Finite Group Databasebeta

← #10848867

Subgroups — #10848867

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×#5733C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic, complemented
3C2×#4712C211yesmaximal, characteristic, complemented
4#74244C211yesmaximal, characteristic, complemented
5C3×#1993C211yesmaximal, characteristic, complemented
6#69830C211yesmaximal, characteristic, complemented
7#69551C211yesmaximal, characteristic, complemented
8#68909C212yesmaximal, complemented
9#68897C212yesmaximal, complemented
10#68887C212yesmaximal, complemented
11#68886C212yesmaximal, complemented
12#25614—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#947—21no—
15C2×#946—21no—
16C22×D24C2212yes—
17C22×#204C2212yes—
18#5733—24no—
19#5733—24no—
20C2×#898—21no—
21C2×#897C2211yescharacteristic
22C2×#895C2211yescharacteristic
23C2×#895C2212yes—
24C2×#894—21no—
25C2×#893C2211yescharacteristic
26C2×#887C2211yescharacteristic
27C2×#875C2211yescharacteristic
28C2×#874C2211yescharacteristic
29C2×#872C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C3×#382C2211yescharacteristic
32C2×(C2×D3)⋊C8C2212yes—
33#5088—24no—
34#4988C2212yes—
35C2×#810C2212yes—
36C2×#809C2212yes—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×C4×C3⋊C8C2211yescharacteristic
40#4712—24no—
41#4672—21no—
42#4672—21no—
43#4671—21no—
44#4671—21no—
45C3×#340C2212yes—
46C3×#340C2212yes—
47#4550C2212yes—
48C2×#542—31no—
49C2×#496—31no—
50C2×#486—31no—
51#2619—31no—
52#2000—32no—
53#1999—32no—
54#1993D311yescharacteristic, complemented
55#1958—31no—
56#1880—32no—
57#1879—32no—
58#1750—31no—
59C22×C3×Q8—21no—
60C22×D12—21no—
61C22×D12—22no—
62C22×D12—21no—
63C22×D12—21no—
64C22×D12C2312yes—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×C4×D3C2311yescharacteristic
68C22×Dic6C2311yescharacteristic
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8D414yescomplemented
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77#947—28no—
78#946—28no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4D411yescharacteristic
88C2×(C2×D3)⋊C4D411yescharacteristic
89C2×C4⋊Dic3—21no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4D411yescharacteristic
92C2×Dic3⋊C4D411yescharacteristic
93C2×C4×Dic3—21no—
94C2×C4×Dic3C2311yescharacteristic
95C22×C3⋊C8C2312yes—
96C2×D24—24no—
97C2×D24—24no—
98C2×D24—42no—
99C2×#204—24no—
100C2×#204—24no—
101C2×#204—42no—
102#898—28no—
103#897—22no—
104#897—22no—
105#895—22no—
106#895—22no—
107#895—42no—
108#895—42no—
109#894—28no—
110#893—22no—
111#893—22no—
112#887—22no—
113#887—22no—
114#875—22no—
115#875D414yescomplemented
116#874—22no—
117#874—22no—
118#872—22no—
119#872—22no—
120C3×#97—22no—
121C3×#95—21no—
122C3×#95—21no—
123(C2×D3)⋊C8—24no—
124#810—42no—
125#809—24no—
126#809—24no—
127#809—42no—
128C4×C3⋊C8—22no—
129C22×SD16—32no—
130C22×D8—32no—
131#542—64no—
132#542—64no—
133C2×C4⋊Q8C2×D311yescharacteristic
134C2×#127—31no—
135C2×#126—61no—
136C2×#124—31no—
137C2×#124—61no—
138C2×#123—61no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—61no—
141C2×#121—31no—
142C2×#121—32no—
143C2×C4×Q8—31no—
144C2×C4×D4—31no—
145C2×#117—31no—
146#496—64no—
147#486—64no—
148C2×#103—32no—
149C2×#102—32no—
150C2×#102—32no—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×#97C2×D311yescharacteristic
154C2×C4×C8—31no—
155#394—61no—
156#394—61no—
157#392—61no—
158#392—61no—
159#390—32no—
160#382C2×D311yescharacteristic
161#340C2×D312yes—
