Finite Group Databasebeta

← #10822286

Subgroups — #10822286

Order 768.

2098 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#214C211yesmaximal, characteristic, complemented
2C22×#932C211yesmaximal, characteristic
3C2×#5526C211yesmaximal, characteristic, complemented
4C3×#2309C211yesmaximal, characteristic, complemented
5C2×#5004C211yesmaximal, characteristic, complemented
6C2×#5002C211yesmaximal, characteristic, complemented
7#71579C211yesmaximal, characteristic, complemented
8#69661C214yesmaximal, complemented
9#69659C214yesmaximal, complemented
10#23424—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×#214—24no—
17C22×#214—24no—
18C22×#214—24no—
19C22×#214C2214yes—
20C22×#214C2212yes—
21C2×#932—24no—
22C2×#932—24no—
23C2×#932—24no—
24C22×#213C2212yes—
25C22×#208C2211yescharacteristic
26C23×C3⋊C8C2211yescharacteristic
27#5526—24no—
28#5526—24no—
29C2×#897—22no—
30C2×#886—22no—
31C2×#875C2212yescomplemented
32C2×#869C2212yescomplemented
33C3×#390—22no—
34C3×#390C2214yes—
35C3×#379—22no—
36#5004—24no—
37#5002—24no—
38C2×#810C2214yescomplemented
39C2×#810C2212yescomplemented
40C2×#809C2212yescomplemented
41#4524C2214yescomplemented
42C22×AGL(1,ℤ/8)—31no—
43C23×SD16—31no—
44C2×#536—31no—
45C2×#419—31no—
46C2×#416—31no—
47#2309D311yescharacteristic, complemented
48#2013—34no—
49#2012—34no—
50#1729—31no—
51C25×C3—21no—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4—21no—
57C22×C3×D4—22no—
58C22×C3×D4D414yes—
59C22×C3×D4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67—21no—
62C23×Dic3—21no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235D414yes—
69C22×D12C2311yescharacteristic
70C22×C4×D3—21no—
71C22×Dic6—22no—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—21no—
76C22×Dic6—22no—
77C22×Dic6D414yes—
78C22×Dic6C2312yes—
79C22×Dic6C2311yescharacteristic
80C3×#117—22no—
81C3×#117—22no—
82C3×#117—22no—
83C3×#117—22no—
84C2×C3×C4⋊C4D412yes—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—28no—
88C2×C3×C22⋊C4D412yes—
89C2×C3×C42—21no—
90C2×C3×C42C2312yes—
91C2×#214—232no—
92C2×#214—24no—
93C2×#214—28no—
94C2×#214—24no—
95C2×#214—28no—
96C2×#214—24no—
97C2×#214—28no—
98#932—232no—
99C2×#213—24no—
100C2×#213—24no—
101C2×#213—24no—
102C2×(C2×D3)⋊C4—24no—
103C2×C4⋊Dic3—24no—
104C2×#208—22no—
105C2×#208—22no—
106C2×#208D414yes—
107C22×C3⋊C8—22no—
108C22×C3⋊C8—22no—
109C22×C3⋊C8D414yes—
110C22×C3⋊C8C2312yes—
111C22×C3⋊C8C2312yes—
112#897—216no—
113#886—216no—
114#875—24no—
115#875—24no—
116#869—24no—
117#869—24no—
118#810—24no—
119#810—28no—
120#809—24no—
121C22×#41—31no—
122C23×Q8—31no—
123C23×D4C2×D311yescharacteristic
124C2×AGL(1,ℤ/8)—64no—
125C2×AGL(1,ℤ/8)—64no—
126C2×AGL(1,ℤ/8)—64no—
127C22×SD16—34no—
128C22×SD16—32no—
129C22×SD16—64no—
130C22×SD16—64no—
131C22×SD16—64no—
132C22×D8—32no—
133C22×M16—31no—
134C23×C8—31no—
135#536—64no—
136#536—64no—
137C2×C4⋊Q8—32no—
138C2×#124—32no—
139C2×#123—62no—
140C2×C22⋊Q8—62no—
141C2×#117C2×D311yescharacteristic
142C2×#117C2×D311yescharacteristic
143#419—64no—
144#416—64no—
145C2×#103—32no—
146C2×#103—34no—
147C2×#102—32no—
148#390—22no—
149#390C2×D314yes—
150#379—22no—
151#336—34no—
152C24×C3—22no—
153C24×C3—24no—
154C24×C3—42no—
155C24×C3—42no—
156C24×C3—84no—
157C24×C3C2×D411yescharacteristic
158C23×D3—21no—
159C2×C3×D4—28no—
160C2×C3×D4—28no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—42no—
168C2×C3×D4—44no—
169C2×C3×D4—44no—
170C2×C3×D4—42no—
171C2×C3×D4C2×D414yes—
172C22×C3×C4—22no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D414yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4#4112yes—
181C22×C3×C4C2411yescharacteristic
182C2×#67—24no—
183C2×#67—42no—
184C2×#67—42no—
185C22×Dic3—24no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—42no—
189C22×Dic3—42no—
