Finite Group Databasebeta

← #10865180

Subgroups — #10865180

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
1#82573C211yesmaximal, characteristic, complemented
2#82564C211yesmaximal, characteristic
3#79242C211yesmaximal, characteristic, complemented
4#78877C211yesmaximal, characteristic, complemented
5C3×#1860C211yesmaximal, characteristic, complemented
6#70672C211yesmaximal, characteristic, complemented
7#70388C211yesmaximal, characteristic
8#70371C211yesmaximal, characteristic, complemented
9#70307C211yesmaximal, characteristic
10#69551C211yesmaximal, characteristic, complemented
11#69543C211yesmaximal, characteristic, complemented
12#68999C211yesmaximal, characteristic, complemented
13#68983C211yesmaximal, characteristic, complemented
14#68759C211yesmaximal, characteristic, complemented
15#68741C211yesmaximal, characteristic, complemented
16#26851—31nomaximal, Sylow
17C3×#520C2211yescharacteristic
18#5825C2211yescharacteristic
19C2×#949C2211yescharacteristic
20#5521C2211yescharacteristic
21#5500C2211yescharacteristic
22C2×#875C2211yescharacteristic
23C3×#428C2211yescharacteristic
24C2×C3×#104C2211yescharacteristic
25C3×#417C2211yescharacteristic
26C3×#382C2211yescharacteristic
27#5234C2211yescharacteristic
28#5233C2211yescharacteristic
29C2×#824C2211yescharacteristic
30#5131C2211yescharacteristic
31C2×#823C2211yescharacteristic
32C2×#822C2211yescharacteristic
33#5122C2211yescharacteristic
34#5120C2211yescharacteristic
35#5119C2211yescharacteristic
36#5026—21no—
37C4×#215—21no—
38#5014—21no—
39C4×#213—21no—
40#5004C2211yescharacteristic
41#4998C2211yescharacteristic
42#4997C2211yescharacteristic
43#4996—21no—
44#4996—21no—
45#4971—21no—
46#4970—21no—
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#809C2211yescharacteristic
50C2×#809C2211yescharacteristic
51#4964C2211yescharacteristic
52#4964C2211yescharacteristic
53C2×C4×C3⋊C8C2211yescharacteristic
54#4891—21no—
55#4890—21no—
56#4871—21no—
57#4870—21no—
58#4797—21no—
59#4792—21no—
60#4768—21no—
61#4765—21no—
62C3×#343C2211yescharacteristic
63C3×#339C2211yescharacteristic
64#4558C2211yescharacteristic
65#4530C2211yescharacteristic
66#4523C2211yescharacteristic
67#4523C2211yescharacteristic
68#3137—31no—
69#3097—31no—
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#1860D311yescharacteristic, complemented
82#1772—31no—
83C2×C3×#41C2311yescharacteristic
84C22×D12C2311yescharacteristic
85C22×Dic6C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C3×C4×Q8—21no—
89C3×C4×D4—21no—
90C3×C4×D4—41no—
91C3×#117—21no—
92C3×#117C2311yescharacteristic
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97#949—22no—
98#949—22no—
99#944—21no—
100C2×#215—21no—
101#936—21no—
102#935—41no—
103C2×#213—21no—
104C2×(C2×D3)⋊C4—41no—
105#926C2311yescharacteristic
106#926C2311yescharacteristic
107#924—21no—
108#924—21no—
109#924—41no—
110C2×#208C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114#893—21no—
115#893—21no—
116#890—21no—
117#890—21no—
118#888—21no—
119#888—21no—
120D3×C22⋊C4—41no—
121#875—22no—
122#875—22no—
123C4×D12—21no—
124C4×D12—21no—
125C3×#107—21no—
126C3×#106—21no—
127C3×#104—22no—
128C3×#103—21no—
129C3×#102—21no—
130C3×#95—41no—
131#835D411yescharacteristic
132#835D411yescharacteristic
133#832D411yescharacteristic
134#832D411yescharacteristic
135#830—21no—
136#830—21no—
137#824—22no—
138#823—22no—
139#822—22no—
140#821—21no—
141#821—21no—
142#816—21no—
143#816—21no—
144#815—21no—
145Dic3⋊C8—21no—
146Dic3⋊C8—21no—
147C8×Dic3—21no—
148#810—21no—
149#810—21no—
150#810—21no—
151#810—21no—
152#810—21no—
153#810—21no—
154#809—21no—
155#809—21no—
156#809—21no—
157#809—21no—
158#808—21no—
159#808—21no—
160#808D411yescharacteristic
161#808D411yescharacteristic
162#807—21no—
