Finite Group Databasebeta

← #10900843

Subgroups — #10900843

Order 768.

1378 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic
2#82075C211yesmaximal, characteristic, complemented
3C2×#4733C212yesmaximal, complemented
4C3×#1817C211yesmaximal, characteristic, complemented
5#69960C212yesmaximal, complemented
6#68688C211yesmaximal, characteristic, complemented
7#68677C212yesmaximal, complemented
8#68676C211yesmaximal, characteristic, complemented
9#68631C211yesmaximal, characteristic, complemented
10#68630C212yesmaximal, complemented
11#68629C211yesmaximal, characteristic, complemented
12#29629—31nomaximal, Sylow
13C22×#235C2212yes—
14C22×C3×C42C2211yescharacteristic
15C2×#930—21no—
16C2×#930—21no—
17C2×#924C2211yescharacteristic
18C22×Dic12C2212yes—
19C22×#204C2212yes—
20#5477—28no—
21C2×#875C2212yes—
22C2×#874C2211yescharacteristic
23C2×C4×D12—21no—
24C2×C4×D12—21no—
25C2×#869C2211yescharacteristic
26C2×C3×C22⋊C8C2212yes—
27#5212C2211yescharacteristic
28#5211—21no—
29#5210—21no—
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35C2×#816C2212yes—
36#4933—21no—
37#4933—21no—
38#4929C2211yescharacteristic
39#4929C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42#4733—28no—
43C3×#336C2211yescharacteristic
44C3×#336C2212yes—
45C3×#336C2211yescharacteristic
46C2×#532—31no—
47C2×#452—32no—
48#2855—31no—
49#2043—31no—
50#2042—32no—
51#2041—31no—
52#2016—31no—
53#2015—32no—
54#2014—31no—
55#1817D311yescharacteristic, complemented
56#1788—32no—
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C2×#235—24no—
62C2×#235—42no—
63C2×#235—42no—
64C2×#235—42no—
65C2×#235—42no—
66C2×#235D418yes—
67C22×D12—21no—
68C22×D12—21no—
69C22×D12C2312yes—
70C22×C4×D3—22no—
71C22×Dic6C2311yescharacteristic
72C22×Dic6C2312yes—
73C22×C3×C8C2312yes—
74C22×C3×C8C2312yes—
75C2×C3×C42—24no—
76C2×C3×C42D411yescharacteristic
77C2×C3×C42D411yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×#217—21no—
81C2×#217—21no—
82#930—28no—
83#930—28no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2311yescharacteristic
91#924—24no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C2×Dic12—24no—
96C2×Dic12—42no—
97C2×Dic12—42no—
98C2×#204—24no—
99C2×#204—42no—
100C2×#204—42no—
101#875—24no—
102#875—24no—
103#874—24no—
104C4×D12—28no—
105C4×D12—28no—
106#869—24no—
107C3×C22⋊C8—24no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—21no—
114C3⋊#95—21no—
115#821—24no—
116#821—42no—
117#821—42no—
118#816—24no—
119#816—42no—
120#816—42no—
121C22×#41—32no—
122C22×Q16—32no—
123C22×SD16—32no—
124#532—68no—
125C2×C4⋊Q8—31no—
126C2×#127—31no—
127C2×#124—32no—
128C2×C22⋊Q8—31no—
129C2×#121—61no—
130C2×#121—61no—
131C2×C4×D4—61no—
132C2×C4×D4—61no—
133C22×C42C2×D311yescharacteristic
134#452—68no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#103—32no—
138C2×#102—32no—
139C2×#102—32no—
140C2×#102—32no—
141C2×C22⋊C8C2×D312yes—
142#391—31no—
143#391—31no—
144#389—31no—
145#389—31no—
146#387—61no—
147#387—61no—
148#385—61no—
149#381—61no—
150#380—31no—
151#336C2×D311yescharacteristic
152#336C2×D312yes—
153#336C2×D311yescharacteristic
154C24×C3C2×D411yescharacteristic
155C23×D3—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—41no—
164C22×C3×C4—41no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2411yescharacteristic
172C2×#67—28no—