162#340C2×D312yes—
163#336—32no—
164C23×D3—21no—
165C23×D3—22no—
166C23×D3C2×D411yescharacteristic
167C2×C3×Q8—24no—
168C2×C3×Q8—82no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—21no—
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178C22×Dic3C2×D411yescharacteristic
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—21no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—24no—
189C2×D12—44no—
190C2×D12—42no—
191C2×D12—42no—
192C2×D12—82no—
193C2×D12—82no—
194C2×D12—82no—
195C2×D12—82no—
196C2×D12—82no—
197C2×D12C2×D412yes—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—24no—
203C2×C4×D3—82no—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6—21no—
207C2×Dic6C2×D412yes—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8#4114yes—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4C2×D412yes—
217C3×C4⋊C4C2×D412yes—
218C3×C42C2×D412yes—
219C3×C42C2×D412yes—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—22no—
224(C2×D3)⋊C4—22no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—44no—
228(C2×D3)⋊C4—82no—
229(C2×D3)⋊C4—82no—
230C4⋊Dic3—24no—
231Dic3⋊C4—24no—
232Dic3⋊C4—22no—
233Dic3⋊C4—22no—
234C4×Dic3—24no—
235C4×Dic3—21no—
236C4×Dic3—21no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240D24—48no—
241#204—48no—
242C22×Q8—21no—
243C22×Q8—31no—
244C22×D4—31no—
245C22×D4—32no—
246C22×D4—61no—
247C22×D4—62no—
248C22×D4—61no—
249C22×D4—61no—
250C23×C4—31no—
251C23×C4—61no—
252C2×SD16—64no—
253C2×SD16—64no—
254C2×SD16—122no—
255C2×D8—64no—
256C2×D8—64no—
257C2×D8—122no—
258C22×C8—32no—
259C22×C8C22×D312yes—
260C4⋊Q8—22no—
261C4⋊Q8D1214yescomplemented
262#127—62no—
263#127—62no—
264#126—68no—
265#124—34no—
266#124—68no—
267#124—62no—
268#123—68no—
269C22⋊Q8—62no—
270C22⋊Q8—68no—
271C22⋊Q8—62no—
272#121—62no—
273#121—62no—
274#121—122no—
275#121—122no—
276C4×Q8—62no—
277C4×Q8—62no—
278C4×D4—62no—
279C4×D4—62no—
280#117—62no—
281#117—62no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4C22×D312yes—
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—31no—
291C2×C22⋊C4—31no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300C2×C42—31no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#103—122no—
304#102—64no—
305#102—64no—
306#102—122no—
307C22⋊C8—64no—
308#97—22no—
309C4×C8—62no—
310#95—21no—
311#95—21no—
312C23×C3C22×D411yescharacteristic
313C22×D3—21no—
314C22×D3—22no—
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—42no—
330C3×Q8—84no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4#12012yes—
341C2×C3×C4#12012yes—
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C2×#4112yes—
345C2×C3×C4H(5,𝔽2)12yes—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—22no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356D12—24no—
357D12—42no—
358D12—42no—
359D12—42no—
360D12—42no—
361D12—42no—
362D12—84no—
363D12—84no—
364D12—84no—
365D12—88no—
366C4×D3—42no—
367C4×D3—42no—
368C4×D3—84no—
369Dic6—24no—
370Dic6—42no—
371C3×C8—24no—
372C3⋊C8—42no—
373C24—31no—
374C24—61no—
375C24—62no—
376C2×Q8—24no—
377C2×Q8—32no—
378C2×Q8—61no—
379C2×Q8—62no—
380C2×Q8—62no—
381C2×Q8—82no—
382C2×D4—32no—
383C2×D4—64no—
384C2×D4—61no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—62no—
392C2×D4—62no—
393C2×D4—124no—
394C2×D4—122no—
395C2×D4—122no—
396C2×D4—242no—
397C2×D4—242no—
398C2×D4—242no—
399C2×D4—242no—
400C2×D4—242no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—64no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—242no—
