Finite Group Databasebeta

← #10778189

Subgroups — #10778189

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83296C211yesmaximal, characteristic
2#83294C211yesmaximal, characteristic, complemented
3#79329C211yesmaximal, characteristic, complemented
4#79302C211yesmaximal, characteristic, complemented
5C3×#1553C211yesmaximal, characteristic, complemented
6#70820C211yesmaximal, characteristic, complemented
7#70746C211yesmaximal, characteristic, complemented
8#68469C211yesmaximal, characteristic, complemented
9#68467C211yesmaximal, characteristic, complemented
10#68453C211yesmaximal, characteristic, complemented
11#68451C211yesmaximal, characteristic, complemented
12#68411C211yesmaximal, characteristic, complemented
13#68408C211yesmaximal, characteristic, complemented
14#68401C211yesmaximal, characteristic, complemented
15#68398C211yesmaximal, characteristic, complemented
16#19182—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C2×#950—21no—
19C4×#235C2211yescharacteristic
20#5470C2211yescharacteristic
21C3×#432C2211yescharacteristic
22C3×#431C2211yescharacteristic
23#5233—21no—
24#5093C2211yescharacteristic
25#5090C2211yescharacteristic
26#5083C2211yescharacteristic
27#5078C2211yescharacteristic
28#5076C2211yescharacteristic
29#5075C2211yescharacteristic
30#5072C2211yescharacteristic
31#5070C2211yescharacteristic
32#5057—21no—
33#5054—21no—
34#5042—21no—
35#5042—21no—
36#5018—21no—
37#5018—21no—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39C4×#208C2211yescharacteristic
40#4851C2211yescharacteristic
41#4849C2211yescharacteristic
42#4846C2211yescharacteristic
43#4841C2211yescharacteristic
44#4838C2211yescharacteristic
45#4838C2211yescharacteristic
46#4830C2211yescharacteristic
47#4830C2211yescharacteristic
48#4733—21no—
49#4729—21no—
50#4729—21no—
51#4723—21no—
52#4707—21no—
53#4705—21no—
54C4×Dic12—21no—
55C4×#204—21no—
56C3×#332C2211yescharacteristic
57C3×#332C2211yescharacteristic
58C3×#331C2211yescharacteristic
59C3×#331C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4482C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3182—31no—
69#3107—31no—
70#3035—31no—
71#2971—31no—
72#1701—31no—
73#1699—31no—
74#1693—31no—
75#1685—31no—
76#1682—31no—
77#1680—31no—
78#1674—31no—
79#1666—31no—
80#1565—31no—
81#1553D311yescharacteristic, complemented
82#1552—31no—
83C2×C3×#41—21no—
84C2×#235D411yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×#127C2311yescharacteristic
87C3×#124—21no—
88C3×#121—21no—
89C3×#121—21no—
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C2×C3×C42C2311yescharacteristic
93#950—44no—
94C2×#216—21no—
95C2×#215—21no—
96C2×#214—21no—
97C2×#214—21no—
98C2×#214—21no—
99C2×#213—21no—
100C4×#67—21no—
101C4×#67—21no—
102#926—21no—
103#926D411yescharacteristic
104#924—21no—
105#924—21no—
106C2×#208D411yescharacteristic
107C2×#208D411yescharacteristic
108C22×C3⋊C8—21no—
109C2×Dic12—21no—
110C2×#204—21no—
111C2×ΓL(1,𝔽25)—21no—
112C2×ΓL(1,𝔽25)—21no—
113C4×D12D411yescharacteristic
114C4×D12C2311yescharacteristic
115#872—21no—
116C42×D3—21no—
117#870—21no—
118#869C2311yescharacteristic
119#869C2311yescharacteristic
120C4×Dic6—21no—
121C4×Dic6—21no—
122C4×Dic6D411yescharacteristic
123C4×Dic6C2311yescharacteristic
124C3×C4⋊C8C2311yescharacteristic
125C3×C4⋊C8C2311yescharacteristic
126C3×C4⋊C8C2311yescharacteristic
127C3×C4⋊C8C2311yescharacteristic
128C3×C22⋊C8—21no—
129C3×C22⋊C8—21no—
130C3×#97—21no—
131C3×C4×C8—21no—
132#835—21no—
133#835—21no—
134#832—21no—
135#832—21no—
136#832—21no—
137#832—21no—
138#821—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141#818—21no—
142#817—21no—
143#816—21no—
144#816—21no—
145#816—21no—
146Dic3⋊C8—21no—
147Dic3⋊C8—21no—
148C8×Dic3—21no—
149C8×Dic3—21no—
150#810—21no—
151#810—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157#803D411yescharacteristic
158#803D411yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160C4×C3⋊C8C2311yescharacteristic
161C2×#135—61no—
162#533—31no—
