Finite Group Databasebeta

← #10829964

Subgroups — #10829964

Order 768.

1394 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5718C211yesmaximal, characteristic
2C2×C3×#460C211yesmaximal, characteristic, complemented
3#82482C211yesmaximal, characteristic, complemented
4C2×C4×#215C211yesmaximal, characteristic, complemented
5#72056C212yesmaximal, complemented
6#72047C212yesmaximal, complemented
7#71764C211yesmaximal, characteristic, complemented
8#70479C211yesmaximal, characteristic, complemented
9#70472C211yesmaximal, characteristic, complemented
10#70211C212yesmaximal, complemented
11#70169C212yesmaximal, complemented
12#24188—31nomaximal, Sylow
13C22×#237—21no—
14C22×C3×SD16C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C22×#215C2211yescharacteristic
18C2×#940C2212yes—
19C2×#939—22no—
20C2×#938—21no—
21C2×D4×Dic3C2212yes—
22#5718—24no—
23#5718—24no—
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#460—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#102C2212yes—
29#5229C2212yes—
30C2×#835C2211yescharacteristic
31C2×#832C2212yes—
32C2×#832C2212yes—
33C2×#821C2212yes—
34C4×#215—24no—
35#4982—22no—
36#4972C2212yes—
37C2×#809C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4935C2211yescharacteristic
40#4929—21no—
41#4928—21no—
42C2×C4×C3⋊C8C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#550—31no—
47C2×#460D311yescharacteristic, complemented
48C2×C4×SD16—31no—
49#2582—31no—
50#2041—32no—
51#2014—32no—
52#1984—31no—
53#1957—31no—
54#1877—32no—
55#1864—32no—
56#1802—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4C2312yes—
61C22×#67—22no—
62C22×#67—22no—
63C23×Dic3—22no—
64C2×#237—28no—
65C2×#237—44no—
66C2×#237—44no—
67C22×D12C2311yescharacteristic
68C22×C4×D3—21no—
69C22×Dic6D411yescharacteristic
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C2×C3×SD16—42no—
73C22×C3×C8C2312yes—
74C3×#127—22no—
75C3×#127D414yes—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C42—21no—
81C2×C3×C42C2311yescharacteristic
82C2×#217—22no—
83C2×#215—22no—
84C2×#215—22no—
85C2×#215—22no—
86#940—24no—
87#940—24no—
88#939—216no—
89#938—28no—
90D4×Dic3—24no—
91D4×Dic3—24no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3D411yescharacteristic
96C2×C4⋊Dic3C2312yes—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4D412yes—
100C2×C4×Dic3C2312yes—
101C22×C3⋊C8C2311yescharacteristic
102C22×C3⋊C8C2311yescharacteristic
103C4×D12—22no—
104C4×D12D414yes—
105C4×Dic6—22no—
106C4×Dic6—22no—
107C3×C4⋊C8—22no—
108C3×#102—24no—
109#835—22no—
110#832—24no—
111#832—42no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114C3⋊#95—22no—
115#821—42no—
116#809—22no—
117#808—22no—
118C4×C3⋊C8—22no—
119C22×#41—61no—
120C22×SD16—31no—
121C22×SD16C2×D312yes—
122#550—64no—
123#550—64no—
124C2×#127—32no—
125C2×#127C2×D311yescharacteristic
126C2×#121—61no—
127C2×#121—62no—
128C2×C4×Q8—31no—
129C2×C4×Q8C2×D311yescharacteristic
130C2×C4×D4—32no—
131C2×C4×D4—31no—
132#460—24no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#103—31no—
137C2×#102—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#102—32no—
141C2×#102C2×D312yes—
142C2×C4×C8—31no—
143#398—62no—
144#391—32no—
145#391—61no—
146#389—32no—
147#389—61no—
148#384—31no—
149#340—32no—
150#336—32no—
151#336—32no—
152C24×C3—22no—
153C23×D3—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8#4112yescomplemented
160C2×C3×D4—24no—
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—42no—
169C2×C3×D4—81no—
170C2×C3×D4—81no—
171C2×C3×D4C2×D414yes—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C2×#67—28no—
178C2×#67—28no—
179C2×#67—44no—
180C2×#67—44no—
181C2×#67—82no—
182C2×#67—82no—
183C22×Dic3—28no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—44no—
