Finite Group Databasebeta

← #10829885

Subgroups — #10829885

Order 768.

1050 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic, complemented
3#83297C211yesmaximal, characteristic
4#82455C211yesmaximal, characteristic, complemented
5#71867C212yesmaximal, complemented
6#71863C212yesmaximal, complemented
7#71840C211yesmaximal, characteristic, complemented
8#71816C211yesmaximal, characteristic, complemented
9#71667C212yesmaximal, complemented
10#71664C212yesmaximal, complemented
11#71637C211yesmaximal, characteristic, complemented
12#24181—31nomaximal, Sylow
13C3×#559—21no—
14C3×#557—21no—
15C3×#556—21no—
16C3×#556—22no—
17C3×#554—21no—
18C3×#552—22no—
19C3×#539—22no—
20C3×#536C2212yes—
21C3×#533C2211yescharacteristic
22C3×#532C2211yescharacteristic
23C3×#529—22no—
24C3×#520C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×#949C2212yes—
27C2×#947C2211yescharacteristic
28C2×C4×D12C2211yescharacteristic
29C2×#837C2212yes—
30C2×#837C2212yes—
31#5232C2212yes—
32#5229C2211yescharacteristic
33C2×#836C2212yes—
34#5087—21no—
35#5084—21no—
36#5083—21no—
37#5072—21no—
38#5066—22no—
39#5057—22no—
40#5001C2212yes—
41#4995C2212yes—
42#4989—22no—
43#4988C2211yescharacteristic
44#4975C2211yescharacteristic
45#4940—22no—
46#4932C2211yescharacteristic
47#4924C2211yescharacteristic
48#4924C2211yescharacteristic
49#4532C2212yes—
50#4532C2212yes—
51#4530C2212yes—
52#3507D311yescharacteristic, complemented
53#3077—31no—
54#3036—31no—
55#2326—31no—
56#2038—32no—
57#2003—32no—
58#1995—31no—
59#1963—31no—
60#1883—32no—
61#1859—32no—
62#1801—31no—
63C2×C3×#41C2312yes—
64C22×C3×Q8C2311yescharacteristic
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D411yescharacteristic
69C3×C4⋊Q8D411yescharacteristic
70C3×C4⋊Q8D411yescharacteristic
71C3×#127D411yescharacteristic
72C3×#126—24no—
73C3×#126—24no—
74C3×#125—22no—
75C3×#125—24no—
76C3×#125D412yes—
77C3×#124—22no—
78C3×#124—22no—
79C3×#124D412yes—
80C3×#123—24no—
81C3×#123—24no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8—21no—
84C3×C22⋊Q8—21no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×#121—21no—
88C3×#121—21no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×Q8—21no—
95C3×C4×Q8—21no—
96C3×C4×Q8—21no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4—24no—
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×#117—24no—
103C3×#117—22no—
104C3×#117C2312yes—
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4D412yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C42D412yes—
109C2×C3×C42C2311yescharacteristic
110#949—24no—
111#949—42no—
112#947—22no—
113#947—22no—
114C2×#215—22no—
115C2×#213—22no—
116C2×(C2×D3)⋊C4—22no—
117C2×(C2×D3)⋊C4—21no—
118C2×(C2×D3)⋊C4—22no—
119C2×C4⋊Dic3C2311yescharacteristic
120C2×Dic3⋊C4—22no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C2×#208C2312yes—
124C2×#208C2312yes—
125C4×D12—21no—
126C4×D12—21no—
127C4×D12—21no—
128C4×D12—21no—
129#837—24no—
130#837—42no—
131#836—42no—
132#835—22no—
133#832—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—22no—
137#809—22no—
138#809—22no—
139#808—22no—
140#807—22no—
141C4⋊(C3⋊C8)—21no—
142C4⋊(C3⋊C8)—21no—
143C4⋊(C3⋊C8)—21no—
144C4⋊(C3⋊C8)—21no—
145C2×AGL(1,ℤ/8)—32no—
146#559—21no—
147#557—21no—
148#556—21no—
149#556—22no—
150#554—21no—
151#552—22no—
152#539—22no—
153#536C2×D312yes—
154#533C2×D311yescharacteristic
155#532C2×D311yescharacteristic
156#529—22no—
157C2×#124—31no—
158#520C2×D312yes—
159C2×C4×Q8C2×D311yescharacteristic
160C2×C4×D4—31no—
161#478—62no—
162#476—61no—
163#474—61no—
164#463—62no—
165#461—61no—
166#459—61no—
167#428—32no—
168#423—31no—
169#423—31no—
170C2×#104—32no—
171C2×#104—32no—
172#418—32no—
173#417—32no—
174C2×#101—32no—
175#391—31no—
176#390—31no—
177#388—62no—
178#385—62no—
179#384—31no—
180#376—31no—
181#343—32no—
182#339—32no—
183#339—32no—
184C23×D3—21no—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—22no—
188C3×#41—22no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8C2×D412yes—
198C2×C3×D4—22no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4C2×D412yes—
202C22×C3×C4—24no—
203C22×C3×C4—22no—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12#4112yes—
218C2×C4×D3—41no—
219C2×C4×D3—41no—
220C2×C4×D3—82no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4C2×D412yes—
235C3×C4⋊C4C2×D412yes—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C42—24no—
243C3×C42—22no—
244C3×C42—22no—
245C3×C42—22no—
246C3×C42C2×D411yescharacteristic
247C3×C42C2×D411yescharacteristic
248C3×C42C2×D411yescharacteristic
249C3×C42C2×D411yescharacteristic
250#215—44no—
251#213—44no—
252(C2×D3)⋊C4—28no—
253(C2×D3)⋊C4—41no—
254(C2×D3)⋊C4—41no—
255(C2×D3)⋊C4—82no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258Dic3⋊C4—82no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3#4112yes—
263#208—22no—
264#208—22no—
265#208—22no—
266#208—22no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—22no—
269C2×#41C22×D312yes—
270C22×Q8C22×D311yescharacteristic
271C22×D4—31no—
272C23×C4—61no—
273AGL(1,ℤ/8)—64no—
274AGL(1,ℤ/8)—122no—
