Finite Group Databasebeta

← #10883498

Subgroups — #10883498

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#214C211yesmaximal, characteristic, complemented
2#82307C211yesmaximal, characteristic
3#79067C211yesmaximal, characteristic, complemented
4#79041C211yesmaximal, characteristic, complemented
5C3×#1800C211yesmaximal, characteristic, complemented
6#70528C211yesmaximal, characteristic, complemented
7#70516C211yesmaximal, characteristic, complemented
8#70091C212yesmaximal, complemented
9#70088C212yesmaximal, complemented
10#69011C212yesmaximal, complemented
11#69007C212yesmaximal, complemented
12#28175—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C22×#214C2211yescharacteristic
15C2×#932C2212yes—
16C2×#909C2212yes—
17C2×#907C2212yes—
18#5477C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C3×#423C2211yescharacteristic
21C3×#423C2211yescharacteristic
22C2×C3×#100C2212yes—
23C2×#832C2211yescharacteristic
24C2×#824C2212yes—
25#5133C2212yes—
26#5129C2212yes—
27#5019—21no—
28C4×#214—21no—
29C4×#214—21no—
30C4×#214—21no—
31C4×#214—21no—
32#5016—22no—
33#5015—21no—
34C2×#810C2211yescharacteristic
35#4966C2212yes—
36C2×#808C2211yescharacteristic
37#4924C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39#4849—21no—
40#4848—21no—
41#4845—22no—
42#4838—21no—
43#4838—21no—
44#4834—21no—
45#4834—21no—
46C3×#343C2212yes—
47#4558C2212yes—
48#4558C2212yes—
49#4554C2212yes—
50#3037—31no—
51#3036—31no—
52C2×C4×SD16—31no—
53#2931—31no—
54#2079—32no—
55#2019—32no—
56#1908—32no—
57#1907—32no—
58#1837—31no—
59#1801—31no—
60#1800D311yescharacteristic, complemented
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C2×#235C2312yes—
64C22×Dic6C2311yescharacteristic
65C2×C3×M16C2312yes—
66C2×C3×M16C2312yes—
67C3×C4×D4—21no—
68C3×C4×D4—21no—
69C3×C4×D4—21no—
70C3×C4×D4—21no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#214—22no—
75C2×#214—22no—
76C2×#214—22no—
77C2×#214—21no—
78C2×#214—21no—
79#932—24no—
80#932—42no—
81C2×#213—22no—
82#930—21no—
83#930—21no—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4—21no—
86C2×C4×Dic3—21no—
87C2×#208C2312yes—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2311yescharacteristic
90#909—24no—
91#909—42no—
92#907—42no—
93#907—42no—
94C2×Dic12—22no—
95C2×#204—22no—
96C2×ΓL(1,𝔽25)—22no—
97C2×D3×C8—22no—
98#875D412yes—
99#874D411yescharacteristic
100C4×D12—21no—
101C4×D12—21no—
102#869D411yescharacteristic
103C4×Dic6—21no—
104C4×Dic6—21no—
105C4×Dic6—21no—
106C4×Dic6—21no—
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#824—42no—
115#821—22no—
116(C2×D3)⋊C8—22no—
117(C2×D3)⋊C8—22no—
118#816—22no—
119#810—22no—
120#810—21no—
121#810—21no—
122#809—22no—
123#808—21no—
124#808—21no—
125C4⋊(C3⋊C8)—21no—
126C4⋊(C3⋊C8)—21no—
127C4×C3⋊C8D411yescharacteristic
128C4×C3⋊C8D411yescharacteristic
129C4×C3⋊C8D411yescharacteristic
130C4×C3⋊C8D411yescharacteristic
131C2×#137—32no—
132C2×AGL(1,ℤ/8)—32no—
133C22×SD16—31no—
134C2×#131—32no—
135#532—31no—
136C2×C4×Q8—31no—
137C2×C4×D4C2×D311yescharacteristic
138#464—62no—
139#463—62no—
140#462—61no—
141#461—61no—
142#460—61no—
143#459—61no—
144C4×SD16—61no—
145C4×SD16—61no—
146C4×SD16—61no—
147C4×SD16—61no—
148#436—61no—
149#436—61no—
150C8×D4—61no—
151C8×D4—61no—
152C2×#106—31no—
153#423—31no—
154#423C2×D311yescharacteristic
155#423C2×D311yescharacteristic
156#417—32no—
157#417—32no—
158C2×#103—31no—
159C2×#102—31no—
160C2×#101—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—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4#4112yes—
175C22×C3×C4—21no—
176C22×C3×C4—41no—
177C22×C3×C4—41no—