187C22×Dic3—44no—
188C22×Dic3C2×D412yes—
189C22×Dic3C2×D412yes—
190#237—432no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12C2×D412yes—
197C2×C4×D3—24no—
198C2×C4×D3—42no—
199C2×C4×D3—42no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6SD1614yescomplemented
203C3×SD16—48no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—42no—
210C3×C42—42no—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213#217—28no—
214#215—28no—
215(C2×D3)⋊C4—24no—
216C4⋊Dic3—22no—
217C4⋊Dic3—21no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—42no—
221C4⋊Dic3—81no—
222C4⋊Dic3C2×D412yes—
223Dic3⋊C4—24no—
224Dic3⋊C4—82no—
225C4×Dic3—22no—
226C4×Dic3—22no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8#4112yescomplemented
233C2×#41—68no—
234C2×#41—124no—
235C2×#41—124no—
236C22×Q8—31no—
237C22×Q8C22×D311yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—31no—
241C22×D4—62no—
242C22×D4—62no—
243C22×D4C22×D312yes—
244C23×C4—62no—
245C23×C4—61no—
246C2×SD16—24no—
247C2×SD16—24no—
248C2×SD16—42no—
249C2×SD16—62no—
250C2×SD16—62no—
251C2×SD16—62no—
252C22×C8—31no—
253C22×C8—31no—
254C22×C8C22×D312yes—
255#127—22no—
256#127—64no—
257#127—64no—
258#127#6714yes—
259#121—68no—
260#121—616no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—62no—
264C4×Q8—62no—
265C4×D4—34no—
266C4×D4—64no—
267C4×D4—64no—
268C4×D4—62no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C22⋊C4—62no—
279C2×C22⋊C4—61no—
280C2×C42—21no—
281C2×C42—32no—
282C2×C42C22×D311yescharacteristic
283#106—62no—
284C4⋊C8—22no—
285#103—62no—
286#102—24no—
287#102—62no—
288#102—64no—
289#102—122no—
290#102—122no—
291C4×C8—62no—
292#95—62no—
293#95—62no—
294#95—62no—
295C23×C3—24no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3C22×D411yescharacteristic
303C22×D3—22no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—41no—
309C22×D3—41no—
310C3×Q8—22no—
311C3×Q8—22no—
312C3×Q8—41no—
313C3×Q8—41no—
314C3×D4—24no—
315C3×D4—24no—
316C3×D4—42no—
317C3×D4—42no—
318C3×D4—48no—
319C3×D4—84no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4#12012yes—
329C2×C3×C4#12012yes—
330C2×C3×C4C2×SD1612yes—
331C2×C3×C4AGL(1,ℤ/8)12yes—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4H(5,𝔽2)11yescharacteristic
337C2×C3×C42−1+411yescharacteristic
338#67—416no—
339#67—88no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—416no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3C2×SD1614yes—
353D12—22no—
354D12—22no—
355D12—41no—
356D12—41no—
357C4×D3—48no—
358Dic6—28no—
359C3×C8—42no—
360C3⋊C8—22no—
361C3⋊C8—41no—
362C24—22no—
363C24—61no—
364#41—1232no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—34no—
371C2×Q8—62no—
372C2×Q8—62no—
373C2×Q8#23712yescomplemented
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—32no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—68no—
385C2×D4—68no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—81no—
392C2×D4—81no—
393C2×D4—124no—
394C2×D4—124no—
395C2×D4—242no—
396C2×D4—242no—
397C2×D4C2×#6714yes—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—32no—
401C22×C4—32no—
402C22×C4—68no—
403C22×C4—61no—
404C22×C4—64no—
405C22×C4—62no—
406C22×C4—124no—
407C22×C4—124no—
408C22×C4—122no—
409C22×C4—122no—
410C22×C4D3×D411yescharacteristic
411C22×C4D3×D411yescharacteristic
412C22×C4C23×D311yescharacteristic
413SD16—48no—
414SD16—68no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—32no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—32no—
427C4⋊C4—42no—
428C4⋊C4—64no—
429C4⋊C4—62no—
430C4⋊C4—61no—
431C4⋊C4—62no—
432C4⋊C4—62no—
433C4⋊C4—122no—
434C4⋊C4—241no—
435C4⋊C4—242no—
436C22⋊C4—68no—
437C22⋊C4—64no—
438C42—42no—
439C42—62no—
