Finite Group Databasebeta

← #10829946

Subgroups — #10829946

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
2C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
3#82376C211yesmaximal, characteristic, complemented
4C2×#5015C211yesmaximal, characteristic
5#72055C212yesmaximal, complemented
6#71626C211yesmaximal, characteristic, complemented
7#70584C212yesmaximal, complemented
8#70279C212yesmaximal, complemented
9#70208C212yesmaximal, complemented
10#69788C211yesmaximal, characteristic, complemented
11#69772C211yesmaximal, characteristic, complemented
12#24186—31nomaximal, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#947C2212yes—
17C22×#213C2212yes—
18C2×#924—21no—
19#5585—24no—
20#5585—24no—
21C2×#894—22no—
22C2×#893C2212yes—
23C2×#875—21no—
24C2×#874C2211yescharacteristic
25C2×C4×D12—21no—
26C2×C4×Dic6C2211yescharacteristic
27C3×C4×SD16—24no—
28C2×C3×#106C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C2×C3×C4×C8C2211yescharacteristic
32C2×#835C2212yes—
33#5221C2212yes—
34#5212—21no—
35C2×#821C2212yes—
36C2×#821C2212yes—
37#5015—24no—
38#4972C2212yes—
39C2×#809C2212yes—
40#4948—22no—
41#4935C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#559—31no—
47C2×#459—31no—
48C2×C4×SD16D311yescharacteristic, complemented
49#2538—31no—
50#2042—32no—
51#2012—32no—
52#1984—31no—
53#1957—31no—
54#1879—32no—
55#1864—32no—
56#1803—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×D12—21no—
61C22×D12—21no—
62C22×D12C2312yes—
63C22×C4×D3—22no—
64C22×Dic6D411yescharacteristic
65C2×C3×SD16—22no—
66C2×C3×SD16—22no—
67C2×C3×SD16—22no—
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×C4×Q8—22no—
71C3×C4×Q8D414yes—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80C2×#217—22no—
81C2×#217—22no—
82#947—24no—
83#947—24no—
84C2×#213—24no—
85C2×#213—24no—
86C2×#213—42no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3C2312yes—
94#924—44no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4D412yes—
97C2×C4×Dic3C2312yes—
98C22×C3⋊C8C2312yes—
99#894—216no—
100#893—24no—
101#893—24no—
102#875—28no—
103#874—22no—
104#874D414yes—
105C4×D12—28no—
106C4×Dic6—22no—
107C4×Dic6—22no—
108C3×#106—22no—
109C3×#103—22no—
110C3×#102—22no—
111C3×C4×C8—22no—
112#835—42no—
113C3⋊#95—22no—
114#821—24no—
115#821—42no—
116#809—24no—
117C4⋊(C3⋊C8)—22no—
118C22×SD16C2×D311yescharacteristic
119C22×D8—32no—
120#559—64no—
121#559—64no—
122C2×#127—31no—
123C2×#124—32no—
124C2×#124—61no—
125C2×#123—62no—
126C2×C22⋊Q8—61no—
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—24no—
134C2×#106C2×D311yescharacteristic
135C2×C4⋊C8—31no—
136C2×#103—32no—
137C2×#103C2×D311yescharacteristic
138C2×#102—32no—
139C2×#102—32no—
140C2×#102—32no—
141C2×#102C2×D311yescharacteristic
142C2×C4×C8C2×D311yescharacteristic
143#396—62no—
144#390—32no—
145#389—32no—
146#384—31no—
147#380—61no—
148#340—32no—
149#336—32no—
150#336—32no—
151C24×C3—21no—
152C23×D3—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8C2×D412yes—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4#4112yes—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4—42no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—22no—
175C22×Dic3C2×D412yes—
176C22×Dic3C2×D412yes—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—24no—
182C2×D12—22no—
183C2×D12—24no—
184C2×D12—42no—
185C2×D12—42no—
186C2×D12—81no—
187C2×D12—81no—
188C2×D12C2×D414yes—
189C2×C4×D3—28no—
190C2×C4×D3—82no—
191C2×C4×D3—82no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C3×SD16—28no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8#4112yes—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4C2×D412yes—
207C3×C22⋊C4—42no—
208C3×C42—24no—
209C3×C42C2×D412yes—
210C3×C42C2×D412yes—
211#217—44no—
212#217—82no—
213#213—48no—
214(C2×D3)⋊C4—28no—
215(C2×D3)⋊C4—28no—
216(C2×D3)⋊C4—28no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—42no—
221C4⋊Dic3—81no—
222Dic3⋊C4—24no—
223Dic3⋊C4—44no—
224C4×Dic3—22no—
225C4×Dic3—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C22×Q8—31no—
230C22×Q8C22×D311yescharacteristic
231C22×D4—32no—
232C22×D4—61no—
233C22×D4—61no—
234C22×D4C22×D311yescharacteristic
235C23×C4—21no—
236C23×C4—62no—
237C2×SD16—22no—
238C2×SD16—22no—
239C2×SD16—22no—
240C2×D8—64no—
241C2×D8—64no—
242C2×D8—122no—
243C22×C8—32no—
244C22×C8C22×D311yescharacteristic
