Finite Group Databasebeta

← #10883499

Subgroups — #10883499

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
1#82441C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic, complemented
3#79055C211yesmaximal, characteristic
4#79041C211yesmaximal, characteristic, complemented
5C3×#1801C211yesmaximal, characteristic, complemented
6#70527C211yesmaximal, characteristic, complemented
7#70516C211yesmaximal, characteristic, complemented
8#70123C212yesmaximal, complemented
9#70117C212yesmaximal, complemented
10#69011C212yesmaximal, complemented
11#69006C212yesmaximal, complemented
12#28175—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#942C2212yes—
15C22×#215C2211yescharacteristic
16C2×#908C2212yes—
17C2×#907C2212yes—
18#5477C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C3×#423C2211yescharacteristic
21C3×#423C2211yescharacteristic
22C2×C3×#101C2212yes—
23C2×#835C2211yescharacteristic
24#5133C2212yes—
25C2×#823C2212yes—
26#5129C2212yes—
27#5031—21no—
28#5028—22no—
29#5027—21no—
30C4×#215—21no—
31C4×#215—21no—
32C4×#215—21no—
33C4×#215—21no—
34#4966C2212yes—
35C2×#809C2211yescharacteristic
36C2×#808C2211yescharacteristic
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#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#1801D311yescharacteristic, complemented
60#1800—31no—
61C22×C3×Q8C2311yescharacteristic
62C2×#235C2312yes—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C2×C3×M16C2312yes—
66C2×C3×M16C2312yes—
67C3×C4×Q8—21no—
68C3×C4×Q8—21no—
69C3×C4×Q8—21no—
70C3×C4×Q8—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75C2×#216—22no—
76#942—24no—
77#942—42no—
78C2×#215—22no—
79C2×#215—22no—
80C2×#215—22no—
81C2×#215—21no—
82C2×#215—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×#101—42no—
112#835—21no—
113#835—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#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×Q8C2×D311yescharacteristic
137C2×C4×D4—31no—
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×#101C2×D312yes—
161C2×#100—32no—
162#408—32no—
163C2×C4×C8—31no—
164#343—32no—
165#343C2×D312yes—
166#341—32no—
167#341—32no—
168C23×D3—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8#4112yes—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4#4111yescharacteristic
178C22×C3×C4#4111yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C2×#67—22no—
181C22×Dic3—21no—
182#235—22no—
183#235—22no—
184#235—22no—
185#235—22no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12C2×D412yes—
194C2×D12#4112yes—
195C2×C4×D3—22no—
196C2×C4×D3—41no—
197C2×C4×D3—41no—
198C2×C4×D3—82no—
199C2×Dic6C2×D412yes—
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×C4⋊C4—41no—
209C3×C4⋊C4—41no—
210C3×C42—41no—
211C3×C42—41no—
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214C3×C42C2×D411yescharacteristic
215C3×C42C2×D411yescharacteristic
216#216—44no—
217#215—24no—
218#215—24no—
219#215—44no—
220(C2×D3)⋊C4—22no—
221(C2×D3)⋊C4—22no—
222(C2×D3)⋊C4—41no—
223(C2×D3)⋊C4—41no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
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
239D24—44no—
240#204—44no—
241ΓL(1,𝔽25)—44no—
242D3×C8—44no—
243C2×#41—32no—
244C22×Q8C22×D311yescharacteristic
245C22×D4—31no—
246C23×C4—61no—
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—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×D4—61no—
278C4×D4—61no—
279C4×D4—61no—
280C4×D4—61no—
281C4×D4—61no—
282C4×D4—61no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C22⋊C4—61no—
287C2×C42—21no—
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—62no—
298#103—62no—
299#103—61no—
300#103—61no—
301#102—61no—
302#102—61no—
303#102—62no—
304#102—62no—
305#101—42no—
306#100—122no—
307C22⋊C8—62no—
308C22⋊C8—62no—
309C4×C8—31no—
310C4×C8—31no—
311C4×C8—31no—
312C4×C8—31no—
313C23×C3C2×#4111yescharacteristic
314C22×D3—22no—
315C22×D3—22no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—81no—
