Finite Group Databasebeta

← #10883496

Subgroups — #10883496

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×C3×C4×SD16C211yesmaximal, characteristic, complemented
2#82441C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic
4#78458C211yesmaximal, characteristic, complemented
5#72059C212yesmaximal, complemented
6#71637C211yesmaximal, characteristic, complemented
7#71617C211yesmaximal, characteristic, complemented
8#70593C212yesmaximal, complemented
9#70286C212yesmaximal, complemented
10#70218C212yesmaximal, complemented
11#69791C211yesmaximal, characteristic, complemented
12#28175—31nomaximal, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#942C2212yes—
17C2×#932C2212yes—
18C2×#901C2212yes—
19#5477C2211yescharacteristic
20C3×C4×SD16—21no—
21C3×C4×SD16—21no—
22C3×C4×SD16—21no—
23C3×C4×SD16—21no—
24C2×C3×#106C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C2×#836C2212yes—
29C2×#833C2212yes—
30#5110C2212yes—
31#5031—21no—
32#5028—22no—
33#5027—21no—
34#5019—21no—
35#5016—22no—
36#5015—21no—
37#4966C2212yes—
38#4966C2212yes—
39#4924C2211yescharacteristic
40#4924C2211yescharacteristic
41#4924C2211yescharacteristic
42#4688—21no—
43#4688—21no—
44C8×D12—21no—
45C8×D12—21no—
46#4552C2212yes—
47#4552C2212yes—
48#4530C2212yes—
49#4530C2212yes—
50#3037—31no—
51#3036—31no—
52C2×C4×SD16D311yescharacteristic, complemented
53#2931—31no—
54#2079—32no—
55#2019—32no—
56#1908—32no—
57#1907—32no—
58#1837—31no—
59#1801—31no—
60#1800—31no—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C2×#235C2312yes—
65C2×C3×SD16—22no—
66C2×C3×SD16—22no—
67C2×C3×SD16—21no—
68C2×C3×SD16—21no—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×C4×Q8—21no—
72C3×C4×Q8—21no—
73C3×C4×Q8—21no—
74C3×C4×Q8—21no—
75C3×C4×D4—21no—
76C3×C4×D4—21no—
77C3×C4×D4—21no—
78C3×C4×D4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C42—21no—
84C2×C3×C42C2311yescharacteristic
85C2×#216—22no—
86#942—24no—
87#942—42no—
88C2×#215—22no—
89C2×#214—22no—
90#932—24no—
91#932—42no—
92C2×#213—22no—
93#930—21no—
94#930—21no—
95C2×#208C2312yes—
96C2×#208C2312yes—
97C2×#208C2312yes—
98#901—42no—
99#901—42no—
100C2×ΓL(1,𝔽25)—22no—
101C2×D3×C8—22no—
102#875D412yes—
103#874D411yescharacteristic
104C4×D12—21no—
105C4×D12—21no—
106#869D411yescharacteristic
107C3×#106—21no—
108C3×#106—21no—
109C3×#103—21no—
110C3×#103—21no—
111C3×#102—21no—
112C3×#102—21no—
113C3×C4×C8D411yescharacteristic
114C3×C4×C8D411yescharacteristic
115C3×C4×C8D411yescharacteristic
116C3×C4×C8D411yescharacteristic
117#836—42no—
118#833—42no—
119(C2×D3)⋊C8—22no—
120(C2×D3)⋊C8—22no—
121#810—22no—
122#810—22no—
123#809—22no—
124#809—22no—
125C4⋊(C3⋊C8)—21no—
126C4⋊(C3⋊C8)—21no—
127C4⋊(C3⋊C8)—21no—
128C4⋊(C3⋊C8)—21no—
129C4⋊(C3⋊C8)—21no—
130C4⋊(C3⋊C8)—21no—
131C2×#137—32no—
132C2×AGL(1,ℤ/8)—32no—
133C22×SD16C2×D311yescharacteristic
134C2×#131—32no—
135#532—31no—
136C2×C4×Q8C2×D311yescharacteristic
137C2×C4×D4C2×D311yescharacteristic
138#464—62no—
139#463—62no—
140#462—61no—
141#461—61no—
142#460—61no—
143#459—61no—
144C4×SD16—21no—
145C4×SD16—21no—
146C4×SD16—21no—
147C4×SD16—21no—
148#436—61no—
149#436—61no—
150C8×D4—61no—
151C8×D4—61no—
152C2×#106C2×D311yescharacteristic
153#423—31no—
154#423—31no—
155#423—31no—
156#417—32no—
157#417—32no—
158C2×#103C2×D311yescharacteristic
