Finite Group Databasebeta

← #10847384

Subgroups — #10847384

Order 768.

1474 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic, complemented
2C2×#5115C212yesmaximal, complemented
3#82080C211yesmaximal, characteristic, complemented
4C2×C3×#336C211yesmaximal, characteristic, complemented
5#71973C212yesmaximal, complemented
6#68678C211yesmaximal, characteristic, complemented
7#68674C211yesmaximal, characteristic, complemented
8#68673C212yesmaximal, complemented
9#68670C212yesmaximal, complemented
10#68632C211yesmaximal, characteristic, complemented
11#68629C211yesmaximal, characteristic, complemented
12#25496—31nomaximal, Sylow
13C22×#235—21no—
14C23×C3×C8C2212yes—
15C22×C3×C42C2211yescharacteristic
16C2×#930—21no—
17C2×#930C2211yescharacteristic
18C2×#930C2212yes—
19C2×#929—21no—
20C22×D24C2212yes—
21#5477—28no—
22C2×#875—21no—
23C2×#874C2211yescharacteristic
24C2×C4×D12—21no—
25C2×C4×D12C2212yes—
26C2×#869C2211yescharacteristic
27#5213—21no—
28#5212C2211yescharacteristic
29#5115—28no—
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#821C2212yes—
34C2×#818C2212yes—
35#4940—21no—
36#4940—21no—
37#4939C2211yescharacteristic
38#4939C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#532—31no—
47C2×#466—32no—
48#2854—31no—
49#2057—31no—
50#2041—31no—
51#2023—31no—
52#2011—31no—
53#1893—32no—
54#1879—32no—
55#1862—32no—
56C2×#336D311yescharacteristic, complemented
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C22×#67—21no—
62C2×#235—28no—
63C2×#235—44no—
64C2×#235—44no—
65C22×D12—21no—
66C22×D12—21no—
67C22×D12C2311yescharacteristic
68C22×D12C2312yes—
69C22×C4×D3—22no—
70C22×Dic6—21no—
71C22×C3×C8—24no—
72C22×C3×C8—24no—
73C22×C3×C8—42no—
74C22×C3×C8C2312yes—
75C22×C3×C8C2312yes—
76C2×C3×C42—24no—
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×#217—21no—
82#930—28no—
83#930—24no—
84#930—24no—
85#930D418yes—
86#929—44no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×D24—24no—
102C2×D24—24no—
103C2×D24—42no—
104#875—28no—
105#874—24no—
106C4×D12—28no—
107C4×D12—24no—
108C4×D12—24no—
109#869—24no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#821—24no—
114#821—24no—
115#821—24no—
116#821—42no—
117#818—24no—
118C22×#41—61no—
119C22×D8—32no—
120C23×C8C2×D312yes—
121#532—68no—
122C2×C4⋊Q8—31no—
123C2×#127—31no—
124C2×#124—61no—
125C2×#123—61no—
126C2×#121—32no—
127C2×#121—31no—
128C2×#121—61no—
129C2×C4×D4—32no—
130C2×C4×D4—61no—
131C22×C42C2×D311yescharacteristic
132#466—68no—
133C2×#107—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#102—32no—
138#390—31no—
139#390—31no—
140#389—31no—
141#389—31no—
142#388—61no—
143#388—61no—
144#386—61no—
145#380—31no—
146#340—32no—
147#340—32no—
148#336C2×D311yescharacteristic
149#336C2×D312yes—
150#336C2×D311yescharacteristic
151C24×C3C2×D411yescharacteristic
152C23×D3—22no—
153C23×D3—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—41no—
162C22×C3×C4D814yescomplemented
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D414yes—
169C22×C3×C4C2411yescharacteristic
170C2×#67—28no—
171C2×#67—44no—
172C2×#67—44no—
173C2×#67—42no—
174C2×#67—42no—
175C2×#67—42no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180#235—432no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—24no—
186C2×D12—24no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—42no—
193C2×D12—42no—
194C2×D12—42no—
195C2×D12—42no—
196C2×D12C2×D414yes—
197C2×C4×D3—28no—
198C2×C4×D3—44no—
199C2×C4×D3—44no—
200C2×Dic6—24no—
