Finite Group Databasebeta

← #10829944

Subgroups — #10829944

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×#5713C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic
3C2×C4×#204C211yesmaximal, characteristic, complemented
4#74369C211yesmaximal, characteristic, complemented
5C3×#1957C211yesmaximal, characteristic, complemented
6#70214C212yesmaximal, complemented
7#70178C212yesmaximal, complemented
8#70172C212yesmaximal, complemented
9#70170C212yesmaximal, complemented
10#69788C211yesmaximal, characteristic, complemented
11#69588C211yesmaximal, characteristic, complemented
12#24186—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×#936—21no—
15C2×#936C2212yes—
16C2×#935—22no—
17C2×D4×Dic3—21no—
18C2×D4×Dic3C2212yes—
19C22×#213C2212yes—
20C22×#204C2211yescharacteristic
21#5713—24no—
22#5713—24no—
23C2×#897—21no—
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#384C2211yescharacteristic
31C2×#832C2212yes—
32C2×#821C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#816C2211yescharacteristic
35#5072—24no—
36#4994—22no—
37#4988C2212yes—
38#4987—21no—
39#4984C2212yes—
40C2×#810C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C4×#204—24no—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#559—31no—
47C2×#459—31no—
48C2×C4×SD16—31no—
49#2538—31no—
50#2042—32no—
51#2012—32no—
52#1984—31no—
53#1957D311yescharacteristic, complemented
54#1879—32no—
55#1864—32no—
56#1803—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C23×Dic3—22no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×Dic6D411yescharacteristic
64C22×Dic6C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×#127—22no—
68C3×#127D414yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C42C2311yescharacteristic
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—22no—
76#936—28no—
77#936—24no—
78#936—24no—
79#935—216no—
80D4×Dic3—28no—
81D4×Dic3—24no—
82D4×Dic3—24no—
83C2×#213—24no—
84C2×#213—24no—
85C2×#213—42no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3D411yescharacteristic
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4D412yes—
95C2×C4×Dic3—21no—
96C2×C4×Dic3C2312yes—
97C22×C3⋊C8C2312yes—
98C2×#204—22no—
99C2×#204—22no—
100C2×#204—22no—
101#897—44no—
102C4×D12—22no—
103C4×D12—22no—
104C4×Dic6—22no—
105C4×Dic6—22no—
106C4×Dic6—22no—
107C4×Dic6D414yes—
108C3×#102—24no—
109C3×#102—42no—
110C3×C4×C8—22no—
111#832—24no—
112C3⋊#95—22no—
113#821—22no—
114#817—22no—
115#816—22no—
116#810—42no—
117C4⋊(C3⋊C8)—22no—
118C22×SD16—31no—
119C22×D8—32no—
120#559—64no—
121#559—64no—
122C2×#127C2×D311yescharacteristic
123C2×#124—32no—
124C2×#124—61no—
125C2×#123—62no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8—31no—
129C2×C4×D4—32no—
130C2×C4×D4—31no—
131C2×C4×D4—61no—
132#459—64no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8—31no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—32no—
139C2×#102—31no—
140C2×#102C2×D312yes—
141C2×#102C2×D312yes—
142C2×C4×C8C2×D311yescharacteristic
143#396—62no—
144#390—32no—
145#389—32no—
146#384C2×D311yescharacteristic
147#380—61no—
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—22no—
152C23×D3—21no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4—81no—
163C2×C3×D4—81no—
164C2×C3×D4C2×D414yes—
165C22×C3×C4—22no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—28no—
172C22×Dic3—21no—
173C22×Dic3—82no—
