Finite Group Databasebeta

← #10829942

Subgroups — #10829942

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×C3×#459C211yesmaximal, characteristic
3#82376C211yesmaximal, characteristic, complemented
4C2×C4×#214C211yesmaximal, characteristic, complemented
5#72045C212yesmaximal, complemented
6#72037C212yesmaximal, complemented
7#71764C211yesmaximal, characteristic, complemented
8#70485C211yesmaximal, characteristic, complemented
9#70472C211yesmaximal, characteristic, complemented
10#70216C212yesmaximal, complemented
11#70169C212yesmaximal, complemented
12#24186—31nomaximal, Sylow
13C22×C3×D8C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#936—21no—
17C2×#936C2212yes—
18C2×#935—22no—
19C2×D4×Dic3—21no—
20C2×D4×Dic3C2212yes—
21C22×#214C2211yescharacteristic
22C2×#924—21no—
23#5713—24no—
24#5713—24no—
25C2×C4×Dic6C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C3×#459—24no—
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#102C2212yes—
30#5221C2212yes—
31C2×#832C2212yes—
32C2×#832C2211yescharacteristic
33C2×#832C2212yes—
34#5212—21no—
35C2×#816C2212yes—
36C4×#214—24no—
37#4972C2212yes—
38C2×#810C2211yescharacteristic
39C2×#808C2211yescharacteristic
40#4948—22no—
41#4935C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#559—31no—
47C2×#459D311yescharacteristic
48C2×C4×SD16—31no—
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×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×Dic3—22no—
63C22×Dic6D411yescharacteristic
64C22×Dic6C2311yescharacteristic
65C2×C3×D8—24no—
66C2×C3×D8—24no—
67C2×C3×D8—42no—
68C22×C3×C8C2312yes—
69C3×#127—22no—
70C3×#127D414yes—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C42C2311yescharacteristic
76C2×#217—22no—
77C2×#217—22no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81#936—28no—
82#936—24no—
83#936—24no—
84#935—216no—
85D4×Dic3—28no—
86D4×Dic3—24no—
87D4×Dic3—24no—
88C2×#214—22no—
89C2×#214—22no—
90C2×#214—22no—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3D411yescharacteristic
94C2×C4⋊Dic3C2312yes—
95C2×C4⋊Dic3C2311yescharacteristic
96#924—44no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4D412yes—
100C2×C4×Dic3—21no—
101C2×C4×Dic3C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104C4×Dic6—22no—
105C4×Dic6—22no—
106C4×Dic6—22no—
107C4×Dic6D414yes—
108C3×C4⋊C8—22no—
109C3×#102—24no—
110#832—22no—
111#832—24no—
112#832—42no—
113C3⋊#95—22no—
114#816—42no—
115#810—22no—
116#808—22no—
117C4×C3⋊C8—22no—
118C22×SD16—31no—
119C22×D8C2×D312yes—
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—61no—
131C2×C4×D4C2×D311yescharacteristic
132#459—24no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—32no—
139C2×#102—31no—
140C2×#102—32no—
141C2×#102C2×D312yes—
142C2×C4×C8—31no—
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—
152C24×C3—22no—
153C2×C3×D4—24no—
154C2×C3×D4—24no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—21no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C2×C3×D4—81no—
168C2×C3×D4—81no—
169C2×C3×D4C2×D414yes—
170C2×C3×D4#4112yes—
171C22×C3×C4—21no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—28no—
181C22×Dic3—22no—
182C22×Dic3—82no—
183C22×Dic3—82no—
184C22×Dic3C2×D412yes—
185C22×Dic3C2×D412yes—
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—
195C3×D8—48no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C22⋊C4—42no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204#217—28no—
205#217—28no—
206#217—28no—
207#217—44no—
208#217—82no—
209#214—28no—
210C4⋊Dic3—22no—
211C4⋊Dic3—21no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—42no—
215C4⋊Dic3—81no—
216C4⋊Dic3C2×D412yes—
217Dic3⋊C4—24no—
218Dic3⋊C4—24no—
219Dic3⋊C4—44no—
220C4×Dic3—24no—
221C4×Dic3—22no—
222C4×Dic3—22no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8#4112yes—
229C22×Q8—31no—
230C22×Q8—31no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C22×D311yescharacteristic
234C22×D4C22×D312yes—
235C23×C4—21no—
236C23×C4—62no—
237C2×SD16—62no—
238C2×SD16—62no—
239C2×SD16—62no—
240C2×D8—24no—
241C2×D8—24no—
242C2×D8—42no—
243C22×C8—31no—
244C22×C8—31no—
245C22×C8C22×D312yes—
246#127—22no—
247#127#6714yes—
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—22no—
258C4×D4—22no—
259C4×D4—64no—
260C4×D4—68no—
261C4×D4—64no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
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—32no—
278C2×C42—61no—
279C2×C42C22×D311yescharacteristic
280#106—62no—
281C4⋊C8—22no—
282#103—62no—
283#103—122no—
284#102—24no—
285#102—62no—
286#102—64no—
287#102—122no—
288C4×C8—62no—
289#95—62no—
290C23×C3—24no—
291C23×C3—22no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3—42no—
302C23×C3—42no—
303C23×C3—41no—
304C23×C3C22×D411yescharacteristic