163#807—21no—
164#807D411yescharacteristic
165#807D411yescharacteristic
166#803—21no—
167C4×C3⋊C8—22no—
168C4×C3⋊C8—21no—
169C2×AGL(1,ℤ/8)—31no—
170#544—31no—
171#536—31no—
172C2×#124—31no—
173#520C2×D311yescharacteristic
174#518—31no—
175#491—61no—
176#489—61no—
177#488—61no—
178#487—61no—
179#485—61no—
180#485—61no—
181#484—61no—
182#483—61no—
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#427—31no—
196#424—31no—
197C2×#104C2×D311yescharacteristic
198#419—31no—
199#419—31no—
200#418—31no—
201#417C2×D311yescharacteristic
202#416—31no—
203C2×#103—31no—
204C2×#103—31no—
205C2×#102—31no—
206C2×#102—31no—
207C2×#101—31no—
208C2×#100—31no—
209#408—31no—
210#405—31no—
211#405—31no—
212C2×C4×C8—31no—
213#382C2×D311yescharacteristic
214#343—31no—
215#343C2×D311yescharacteristic
216#341—31no—
217#339—31no—
218#339—31no—
219#339C2×D311yescharacteristic
220C23×D3—21no—
221C3×#41—21no—
222C3×#41—21no—
223C3×#41—21no—
224C3×#41—21no—
225C2×C3×Q8C2×D411yescharacteristic
226C2×C3×D4—21no—
227C2×C3×D4C2×D411yescharacteristic
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—41no—
233C22×C3×C4C2411yescharacteristic
234C22×Dic3—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12#4112yes—
241C2×C4×D3—41no—
242C2×C4×D3—41no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—41no—
250C3×M16—21no—
251C3×M16—21no—
252C3×M16—21no—
253C3×M16—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—41no—
261C3×C4⋊C4C2×D411yescharacteristic
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C22⋊C4—21no—
264C3×C22⋊C4—41no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42—41no—
268C3×C42#4112yes—
269#217—21no—
270#217—21no—
271#217—41no—
272#217—41no—
273#215—24no—
274#213—24no—
275(C2×D3)⋊C4—41no—
276(C2×D3)⋊C4—41no—
277(C2×D3)⋊C4—44no—
278C4⋊Dic3—41no—
279C4⋊Dic3C2×D411yescharacteristic
280C4⋊Dic3C2×D411yescharacteristic
281C4⋊Dic3C2×D411yescharacteristic
282C4⋊Dic3C2×D411yescharacteristic
283Dic3⋊C4—41no—
284Dic3⋊C4—41no—
285Dic3⋊C4—41no—
286Dic3⋊C4—41no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289#208—21no—
290#208—21no—
291#208—21no—
292#208#4112yes—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8C2×D411yescharacteristic
300C2×C3⋊C8C2×D411yescharacteristic
301C2×C3⋊C8C2×D411yescharacteristic
302C2×C3⋊C8C2×D411yescharacteristic
303C2×#41C22×D311yescharacteristic
304C22×Q8—31no—
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×M16C22×D311yescharacteristic
313C2×M16C22×D311yescharacteristic
314C22×C8—31no—
315C22×C8—31no—
316C4⋊Q8—61no—
317C4⋊Q8—61no—
318C4⋊Q8—61no—
319#125—61no—
320#125—61no—
321#124—62no—
322#124—61no—
323#124—62no—
324#123—121no—
325C22⋊Q8—61no—
326C22⋊Q8—61no—
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#117—31no—
338#117C22×D311yescharacteristic
339C2×C4⋊C4—21no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—121no—
343C2×C22⋊C4—121no—
344C2×C42C22×D311yescharacteristic
345#108—62no—
346#107—21no—
347#107—31no—
348#107—31no—
349#107—61no—
350#107—61no—
351#106—21no—
352#106—31no—
353#106—31no—
354#106—61no—
355#106—61no—
356C4⋊C8—61no—
357C4⋊C8—61no—
358#104—22no—
359#103—21no—
360#103—31no—
361#103—31no—
362#103—61no—
363#103—61no—
364#103—61no—
365#103—61no—
366#103—61no—
367#103—61no—
368#103—61no—
369#103—61no—
370#102—21no—
371#102—31no—
372#102—31no—
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—61no—
382C22⋊C8—61no—
383#97—61no—
384#97—61no—
385C4×C8—61no—
386C4×C8—61no—
387C4×C8—62no—
388#95—41no—
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×D4—21no—
399C3×D4—21no—
400C3×D4—41no—
401C3×D4—41no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—22no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—81no—
413C2×C3×C4—81no—
414C2×C3×C4—161no—