174C22×Dic3—82no—
175C22×Dic3C2×D412yes—
176C22×Dic3C2×D412yes—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12#4112yes—
183C2×C4×D3—42no—
184C2×C4×D3—42no—
185C2×C4×D3—42no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—21no—
194C2×Dic6C2×D412yes—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8#4112yes—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—81no—
205C3×C42C2×D412yes—
206C3×C42C2×D412yes—
207#217—28no—
208#217—28no—
209#217—28no—
210#213—48no—
211(C2×D3)⋊C4—44no—
212(C2×D3)⋊C4—42no—
213(C2×D3)⋊C4—82no—
214C4⋊Dic3—22no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3—44no—
219C4⋊Dic3C2×D412yes—
220Dic3⋊C4—24no—
221Dic3⋊C4—24no—
222C4×Dic3—24no—
223C4×Dic3—22no—
224C4×Dic3—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228#204—28no—
229C22×Q8—31no—
230C22×Q8—31no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—31no—
234C22×D4C22×D312yes—
235C23×C4—62no—
236C23×C4—61no—
237C2×SD16—62no—
238C2×SD16—62no—
239C2×SD16—62no—
240C2×D8—64no—
241C2×D8—64no—
242C2×D8—122no—
243C22×C8—32no—
244C22×C8C22×D311yescharacteristic
245C22×C8C22×D311yescharacteristic
246#127—22no—
247#127D1214yes—
248#124—64no—
249#124—68no—
250#124—64no—
251#123—616no—
252C22⋊Q8—124no—
253C4×Q8—34no—
254C4×Q8—62no—
255C4×Q8—62no—
256C4×Q8—62no—
257C4×D4—64no—
258C4×D4—68no—
259C4×D4—64no—
260C4×D4—62no—
261C4×D4—62no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4C22×D312yes—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—62no—
277C2×C42—32no—
278C2×C42—61no—
279C2×C42C22×D311yescharacteristic
280#106—62no—
281C4⋊C8—62no—
282#103—62no—
283#103—122no—
284#102—24no—
285#102—42no—
286#102—62no—
287#102—64no—
288C4×C8—22no—
289#95—62no—
290C23×C3—24no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3C22×D411yescharacteristic
298C22×D3—22no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C3×D4—24no—
306C3×D4—24no—
307C3×D4—42no—
308C3×D4—42no—
309C3×D4—48no—
310C3×D4—84no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4#12012yes—
317C2×C3×C4#12012yes—
318C2×C3×C4#13512yes—
319C2×C3×C4AGL(1,ℤ/8)12yes—
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4H(5,𝔽2)11yescharacteristic
325C2×C3×C42−1+411yescharacteristic
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—88no—
342D12—22no—
343D12—22no—
344D12—41no—
345D12—41no—
346C4×D3—84no—
347Dic6—22no—
348Dic6—22no—
349Dic6—41no—
350Dic6—41no—
351Dic6—44no—
352C3×C8—22no—
353C3×C8—41no—
354C3⋊C8—42no—
355C24—22no—
356C24—61no—
357C2×Q8—32no—
358C2×Q8—62no—
359C2×Q8—62no—
360C2×Q8—62no—
361C2×Q8—61no—
362C2×Q8—61no—
363C2×Q8—61no—
364C2×Q8—61no—
365C2×Q8—61no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—32no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—61no—
377C2×D4—61no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4C2×D1214yes—
384C22×C4—22no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—68no—
390C22×C4—61no—
391C22×C4—122no—
392C22×C4—122no—
393C22×C4—122no—
394C22×C4—242no—
395C22×C4—242no—
396C22×C4D3×D411yescharacteristic
397C22×C4D3×D411yescharacteristic
398C22×C4C23×D311yescharacteristic
399SD16—68no—
400D8—128no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—62no—
407C2×C8—62no—
408C2×C8—62no—
409C2×C8#23712yes—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—32no—
413C4⋊C4—42no—
414C4⋊C4—62no—
415C4⋊C4—64no—
416C4⋊C4—64no—