414C22×C4D3×D411yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4C23×D311yescharacteristic
417SD16—128no—
418D8—128no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8#23914yes—
425C4⋊C4—24no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—64no—
431C4⋊C4—64no—
432C4⋊C4—62no—
433C4⋊C4—62no—
434C4⋊C4C2×D1212yes—
435C4⋊C4C2×D1212yes—
436C22⋊C4—64no—
437C22⋊C4—64no—
438C22⋊C4—64no—
439C22⋊C4—62no—
440C22⋊C4—62no—
441C22⋊C4—64no—
442C22⋊C4—64no—
443C22⋊C4—124no—
444C22⋊C4—242no—
445C22⋊C4—242no—
446C42—64no—
447C42—61no—
448C42—61no—
449C42C2×D1212yes—
450C42D3×D412yes—
451C22×C3C2×#12011yescharacteristic
452C22×C3#53211yescharacteristic
453C22×C3#53511yescharacteristic
454C22×C3#57612yes—
455C22×C3#57712yes—
456C2×D3—42no—
457C2×D3—41no—
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—41no—
462C2×D3—41no—
463C2×D3—42no—
464C2×D3—82no—
465C2×D3—82no—
466C2×D3—82no—
467C2×D3—82no—
468C2×D3—82no—
469C2×D3—82no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—82no—
478C3×C4#54612yes—
479C3×C4#55012yes—
480Dic3—42no—
481Dic3—41no—
482Dic3—41no—
483Dic3—41no—
484C23—61no—
485C23—61no—
486C23—62no—
487C23—61no—
488C23—62no—
489C23—62no—
490C23—121no—
491C23—121no—
492C23—122no—
493C23—122no—
494C23—122no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—122no—
499C23—122no—
500C23—122no—
501C23C2×D3×D411yescharacteristic
502Q8—64no—
503Q8—84no—
504Q8—122no—
505D4—64no—
506D4—122no—
507D4—122no—
508D4—122no—
509D4—122no—
510D4—122no—
511D4—244no—
512D4—244no—
513D4—244no—
514D4—248no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—244no—
537C2×C4#88212yes—
538C2×C4#88212yes—
539C2×C4C22×D1211yescharacteristic
540C2×C4C2×D3×D411yescharacteristic
541C2×C4C2×#23912yes—
542C2×C4#100912yes—
543C8—24no—
544C8—122no—
545C2×C3#243111yescharacteristic
546C2×C3#300811yescharacteristic
547C2×C3#301011yescharacteristic
548C2×C3#317912yes—
549C2×C3#322412yes—
550D3—81no—
551D3—82no—
552D3—162no—
553C22—122no—
554C22—122no—
555C22—122no—
556C22—121no—
557C22—121no—
558C22—122no—
559C22—121no—
560C22—122no—
561C22—242no—
562C22—242no—
563C22—242no—
564C22—242no—
565C22—242no—
566C22—242no—
567C22—242no—
568C22—242no—
569C22—242no—
570C22C2×#88211yescharacteristic
571C22#567411yescharacteristic
572C22#572511yescharacteristic
573C22#577412yes—
574C22#577812yes—
575C4—22no—
576C4—22no—
577C4—42no—
578C4—42no—
579C4—82no—
580C4—121no—
581C4—121no—
582C4—122no—
583C4—121no—
584C4#557712yes—
585C4#563012yes—
586C3#2561411yescharacteristic, complemented, Sylow
587C2—241no—
588C2—242no—
589C2—482no—
590C2#7662611yescharacteristic
591C2#7892811yescharacteristic
592C2#7896011yescharacteristic
593C2#8168812yes—
594C2#8196112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
hypercenter#1993D3
hypocenterC3#25614
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
1. C2×#5733C2semidirect
2. C2×#5088C2semidirect
3. C2×#4712C2semidirect
4. #74244C2semidirect
5. C3×#1993C2semidirect
6. #69830C2semidirect
7. #69551C2semidirect
8. #68909C2semidirect
9. #68897C2semidirect
10. #68887C2semidirect
11. #68886C2semidirect
54. #1993D3semidirect
71. C3×C4⋊Q8D4semidirect
115. #875D4semidirect
261. C4⋊Q8D12semidirect
586. C3#25614semidirect

Maximal subgroups

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

subgroupindex
C2×#57332
C2×#50882
C2×#47122
#742442
C3×#19932
#698302
#695512
#689092
#688972
#688872
#688862
#256143