319C22×D3—81no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—41no—
323C3×Q8—41no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—81no—
331C2×C3×C4#12111yescharacteristic
332C2×C3×C4#12111yescharacteristic
333C2×C3×C4#12111yescharacteristic
334C2×C3×C4#12111yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4H(5,𝔽2)11yescharacteristic
340C2×C3×C4H(5,𝔽2)11yescharacteristic
341#67—42no—
342#67—42no—
343C2×Dic3—22no—
344C2×Dic3—41no—
345C2×Dic3—81no—
346D12—22no—
347D12—22no—
348D12—22no—
349D12—22no—
350D12—42no—
351D12—41no—
352D12—41no—
353D12—42no—
354C4×D3—42no—
355C4×D3—42no—
356C4×D3—82no—
357C4×D3—82no—
358Dic6—22no—
359Dic6—22no—
360C3×C8—42no—
361C3×C8—42no—
362C3⋊C8—21no—
363C3⋊C8—21no—
364C3⋊C8—21no—
365C3⋊C8—21no—
366C3⋊C8—42no—
367C24—61no—
368#41—62no—
369#41—62no—
370#41—62no—
371#41—62no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—32no—
378C2×Q8#23912yes—
379C2×D4—32no—
380C2×D4—32no—
381C2×D4—62no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—61no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—62no—
392C22×C4—61no—
393C22×C4—121no—
394C22×C4—121no—
395C22×C4—242no—
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—41no—
427C4⋊C4—41no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C22⋊C4—62no—
433C22⋊C4—62no—
434C22⋊C4—121no—
435C22⋊C4—121no—
436C42—41no—
437C42—41no—
438C42D3×D411yescharacteristic
439C42D3×D411yescharacteristic
440C42D3×D411yescharacteristic
441C42D3×D411yescharacteristic
442C22×C3—21no—
443C22×C3—21no—
444C22×C3C2×#12111yescharacteristic
445C22×C3#55311yescharacteristic
446C22×C3#55311yescharacteristic
447C2×D3—42no—
448C2×D3—42no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—82no—
452C2×D3—81no—
453C2×D3—81no—
454C2×D3—82no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—42no—
459C3×C4—81no—
460C3×C4#47112yes—
461C3×C4#52511yescharacteristic
462C3×C4#53511yescharacteristic
463C3×C4#54611yescharacteristic
464C3×C4#54611yescharacteristic
465Dic3—42no—
466Dic3—81no—
467C23—62no—
468C23—62no—
469C23—121no—
470C23—121no—
471C23—241no—
472C23—241no—
473C23C2×#23511yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—41no—
477Q8—41no—
478Q8—62no—
479Q8—62no—
480D4—62no—
481D4—62no—
482D4—62no—
483D4—62no—
484D4—122no—
485D4—122no—
486D4—122no—
487D4—122no—
488D4—121no—
489D4—121no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—62no—
497C2×C4—81no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4—242no—
502C2×C4—242no—
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#253511yescharacteristic
524C2×C3C2×#47111yescharacteristic
525C2×C3#306711yescharacteristic
526D3—82no—
527D3—82no—
528C22—21no—
529C22—21no—
530C22—122no—
531C22—122no—
532C22—122no—
533C22—122no—
534C22—242no—
535C22—242no—
536C22—241no—
537C22—241no—
538C22C2×#88411yescharacteristic
539C22#551411yescharacteristic
540C22#553011yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—42no—
545C4—81no—
546C4—122no—
547C4—241no—
548C4#475412yes—
549C4#560711yescharacteristic
550C4#561211yescharacteristic
551C4#562711yescharacteristic
552C4#563011yescharacteristic
553C3#2817511yescharacteristic, complemented, Sylow
554C2—21no—
555C2—21no—
556C2—242no—
557C2—242no—
558C2#7393311yescharacteristic
559C2C2×#475411yescharacteristic
560C2#7871911yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1801C2
socleC22×C3C2×#121
prime radical#79055C2
hypercenter#1801D3
hypocenterC3#28175
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #82441C2semidirect
2. C2×C4×#215C2semidirect
4. #79041C2semidirect
5. C3×#1801C2semidirect
6. #70527C2semidirect
7. #70516C2semidirect
8. #70123C2semidirect
9. #70117C2semidirect
10. #69011C2semidirect
11. #69006C2semidirect
59. #1801D3semidirect
553. C3#28175semidirect

Maximal subgroups

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

subgroupindex
#824412
C2×C4×#2152
#790552
#790412
C3×#18012
#705272
#705162
#701232
#701172
#690112
#690062
#281753