417C4⋊C4—61no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C4⋊C4—81no—
421C4⋊C4—124no—
422C22⋊C4—68no—
423C22⋊C4—68no—
424C22⋊C4—68no—
425C22⋊C4—124no—
426C22⋊C4—122no—
427C22⋊C4—242no—
428C42—64no—
429C42—62no—
430C42—62no—
431C42C2×D1212yes—
432C42D3×D412yes—
433C22×C3—44no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—82no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3—82no—
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—42no—
448C2×D3—42no—
449C2×D3—42no—
450C2×D3—81no—
451C2×D3—81no—
452C2×D3—81no—
453C2×D3—81no—
454C3×C4—22no—
455C3×C4—82no—
456C3×C4#44914yes—
457C3×C4#52511yescharacteristic
458C3×C4#53711yescharacteristic
459C3×C4#54611yescharacteristic
460C3×C4#54711yescharacteristic
461Dic3—42no—
462Dic3—42no—
463Dic3—42no—
464Dic3—42no—
465Dic3—81no—
466C23—24no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—62no—
474C23—121no—
475C23—121no—
476C23—121no—
477C23—121no—
478C23—121no—
479C23—121no—
480C23C2×D3×D411yescharacteristic
481Q8—62no—
482Q8—62no—
483Q8—124no—
484Q8—121no—
485Q8—121no—
486D4—24no—
487D4—24no—
488D4—48no—
489D4—42no—
490D4—42no—
491D4—62no—
492D4—62no—
493D4—84no—
494D4—121no—
495D4—121no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—248no—
517C2×C4—244no—
518C2×C4#88212yes—
519C2×C4#88212yes—
520C2×C4#91112yes—
521C2×C4#91612yes—
522C2×C4C22×D1211yescharacteristic
523C2×C4C2×D3×D411yescharacteristic
524C2×C4C2×#23711yescharacteristic
525C2×C4#100411yescharacteristic
526C2×C4D3×#4111yescharacteristic
527C2×C4#101011yescharacteristic
528C8—22no—
529C8—41no—
530C8—122no—
531C2×C3—84no—
532C2×C3#240711yescharacteristic
533C2×C3C2×#44911yescharacteristic
534C2×C3#301211yescharacteristic
535C2×C3#336111yescharacteristic
536C2×C3#337311yescharacteristic
537C2×C3#339411yescharacteristic
538C2×C3#339911yescharacteristic
539D3—82no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—122no—
548C22—122no—
549C22—122no—
550C22—241no—
551C22—241no—
552C22—241no—
553C22—241no—
554C22C2×#88211yescharacteristic
555C22#567711yescharacteristic
556C22#571811yescharacteristic
557C22C2×#91111yescharacteristic
558C22#575711yescharacteristic
559C22C2×#91611yescharacteristic
560C22#576211yescharacteristic
561C4—22no—
562C4—82no—
563C4—122no—
564C4—122no—
565C4—122no—
566C4—122no—
567C4—241no—
568C4#478114yes—
569C4#555411yescharacteristic
570C4#555511yescharacteristic
571C4#560611yescharacteristic
572C4#560811yescharacteristic
573C3#2418611yescharacteristic, complemented, Sylow
574C2—84no—
575C2—242no—
576C2#7643511yescharacteristic
577C2C2×#478111yescharacteristic
578C2#7874711yescharacteristic
579C2#8158511yescharacteristic
580C2#8158711yescharacteristic
581C2#8175611yescharacteristic
582C2#8178011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1957C2
socleC23×C3C22×D4
prime radicalC2×#5072C2
hypercenter#1957D3
hypocenterC3#24186
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
1. C2×#5713C2semidirect
3. C2×C4×#204C2semidirect
4. #74369C2semidirect
5. C3×#1957C2semidirect
6. #70214C2semidirect
7. #70178C2semidirect
8. #70172C2semidirect
9. #70170C2semidirect
10. #69788C2semidirect
11. #69588C2semidirect
53. #1957D3semidirect
573. C3#24186semidirect

Maximal subgroups

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

subgroupindex
C2×#57132
C2×#50722
C2×C4×#2042
#743692
C3×#19572
#702142
#701782
#701722
#701702
#697882
#695882
#241863