Finite Group Databasebeta

← #10848866

Subgroups — #10848866

Order 768.

1216 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×#5534C211yesmaximal, characteristic, complemented
2C3×#2619C211yesmaximal, characteristic, complemented
3C2×#5074C211yesmaximal, characteristic, complemented
4C2×#5070C211yesmaximal, characteristic, complemented
5#71685C212yesmaximal, complemented
6#71684C212yesmaximal, complemented
7#71582C211yesmaximal, characteristic, complemented
8#69660C212yesmaximal, complemented
9#69657C212yesmaximal, complemented
10#69555C211yesmaximal, characteristic, complemented
11#69551C211yesmaximal, characteristic, complemented
12#25614—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×#121C2212yes—
15C2×C3×#117C2211yescharacteristic
16C22×#214C2212yes—
17C22×#213C2212yes—
18C2×#930C2211yescharacteristic
19C2×#924C2211yescharacteristic
20#5534—24no—
21#5534—24no—
22C2×#898—21no—
23C2×#896—21no—
24C2×#885—21no—
25C2×#883—21no—
26C2×#875C2211yescharacteristic
27C2×C4×D12C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C3×#394—21no—
31C3×#394—21no—
32C3×#392—21no—
33C3×#392—21no—
34C3×#390C2212yes—
35C3×#382C2211yescharacteristic
36C2×#832C2212yes—
37C2×#832C2212yes—
38C2×#830C2212yes—
39#5074—24no—
40#5070—24no—
41C2×#810C2212yes—
42C2×#809C2212yes—
43C2×#803C2211yescharacteristic
44C2×C4×C3⋊C8C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#542—31no—
49C2×#496—31no—
50C2×#486—31no—
51#2619D311yescharacteristic, complemented
52#2000—32no—
53#1999—32no—
54#1993—31no—
55#1958—31no—
56#1880—32no—
57#1879—32no—
58#1750—31no—
59C22×C3×D4—22no—
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4C2312yes—
63C23×C3×C4C2311yescharacteristic
64C22×#67—21no—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×Dic6—21no—
68C22×Dic6C2311yescharacteristic
69C3×#127—22no—
70C3×#127—22no—
71C3×#121—42no—
72C3×#121—42no—
73C3×#117—22no—
74C3×#117—22no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4D411yescharacteristic
82C2×C3×C22⋊C4D411yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86C2×#214—24no—
87C2×#214—24no—
88C2×#214—42no—
89C2×#213—24no—
90C2×#213—24no—
91C2×#213—42no—
92#930—22no—
93#930—22no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101#924—22no—
102#924—22no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C22×C3⋊C8C2312yes—
107C22×C3⋊C8C2312yes—
108#898—28no—
109#896—28no—
110#885—28no—
111#883—28no—
112#875—22no—
113#875D414yescomplemented
114C4×D12—22no—
115C4×D12—22no—
116#869—22no—
117#869D414yescomplemented
118C4×Dic6—22no—
119C4×Dic6—22no—
120C3×#95—21no—
121C3×#95—21no—
122#832—24no—
123#832—24no—
124#830—24no—
125#810—42no—
126#809—42no—
127#803—22no—
128C4×C3⋊C8—22no—
129C22×SD16—32no—
130C22×D8—32no—
131#542—64no—
132#542—64no—
133C2×C4⋊Q8—31no—
134C2×#127C2×D311yescharacteristic
135C2×#126—61no—
136C2×#124—31no—
137C2×#124—61no—
138C2×#123—61no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—61no—
141C2×#121—31no—
142C2×#121C2×D312yes—
143C2×C4×Q8—31no—
144C2×C4×D4—31no—
145C2×#117C2×D311yescharacteristic
146#496—64no—
147#486—64no—
148C2×#103—32no—
149C2×#102—32no—
150C2×#102—32no—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×#97—31no—
154C2×C4×C8—31no—
155#394—21no—
156#394—21no—
157#392—21no—
158#392—21no—
159#390C2×D312yes—
160#382C2×D311yescharacteristic
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3—22no—
165C24×C3C2×D411yescharacteristic
166C23×D3—21no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—44no—
174C2×C3×D4—42no—
175C2×C3×D4—42no—
176C2×C3×D4—82no—
177C2×C3×D4—82no—
178C2×C3×D4—82no—
179C2×C3×D4—82no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C2×#67—24no—
192C2×#67—82no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12—21no—
199C2×D12C2×D412yes—
200C2×C4×D3—24no—
201C2×C4×D3—82no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6—21no—
205C2×Dic6—24no—
206C2×Dic6—82no—
207C2×Dic6C2×D412yes—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—44no—
215C3×C22⋊C4—82no—
216C3×C22⋊C4—82no—
217C3×C42—21no—
218C3×C42—21no—
219C3×C42C2×D412yes—
220C3×C42C2×D412yes—
221#217—24no—
222#214—48no—
223#213—48no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—24no—
228C4⋊Dic3—24no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3C2×D412yes—
232C4⋊Dic3C2×D412yes—
233Dic3⋊C4—24no—
234Dic3⋊C4—24no—
235C4×Dic3—24no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8#4114yes—
242C22×Q8—31no—
243C22×Q8—61no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—31no—
248C22×D4—61no—
249C22×D4C22×D312yes—
250C23×C4—61no—
251C23×C4C22×D311yescharacteristic
252C2×SD16—64no—
253C2×SD16—64no—
254C2×SD16—122no—
255C2×D8—64no—
256C2×D8—64no—
257C2×D8—122no—
258C22×C8—32no—
259C22×C8—32no—
260C4⋊Q8—34no—
261C4⋊Q8—62no—
262#127—22no—
263#127—22no—
264#126—68no—
265#124—34no—
266#124—68no—
267#124—62no—
268#123—68no—
269C22⋊Q8—68no—
270C22⋊Q8—62no—
271C22⋊Q8—62no—
272#121—42no—
273#121—42no—
274#121—62no—
275#121—62no—
276C4×Q8—62no—