178C22×C3×C4—82no—
179C22×C3×C4#4111yescharacteristic
180C22×C3×C4#4111yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C2×#67—22no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185#235—22no—
186#235—22no—
187#235—22no—
188#235—22no—
189C2×D12—21no—
190C2×D12—21no—
191C2×D12C2×D412yes—
192C2×C4×D3—22no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6—21no—
198C2×Dic6C2×D412yes—
199C2×Dic6#4112yes—
200C3×M16—22no—
201C3×M16—22no—
202C3×M16—22no—
203C3×M16—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C22⋊C4—41no—
209C3×C22⋊C4—41no—
210C3×C42C2×D411yescharacteristic
211C3×C42C2×D411yescharacteristic
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214#214—24no—
215#214—24no—
216#214—44no—
217#213—44no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—22no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224Dic3⋊C4—41no—
225Dic3⋊C4—41no—
226C4×Dic3—41no—
227C4×Dic3—41no—
228#208—22no—
229#208—22no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8C2×D411yescharacteristic
236C2×C3⋊C8C2×D411yescharacteristic
237C2×C3⋊C8C2×D411yescharacteristic
238C2×C3⋊C8C2×D411yescharacteristic
239Dic12—44no—
240#204—44no—
241ΓL(1,𝔽25)—44no—
242D3×C8—44no—
243C2×#41—32no—
244C22×Q8—31no—
245C22×D4C22×D311yescharacteristic
246C23×C4—21no—
247#137—64no—
248#137—122no—
249AGL(1,ℤ/8)—64no—
250AGL(1,ℤ/8)—122no—
251C2×Q16—62no—
252C2×SD16—62no—
253C2×SD16—61no—
254C2×SD16—61no—
255C2×SD16—62no—
256C2×SD16—62no—
257C2×SD16—62no—
258C2×D8—62no—
259#131—122no—
260#131—122no—
261C2×M16—32no—
262C2×M16—62no—
263C2×M16C22×D312yes—
264C2×M16C22×D312yes—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—62no—
268C4⋊Q8—31no—
269#127—31no—
270#124—32no—
271#121—61no—
272#121—61no—
273C4×Q8—61no—
274C4×Q8—61no—
275C4×Q8—61no—
276C4×Q8—61no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—61no—
282C4×D4—61no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C22⋊C4—21no—
287C2×C42—61no—
288C2×C42C22×D311yescharacteristic
289#106—61no—
290#106—61no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—61no—
296C4⋊C8—61no—
297#103—61no—
298#103—61no—
299#103—62no—
300#103—62no—
301#102—62no—
302#102—62no—
303#102—61no—
304#102—61no—
305#101—122no—
306#100—42no—
307C22⋊C8—62no—
308C22⋊C8—62no—
309C4×C8—31no—
310C4×C8—31no—
311C4×C8—31no—
312C4×C8—31no—
313C23×C3—22no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—81no—
317C23×C3—81no—
318C23×C3C2×#4111yescharacteristic
319C22×D3—22no—
320C3×D4—22no—
321C3×D4—22no—
322C3×D4—41no—
323C3×D4—41no—
324C2×C3×C4—22no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—82no—
328C2×C3×C4—82no—
329C2×C3×C4—81no—
330C2×C3×C4#12111yescharacteristic
331C2×C3×C4#12111yescharacteristic
332C2×C3×C4#12111yescharacteristic
333C2×C3×C4#12111yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4H(5,𝔽2)11yescharacteristic
339C2×C3×C4H(5,𝔽2)11yescharacteristic
340#67—42no—
341#67—42no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—81no—
348D12—22no—
349D12—22no—
350D12—42no—
351D12—42no—
352C4×D3—42no—
353C4×D3—42no—
354Dic6—22no—
355Dic6—22no—
356Dic6—22no—
357Dic6—22no—
358Dic6—41no—
359Dic6—41no—
360C3×C8—42no—
361C3×C8—42no—
362C3⋊C8—21no—
363C3⋊C8—21no—
364C3⋊C8—21no—
365C3⋊C8—21no—
366C3⋊C8—42no—
367C24—21no—
368#41—62no—
369#41—62no—
370#41—62no—
371#41—62no—
372C2×Q8—32no—
373C2×Q8—32no—
374C2×Q8—61no—
375C2×Q8—61no—
376C2×Q8—61no—
377C2×Q8—61no—
378C2×Q8—61no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—32no—
385C2×D4—62no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4#23912yes—