201C2×Dic6—42no—
202C2×Dic6—42no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—48no—
209C2×C3×C8#4114yescomplemented
210C3×C42—22no—
211C3×C42—22no—
212C3×C42—22no—
213C3×C42—22no—
214#217—42no—
215(C2×D3)⋊C4—28no—
216(C2×D3)⋊C4—28no—
217(C2×D3)⋊C4—42no—
218(C2×D3)⋊C4—42no—
219(C2×D3)⋊C4—42no—
220(C2×D3)⋊C4—42no—
221(C2×D3)⋊C4—42no—
222C4⋊Dic3—24no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3C2×D414yes—
229Dic3⋊C4—42no—
230Dic3⋊C4—42no—
231D24—48no—
232C2×#41—68no—
233C2×#41—124no—
234C2×#41—124no—
235C22×Q8—61no—
236C22×D4—32no—
237C22×D4—31no—
238C22×D4—62no—
239C22×D4—61no—
240C22×D4—61no—
241C22×D4—61no—
242C23×C4—62no—
243C23×C4D1211yescharacteristic
244C23×C4D1211yescharacteristic
245C23×C4C22×D311yescharacteristic
246C2×D8—64no—
247C2×D8—64no—
248C2×D8—122no—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—42no—
252C22×C8C22×D312yes—
253C22×C8C22×D312yes—
254C4⋊Q8—64no—
255#127—64no—
256#124—68no—
257#123—124no—
258#121—38no—
259#121—68no—
260#121—64no—
261#121—64no—
262C4×D4—64no—
263C4×D4—64no—
264C4×D4—68no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C42—24no—
281C2×C42D1211yescharacteristic
282C2×C42D1211yescharacteristic
283C2×C42C22×D311yescharacteristic
284C2×C42C22×D311yescharacteristic
285#107—64no—
286#102—64no—
287#102—64no—
288#102—64no—
289#102—122no—
290#95—61no—
291#95—61no—
292#95—61no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3#12012yes—
298C23×C3C2×D812yes—
299C23×C3AGL(1,ℤ/8)12yes—
300C23×C3C22×D411yescharacteristic
301C22×D3—24no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—41no—
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—41no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—28no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—44no—
330C2×C3×C4#12012yes—
331C2×C3×C4C2×D812yes—
332C2×C3×C4C2×D812yes—
333C2×C3×C4#13512yes—
334C2×C3×C4C22×D412yes—
335C2×C3×C4C2×#4112yes—
336C2×C3×C4H(5,𝔽2)11yescharacteristic
337C2×C3×C42−1+411yescharacteristic
338#67—416no—
339#67—84no—
340C2×Dic3—24no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352D12—24no—
353D12—24no—
354D12—48no—
355D12—48no—
356D12—42no—
357D12—42no—
358D12—42no—
359D12—42no—
360C4×D3—416no—
361Dic6—48no—
362C3×C8—24no—
363C3×C8—42no—
364C24—62no—
365C24—61no—
366C24C2×D1211yescharacteristic
367#41—1232no—
368C2×Q8—64no—
369C2×Q8—122no—
370C2×Q8—122no—
371C2×D4—34no—
372C2×D4—64no—
373C2×D4—64no—
374C2×D4—68no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—62no—
381C2×D4—61no—
382C2×D4—62no—
383C2×D4—61no—
384C2×D4—122no—
385C2×D4—122no—
386C2×D4—124no—
387C2×D4—124no—
388C2×D4—122no—
389C2×D4—122no—
390C2×D4—122no—
391C2×D4—122no—
392C2×D4—122no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—68no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—124no—
407C22×C4—124no—
408C22×C4D2414yescomplemented
409C22×C4C2×D1211yescharacteristic
410C22×C4C2×D1211yescharacteristic
411C22×C4C2×D1211yescharacteristic
412C22×C4C2×D1211yescharacteristic
413C22×C4C2×D1211yescharacteristic
414C22×C4D3×D414yes—
415C22×C4C23×D311yescharacteristic
416D8—128no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—48no—
423C2×C8#23514yescomplemented
424C4⋊C4—34no—
425C4⋊C4—64no—
426C4⋊C4—62no—
427C4⋊C4—62no—
428C4⋊C4—62no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—122no—
432C4⋊C4—122no—
433C22⋊C4—68no—
434C22⋊C4—68no—
435C22⋊C4—122no—
436C22⋊C4—122no—
437C22⋊C4—122no—
438C22⋊C4—122no—
439C22⋊C4—122no—