277C4×Q8—62no—
278C4×D4—62no—
279C4×D4—62no—
280#117—22no—
281#117—22no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4#6711yescharacteristic
288C2×C4⋊C4#6711yescharacteristic
289C2×C4⋊C4C22×D312yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4#6711yescharacteristic
299C2×C22⋊C4#6711yescharacteristic
300C2×C42—61no—
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#103—122no—
304#102—64no—
305#102—64no—
306#102—122no—
307C22⋊C8—64no—
308#97—62no—
309C4×C8—62no—
310#95—21no—
311#95—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3C22×D411yescharacteristic
325C22×D3—22no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—42no—
329C22×D3—42no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—84no—
335C3×D4—84no—
336C3×D4—88no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4#12012yes—
352C2×C3×C4#12012yes—
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C2×#4112yes—
356C2×C3×C4H(5,𝔽2)12yes—
357#67—84no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365D12—24no—
366D12—42no—
367C4×D3—84no—
368Dic6—24no—
369Dic6—42no—
370Dic6—84no—
371C3⋊C8—24no—
372C3⋊C8—42no—
373C24—22no—
374C24—61no—
375C24C2×#6711yescharacteristic
376C2×Q8—32no—
377C2×Q8—64no—
378C2×Q8—61no—
379C2×Q8—62no—
380C2×Q8—62no—
381C2×Q8—242no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—32no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—64no—
393C2×D4—61no—
394C2×D4—62no—
395C2×D4—62no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4—82no—
399C2×D4—82no—
400C2×D4—242no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—64no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—242no—
411C22×C4C2×#6711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413C22×C4C2×#6711yescharacteristic
414C22×C4C2×#6711yescharacteristic
415C22×C4C2×#6711yescharacteristic
416C22×C4C23×D311yescharacteristic
417SD16—128no—
418D8—128no—
419C2×C8—34no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—62no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—32no—
430C4⋊C4—32no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4—64no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—44no—
439C22⋊C4—64no—
440C22⋊C4—64no—
441C22⋊C4—64no—
442C22⋊C4—64no—
443C22⋊C4—64no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C42—21no—
447C42—21no—
448C42—64no—
449C42D3×D412yes—
450C42D3×D412yes—
451C22×C3—42no—
452C22×C3—41no—
453C22×C3—42no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—82no—
457C22×C3—82no—
458C22×C3—82no—
459C22×C3—82no—
460C22×C3—82no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3C2×#12011yescharacteristic
464C22×C3#53211yescharacteristic
465C22×C3#53511yescharacteristic
466C22×C3#57612yes—
467C22×C3#57712yes—
468C2×D3—42no—
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—82no—
479C3×C4#54612yes—
480C3×C4#55012yes—
481Dic3—42no—
482Dic3—42no—
483Dic3—42no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—62no—
497C23—121no—
498C23—121no—
499C23—122no—
500C23—122no—
501C23C22×#6711yescharacteristic
502Q8—64no—
503Q8—122no—
504Q8—244no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—64no—
510D4—84no—
511D4—84no—
512D4—88no—
513D4—122no—
514D4—244no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—244no—
537C2×C4#93712yes—
538C2×C4#93712yes—
539C2×C4C2×D3×D411yescharacteristic
540C2×C4C2×D3×D411yescharacteristic
541C2×C4#100412yes—
542C2×C4D3×#4112yes—
543C8—64no—
544C8—122no—
545C2×C3—81no—
546C2×C3—162no—
547C2×C3#243111yescharacteristic
548C2×C3#300811yescharacteristic
549C2×C3#301011yescharacteristic
550C2×C3#317912yes—
551C2×C3#322412yes—
552D3—82no—
553C22—41no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—42no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—122no—
566C22—122no—
567C22—122no—
568C22—242no—
569C22—242no—
570C22C2×#93711yescharacteristic
571C22#580511yescharacteristic
572C22#583211yescharacteristic
573C22#583512yes—
574C22#583612yes—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—82no—
581C4—122no—
582C4—122no—
583C4—122no—
584C4#559912yes—
585C4#561212yes—
586C3#2561411yescharacteristic, complemented, Sylow
587C2—81no—
588C2—162no—
589C2—242no—
590C2#8265211yescharacteristic
591C2#8338311yescharacteristic
592C2#8338912yes—
593C2#8339011yescharacteristic
594C2#8343312yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2619C2
socleC23×C3C22×D4
hypercenter#2619D3
hypocenterC3#25614
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. C2×#5534C2semidirect
2. C3×#2619C2semidirect
3. C2×#5074C2semidirect
4. C2×#5070C2semidirect
5. #71685C2semidirect
6. #71684C2semidirect
7. #71582C2semidirect
8. #69660C2semidirect
9. #69657C2semidirect
10. #69555C2semidirect
11. #69551C2semidirect
51. #2619D3semidirect
113. #875D4semidirect
117. #869D4semidirect
586. C3#25614semidirect

Maximal subgroups

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

subgroupindex
C2×#55342
C3×#26192
C2×#50742
C2×#50702
#716852
#716842
#715822
#696602
#696572
#695552
#695512
#256143