Finite Group Databasebeta

← #10829948

Subgroups — #10829948

Order 768.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic, complemented
2C3×#2538C211yesmaximal, characteristic, complemented
3C2×C4×#214C211yesmaximal, characteristic, complemented
4C2×#5015C211yesmaximal, characteristic
5#71634C211yesmaximal, characteristic, complemented
6#69659C212yesmaximal, complemented
7#69657C212yesmaximal, complemented
8#69628C212yesmaximal, complemented
9#69626C212yesmaximal, complemented
10#69588C211yesmaximal, characteristic, complemented
11#69580C211yesmaximal, characteristic, complemented
12#24186—31nomaximal, Sylow
13C2×C3×C22⋊Q8—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#947—21no—
17C22×#214C2211yescharacteristic
18C22×#213C2212yes—
19#5585—24no—
20#5585—24no—
21C2×#894—22no—
22C2×#893—21no—
23C2×#875C2212yes—
24C2×#874C2211yescharacteristic
25C2×C4×D12C2212yes—
26C2×C4×Dic6C2211yescharacteristic
27C3×#396—22no—
28C3×#390C2212yes—
29C3×#389C2212yes—
30C3×#384C2211yescharacteristic
31C3×#380—21no—
32C2×#832C2211yescharacteristic
33C4×#214—24no—
34#5015—24no—
35C2×#810C2211yescharacteristic
36C2×#810C2212yes—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#808C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#559—31no—
47C2×#459—31no—
48C2×C4×SD16—31no—
49#2538D311yescharacteristic, complemented
50#2042—32no—
51#2012—32no—
52#1984—31no—
53#1957—31no—
54#1879—32no—
55#1864—32no—
56#1803—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×D12—21no—
61C22×D12—21no—
62C22×D12C2312yes—
63C22×C4×D3—22no—
64C22×Dic6C2311yescharacteristic
65C3×C22⋊Q8—44no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4D412yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42C2312yes—
81C2×C3×C42C2311yescharacteristic
82#947—28no—
83C2×#214—22no—
84C2×#214—22no—
85C2×#214—22no—
86C2×#213—24no—
87C2×#213—24no—
88C2×#213—42no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×Dic3⋊C4—21no—
94C2×C4×Dic3—21no—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2311yescharacteristic
98#894—216no—
99#893—28no—
100#875—24no—
101#875—24no—
102#874—22no—
103#874D414yes—
104C4×D12—24no—
105C4×D12—24no—
106C4×Dic6—22no—
107C4×Dic6D414yes—
108C3×#95—22no—
109#832—22no—
110#810—22no—
111#810—42no—
112#809—24no—
113#809—24no—
114#809—42no—
115#808—22no—
116C4⋊(C3⋊C8)—22no—
117C4×C3⋊C8—22no—
118C22×SD16—31no—
119C22×D8—32no—
120#559—64no—
121#559—64no—
122C2×#127—31no—
123C2×#124—32no—
124C2×#124—61no—
125C2×#123—62no—
126C2×C22⋊Q8—21no—
127C2×C4×Q8—31no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4—32no—
130C2×C4×D4—61no—
131C2×C4×D4C2×D311yescharacteristic
132#459—64no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8—31no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#102—32no—
141C2×#102—32no—
142C2×C4×C8—31no—
143#396—22no—
144#390C2×D312yes—
145#389C2×D312yes—
146#384C2×D311yescharacteristic
147#380—21no—
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—21no—
152C23×D3—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4#4112yes—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—21no—
173C22×Dic3—21no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—24no—
179C2×D12—22no—
180C2×D12—24no—
181C2×D12—42no—
182C2×D12—42no—
183C2×D12—81no—
184C2×D12—81no—
185C2×D12C2×D414yes—
186C2×C4×D3—28no—
187C2×C4×D3—82no—
188C2×C4×D3—82no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6C2×D412yes—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—44no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—81no—
204C3×C22⋊C4—44no—
205C3×C22⋊C4—42no—
206C3×C22⋊C4—82no—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42C2×D412yes—
210C3×C42C2×D412yes—
211#214—28no—
212#213—48no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—28no—
215(C2×D3)⋊C4—28no—
216C4⋊Dic3—21no—
217C4⋊Dic3C2×D412yes—
218Dic3⋊C4—24no—
219C4×Dic3—24no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8#4112yes—
229C22×Q8—31no—
230C22×Q8#6711yescharacteristic
231C22×D4—32no—
232C22×D4—61no—
233C22×D4—61no—
234C22×D4C22×D311yescharacteristic
235C23×C4—21no—
236C23×C4—62no—
237C2×SD16—62no—
238C2×SD16—62no—
239C2×SD16—62no—
240C2×D8—64no—
241C2×D8—64no—
242C2×D8—122no—
243C22×C8—31no—
244C22×C8—32no—
245C22×C8—31no—
246#127—34no—
247#127—62no—
248#124—64no—
249#124—64no—
250#124—68no—
251#123—616no—
252C22⋊Q8—44no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—34no—
256C4×Q8—62no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—64no—
260C4×D4—64no—
261C4×D4—68no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4#6711yescharacteristic
268C2×C4⋊C4#6712yes—
269C2×C4⋊C4C22×D311yescharacteristic
270C2×C4⋊C4C22×D312yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C42—61no—
278C2×C42C22×D312yes—
279C2×C42C22×D311yescharacteristic
280#106—62no—
281C4⋊C8—62no—
282#103—62no—
283#103—122no—
284#102—64no—
285#102—64no—
286#102—62no—
287#102—122no—
288C4×C8—62no—
289#95—22no—
290C23×C3—22no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3C22×D411yescharacteristic