440C42—62no—
441C42D3×D412yes—
442C42C2×#6712yes—
443C22×C3—44no—
444C22×C3—44no—
445C22×C3—44no—
446C22×C3—82no—
447C22×C3—82no—
448C22×C3—82no—
449C22×C3—82no—
450C22×C3C2×#12011yescharacteristic
451C22×C3#54811yescharacteristic
452C22×C3#55011yescharacteristic
453C22×C3C22×SD1611yescharacteristic
454C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
455C22×C3#57611yescharacteristic
456C22×C3#57711yescharacteristic
457C2×D3—42no—
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—81no—
461C2×D3—81no—
462C2×D3—81no—
463C2×D3—81no—
464C3×C4—22no—
465C3×C4—42no—
466C3×C4—81no—
467C3×C4#45014yes—
468C3×C4#52511yescharacteristic
469C3×C4#53711yescharacteristic
470C3×C4#54611yescharacteristic
471C3×C4#54711yescharacteristic
472Dic3—24no—
473Dic3—42no—
474Dic3—42no—
475Dic3—82no—
476C23—24no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—62no—
484C23—121no—
485C23—121no—
486C23—121no—
487C23—121no—
488C23—121no—
489C23—121no—
490C23C2×D3×D411yescharacteristic
491Q8—22no—
492Q8—22no—
493Q8—41no—
494Q8—41no—
495Q8—68no—
496D4—24no—
497D4—24no—
498D4—48no—
499D4—42no—
500D4—42no—
501D4—62no—
502D4—62no—
503D4—84no—
504D4—1216no—
505D4—121no—
506D4—121no—
507D4—248no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—34no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—1216no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—128no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4D3×SD1612yes—
531C2×C4#91412yes—
532C2×C4#93712yes—
533C2×C4#93712yes—
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×#23711yescharacteristic
536C2×C4#100411yescharacteristic
537C2×C4D3×#4111yescharacteristic
538C2×C4#101011yescharacteristic
539C2×C4C22×#6711yescharacteristic
540C8—42no—
541C8—62no—
542C8—121no—
543C2×C3—84no—
544C2×C3#240711yescharacteristic
545C2×C3C2×#45011yescharacteristic
546C2×C3#301211yescharacteristic
547C2×C3#336211yescharacteristic
548C2×C3#336811yescharacteristic
549C2×C3#339211yescharacteristic
550C2×C3#339911yescharacteristic
551D3—82no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—122no—
560C22—122no—
561C22—122no—
562C22—241no—
563C22—241no—
564C22—241no—
565C22—241no—
566C22#559811yescharacteristic
567C22#559911yescharacteristic
568C22C2×D3×SD1611yescharacteristic
569C22C2×#91411yescharacteristic
570C22#577411yescharacteristic
571C22#577511yescharacteristic
572C22C2×#93711yescharacteristic
573C4—22no—
574C4—42no—
575C4—64no—
576C4—81no—
577C4—122no—
578C4—122no—
579C4—242no—
580C4#516914yes—
581C4#560611yescharacteristic
582C4#560811yescharacteristic
583C4#580611yescharacteristic
584C4#582411yescharacteristic
585C3#2418811yescharacteristic, complemented, Sylow
586C2—84no—
587C2—242no—
588C2#8036911yescharacteristic
589C2#8037111yescharacteristic
590C2#8037811yescharacteristic
591C2#8037911yescharacteristic
592C2#8270111yescharacteristic
593C2C2×#516911yescharacteristic
594C2#8394911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#460C2
socleC23×C3C22×D4
prime radicalC2×#5718C2
hypercenterC2×#460D3
hypocenterC3#24188
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. C2×C3×#460C2semidirect
3. #82482C2semidirect
4. C2×C4×#215C2semidirect
5. #72056C2semidirect
6. #72047C2semidirect
7. #71764C2semidirect
8. #70479C2semidirect
9. #70472C2semidirect
10. #70211C2semidirect
11. #70169C2semidirect
47. C2×#460D3semidirect
159. C2×C3×Q8#41semidirect
202. C2×Dic6SD16semidirect
232. C2×C3⋊C8#41semidirect
373. C2×Q8#237semidirect
585. C3#24188semidirect

Maximal subgroups

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

subgroupindex
C2×#57182
C2×C3×#4602
#824822
C2×C4×#2152
#720562
#720472
#717642
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