Finite Group Databasebeta

← #10883488

Subgroups — #10883488

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82307C211yesmaximal, characteristic, complemented
2C2×C4×#213C211yesmaximal, characteristic, complemented
3#79055C211yesmaximal, characteristic, complemented
4#79041C211yesmaximal, characteristic, complemented
5C3×#1800C211yesmaximal, characteristic, complemented
6#70527C211yesmaximal, characteristic, complemented
7#70516C211yesmaximal, characteristic, complemented
8#70090C212yesmaximal, complemented
9#70088C212yesmaximal, complemented
10#69010C212yesmaximal, complemented
11#69006C212yesmaximal, complemented
12#28174—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#932C2212yes—
15C22×#213C2211yescharacteristic
16C2×#908C2212yes—
17C2×#907C2212yes—
18#5477C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C3×#423C2211yescharacteristic
21C3×#423C2211yescharacteristic
22C2×C3×#100C2212yes—
23C2×#832C2211yescharacteristic
24#5133C2212yes—
25C2×#823C2212yes—
26#5129C2212yes—
27#5019—21no—
28#5016—22no—
29#5015—21no—
30C4×#213—21no—
31C4×#213—21no—
32C4×#213—21no—
33C4×#213—21no—
34#4966C2212yes—
35C2×#809C2211yescharacteristic
36C2×#807C2211yescharacteristic
37#4924C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39#4844—22no—
40#4843—21no—
41#4841—21no—
42#4838—21no—
43#4838—21no—
44#4834—21no—
45#4834—21no—
46C3×#343C2212yes—
47#4558C2212yes—
48#4558C2212yes—
49#4554C2212yes—
50#3036—31no—
51#3036—31no—
52C2×C4×D8—31no—
53#2931—31no—
54#2078—32no—
55#2018—32no—
56#1907—32no—
57#1907—32no—
58#1837—31no—
59#1800—31no—
60#1800D311yescharacteristic, complemented
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C2×#235C2312yes—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C2×C3×M16C2312yes—
67C2×C3×M16C2312yes—
68C3×C4×D4—21no—
69C3×C4×D4—21no—
70C3×C4×D4—21no—
71C3×C4×D4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4—21no—
74C2×C3×C42C2311yescharacteristic
75C2×#214—22no—
76#932—24no—
77#932—42no—
78C2×#213—22no—
79C2×#213—22no—
80C2×#213—22no—
81C2×#213—21no—
82C2×#213—21no—
83#930—21no—
84#930—21no—
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×#208C2312yes—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2311yescharacteristic
90#908—24no—
91#908—42no—
92#907—42no—
93#907—42no—
94C2×D24—22no—
95C2×#204—22no—
96C2×ΓL(1,𝔽25)—22no—
97C2×D3×C8—22no—
98#875D412yes—
99#874D411yescharacteristic
100C4×D12—21no—
101C4×D12—21no—
102C4×D12—21no—
103C4×D12—21no—
104C4×D12—21no—
105C4×D12—21no—
106#869D411yescharacteristic
107C3×C4⋊C8—21no—
108C3×C4⋊C8—21no—
109C3×C4⋊C8—21no—
110C3×C4⋊C8—21no—
111C3×#100—42no—
112#832—21no—
113#832—21no—
114#823—42no—
115#821—22no—
116(C2×D3)⋊C8—22no—
117(C2×D3)⋊C8—22no—
118#816—22no—
119#810—22no—
120#809—22no—
121#809—21no—
122#809—21no—
123#807—21no—
124#807—21no—
125C4⋊(C3⋊C8)—21no—
126C4⋊(C3⋊C8)—21no—
127C4×C3⋊C8D411yescharacteristic
128C4×C3⋊C8D411yescharacteristic
129C4×C3⋊C8D411yescharacteristic
130C4×C3⋊C8D411yescharacteristic
131C2×AGL(1,ℤ/8)—32no—
132C2×AGL(1,ℤ/8)—32no—
133C22×D8—31no—
134C2×#131—32no—
135#532—31no—
136C2×C4×D4—31no—
137C2×C4×D4C2×D311yescharacteristic
138#463—62no—
139#463—62no—
140#461—61no—
141#461—61no—
142#459—61no—
143#459—61no—
144C4×D8—61no—
145C4×D8—61no—
146C4×D8—61no—
147C4×D8—61no—
148#436—61no—
149#436—61no—
150C8×D4—61no—
151C8×D4—61no—
152C2×#107—31no—
153#423—31no—
154#423C2×D311yescharacteristic
155#423C2×D311yescharacteristic
156#417—32no—
157#417—32no—
158C2×#102—31no—
159C2×#102—31no—
160C2×#100—32no—
161C2×#100C2×D312yes—
162#408—32no—
163C2×C4×C8—31no—
164#343—32no—
165#343C2×D312yes—
166#341—32no—
167#341—32no—
168C24×C3—21no—
169C23×D3—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4#4112yes—