190#235—28no—
191#235—28no—
192#235—44no—
193#235—44no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12C2×D412yes—
200C2×C4×D3—24no—
201C2×C4×D3—42no—
202C2×C4×D3—42no—
203C2×Dic6—28no—
204C2×Dic6—28no—
205C2×Dic6—22no—
206C2×Dic6—21no—
207C2×Dic6—22no—
208C2×Dic6—22no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—22no—
212C2×Dic6—21no—
213C2×Dic6—24no—
214C2×Dic6—21no—
215C2×Dic6—22no—
216C2×Dic6—44no—
217C2×Dic6—44no—
218C2×Dic6—42no—
219C2×Dic6—42no—
220C2×Dic6C2×D412yes—
221C2×Dic6C2×D414yes—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—88no—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—42no—
231#214—216no—
232#214—232no—
233#213—216no—
234(C2×D3)⋊C4—216no—
235C4⋊Dic3—216no—
236#208—28no—
237C2×C3⋊C8—28no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8C2×D414yes—
243C2×C3⋊C8C2×D414yes—
244C25—21no—
245C2×#41—34no—
246C2×#41—62no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—62no—
251C22×Q8—34no—
252C22×Q8—32no—
253C22×Q8—31no—
254C22×Q8—62no—
255C22×Q8—61no—
256C22×Q8—62no—
257C22×Q8—62no—
258C22×Q8—62no—
259C22×Q8—62no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—21no—
266C22×D4—31no—
267C22×D4—61no—
268C22×D4D1214yes—
269C22×D4C22×D312yes—
270C23×C4—61no—
271C23×C4—61no—
272C23×C4C22×D311yescharacteristic
273AGL(1,ℤ/8)—632no—
274C2×SD16—68no—
275C2×SD16—64no—
276C2×SD16—68no—
277C2×SD16—64no—
278C2×SD16—632no—
279C2×SD16—68no—
280C2×SD16—64no—
281C2×D8—64no—
282C2×D8—64no—
283C2×D8—64no—
284C2×M16—34no—
285C2×M16—62no—
286C2×M16—62no—
287C22×C8—34no—
288C22×C8—32no—
289C22×C8—32no—
290C22×C8—62no—
291C22×C8—62no—
292C4⋊Q8—64no—
293C4⋊Q8—64no—
294#124—64no—
295#124—64no—
296#123—616no—
297C22⋊Q8—616no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302C2×C4⋊C4—64no—
303C2×C4⋊C4#6712yes—
304C2×C4⋊C4C22×D312yes—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—64no—
308C2×C22⋊C4#6712yes—
309C2×C42—21no—
310C2×C42C22×D312yes—
311#103—64no—
312#103—68no—
313#102—64no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—48no—
319C23×C3—48no—
320C23×C3—88no—
321C23×C3—88no—
322C23×C3—88no—
323C23×C3#12012yes—
324C23×C3#12112yes—
325C23×C3#12712yes—
326C23×C3C22×D411yescharacteristic
327C22×D3—22no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—44no—
331C3×D4—24no—
332C3×D4—24no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—48no—
336C2×C3×C4—24no—
337C2×C3×C4—28no—
338C2×C3×C4—42no—
339C2×C3×C4—44no—
340C2×C3×C4#12014yes—
341C2×C3×C4#12014yes—
342C2×C3×C4#12012yes—
343C2×C3×C4#12114yes—
344C2×C3×C4#12114yes—
345C2×C3×C4#12112yes—
346C2×C3×C4#12712yes—
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353#67—48no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—42no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—41no—
360C2×Dic3—48no—
361D12—22no—
362D12—22no—
363D12—41no—
364D12—41no—
365C4×D3—48no—
366Dic6—24no—
367Dic6—22no—
368Dic6—24no—
369Dic6—22no—
370Dic6—42no—
371Dic6—41no—
372Dic6—42no—
373Dic6—41no—
374Dic6—48no—
375C3⋊C8—24no—
376C3⋊C8—24no—
377C24—22no—
378C24—24no—
379C24—42no—
380C24—42no—
381C24—61no—
382C24—84no—
383C24C2×#6711yescharacteristic
384#41—68no—
385#41—68no—
386#41—124no—
387#41—124no—
388C2×Q8—34no—
389C2×Q8—32no—
390C2×Q8—64no—
391C2×Q8—62no—
392C2×Q8—61no—
393C2×Q8—68no—
394C2×Q8—68no—
395C2×Q8—62no—
396C2×Q8—61no—
397C2×Q8—62no—
398C2×Q8—61no—
399C2×Q8—62no—
400C2×Q8—61no—
401C2×Q8—62no—
402C2×Q8—61no—
403C2×Q8—124no—
404C2×Q8—124no—
405C2×Q8—122no—
406C2×Q8—122no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—32no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—64no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—122no—
427C2×D4—122no—
428C2×D4C2×D1214yes—
429C22×C4—22no—
430C22×C4—64no—
431C22×C4—62no—
432C22×C4—61no—
433C22×C4—64no—
434C22×C4—122no—