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C42−1+411yescharacteristic
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428D12—22no—
429D12—22no—
430D12—41no—
431D12—41no—
432C4×D3—161no—
433C4×D3—161no—
434Dic6—41no—
435Dic6—41no—
436Dic6—41no—
437Dic6—41no—
438Dic6—161no—
439C3×C8—41no—
440C3×C8—41no—
441C3⋊C8—22no—
442C3⋊C8—21no—
443C3⋊C8—21no—
444C3⋊C8—21no—
445C3⋊C8—21no—
446C3⋊C8—41no—
447C24—61no—
448#41—21no—
449#41—21no—
450#41—21no—
451#41—21no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
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—21no—
472C22×C4—41no—
473C22×C4—61no—
474C22×C4—121no—
475C22×C4—121no—
476C22×C4C23×D311yescharacteristic
477SD16—64no—
478D8—64no—
479M16—21no—
480M16—21no—
481M16—21no—
482M16—21no—
483M16—32no—
484M16—61no—
485M16—61no—
486M16—61no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—31no—
490C2×C8—31no—
491C2×C8—31no—
492C2×C8—31no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—31no—
504C4⋊C4—31no—
505C4⋊C4—31no—
506C4⋊C4—31no—
507C4⋊C4—41no—
508C4⋊C4—121no—
509C4⋊C4—121no—
510C4⋊C4—121no—
511C4⋊C4—121no—
512C4⋊C4—121no—
513C4⋊C4D3×D411yescharacteristic
514C4⋊C4D3×D411yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—41no—
517C22⋊C4—61no—
518C22⋊C4—61no—
519C22⋊C4—121no—
520C22⋊C4—121no—
521C22⋊C4—124no—
522C22⋊C4—121no—
523C22⋊C4—121no—
524C42—21no—
525C42—21no—
526C42—41no—
527C42—61no—
528C42—61no—
529C42#23512yes—
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—42no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—41no—
547C3×C4—81no—
548C3×C4—81no—
549C3×C4#52511yescharacteristic
550C3×C4#53211yescharacteristic
551C3×C4#54711yescharacteristic
552C3×C4#54811yescharacteristic
553Dic3—41no—
554Dic3—41no—
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—121no—
568Q8—121no—
569Q8—121no—
570Q8—121no—
571Q8—481no—
572D4—21no—
573D4—21no—
574D4—41no—
575D4—41no—
576D4—62no—
577D4—62no—
578D4—121no—
579D4—121no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—161no—
600C2×C4—481no—
601C2×C4—481no—
602C2×C4C2×#23511yescharacteristic
603C2×C4C2×D3×D411yescharacteristic
604C2×C4C2×D3×D411yescharacteristic
605C2×C4#100711yescharacteristic
606C2×C4D3×#4111yescharacteristic
607C2×C4D3×#4111yescharacteristic
608C8—41no—
609C8—41no—
610C8—62no—
611C8—61no—
612C8—61no—
613C8—61no—
614C8—61no—
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#555611yescharacteristic
634C22#559011yescharacteristic
635C22#564011yescharacteristic
636C4—42no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—81no—
641C4—81no—
642C4—121no—
643C4—121no—
644C4—121no—
645C4—121no—
646C4—241no—
647C4#550711yescharacteristic
648C4#554811yescharacteristic
649C4#563311yescharacteristic
650C4#565611yescharacteristic
651C3#2685111yescharacteristic, complemented, Sylow
652C2—21no—
653C2—21no—
654C2—81no—
655C2—242no—
656C2#7443111yescharacteristic
657C2#8028911yescharacteristic
658C2#8029611yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1860C2
socleC22×C3D42
prime radicalC2×#949C22
hypercenter#1860D3
hypocenterC3#26851
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1860

Complements

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

NH ≅ G/Ntype
1. #82573C2semidirect
3. #79242C2semidirect
4. #78877C2semidirect
5. C3×#1860C2semidirect
6. #70672C2semidirect
8. #70371C2semidirect
10. #69551C2semidirect
11. #69543C2semidirect
12. #68999C2semidirect
13. #68983C2semidirect
14. #68759C2semidirect
15. #68741C2semidirect
81. #1860D3semidirect
651. C3#26851semidirect

Maximal subgroups

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

subgroupindex
#825732
#825642
#792422
#788772
C3×#18602
#706722
#703882
#703712
#703072
#695512
#695432
#689992
#689832
#687592
#687412
#268513