163#532C2×D311yescharacteristic
164C4×#41—31no—
165#500—31no—
166#492—31no—
167#487—31no—
168#486—31no—
169#485—61no—
170#475—31no—
171#474—31no—
172#468—61no—
173#468—61no—
174#464—61no—
175#463—61no—
176#462—31no—
177#462—31no—
178#461—31no—
179#461—31no—
180#460—31no—
181#459—31no—
182#452—61no—
183C8×Q8—31no—
184C8×Q8—31no—
185#440—61no—
186#440—61no—
187C4×Q16—61no—
188C4×SD16—61no—
189#436—31no—
190#436—31no—
191#436—61no—
192#436—61no—
193#432—61no—
194#432C2×D311yescharacteristic
195#431C2×D311yescharacteristic
196C2×C4⋊C8—31no—
197#416—61no—
198C4×M16—31no—
199C4×M16—61no—
200#335—31no—
201#335—31no—
202#334—31no—
203#334—31no—
204#332—31no—
205#332—31no—
206#332—31no—
207#332—31no—
208#332C2×D311yescharacteristic
209#332C2×D311yescharacteristic
210#331C2×D311yescharacteristic
211#331C2×D311yescharacteristic
212C3×#41—42no—
213C3×#41—42no—
214C2×C3×Q8—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—41no—
219C22×C3×C4—41no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4#4111yescharacteristic
223C2×#67—21no—
224#235—22no—
225#235—41no—
226C2×D12C2×D411yescharacteristic
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6C2×D411yescharacteristic
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C22⋊C4—41no—
240C3×C22⋊C4—41no—
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C3×C42#4111yescharacteristic
244C3×C42C2411yescharacteristic
245#217—21no—
246#217—41no—
247#216—42no—
248#215—42no—
249#214—42no—
250#214—42no—
251#214—42no—
252#213—42no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3C2×D411yescharacteristic
260C4⋊Dic3C2×D411yescharacteristic
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263Dic3⋊C4—41no—
264Dic3⋊C4—41no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—41no—
269#208—22no—
270#208—41no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—42no—
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8C2×D411yescharacteristic
277C2×C3⋊C8C2×D411yescharacteristic
278Dic12—42no—
279#204—42no—
280ΓL(1,𝔽25)—42no—
281ΓL(1,𝔽25)—42no—
282C2×#41—21no—
283C2×#41—31no—
284#135—124no—
285C2×Q16—61no—
286C2×Q16—61no—
287C2×SD16—61no—
288C2×SD16—61no—
289C2×SD16—61no—
290C2×SD16—61no—
291C2×SD16—61no—
292C2×D8—61no—
293C2×M16—31no—
294C2×M16—31no—
295C2×M16—61no—
296C2×M16—61no—
297C22×C8—61no—
298C4⋊Q8—31no—
299C4⋊Q8—31no—
300C4⋊Q8C22×D311yescharacteristic
301#127C22×D311yescharacteristic
302#125—61no—
303#124—21no—
304C22⋊Q8—61no—
305C22⋊Q8—61no—
306#121—21no—
307#121—21no—
308C4×Q8—31no—
309C4×Q8—31no—
310C4×Q8—61no—
311C4×Q8—61no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—31no—
315C4×D4—31no—
316C4×D4—61no—
317C4×D4—61no—
318#117—31no—
319#117—61no—
320#117—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—61no—
324#107—61no—
325#107—61no—
326#106—61no—
327C4⋊C8—31no—
328C4⋊C8—31no—
329C4⋊C8—61no—
330C4⋊C8—61no—
331C4⋊C8—61no—
332C4⋊C8C22×D311yescharacteristic
333C4⋊C8C22×D311yescharacteristic
334C4⋊C8C22×D311yescharacteristic
335C4⋊C8C22×D311yescharacteristic
336#103—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—61no—
341#103—61no—
342#103—61no—
343#102—61no—
344#102—61no—
345#102—61no—
346#102—61no—
347#102—61no—
348C22⋊C8—21no—
349C22⋊C8—21no—
350C22⋊C8—61no—
351C22⋊C8—61no—
352#97—21no—
353#97—31no—
354#97—31no—
355C4×C8—21no—
356C4×C8—31no—
357C4×C8—31no—
358C4×C8—61no—
359C4×C8—61no—
360C23×C3—41no—
361C23×C3#12111yescharacteristic
362C22×D3—21no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×D4—41no—
366C3×D4—41no—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—81no—
370C3×D4—82no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—82no—
378C2×C3×C4#12111yescharacteristic
379C2×C3×C4#12111yescharacteristic
380C2×C3×C4#12111yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384#67—42no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390D12—21no—
391D12—21no—
392C4×D3—42no—
393C4×D3—41no—
394C4×D3—41no—
395C4×D3—81no—
396Dic6—21no—
397Dic6—41no—
398Dic6—41no—
399Dic6—41no—