435C22×C4—122no—
436C22×C4—122no—
437C22×C4—122no—
438C22×C4C2×D1212yes—
439C22×C4#23512yes—
440C22×C4D3×D412yes—
441C22×C4D3×D412yes—
442C22×C4D3×D412yes—
443C22×C4D3×D412yes—
444C22×C4C2×#6714yes—
445C22×C4C2×#6711yescharacteristic
446C22×C4C23×D311yescharacteristic
447SD16—632no—
448SD16—616no—
449D8—616no—
450M16—68no—
451C2×C8—34no—
452C2×C8—34no—
453C2×C8—68no—
454C2×C8—62no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8—62no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—616no—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—616no—
465C22⋊C4—88no—
466C42—22no—
467C42—22no—
468C42—42no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—44no—
473C22×C3—48no—
474C22×C3—816no—
475C22×C3—84no—
476C22×C3—84no—
477C22×C3#39214yes—
478C22×C3C2×#12011yescharacteristic
479C22×C3C2×#12111yescharacteristic
480C22×C3C2×#12711yescharacteristic
481C22×C3#57714yes—
482C2×D3—42no—
483C2×D3—44no—
484C2×D3—82no—
485C2×D3—82no—
486C3×C4—44no—
487C3×C4—82no—
488C3×C4#39214yes—
489C3×C4C2×#12011yescharacteristic
490C3×C4C2×#12011yescharacteristic
491C3×C4C2×#12111yescharacteristic
492C3×C4C2×#12111yescharacteristic
493Dic3—44no—
494Dic3—42no—
495C23—24no—
496C23—24no—
497C23—48no—
498C23—42no—
499C23—42no—
500C23—48no—
501C23—62no—
502C23—88no—
503C23—88no—
504C23—88no—
505C23—121no—
506C23—121no—
507C23—124no—
508C23#93012yes—
509C23#93712yes—
510C23#94012yes—
511C23C22×#6711yescharacteristic
512Q8—64no—
513Q8—62no—
514Q8—64no—
515Q8—62no—
516Q8—128no—
517Q8—122no—
518Q8—121no—
519Q8—122no—
520Q8—121no—
521D4—24no—
522D4—24no—
523D4—48no—
524D4—42no—
525D4—42no—
526D4—62no—
527D4—62no—
528D4—128no—
529D4—121no—
530D4—121no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—64no—
536C2×C4—62no—
537C2×C4—128no—
538C2×C4—121no—
539C2×C4—122no—
540C2×C4—121no—
541C2×C4—122no—
542C2×C4—128no—
543C2×C4#88214yes—
544C2×C4#88214yes—
545C2×C4#88414yes—
546C2×C4#88414yes—
547C2×C4#93012yes—
548C2×C4#93712yes—
549C2×C4#94012yes—
550C2×C4C22×D1211yescharacteristic
551C2×C4C2×#23511yescharacteristic
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×D3×D411yescharacteristic
555C2×C4C2×D3×D411yescharacteristic
556C8—64no—
557C8—64no—
558C2×C3—24no—
559C2×C3—84no—
560C2×C3C2×#39211yescharacteristic
561C2×C3#301012yes—
562C2×C3#304212yes—
563C2×C3#315412yes—
564D3—82no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—48no—
569C22—44no—
570C22—816no—
571C22—84no—
572C22—84no—
573C22—124no—
574C22—122no—
575C22—242no—
576C22—242no—
577C22#496014yes—
578C22C2×#93011yescharacteristic
579C22C2×#93711yescharacteristic
580C22C2×#94011yescharacteristic
581C22#583614yes—
582C4—44no—
583C4—82no—
584C4—122no—
585C4—124no—
586C4#496014yes—
587C4C2×#88211yescharacteristic
588C4C2×#88211yescharacteristic
589C4C2×#88411yescharacteristic
590C4C2×#88411yescharacteristic
591C3#2342411yescharacteristic, complemented, Sylow
592C2—24no—
593C2—84no—
594C2—242no—
595C2C2×#496011yescharacteristic
596C2#8254412yes—
597C2#8338112yes—
598C2#8339012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2309C2
socleC23×C3C22×D4
prime radicalC22×#932C2
hypercenter#2309D3
hypocenterC3#23424
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
1. C23×#214C2semidirect
3. C2×#5526C2semidirect
4. C3×#2309C2semidirect
5. C2×#5004C2semidirect
6. C2×#5002C2semidirect
7. #71579C2semidirect
8. #69661C2semidirect
9. #69659C2semidirect
31. C2×#875C22semidirect
32. C2×#869C22semidirect
38. C2×#810C22semidirect
39. C2×#810C22semidirect
40. C2×#809C22semidirect
41. #4524C22semidirect
47. #2309D3semidirect
591. C3#23424semidirect

Maximal subgroups

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

subgroupindex
C23×#2142
C22×#9322
C2×#55262
C3×#23092
C2×#50042
C2×#50022
#715792
#696612
#696592
#234243