305C3×D4—22no—
306C3×D4—22no—
307C3×D4—24no—
308C3×D4—24no—
309C3×D4—42no—
310C3×D4—41no—
311C3×D4—42no—
312C3×D4—41no—
313C3×D4—48no—
314C3×D4—84no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—84no—
321C2×C3×C4#12012yes—
322C2×C3×C4#12012yes—
323C2×C3×C4#13512yes—
324C2×C3×C4AGL(1,ℤ/8)12yes—
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4H(5,𝔽2)11yescharacteristic
330C2×C3×C42−1+411yescharacteristic
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—88no—
347Dic6—22no—
348Dic6—22no—
349Dic6—41no—
350Dic6—41no—
351Dic6—44no—
352C3×C8—42no—
353C3⋊C8—22no—
354C3⋊C8—41no—
355C24—21no—
356C24—22no—
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—21no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4#23712yes—
383C2×D4C2×#6714yes—
384C22×C4—21no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—68no—
393C22×C4—62no—
394C22×C4—242no—
395C22×C4—242no—
396C22×C4D3×D411yescharacteristic
397C22×C4D3×D411yescharacteristic
398C22×C4C23×D311yescharacteristic
399SD16—68no—
400D8—48no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—32no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—61no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—32no—
413C4⋊C4—62no—
414C4⋊C4—64no—
415C4⋊C4—64no—
416C4⋊C4—61no—
417C4⋊C4—62no—
418C4⋊C4—62no—
419C4⋊C4—124no—
420C4⋊C4—122no—
421C4⋊C4—241no—
422C22⋊C4—42no—
423C22⋊C4—68no—
424C22⋊C4—68no—
425C22⋊C4—68no—
426C22⋊C4—124no—
427C22⋊C4—242no—
428C42—64no—
429C42—62no—
430C42—62no—
431C42D3×D412yes—
432C42C2×#6712yes—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—82no—
440C22×C3—82no—
441C22×C3—81no—
442C22×C3—82no—
443C22×C3—81no—
444C22×C3—82no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3C2×#12011yescharacteristic
448C22×C3#54811yescharacteristic
449C22×C3#55011yescharacteristic
450C22×C3C2×#13511yescharacteristic
451C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
452C22×C3#57611yescharacteristic
453C22×C3#57711yescharacteristic
454C3×C4—22no—
455C3×C4—81no—
456C3×C4#44914yes—
457C3×C4#52511yescharacteristic
458C3×C4#53711yescharacteristic
459C3×C4#54611yescharacteristic
460C3×C4#54711yescharacteristic
461Dic3—42no—
462Dic3—42no—
463Dic3—42no—
464Dic3—42no—
465Dic3—82no—
466C23—24no—
467C23—22no—
468C23—41no—
469C23—42no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23C2×D3×D411yescharacteristic
481Q8—62no—
482Q8—62no—
483Q8—124no—
484Q8—121no—
485Q8—121no—
486D4—22no—
487D4—24no—
488D4—24no—
489D4—22no—
490D4—48no—
491D4—41no—
492D4—42no—
493D4—41no—
494D4—42no—
495D4—84no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
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—84no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—248no—
518C2×C4#91112yes—
519C2×C4#91212yes—
520C2×C4#93712yes—
521C2×C4#93712yes—
522C2×C4C2×D3×D411yescharacteristic
523C2×C4C2×#23711yescharacteristic
524C2×C4#100411yescharacteristic
525C2×C4D3×#4111yescharacteristic
526C2×C4#101011yescharacteristic
527C2×C4C22×#6711yescharacteristic
528C8—42no—
529C8—62no—
530C8—121no—
531C2×C3—82no—
532C2×C3—84no—
533C2×C3#240711yescharacteristic
534C2×C3C2×#44911yescharacteristic
535C2×C3#301211yescharacteristic
536C2×C3#336111yescharacteristic
537C2×C3#337311yescharacteristic
538C2×C3#339411yescharacteristic
539C2×C3#339911yescharacteristic
540C22—42no—
541C22—44no—
542C22—44no—
543C22—42no—
544C22—42no—
545C22—44no—
546C22—81no—
547C22—82no—
548C22—82no—
549C22—81no—
550C22—82no—
551C22—81no—
552C22—82no—
553C22—81no—
554C22#559811yescharacteristic
555C22#559911yescharacteristic
556C22C2×#91111yescharacteristic
557C22C2×#91211yescharacteristic
558C22#577411yescharacteristic
559C22#577511yescharacteristic
560C22C2×#93711yescharacteristic
561C4—22no—
562C4—81no—
563C4—122no—
564C4—122no—
565C4—122no—
566C4—122no—
567C4—242no—
568C4#515814yes—
569C4#560611yescharacteristic
570C4#560811yescharacteristic
571C4#580611yescharacteristic
572C4#582411yescharacteristic
573C3#2418611yescharacteristic, complemented, Sylow
574C2—84no—
575C2—82no—
576C2#8031911yescharacteristic
577C2#8032311yescharacteristic
578C2#8032511yescharacteristic
579C2#8032711yescharacteristic
580C2#8270111yescharacteristic
581C2C2×#515811yescharacteristic
582C2#8394911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3#24186
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#5713C2semidirect
3. #82376C2semidirect
4. C2×C4×#214C2semidirect
5. #72045C2semidirect
6. #72037C2semidirect
7. #71764C2semidirect
8. #70485C2semidirect
9. #70472C2semidirect
10. #70216C2semidirect
11. #70169C2semidirect
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×C3×#4592
#823762
C2×C4×#2142
#720452
#720372
#717642
#704852
#704722
#702162
#701692
#241863