159C2×#102C2×D311yescharacteristic
160C2×#101—32no—
161C2×#100—32no—
162#408—32no—
163C2×C4×C8C2×D311yescharacteristic
164#343—32no—
165#343—32no—
166#341—32no—
167#341—32no—
168C24×C3—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—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4#4112yes—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—41no—
185C22×C3×C4—41no—
186C22×C3×C4—82no—
187C22×C3×C4#4111yescharacteristic
188C22×C3×C4#4111yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C2×#67—22no—
191#235—22no—
192#235—22no—
193#235—22no—
194#235—22no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12C2×D412yes—
198C2×C4×D3—22no—
199C2×Dic6C2×D412yes—
200C3×SD16—24no—
201C3×SD16—24no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8C2×D411yescharacteristic
207C2×C3×C8C2×D411yescharacteristic
208C2×C3×C8C2×D411yescharacteristic
209C2×C3×C8C2×D411yescharacteristic
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—41no—
215C3×C4⋊C4—41no—
216C3×C22⋊C4—41no—
217C3×C22⋊C4—41no—
218C3×C42—41no—
219C3×C42—41no—
220C3×C42C2×D411yescharacteristic
221C3×C42C2×D411yescharacteristic
222C3×C42C2×D411yescharacteristic
223C3×C42C2×D411yescharacteristic
224#216—44no—
225#215—44no—
226#214—44no—
227#213—44no—
228(C2×D3)⋊C4—22no—
229(C2×D3)⋊C4—22no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232#208—22no—
233#208—22no—
234#208—22no—
235#208—22no—
236#208—22no—
237#208—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241ΓL(1,𝔽25)—44no—
242D3×C8—44no—
243C2×#41—32no—
244C22×Q8C22×D311yescharacteristic
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—22no—
253C2×SD16—22no—
254C2×SD16—21no—
255C2×SD16—21no—
256C2×SD16—62no—
257C2×SD16—62no—
258C2×D8—62no—
259#131—122no—
260#131—122no—
261C2×M16—32no—
262C2×M16—32no—
263C2×M16—32no—
264C2×M16—62no—
265C22×C8—62no—
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
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—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—61no—
282C4×D4—61no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C22⋊C4—21no—
287C2×C42—21no—
288C2×C42C22×D311yescharacteristic
289#106—21no—
290#106—21no—
291C4⋊C8—61no—
292C4⋊C8—61no—
293C4⋊C8—61no—
294C4⋊C8—61no—
295C4⋊C8—61no—
296C4⋊C8—61no—
297#103—21no—
298#103—21no—
299#103—62no—
300#103—62no—
301#102—21no—
302#102—21no—
303#102—62no—
304#102—62no—
305#101—122no—
306#100—122no—
307C22⋊C8—62no—
308C22⋊C8—62no—
309C4×C8#6711yescharacteristic
310C4×C8#6711yescharacteristic
311C4×C8#6711yescharacteristic
312C4×C8#6711yescharacteristic
313C23×C3—22no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—81no—
317C23×C3—81no—
318C23×C3C2×#4111yescharacteristic
319C22×D3—22no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—41no—
323C3×Q8—41no—
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—41no—
327C3×D4—41no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—82no—
336C2×C3×C4—82no—
337C2×C3×C4—81no—
338C2×C3×C4—81no—
339C2×C3×C4#12111yescharacteristic
340C2×C3×C4#12111yescharacteristic
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4H(5,𝔽2)11yescharacteristic
348C2×C3×C4H(5,𝔽2)11yescharacteristic
349#67—42no—
350#67—42no—
351C2×Dic3—22no—
352D12—22no—
353D12—22no—
354D12—42no—
355D12—42no—
356C4×D3—42no—
357C4×D3—42no—