440C22⋊C4—122no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3#44714yes—
452C22×C3C2×#12011yescharacteristic
453C22×C3#53211yescharacteristic
454C22×C3#53711yescharacteristic
455C22×C3C22×D811yescharacteristic
456C22×C3C2×#13511yescharacteristic
457C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
458C22×C3#57511yescharacteristic
459C2×D3—44no—
460C2×D3—44no—
461C2×D3—44no—
462C2×D3—82no—
463C2×D3—82no—
464C2×D3—82no—
465C2×D3—82no—
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C2×D3—81no—
470C2×D3—81no—
471C2×D3—81no—
472C2×D3—81no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—41no—
478C3×C4#54612yes—
479C3×C4#54712yes—
480Dic3—44no—
481Dic3—81no—
482Dic3—81no—
483Dic3—81no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—64no—
489C23—122no—
490C23—122no—
491C23—122no—
492C23—122no—
493C23—122no—
494C23—122no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23#88212yes—
503C23C2×D2412yes—
504C23#90812yes—
505C23C22×D1211yescharacteristic
506Q8—128no—
507D4—64no—
508D4—64no—
509D4—128no—
510D4—128no—
511D4—1216no—
512D4—122no—
513D4—122no—
514D4—122no—
515D4—122no—
516D4—244no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—28no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—44no—
532C2×C4—64no—
533C2×C4—1216no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4#88212yes—
546C2×C4C2×D2412yes—
547C2×C4C2×D2412yes—
548C2×C4#90412yes—
549C2×C4C2×#23512yes—
550C2×C4C2×D3×D412yes—
551C2×C4#100411yescharacteristic
552C2×C4#100711yescharacteristic
553C8—24no—
554C8—42no—
555C2×C3—24no—
556C2×C3#243011yescharacteristic
557C2×C3C2×#44711yescharacteristic
558C2×C3#300211yescharacteristic
559C2×C3#316911yescharacteristic
560C2×C3#318011yescharacteristic
561C2×C3#324511yescharacteristic
562C2×C3#324711yescharacteristic
563D3—84no—
564D3—161no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—124no—
572C22—124no—
573C22—124no—
574C22—242no—
575C22—242no—
576C22—242no—
577C22—242no—
578C22—241no—
579C22—241no—
580C22—241no—
581C22—241no—
582C22—241no—
583C22—241no—
584C22—241no—
585C22#473214yes—
586C22#547711yescharacteristic
587C22C2×#88211yescharacteristic
588C22#551011yescharacteristic
589C22C22×D2411yescharacteristic
590C22C2×#90411yescharacteristic
591C22C2×#90811yescharacteristic
592C22#574811yescharacteristic
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—124no—
599C4—241no—
600C4—241no—
601C4—241no—
602C4#549012yes—
603C4#550712yes—
604C3#2549611yescharacteristic, complemented, Sylow
605C2—24no—
606C2—244no—
607C2—481no—
608C2#7337611yescharacteristic
609C2#7847011yescharacteristic
610C2#7850011yescharacteristic
611C2C2×#473211yescharacteristic
612C2#7861011yescharacteristic
613C2#7902611yescharacteristic
614C2#7902811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#336C2
socleC23×C3C22×D4
hypercenterC2×#336D3
hypocenterC3#25496
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
1. C2×#5477C2semidirect
2. C2×#5115C2semidirect
3. #82080C2semidirect
4. C2×C3×#336C2semidirect
5. #71973C2semidirect
6. #68678C2semidirect
7. #68674C2semidirect
8. #68673C2semidirect
9. #68670C2semidirect
10. #68632C2semidirect
11. #68629C2semidirect
56. C2×#336D3semidirect
162. C22×C3×C4D8semidirect
209. C2×C3×C8#41semidirect
408. C22×C4D24semidirect
423. C2×C8#235semidirect
604. C3#25496semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
C2×#51152
#820802
C2×C3×#3362
#719732
#686782
#686742
#686732
#686702
#686322
#686292
#254963