389C22×C4—21no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—61no—
393C22×C4—62no—
394C22×C4—61no—
395C22×C4—82no—
396C22×C4#23511yescharacteristic
397C22×C4#23511yescharacteristic
398C22×C4C23×D311yescharacteristic
399Q16—124no—
400SD16—64no—
401SD16—64no—
402SD16—124no—
403SD16—124no—
404D8—124no—
405M16—22no—
406M16—22no—
407M16—22no—
408M16—22no—
409M16—62no—
410M16—62no—
411M16—124no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—31no—
415C2×C8—31no—
416C2×C8—31no—
417C2×C8—31no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—62no—
423C2×C8—124no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—121no—
431C4⋊C4—121no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—62no—
435C22⋊C4—62no—
436C42—121no—
437C42—121no—
438C42D3×D411yescharacteristic
439C42D3×D411yescharacteristic
440C42D3×D411yescharacteristic
441C42D3×D411yescharacteristic
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—42no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3C2×#12111yescharacteristic
450C22×C3#55311yescharacteristic
451C22×C3#55311yescharacteristic
452C2×D3—42no—
453C2×D3—42no—
454C2×D3—42no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—81no—
459C3×C4#47112yes—
460C3×C4#52511yescharacteristic
461C3×C4#53511yescharacteristic
462C3×C4#54611yescharacteristic
463C3×C4#54611yescharacteristic
464Dic3—42no—
465Dic3—42no—
466Dic3—81no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—62no—
471C23—81no—
472C23—81no—
473C23C2×#23511yescharacteristic
474Q8—62no—
475Q8—62no—
476Q8—62no—
477Q8—62no—
478Q8—121no—
479Q8—121no—
480D4—22no—
481D4—22no—
482D4—41no—
483D4—41no—
484D4—62no—
485D4—62no—
486D4—122no—
487D4—122no—
488D4—122no—
489D4—122no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—121no—
503C2×C4—241no—
504C2×C4#88411yescharacteristic
505C2×C4#88411yescharacteristic
506C2×C4#88411yescharacteristic
507C2×C4#88411yescharacteristic
508C2×C4C2×D3×D411yescharacteristic
509C2×C4C2×D3×D411yescharacteristic
510C2×C4#100411yescharacteristic
511C2×C4C2×#23911yescharacteristic
512C2×C4D3×#4111yescharacteristic
513C2×C4#100911yescharacteristic
514C8—42no—
515C8—42no—
516C8—61no—
517C8—61no—
518C8—61no—
519C8—61no—
520C8—122no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—82no—
524C2×C3#253511yescharacteristic
525C2×C3C2×#47111yescharacteristic
526C2×C3#306711yescharacteristic
527D3—82no—
528C22—21no—
529C22—21no—
530C22—42no—
531C22—82no—
532C22—82no—
533C22—81no—
534C22—81no—
535C22—122no—
536C22—122no—
537C22—122no—
538C22C2×#88411yescharacteristic
539C22#551411yescharacteristic
540C22#553011yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—81no—
545C4—122no—
546C4—122no—
547C4—241no—
548C4#474812yes—
549C4#560711yescharacteristic
550C4#561211yescharacteristic
551C4#562711yescharacteristic
552C4#563011yescharacteristic
553C3#2817511yescharacteristic, complemented, Sylow
554C2—21no—
555C2—21no—
556C2—82no—
557C2—242no—
558C2#7393311yescharacteristic
559C2C2×#474811yescharacteristic
560C2#7871911yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1800C2
socleC22×C3C2×#121
prime radical#82307C2
hypercenter#1800D3
hypocenterC3#28175
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. C2×C4×#214C2semidirect
3. #79067C2semidirect
4. #79041C2semidirect
5. C3×#1800C2semidirect
6. #70528C2semidirect
7. #70516C2semidirect
8. #70091C2semidirect
9. #70088C2semidirect
10. #69011C2semidirect
11. #69007C2semidirect
60. #1800D3semidirect
553. C3#28175semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2142
#823072
#790672
#790412
C3×#18002
#705282
#705162
#700912
#700882
#690112
#690072
#281753