358Dic6—22no—
359Dic6—22no—
360C3×C8—21no—
361C3×C8—21no—
362C3×C8—21no—
363C3×C8—21no—
364C3⋊C8—42no—
365C3⋊C8—42no—
366C3⋊C8—42no—
367C24—21no—
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#23512yes—
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#23512yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—62no—
395C22×C4—82no—
396C22×C4#23711yescharacteristic
397C22×C4#23711yescharacteristic
398C22×C4C23×D311yescharacteristic
399Q16—124no—
400SD16—24no—
401SD16—24no—
402SD16—124no—
403SD16—124no—
404D8—124no—
405M16—62no—
406M16—62no—
407M16—62no—
408M16—62no—
409M16—62no—
410M16—62no—
411M16—124no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—62no—
417C2×C8—62no—
418C2×C8—62no—
419C2×C8—124no—
420C2×C8C2×#6711yescharacteristic
421C2×C8C2×#6711yescharacteristic
422C2×C8C2×#6711yescharacteristic
423C2×C8C2×#6711yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—62no—
435C22⋊C4—62no—
436C42—41no—
437C42—41no—
438C42D3×D411yescharacteristic
439C42D3×D411yescharacteristic
440C42C2×#6711yescharacteristic
441C42C2×#6711yescharacteristic
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—42no—
459C3×C4—81no—
460C3×C4—81no—
461C3×C4#47112yes—
462C3×C4#52511yescharacteristic
463C3×C4#53511yescharacteristic
464C3×C4#54611yescharacteristic
465C3×C4#54611yescharacteristic
466Dic3—42no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—62no—
471C23—81no—
472C23—81no—
473C23C2×#23711yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—41no—
477Q8—41no—
478Q8—62no—
479Q8—62no—
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—22no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—62no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4#93811yescharacteristic
505C2×C4#93811yescharacteristic
506C2×C4#93811yescharacteristic
507C2×C4#93811yescharacteristic
508C2×C4C2×#23511yescharacteristic
509C2×C4C2×#23511yescharacteristic
510C2×C4C2×D3×D411yescharacteristic
511C2×C4#100911yescharacteristic
512C2×C4#100911yescharacteristic
513C2×C4C22×#6711yescharacteristic
514C8—21no—
515C8—21no—
516C8—21no—
517C8—21no—
518C8—122no—
519C8—122no—
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—
538C22#556511yescharacteristic
539C22#556511yescharacteristic
540C22C2×#93811yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—42no—
545C4—81no—
546C4—81no—
547C4—122no—
548C4#517712yes—
549C4#555011yescharacteristic
550C4#555011yescharacteristic
551C4#579111yescharacteristic
552C4#583011yescharacteristic
553C3#2817511yescharacteristic, complemented, Sylow
554C2—21no—
555C2—21no—
556C2—82no—
557C2—242no—
558C2#8241511yescharacteristic
559C2C2×#517711yescharacteristic
560C2#8392811yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×SD16C2
socleC22×C3C2×#121
prime radical#82307C2
hypercenterC2×C4×SD16D3
hypocenterC3#28175
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×SD16C2semidirect
2. #82441C2semidirect
4. #78458C2semidirect
5. #72059C2semidirect
6. #71637C2semidirect
7. #71617C2semidirect
8. #70593C2semidirect
9. #70286C2semidirect
10. #70218C2semidirect
11. #69791C2semidirect
52. C2×C4×SD16D3semidirect
553. C3#28175semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SD162
#824412
#823072
#784582
#720592
#716372
#716172
#705932
#702862
#702182
#697912
#281753