173C2×#67—44no—
174C2×#67—44no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180#235—48no—
181#235—48no—
182#235—48no—
183#235—48no—
184C2×D12—22no—
185C2×D12—22no—
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—24no—
189C2×D12—24no—
190C2×D12—22no—
191C2×D12—42no—
192C2×D12—42no—
193C2×D12—42no—
194C2×D12—42no—
195C2×D12C2×D414yes—
196C2×C4×D3—28no—
197C2×C4×D3—44no—
198C2×C4×D3—44no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6—22no—
205C2×Dic6—21no—
206C2×Dic6—22no—
207C2×Dic6—21no—
208C2×Dic6C2×D414yes—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C3×C42—22no—
216C3×C42—22no—
217C3×C42—22no—
218C3×C42—22no—
219#217—42no—
220#217—42no—
221(C2×D3)⋊C4—28no—
222(C2×D3)⋊C4—28no—
223C4⋊Dic3—24no—
224C4⋊Dic3—24no—
225C4⋊Dic3—22no—
226C4⋊Dic3—21no—
227C4⋊Dic3—22no—
228C4⋊Dic3—21no—
229Dic3⋊C4—42no—
230Dic3⋊C4—42no—
231Dic3⋊C4—42no—
232Dic12—48no—
233#204—48no—
234C2×#41—38no—
235C2×#41—64no—
236C2×#41—122no—
237C2×#41—122no—
238C2×#41—122no—
239C2×#41—122no—
240C22×Q8—32no—
241C22×Q8—31no—
242C22×D4—32no—
243C22×D4—62no—
244C22×D4—61no—
245C22×D4—61no—
246C23×C4—62no—
247C23×C4D1211yescharacteristic
248C23×C4D1211yescharacteristic
249C23×C4C22×D311yescharacteristic
250C2×Q16—64no—
251C2×Q16—122no—
252C2×Q16—122no—
253C2×SD16—64no—
254C2×SD16—122no—
255C2×SD16—122no—
256C22×C8C22×D312yes—
257C22×C8C22×D312yes—
258C4⋊Q8—64no—
259#127—64no—
260#124—64no—
261#124—64no—
262C22⋊Q8—64no—
263#121—68no—
264#121—68no—
265C4×D4—68no—
266C4×D4—68no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C42—24no—
280C2×C42D1211yescharacteristic
281C2×C42D1211yescharacteristic
282C2×C42C22×D311yescharacteristic
283C2×C42C22×D311yescharacteristic
284#103—64no—
285#103—122no—
286#103—122no—
287#102—64no—
288#102—122no—
289#102—122no—
290C22⋊C8—24no—
291#95—61no—
292#95—61no—
293#95—61no—
294#95—61no—
295#95—61no—
296#95—61no—
297#95—61no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3#12012yes—
305C23×C3C22×D411yescharacteristic
306C22×D3—24no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—42no—
312C22×D3—42no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—44no—
328C2×C3×C4—44no—
329C2×C3×C4#12012yes—
330C2×C3×C4#13512yes—
331C2×C3×C4#13712yes—
332C2×C3×C4C22×D412yes—
333C2×C3×C4H(5,𝔽2)12yes—
334C2×C3×C4H(5,𝔽2)11yescharacteristic
335C2×C3×C4H(5,𝔽2)11yescharacteristic
336#67—416no—
337C2×Dic3—24no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—41no—
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—
358C4×D3—416no—
359Dic6—24no—
360Dic6—24no—
361Dic6—42no—
362Dic6—42no—
363Dic6—42no—
364Dic6—42no—
365C3×C8—42no—
366C3×C8—42no—
367C24—62no—
368C24C2×D1211yescharacteristic
369#41—128no—
370#41—128no—
371#41—128no—
372#41—128no—
373C2×Q8—34no—
374C2×Q8—62no—
375C2×Q8—62no—
376C2×Q8—62no—
377C2×Q8—62no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×Q8—61no—
381C2×Q8—62no—
382C2×Q8—61no—
383C2×D4—34no—
384C2×D4—64no—
385C2×D4—64no—
386C2×D4—68no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—62no—
392C2×D4—122no—
393C2×D4—122no—
394C2×D4—124no—
395C2×D4—124no—
396C2×D4—122no—
397C2×D4—122no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—62no—
408C22×C4—68no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—124no—
414C22×C4—124no—
415C22×C4C2×D1211yescharacteristic
416C22×C4C2×D1211yescharacteristic