400Dic6—41no—
401Dic6AGL(1,ℤ/8)12yes—
402C3×C8—41no—
403C3×C8—41no—
404C3×C8—41no—
405C3×C8—41no—
406C3⋊C8—21no—
407C3⋊C8—21no—
408C3⋊C8—21no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C3⋊C8#13512yescomplemented
412#41—42no—
413#41—42no—
414#41—62no—
415#41—121no—
416C2×Q8—21no—
417C2×Q8—31no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—31no—
424C2×D4—41no—
425C2×D4—61no—
426C22×C4—41no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4#23511yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433Q16—122no—
434Q16—122no—
435SD16—122no—
436SD16—122no—
437SD16—122no—
438SD16—122no—
439SD16—122no—
440D8—122no—
441M16—62no—
442M16—122no—
443M16—122no—
444M16—121no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—31no—
450C2×C8—31no—
451C2×C8—31no—
452C2×C8—31no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—122no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—31no—
459C4⋊C4—31no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—121no—
467C4⋊C4—121no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—61no—
471C22⋊C4—61no—
472C22⋊C4—61no—
473C22⋊C4—121no—
474C42—61no—
475C42—61no—
476C42—61no—
477C42—121no—
478C42#23511yescharacteristic
479C42D3×D411yescharacteristic
480C42D3×D411yescharacteristic
481C42C23×D311yescharacteristic
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3C2×#12111yescharacteristic
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—81no—
497C3×C4C2×#13511yescharacteristic
498C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
499C3×C4#57511yescharacteristic
500C3×C4#57511yescharacteristic
501Dic3—22no—
502Dic3—41no—
503Dic3—41no—
504Dic3—81no—
505Dic3—81no—
506C23—41no—
507C23—61no—
508C23#88411yescharacteristic
509Q8—32no—
510Q8—41no—
511Q8—41no—
512Q8—61no—
513Q8—121no—
514Q8—121no—
515Q8—121no—
516Q8—121no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—61no—
522D4—61no—
523D4—81no—
524D4—82no—
525D4—122no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—82no—
536C2×C4—122no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—241no—
542C2×C4#88411yescharacteristic
543C2×C4#88411yescharacteristic
544C2×C4#88411yescharacteristic
545C2×C4C2×#23511yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4C2×D3×D411yescharacteristic
548C8—32no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—61no—
554C8—61no—
555C8—61no—
556C8—121no—
557C8—121no—
558C2×C3—41no—
559C2×C3—161no—
560C2×C3#303511yescharacteristic
561C2×C3#305511yescharacteristic
562C2×C3#306911yescharacteristic
563D3—81no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—121no—
571C22—121no—
572C22—121no—
573C22C2×#88411yescharacteristic
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—62no—
579C4—81no—
580C4—121no—
581C4—121no—
582C4—241no—
583C4—241no—
584C4D3×#13511yescharacteristic
585C4#576811yescharacteristic
586C4D3×AGL(1,ℤ/8)11yescharacteristic
587C4#577311yescharacteristic
588C3#1918211yescharacteristic, complemented, Sylow
589C2—41no—
590C2—161no—
591C2—241no—
592C2#7872111yescharacteristic
593C2#7886111yescharacteristic
594C2#7893011yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1553C2
socleC22×C3C2×#121
prime radical#83296C2
hypercenter#1553D3
hypocenterC3#19182
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1553

Complements

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

NH ≅ G/Ntype
2. #83294C2semidirect
3. #79329C2semidirect
4. #79302C2semidirect
5. C3×#1553C2semidirect
6. #70820C2semidirect
7. #70746C2semidirect
8. #68469C2semidirect
9. #68467C2semidirect
10. #68453C2semidirect
11. #68451C2semidirect
12. #68411C2semidirect
13. #68408C2semidirect
14. #68401C2semidirect
15. #68398C2semidirect
81. #1553D3semidirect
411. C3⋊C8#135semidirect
588. C3#19182semidirect

Maximal subgroups

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

subgroupindex
#832962
#832942
#793292
#793022
C3×#15532
#708202
#707462
#684692
#684672
#684532
#684512
#684112
#684082
#684012
#683982
#191823