176C22×C3×C4—21no—
177C22×C3×C4—41no—
178C22×C3×C4—41no—
179C22×C3×C4—82no—
180C22×C3×C4#4111yescharacteristic
181C22×C3×C4#4111yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C2×#67—22no—
184C22×Dic3—21no—
185#235—22no—
186#235—22no—
187#235—22no—
188#235—22no—
189C2×D12—21no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12C2×D412yes—
197C2×D12#4112yes—
198C2×C4×D3—22no—
199C2×C4×D3—41no—
200C2×C4×D3—41no—
201C2×C4×D3—82no—
202C2×Dic6C2×D412yes—
203C3×M16—22no—
204C3×M16—22no—
205C3×M16—22no—
206C3×M16—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C22⋊C4—41no—
212C3×C22⋊C4—41no—
213C3×C42C2×D411yescharacteristic
214C3×C42C2×D411yescharacteristic
215C3×C42C2×D411yescharacteristic
216C3×C42C2×D411yescharacteristic
217#214—44no—
218#213—24no—
219#213—24no—
220#213—44no—
221(C2×D3)⋊C4—22no—
222(C2×D3)⋊C4—22no—
223(C2×D3)⋊C4—41no—
224(C2×D3)⋊C4—41no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229#208—22no—
230#208—22no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8C2×D411yescharacteristic
237C2×C3⋊C8C2×D411yescharacteristic
238C2×C3⋊C8C2×D411yescharacteristic
239C2×C3⋊C8C2×D411yescharacteristic
240D24—44no—
241#204—44no—
242ΓL(1,𝔽25)—44no—
243D3×C8—44no—
244C2×#41—32no—
245C22×D4—31no—
246C22×D4C22×D311yescharacteristic
247C23×C4—21no—
248C23×C4—61no—
249AGL(1,ℤ/8)—64no—
250AGL(1,ℤ/8)—64no—
251AGL(1,ℤ/8)—122no—
252AGL(1,ℤ/8)—122no—
253C2×SD16—62no—
254C2×SD16—62no—
255C2×D8—62no—
256C2×D8—61no—
257C2×D8—61no—
258C2×D8—62no—
259C2×D8—62no—
260C2×D8—62no—
261#131—122no—
262#131—122no—
263C2×M16—32no—
264C2×M16—62no—
265C2×M16C22×D312yes—
266C2×M16C22×D312yes—
267C22×C8—31no—
268C22×C8—31no—
269C22×C8—62no—
270C4⋊Q8—31no—
271#127—31no—
272#124—32no—
273#121—61no—
274#121—61no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—61no—
280C4×D4—61no—
281C4×D4—61no—
282C4×D4—61no—
283C4×D4—61no—
284C4×D4—61no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—61no—
289C2×C42C22×D311yescharacteristic
290#107—61no—
291#107—61no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—61no—
297C4⋊C8—61no—
298#103—62no—
299#103—62no—
300#102—61no—
301#102—61no—
302#102—62no—
303#102—62no—
304#102—61no—
305#102—61no—
306#100—42no—
307#100—122no—
308C22⋊C8—62no—
309C22⋊C8—62no—
310C4×C8—31no—
311C4×C8—31no—
312C4×C8—31no—
313C4×C8—31no—
314C23×C3—22no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—81no—
318C23×C3—81no—
319C23×C3C2×#4111yescharacteristic
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—81no—
325C22×D3—81no—
326C3×D4—22no—
327C3×D4—22no—
328C3×D4—41no—
329C3×D4—41no—
330C2×C3×C4—22no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—82no—
334C2×C3×C4—82no—
335C2×C3×C4—81no—
336C2×C3×C4#12111yescharacteristic
337C2×C3×C4#12111yescharacteristic
338C2×C3×C4#12111yescharacteristic
339C2×C3×C4#12111yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4H(5,𝔽2)11yescharacteristic
345C2×C3×C4H(5,𝔽2)11yescharacteristic
346#67—42no—
347#67—42no—
348C2×Dic3—22no—
349C2×Dic3—41no—
350C2×Dic3—81no—
351D12—22no—
352D12—22no—
353D12—22no—
354D12—22no—
355D12—42no—
356D12—41no—
357D12—41no—
358D12—42no—
359C4×D3—42no—
360C4×D3—42no—
361C4×D3—82no—
362C4×D3—82no—
363Dic6—22no—
364Dic6—22no—
365C3×C8—42no—
366C3×C8—42no—
367C3⋊C8—21no—
368C3⋊C8—21no—
369C3⋊C8—21no—
370C3⋊C8—21no—
371C3⋊C8—42no—
372C24—21no—
373C24—61no—
374#41—62no—
375#41—62no—
376#41—62no—
377#41—62no—
378C2×Q8—32no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—32no—
385C2×D4—32no—
386C2×D4—62no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4#23912yes—
395C22×C4—21no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—82no—
401C22×C4—121no—
402C22×C4—121no—
403C22×C4—242no—
404C22×C4#23511yescharacteristic