417C22×C4C2×D1211yescharacteristic
418C22×C4C2×D1211yescharacteristic
419C22×C4C2×D1211yescharacteristic
420C22×C4D3×D414yes—
421C22×C4C23×D311yescharacteristic
422Q16—128no—
423SD16—128no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C4⋊C4—64no—
431C4⋊C4—64no—
432C4⋊C4—62no—
433C4⋊C4—61no—
434C4⋊C4—62no—
435C4⋊C4—61no—
436C4⋊C4—122no—
437C4⋊C4—122no—
438C4⋊C4—122no—
439C22⋊C4—68no—
440C22⋊C4—68no—
441C22⋊C4—122no—
442C22⋊C4—122no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3C2×#12011yescharacteristic
455C22×C3#53511yescharacteristic
456C22×C3#53511yescharacteristic
457C22×C3C2×#13511yescharacteristic
458C22×C3C2×#13711yescharacteristic
459C22×C3#57511yescharacteristic
460C22×C3#57511yescharacteristic
461C2×D3—44no—
462C2×D3—44no—
463C2×D3—44no—
464C2×D3—82no—
465C2×D3—82no—
466C2×D3—82no—
467C2×D3—82no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4#53412yes—
474C3×C4#53412yes—
475Dic3—44no—
476Dic3—81no—
477Dic3—81no—
478Dic3—81no—
479Dic3—81no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—64no—
487C23—122no—
488C23—122no—
489C23—122no—
490C23—122no—
491C23—122no—
492C23—122no—
493C23#88212yes—
494C23C22×D1211yescharacteristic
495Q8—64no—
496Q8—64no—
497Q8—122no—
498Q8—122no—
499Q8—122no—
500Q8—122no—
501D4—64no—
502D4—64no—
503D4—128no—
504D4—128no—
505D4—1216no—
506D4—122no—
507D4—122no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—64no—
525C2×C4—1216no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4#88212yes—
541C2×C4#90412yes—
542C2×C4#90912yes—
543C2×C4C2×D3×D412yes—
544C2×C4#100912yes—
545C2×C4#100911yescharacteristic
546C2×C4#100911yescharacteristic
547C8—42no—
548C8—42no—
549C2×C3—42no—
550C2×C3#261311yescharacteristic
551C2×C3#300211yescharacteristic
552C2×C3#300711yescharacteristic
553C2×C3#321011yescharacteristic
554C2×C3#321211yescharacteristic
555C2×C3#321611yescharacteristic
556C2×C3#321911yescharacteristic
557D3—84no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—124no—
566C22—124no—
567C22—124no—
568C22—242no—
569C22—242no—
570C22—242no—
571C22—242no—
572C22C2×#88211yescharacteristic
573C22#552511yescharacteristic
574C22#552511yescharacteristic
575C22C2×#90411yescharacteristic
576C22C2×#90911yescharacteristic
577C22#574811yescharacteristic
578C22#574811yescharacteristic
579C4—22no—
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—124no—
585C4—241no—
586C4—241no—
587C4—241no—
588C4—241no—
589C4#565312yes—
590C4#565312yes—
591C3#2962911yescharacteristic, complemented, Sylow
592C2—42no—
593C2—244no—
594C2#7413411yescharacteristic
595C2#7861011yescharacteristic
596C2#7864011yescharacteristic
597C2#7930611yescharacteristic
598C2#7930811yescharacteristic
599C2#7935211yescharacteristic
600C2#7935411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC2×#5477C2
hypercenter#1817D3
hypocenterC3#29629
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #82075C2semidirect
3. C2×#4733C2semidirect
4. C3×#1817C2semidirect
5. #69960C2semidirect
6. #68688C2semidirect
7. #68677C2semidirect
8. #68676C2semidirect
9. #68631C2semidirect
10. #68630C2semidirect
11. #68629C2semidirect
55. #1817D3semidirect
591. C3#29629semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
#820752
C2×#47332
C3×#18172
#699602
#686882
#686772
#686762
#686312
#686302
#686292
#296293