Finite Group Databasebeta

← #10822348

Subgroups — #10822348

Order 768.

2146 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#951C211yesmaximal, characteristic, complemented
2C22×#949C211yesmaximal, characteristic
3C22×#875C211yesmaximal, characteristic, complemented
4C3×#2297C211yesmaximal, characteristic, complemented
5C2×#5004C211yesmaximal, characteristic, complemented
6C2×#5001C211yesmaximal, characteristic, complemented
7#71527C211yesmaximal, characteristic, complemented
8#69689C214yesmaximal, complemented
9#69659C214yesmaximal, complemented
10#23431—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C2×#951—24no—
17C2×#951—24no—
18C2×#951—24no—
19C2×#949—24no—
20C2×#949—24no—
21C2×#949—24no—
22C22×#216C2212yescomplemented
23C22×#215C2212yes—
24C22×#214C2212yescomplemented
25C22×#213C2212yes—
26C22×(C2×D3)⋊C4—22no—
27C22×#208C2211yescharacteristic
28C22×#208C2211yescharacteristic
29C2×#875—24no—
30C2×#875—24no—
31C2×#875—22no—
32C2×#875C2214yes—
33C3×#391C2212yescomplemented
34C3×#390C2212yescomplemented
35C3×#386—22no—
36C3×C4×C22⋊C4—22no—
37#5004—24no—
38#5001—24no—
39C2×#810C2214yescomplemented
40C2×#809C2214yes—
41#4524C2214yescomplemented
42C22×#137—31no—
43C22×AGL(1,ℤ/8)—31no—
44C22×#124—31no—
45C2×#419—31no—
46C2×#418—31no—
47#2297D311yescharacteristic, complemented
48#2015—34no—
49#2012—34no—
50#1728—31no—
51C24×D3—21no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41D414yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4D412yes—
63C23×C3×C4C2311yescharacteristic
64C23×Dic3—21no—
65C22×D12—22no—
66C22×D12—22no—
67C22×D12—22no—
68C22×D12—22no—
69C22×D12—22no—
70C22×D12—21no—
71C22×D12D414yes—
72C22×D12C2312yes—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—22no—
77C22×Dic6—22no—
78C22×Dic6—21no—
79C22×Dic6D414yes—
80C22×Dic6C2312yes—
81C3×#117—22no—
82C3×#117—22no—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42—21no—
89C2×C3×C42D412yes—
90C2×C3×C42C2312yes—
91#951—232no—
92#949—232no—
93C2×#216—24no—
94C2×#216—24no—
95C2×#216—24no—
96C2×#215—24no—
97C2×#215—24no—
98C2×#215—24no—
99C2×#214—24no—
100C2×#214—24no—
101C2×#214—24no—
102C2×#213—24no—
103C2×#213—24no—
104C2×#213—24no—
105C2×(C2×D3)⋊C4—216no—
106C2×(C2×D3)⋊C4—28no—
107C2×#208—22no—
108C2×#208—22no—
109C2×#208—22no—
110C2×#208—22no—
111C2×#208D414yes—
112C2×#208D414yes—
113C22×C3⋊C8C2312yes—
114C22×C3⋊C8C2312yes—
115#875—216no—
116#875—28no—
117#875—28no—
118C3×#95—24no—
119#810—28no—
120#809—28no—
121C22×#41C2×D311yescharacteristic
122C23×Q8—31no—
123C23×D4—31no—
124C2×#137—64no—
125C2×#137—64no—
126C2×#137—64no—
127C2×AGL(1,ℤ/8)—64no—
128C2×AGL(1,ℤ/8)—64no—
129C2×AGL(1,ℤ/8)—64no—
130C22×Q16—32no—
131C22×SD16—32no—
132C22×SD16—32no—
133C22×D8—32no—
134C22×M16—31no—
135C22×M16—31no—
136C2×#124—34no—
137C2×#124—64no—
138C2×#124—64no—
139C2×#124—62no—
140C2×#117C2×D311yescharacteristic
141C22×C22⋊C4—62no—
142C22×C42C2×D311yescharacteristic
143#419—64no—
144#418—64no—
145C2×#103—34no—
146C2×#102—34no—
147#391C2×D312yescomplemented
148#390C2×D312yescomplemented
149#386—22no—
150C4×C22⋊C4—22no—
151#336—34no—
152C24×C3—21no—
153C24×C3C2×D411yescharacteristic
154C23×D3—24no—
155C23×D3—22no—
156C23×D3—48no—
157C23×D3—42no—
158C23×D3—42no—
159C3×#41—28no—
160C3×#41—28no—
161C3×#41—44no—
162C3×#41—44no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—24no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—24no—
179C22×C3×C4—21no—
180C22×C3×C4—24no—
181C22×C3×C4—22no—
182C22×C3×C4—44no—
183C22×C3×C4—42no—
184C22×C3×C4—42no—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4#4112yes—
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—24no—
195C22×Dic3—22no—
196C22×Dic3—42no—
197C22×Dic3—42no—
198C2×D12—28no—
199C2×D12—28no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—24no—