298C22×D3—24no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
305C3×Q8—44no—
306C3×D4—22no—
307C3×D4—22no—
308C3×D4—41no—
309C3×D4—41no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—84no—
325C2×C3×C4#12012yes—
326C2×C3×C4#12012yes—
327C2×C3×C4#13512yes—
328C2×C3×C4AGL(1,ℤ/8)12yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4H(5,𝔽2)11yescharacteristic
334C2×C3×C42−1+411yescharacteristic
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341D12—24no—
342D12—24no—
343D12—42no—
344D12—42no—
345D12—48no—
346D12—84no—
347C4×D3—88no—
348Dic6—22no—
349Dic6—22no—
350Dic6—41no—
351Dic6—41no—
352C3⋊C8—22no—
353C3⋊C8—41no—
354C3⋊C8—42no—
355C24—21no—
356C24—62no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—32no—
361C2×Q8—61no—
362C2×Q8—61no—
363C2×Q8—61no—
364C2×Q8—61no—
365C2×Q8—61no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—34no—
372C2×D4—64no—
373C2×D4—64no—
374C2×D4—62no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—122no—
380C2×D4—122no—
381C2×D4—241no—
382C2×D4—241no—
383C2×D4#23912yes—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—68no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—242no—
393C22×C4—242no—
394C22×C4C2×#6712yes—
395C22×C4C2×#6712yes—
396C22×C4C2×#6711yescharacteristic
397C22×C4C2×#6711yescharacteristic
398C22×C4C23×D311yescharacteristic
399SD16—68no—
400D8—128no—
401C2×C8—32no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—62no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—32no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—64no—
420C4⋊C4—61no—
421C4⋊C4—81no—
422C22⋊C4—44no—
423C22⋊C4—42no—
424C22⋊C4—68no—
425C22⋊C4—68no—
426C22⋊C4—68no—
427C22⋊C4—82no—
428C42—22no—
429C42—22no—
430C42—64no—
431C42D3×D412yes—
432C42D3×D412yes—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—81no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3C2×#12011yescharacteristic
441C22×C3#54811yescharacteristic
442C22×C3#55011yescharacteristic
443C22×C3C2×#13511yescharacteristic
444C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
445C22×C3#57611yescharacteristic
446C22×C3#57711yescharacteristic
447C2×D3—44no—
448C2×D3—44no—
449C2×D3—44no—
450C2×D3—82no—
451C2×D3—82no—
452C2×D3—82no—
453C2×D3—82no—
454C3×C4—22no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—82no—
458C3×C4—81no—
459C3×C4#44914yes—
460C3×C4#52511yescharacteristic
461C3×C4#53711yescharacteristic
462C3×C4#54611yescharacteristic
463C3×C4#54711yescharacteristic
464Dic3—42no—
465Dic3—42no—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—64no—
474C23—122no—
475C23—122no—
476C23—122no—
477C23—122no—
478C23—122no—
479C23—122no—
480C23C22×#6711yescharacteristic
481Q8—44no—
482Q8—62no—
483Q8—62no—
484Q8—121no—
485Q8—121no—
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490D4—64no—
491D4—64no—
492D4—128no—
493D4—122no—
494D4—122no—
495D4—244no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—84no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—248no—
518C2×C4#93212yes—
519C2×C4#93712yes—
520C2×C4#93712yes—
521C2×C4#95012yes—
522C2×C4C2×D3×D411yescharacteristic
523C2×C4C2×D3×D411yescharacteristic
524C2×C4C2×#23911yescharacteristic
525C2×C4#100711yescharacteristic
526C2×C4D3×#4111yescharacteristic
527C2×C4#100911yescharacteristic
528C8—62no—
529C8—121no—
530C8—122no—
531C2×C3—82no—
532C2×C3#240711yescharacteristic
533C2×C3C2×#44911yescharacteristic
534C2×C3#301211yescharacteristic
535C2×C3#336111yescharacteristic
536C2×C3#337311yescharacteristic
537C2×C3#339411yescharacteristic
538C2×C3#339911yescharacteristic
539D3—84no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—124no—
548C22—124no—
549C22—124no—
550C22—242no—
551C22—242no—
552C22—242no—
553C22—242no—
554C22C2×#93211yescharacteristic
555C22C2×#93711yescharacteristic
556C22#581711yescharacteristic
557C22C2×#95011yescharacteristic
558C22#583411yescharacteristic
559C22#583511yescharacteristic
560C22#583711yescharacteristic
561C4—22no—
562C4—42no—
563C4—42no—
564C4—82no—
565C4—81no—
566C4—122no—
567C4—122no—
568C4#503714yes—
569C4#562711yescharacteristic
570C4#562911yescharacteristic
571C4#563011yescharacteristic
572C4#563111yescharacteristic
573C3#2418611yescharacteristic, complemented, Sylow
574C2—82no—
575C2—244no—
576C2C2×#503711yescharacteristic
577C2#8281211yescharacteristic
578C2#8342411yescharacteristic
579C2#8356911yescharacteristic
580C2#8358911yescharacteristic
581C2#8364311yescharacteristic
582C2#8367911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2538C2
socleC23×C3C22×D4
prime radicalC2×#5015C2
hypercenter#2538D3
hypocenterC3#24186
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#5585C2semidirect
2. C3×#2538C2semidirect
3. C2×C4×#214C2semidirect
5. #71634C2semidirect
6. #69659C2semidirect
7. #69657C2semidirect
8. #69628C2semidirect
9. #69626C2semidirect
10. #69588C2semidirect
11. #69580C2semidirect
49. #2538D3semidirect
573. C3#24186semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C3×#25382
C2×C4×#2142
C2×#50152
#716342
#696592
#696572
#696282
#696262
#695882
#695802
#241863