405C22×C4#23511yescharacteristic
406C22×C4C23×D311yescharacteristic
407SD16—124no—
408SD16—124no—
409D8—64no—
410D8—64no—
411D8—124no—
412D8—124no—
413M16—22no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—62no—
418M16—62no—
419M16—124no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—31no—
423C2×C8—31no—
424C2×C8—31no—
425C2×C8—31no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—62no—
431C2×C8—124no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—62no—
441C22⋊C4—62no—
442C22⋊C4—121no—
443C22⋊C4—121no—
444C42D3×D411yescharacteristic
445C42D3×D411yescharacteristic
446C42D3×D411yescharacteristic
447C42D3×D411yescharacteristic
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—42no—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3C2×#12111yescharacteristic
456C22×C3#55311yescharacteristic
457C22×C3#55311yescharacteristic
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—82no—
463C2×D3—81no—
464C2×D3—81no—
465C2×D3—82no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—81no—
470C3×C4#47212yes—
471C3×C4#52511yescharacteristic
472C3×C4#53511yescharacteristic
473C3×C4#54611yescharacteristic
474C3×C4#54611yescharacteristic
475Dic3—42no—
476Dic3—81no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—62no—
481C23—62no—
482C23—81no—
483C23—81no—
484C23—121no—
485C23—121no—
486C23—241no—
487C23—241no—
488C23C2×#23511yescharacteristic
489Q8—62no—
490Q8—62no—
491D4—22no—
492D4—22no—
493D4—41no—
494D4—41no—
495D4—62no—
496D4—62no—
497D4—62no—
498D4—62no—
499D4—122no—
500D4—122no—
501D4—122no—
502D4—122no—
503D4—121no—
504D4—121no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—62no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—121no—
515C2×C4—242no—
516C2×C4—242no—
517C2×C4—241no—
518C2×C4#88411yescharacteristic
519C2×C4#88411yescharacteristic
520C2×C4#88411yescharacteristic
521C2×C4#88411yescharacteristic
522C2×C4C2×D3×D411yescharacteristic
523C2×C4C2×D3×D411yescharacteristic
524C2×C4#100411yescharacteristic
525C2×C4C2×#23911yescharacteristic
526C2×C4D3×#4111yescharacteristic
527C2×C4#100911yescharacteristic
528C8—42no—
529C8—42no—
530C8—61no—
531C8—61no—
532C8—61no—
533C8—61no—
534C8—122no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—82no—
538C2×C3#253511yescharacteristic
539C2×C3C2×#47211yescharacteristic
540C2×C3#306711yescharacteristic
541D3—82no—
542D3—82no—
543C22—21no—
544C22—21no—
545C22—42no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—81no—
550C22—122no—
551C22—122no—
552C22—122no—
553C22—122no—
554C22—242no—
555C22—242no—
556C22—241no—
557C22—241no—
558C22C2×#88411yescharacteristic
559C22#551411yescharacteristic
560C22#553011yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—81no—
565C4—122no—
566C4—241no—
567C4#474912yes—
568C4#560711yescharacteristic
569C4#561211yescharacteristic
570C4#562711yescharacteristic
571C4#563011yescharacteristic
572C3#2817411yescharacteristic, complemented, Sylow
573C2—21no—
574C2—21no—
575C2—82no—
576C2—242no—
577C2—242no—
578C2#7393311yescharacteristic
579C2C2×#474911yescharacteristic
580C2#7871911yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1800C2
socleC22×C3C2×#121
hypercenter#1800D3
hypocenterC3#28174
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
1. #82307C2semidirect
2. C2×C4×#213C2semidirect
3. #79055C2semidirect
4. #79041C2semidirect
5. C3×#1800C2semidirect
6. #70527C2semidirect
7. #70516C2semidirect
8. #70090C2semidirect
9. #70088C2semidirect
10. #69010C2semidirect
11. #69006C2semidirect
60. #1800D3semidirect
572. C3#28174semidirect

Maximal subgroups

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

subgroupindex
#823072
C2×C4×#2132
#790552
#790412
C3×#18002
#705272
#705162
#700902
#700882
#690102
#690062
#281743