245C22×C8C22×D311yescharacteristic
246#127—34no—
247#127—62no—
248#124—68no—
249#124—64no—
250#124—64no—
251#123—616no—
252C22⋊Q8—124no—
253C4×Q8—22no—
254C4×Q8—62no—
255C4×Q8—62no—
256C4×Q8#6714yes—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—68no—
260C4×D4—64no—
261C4×D4—64no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4C22×D311yescharacteristic
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C42—21no—
278C2×C42—32no—
279C2×C42C22×D311yescharacteristic
280#106—22no—
281C4⋊C8—62no—
282#103—22no—
283#103—122no—
284#102—22no—
285#102—64no—
286#102—64no—
287#102—122no—
288C4×C8—22no—
289#95—62no—
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—22no—
306C3×Q8—22no—
307C3×Q8—41no—
308C3×Q8—41no—
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—41no—
312C3×D4—41no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
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—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344D12—24no—
345D12—24no—
346D12—42no—
347D12—42no—
348D12—48no—
349D12—84no—
350C4×D3—88no—
351Dic6—44no—
352C3×C8—22no—
353C3×C8—41no—
354C3⋊C8—42no—
355C24—21no—
356C24—62no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—62no—
363C2×Q8—62no—
364C2×Q8—62no—
365C2×Q8C2×#6712yes—
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#23512yes—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—32no—
388C22×C4—32no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—68no—
393C22×C4—62no—
394C22×C4—242no—
395C22×C4—242no—
396C22×C4D3×D411yescharacteristic
397C22×C4D3×D411yescharacteristic
398C22×C4C23×D311yescharacteristic
399SD16—28no—
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#23512yes—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—62no—
415C4⋊C4—64no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4—124no—
419C4⋊C4—122no—
420C4⋊C4—241no—
421C4⋊C4C2×#6712yes—
422C22⋊C4—42no—
423C22⋊C4—68no—
424C22⋊C4—68no—
425C22⋊C4—68no—
426C22⋊C4—124no—
427C22⋊C4—242no—
428C42—24no—
429C42—62no—
430C42—62no—
431C42D3×D412yes—
432C42C2×#6712yes—
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—81no—
458C3×C4#44914yes—
459C3×C4#52511yescharacteristic
460C3×C4#53711yescharacteristic
461C3×C4#54611yescharacteristic
462C3×C4#54711yescharacteristic
463Dic3—42no—
464Dic3—42no—
465Dic3—82no—
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—
480C23C2×D3×D411yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—41no—
484Q8—41no—
485Q8—124no—
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—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—84no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—248no—
518C2×C4#91412yes—
519C2×C4#91612yes—
520C2×C4#93712yes—
521C2×C4#93712yes—
522C2×C4C2×#23511yescharacteristic
523C2×C4C2×#23511yescharacteristic
524C2×C4C2×D3×D411yescharacteristic
525C2×C4#100911yescharacteristic
526C2×C4#101011yescharacteristic
527C2×C4C22×#6711yescharacteristic
528C8—22no—
529C8—41no—
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—
554C22#555211yescharacteristic
555C22#555311yescharacteristic
556C22C2×#91411yescharacteristic
557C22C2×#91611yescharacteristic
558C22#576911yescharacteristic
559C22#577011yescharacteristic
560C22C2×#93711yescharacteristic
561C4—22no—
562C4—42no—
563C4—42no—
564C4—81no—
565C4—122no—
566C4—122no—
567C4—242no—
568C4#517214yes—
569C4#555011yescharacteristic
570C4#555111yescharacteristic
571C4#579111yescharacteristic
572C4#583311yescharacteristic
573C3#2418611yescharacteristic, complemented, Sylow
574C2—82no—
575C2—244no—
576C2#7962111yescharacteristic
577C2#7962911yescharacteristic
578C2#7963811yescharacteristic
579C2#7964111yescharacteristic
580C2#8240211yescharacteristic
581C2C2×#517211yescharacteristic
582C2#8391211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×SD16C2
socleC23×C3C22×D4
prime radicalC2×#5015C2
hypercenterC2×C4×SD16D3
hypocenterC3#24186
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#5585C2semidirect
2. C2×C3×C4×SD16C2semidirect
3. #82376C2semidirect
5. #72055C2semidirect
6. #71626C2semidirect
7. #70584C2semidirect
8. #70279C2semidirect
9. #70208C2semidirect
10. #69788C2semidirect
11. #69772C2semidirect
48. C2×C4×SD16D3semidirect
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
C2×C3×C4×SD162
#823762
C2×#50152
#720552
#716262
#705842
#702792
#702082
#697882
#697722
#241863