205C2×D12—22no—
206C2×D12—44no—
207C2×D12—44no—
208C2×D12—42no—
209C2×D12—42no—
210C2×D12C2×D414yes—
211C2×Dic6—28no—
212C2×Dic6—28no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—24no—
218C2×Dic6—22no—
219C2×Dic6—44no—
220C2×Dic6—44no—
221C2×Dic6—42no—
222C2×Dic6—42no—
223C2×Dic6C2×D414yes—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C22⋊C4—48no—
227C3×C22⋊C4—42no—
228C3×C42—24no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42—42no—
232#216—216no—
233#215—216no—
234#214—216no—
235#213—216no—
236(C2×D3)⋊C4—232no—
237#208—28no—
238#208—28no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8C2×D414yes—
244C2×C3⋊C8C2×D414yes—
245C25—61no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41D1214yes—
252C22×Q8—34no—
253C22×Q8—32no—
254C22×Q8—62no—
255C22×Q8—61no—
256C22×Q8—62no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×Q8—62no—
260C22×Q8C22×D311yescharacteristic
261C22×D4—21no—
262C22×D4—34no—
263C22×D4—32no—
264C22×D4—62no—
265C22×D4—62no—
266C22×D4—62no—
267C22×D4—62no—
268C22×D4—62no—
269C22×D4—61no—
270C22×D4C22×D311yescharacteristic
271C23×C4—21no—
272C23×C4—61no—
273C23×C4#6712yes—
274C23×C4C22×D311yescharacteristic
275#137—632no—
276AGL(1,ℤ/8)—632no—
277C2×Q16—64no—
278C2×Q16—64no—
279C2×Q16—64no—
280C2×SD16—64no—
281C2×SD16—64no—
282C2×SD16—64no—
283C2×SD16—64no—
284C2×SD16—64no—
285C2×SD16—64no—
286C2×D8—64no—
287C2×D8—64no—
288C2×D8—64no—
289C2×M16—34no—
290C2×M16—34no—
291C2×M16—62no—
292C2×M16—62no—
293C2×M16—62no—
294C2×M16—62no—
295C22×C8—32no—
296C22×C8—32no—
297#124—68no—
298#124—68no—
299#124—616no—
300#117—22no—
301#117—22no—
302C2×C4⋊C4C22×D312yes—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—616no—
306C2×C22⋊C4—68no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—21no—
310C2×C42#6712yes—
311C2×C42C22×D312yes—
312#103—68no—
313#102—68no—
314#95—24no—
315C23×C3—24no—
316C23×C3—22no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—44no—
320C23×C3#12012yes—
321C23×C3#12112yes—
322C23×C3#12712yes—
323C23×C3C22×D411yescharacteristic
324C22×D3—24no—
325C22×D3—416no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—48no—
329C22×D3—48no—
330C22×D3—88no—
331C22×D3—88no—
332C3×Q8—22no—
333C3×Q8—22no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—48no—
339C3×D4—41no—
340C3×D4—41no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—48no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—48no—
349C2×C3×C4—42no—
350C2×C3×C4—44no—
351C2×C3×C4#12014yes—
352C2×C3×C4#12014yes—
353C2×C3×C4#12012yes—
354C2×C3×C4#12114yes—
355C2×C3×C4#12114yes—
356C2×C3×C4#12112yes—
357C2×C3×C4#12712yes—
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×Dic3—24no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—48no—
368D12—24no—
369D12—24no—
370D12—42no—
371D12—42no—
372D12—48no—
373Dic6—24no—
374Dic6—24no—
375Dic6—42no—
376Dic6—42no—
377Dic6—48no—
378C3⋊C8—24no—
379C3⋊C8—24no—
380C24—21no—
381C24—64no—
382C24—62no—
383C24—128no—
384C24—122no—
385C24—122no—
386C24C2×#6711yescharacteristic
387#41—28no—
388#41—28no—
389#41—44no—
390#41—44no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—34no—
397C2×Q8—64no—
398C2×Q8—62no—
399C2×Q8—68no—
400C2×Q8—68no—
401C2×Q8—62no—
402C2×Q8—62no—
403C2×Q8—62no—
404C2×Q8—62no—
405C2×Q8—124no—
406C2×Q8—124no—
407C2×Q8—122no—
408C2×Q8—122no—
409C2×Q8C2×D1212yes—
410C2×D4—24no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—34no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—68no—
420C2×D4—68no—
421C2×D4—64no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—62no—
425C2×D4—62no—
426C2×D4—62no—
427C2×D4—124no—
428C2×D4—124no—
429C2×D4—122no—
430C2×D4—122no—
431C2×D4C2×D1212yes—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—64no—
440C22×C4—62no—
441C22×C4—122no—
442C22×C4—122no—
