—21no—
451#41—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—121no—
459C2×Q8D3×D411yescharacteristic
460C2×D4—21no—
461C2×D4—32no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4D3×D411yescharacteristic
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—41no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—121no—
475C22×C4—121no—
476C22×C4C23×D311yescharacteristic
477SD16—64no—
478D8—64no—
479M16—21no—
480M16—21no—
481M16—32no—
482M16—61no—
483M16—61no—
484M16—61no—
485M16—61no—
486M16—61no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—31no—
491C2×C8—31no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8C2×#6711yescharacteristic
498C2×C8C2×#6711yescharacteristic
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—31no—
502C4⋊C4—31no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—121no—
509C4⋊C4—121no—
510C4⋊C4—121no—
511C4⋊C4C2×#6711yescharacteristic
512C4⋊C4C2×#6711yescharacteristic
513C4⋊C4C2×#6711yescharacteristic
514C4⋊C4C2×#6711yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—61no—
521C22⋊C4—121no—
522C22⋊C4—121no—
523C22⋊C4—124no—
524C42—21no—
525C42—21no—
526C42—32no—
527C42—41no—
528C42—61no—
529C42—61no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—41no—
535C22×C3#54211yescharacteristic
536C22×C3D4211yescharacteristic
537C22×C3#55111yescharacteristic
538C2×D3—42no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—82no—
542C2×D3—161no—
543C3×C4—41no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—41no—
547C3×C4—41no—
548C3×C4—81no—
549C3×C4—81no—
550C3×C4#52511yescharacteristic
551C3×C4#53211yescharacteristic
552C3×C4#54711yescharacteristic
553C3×C4#54811yescharacteristic
554Dic3—42no—
555Dic3—41no—
556Dic3—41no—
557Dic3—81no—
558C23—21no—
559C23—62no—
560C23—121no—
561C23—121no—
562C23—241no—
563C23—241no—
564C23#100411yescharacteristic
565Q8—21no—
566Q8—21no—
567Q8—41no—
568Q8—41no—
569Q8—41no—
570Q8—41no—
571Q8—481no—
572D4—21no—
573D4—21no—
574D4—41no—
575D4—41no—
576D4—62no—
577D4—62no—
578D4—121no—
579D4—121no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—62no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—81no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—161no—
599C2×C4—241no—
600C2×C4—481no—
601C2×C4—481no—
602C2×C4C2×D3×D411yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4D3×#4111yescharacteristic
605C2×C4D3×#4111yescharacteristic
606C2×C4#101011yescharacteristic
607C2×C4C22×#6711yescharacteristic
608C8—21no—
609C8—21no—
610C8—41no—
611C8—61no—
612C8—61no—
613C8—62no—
614C8—121no—
615C8—121no—
616C2×C3—21no—
617C2×C3—21no—
618C2×C3—81no—
619C2×C3#246111yescharacteristic
620C2×C3#329011yescharacteristic
621C2×C3#329011yescharacteristic
622D3—82no—
623C22—21no—
624C22—21no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—122no—
629C22—242no—
630C22—241no—
631C22—241no—
632C22—481no—
633C22#565911yescharacteristic
634C22#566211yescharacteristic
635C22D4×#6711yescharacteristic
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—81no—
642C4—81no—
643C4—122no—
644C4—121no—
645C4—121no—
646C4—241no—
647C4#565511yescharacteristic
648C4#565611yescharacteristic
649C4#583111yescharacteristic
650C4#583211yescharacteristic
651C3#2685111yescharacteristic, complemented, Sylow
652C2—21no—
653C2—21no—
654C2—81no—
655C2—242no—
656C2#8285211yescharacteristic
657C2#8401211yescharacteristic
658C2#8402011yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3097C2
socleC22×C3D42
prime radicalC2×#949C22
hypercenter#3097D3
hypocenterC3#26851
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3097

Complements

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

NH ≅ G/Ntype
1. C3×#3097C2semidirect
2. #83401C2semidirect
4. #79197C2semidirect
6. #71665C2semidirect
7. #70673C2semidirect
8. #70667C2semidirect
9. #70659C2semidirect
10. #70343C2semidirect
11. #70338C2semidirect
12. #70265C2semidirect
13. #70226C2semidirect
15. #69997C2semidirect
69. #3097D3semidirect
651. C3#26851semidirect

Maximal subgroups

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

subgroupindex
C3×#30972
#834012
#825642
#791972
#718672
#716652
#706732
#706672
#706592
#703432
#703382
#702652
#702262
#700032
#699972
#268513