443C22×C4C2×D1212yes—
444C22×C4#23512yes—
445C22×C4D3×D412yes—
446C22×C4D3×D412yes—
447C22×C4D3×D412yes—
448C22×C4D3×D412yes—
449C22×C4C2×#6714yes—
450C22×C4C2×#6711yescharacteristic
451C22×C4C23×D311yescharacteristic
452Q16—616no—
453SD16—616no—
454SD16—616no—
455D8—616no—
456M16—68no—
457M16—68no—
458C2×C8—34no—
459C2×C8—34no—
460C2×C8—62no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8—62no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C22⋊C4—48no—
467C22⋊C4—42no—
468C22⋊C4—632no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—42no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3—42no—
477C22×C3—44no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3#39214yes—
481C22×C3C2×#12011yescharacteristic
482C22×C3C2×#12111yescharacteristic
483C22×C3C2×#12711yescharacteristic
484C22×C3#57514yes—
485C2×D3—44no—
486C2×D3—48no—
487C2×D3—816no—
488C2×D3—84no—
489C2×D3—84no—
490C3×C4—42no—
491C3×C4—44no—
492C3×C4—82no—
493C3×C4#39214yes—
494C3×C4C2×#12011yescharacteristic
495C3×C4C2×#12011yescharacteristic
496C3×C4C2×#12111yescharacteristic
497C3×C4C2×#12111yescharacteristic
498Dic3—44no—
499C23—24no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—44no—
504C23—64no—
505C23—1216no—
506C23—128no—
507C23—122no—
508C23—122no—
509C23—128no—
510C23—248no—
511C23—248no—
512C23#93012yes—
513C23#93712yes—
514C23#94012yes—
515C23C22×#6711yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520Q8—64no—
521Q8—64no—
522Q8—128no—
523Q8—122no—
524Q8—122no—
525D4—22no—
526D4—22no—
527D4—48no—
528D4—41no—
529D4—41no—
530D4—64no—
531D4—64no—
532D4—128no—
533D4—122no—
534D4—122no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—48no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—64no—
546C2×C4—128no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4#88214yes—
550C2×C4#88214yes—
551C2×C4#88414yes—
552C2×C4#88414yes—
553C2×C4#93012yes—
554C2×C4#93712yes—
555C2×C4#94012yes—
556C2×C4C22×D1211yescharacteristic
557C2×C4C2×#23511yescharacteristic
558C2×C4C2×D3×D411yescharacteristic
559C2×C4C2×D3×D411yescharacteristic
560C2×C4C2×D3×D411yescharacteristic
561C2×C4C2×D3×D411yescharacteristic
562C8—64no—
563C8—64no—
564C2×C3—24no—
565C2×C3—82no—
566C2×C3C2×#39211yescharacteristic
567C2×C3#300512yes—
568C2×C3#304012yes—
569C2×C3#315312yes—
570D3—84no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—42no—
576C22—82no—
577C22—82no—
578C22—124no—
579C22—128no—
580C22—2416no—
581C22—244no—
582C22—244no—
583C22#496014yes—
584C22C2×#93011yescharacteristic
585C22C2×#93711yescharacteristic
586C22C2×#94011yescharacteristic
587C22#582214yes—
588C4—42no—
589C4—44no—
590C4—82no—
591C4—124no—
592C4#496014yes—
593C4C2×#88211yescharacteristic
594C4C2×#88211yescharacteristic
595C4C2×#88411yescharacteristic
596C4C2×#88411yescharacteristic
597C3#2343111yescharacteristic, complemented, Sylow
598C2—24no—
599C2—82no—
600C2—244no—
601C2C2×#496011yescharacteristic
602C2#8252712yes—
603C2#8292812yes—
604C2#8328912yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2297C2
socleC23×C3C22×D4
prime radicalC22×#949C2
hypercenter#2297D3
hypocenterC3#23431
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
1. C22×#951C2semidirect
3. C22×#875C2semidirect
4. C3×#2297C2semidirect
5. C2×#5004C2semidirect
6. C2×#5001C2semidirect
7. #71527C2semidirect
8. #69689C2semidirect
9. #69659C2semidirect
22. C22×#216C22semidirect
24. C22×#214C22semidirect
33. C3×#391C22semidirect
34. C3×#390C22semidirect
39. C2×#810C22semidirect
41. #4524C22semidirect
47. #2297D3semidirect
147. #391C2×D3semidirect
148. #390C2×D3semidirect
597. C3#23431semidirect

Maximal subgroups

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

subgroupindex
C22×#9512
C22×#9492
C22×#8752
C3×#22972
C2×#50042
C2×#50012
#715272
#696892
#696592
#234313