275C2×SD16—62no—
276C2×D8—62no—
277C2×M16—32no—
278C2×M16—32no—
279C4⋊Q8—22no—
280C4⋊Q8#6711yescharacteristic
281C4⋊Q8#6711yescharacteristic
282C4⋊Q8#6711yescharacteristic
283#127#6711yescharacteristic
284#126—24no—
285#126—24no—
286#125—22no—
287#125—24no—
288#125#6712yes—
289#124—22no—
290#124—22no—
291#124—62no—
292#124—62no—
293#124#6712yes—
294#123—24no—
295#123—24no—
296C22⋊Q8—24no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301#121—21no—
302#121—21no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—24no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—61no—
317C4×D4—61no—
318C4×D4—61no—
319C4×D4—61no—
320#117—24no—
321#117—22no—
322#117C22×D312yes—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—62no—
326C2×C4⋊C4#6712yes—
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—62no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—62no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42#6712yes—
334C2×C42C22×D311yescharacteristic
335#107—62no—
336#106—62no—
337C4⋊C8—61no—
338C4⋊C8—61no—
339C4⋊C8—61no—
340C4⋊C8—61no—
341#104—64no—
342#104—122no—
343#103—62no—
344#102—62no—
345#102—62no—
346#102—62no—
347#101—122no—
348#95—61no—
349#95—61no—
350#95—62no—
351C23×C3—22no—
352C23×C3C22×D411yescharacteristic
353C22×D3—22no—
354C22×D3—42no—
355C22×D3—42no—
356C22×D3—41no—
357C22×D3—41no—
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—42no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—81no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—42no—
369C3×D4—42no—
370C3×D4—42no—
371C3×D4—81no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—42no—
384C2×C3×C4—84no—
385C2×C3×C4#12011yescharacteristic
386C2×C3×C4#12011yescharacteristic
387C2×C3×C4#12011yescharacteristic
388C2×C3×C4#12011yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×C3×C42−1+411yescharacteristic
395C2×Dic3—22no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—42no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403D12—22no—
404D12—22no—
405D12—41no—
406D12—41no—
407C4×D3—82no—
408C4×D3—82no—
409C3⋊C8—42no—
410C3⋊C8—42no—
411C24—61no—
412#41—22no—
413#41—22no—
414#41—22no—
415#41—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8C2×#6712yes—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—32no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4C2×#6712yes—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—121no—
441C22×C4—121no—
442C22×C4—242no—
443C22×C4C2×#6712yes—
444C22×C4C2×#6712yes—
445C22×C4C2×#6711yescharacteristic
446C22×C4C2×#6711yescharacteristic
447C22×C4C23×D311yescharacteristic
448SD16—124no—
449D8—124no—
450M16—62no—
451M16—62no—
452M16—62no—
453M16—62no—
454C2×C8—62no—
455C2×C8—62no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—242no—
472C4⋊C4C2×#6712yes—
473C4⋊C4C2×#6712yes—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—68no—
481C22⋊C4—121no—
482C22⋊C4—121no—
483C22⋊C4—242no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—32no—
489C42—61no—
490C42—61no—
491C42—61no—
492C42C2×#6711yescharacteristic
493C42C2×#6711yescharacteristic
494C42C2×#6711yescharacteristic
495C42C2×#6711yescharacteristic
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3C2×#12011yescharacteristic
502C22×C3#52511yescharacteristic
503C22×C3#53711yescharacteristic
504C2×D3—44no—
505C2×D3—42no—
506C2×D3—82no—
507C2×D3—81no—
508C2×D3—81no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—42no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4#54611yescharacteristic
519C3×C4#54711yescharacteristic
520C3×C4#54811yescharacteristic
521C3×C4#55011yescharacteristic
522Dic3—42no—
523Dic3—81no—
524Dic3—81no—
525C23—22no—
526C23—62no—
527C23—122no—
528C23—122no—
529C23—121no—
530C23—121no—
531C23C22×#6711yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—42no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—81no—
540D4—22no—
541D4—22no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—62no—
546D4—62no—
547D4—81no—
548D4—121no—
549D4—121no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—62no—
563C2×C4—84no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—122no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—242no—
572C2×C4—242no—
573C2×C4#95311yescharacteristic
574C2×C4#95311yescharacteristic
575C2×C4#95311yescharacteristic
576C2×C4#95311yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4D3×#4111yescharacteristic
579C2×C4#100911yescharacteristic
580C2×C4#101011yescharacteristic
581C2×C4C22×#6711yescharacteristic
582C2×C4C22×#6711yescharacteristic
583C8—122no—
584C8—122no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—82no—
588C2×C3#240711yescharacteristic
589C2×C3#302011yescharacteristic
590C2×C3#302111yescharacteristic
591D3—82no—
592C22—21no—
593C22—21no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—124no—
598C22—122no—
599C22—242no—
600C22—241no—
601C22—241no—
602C22#582811yescharacteristic
603C22#583311yescharacteristic
604C22C2×#95311yescharacteristic
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—122no—
615C4—241no—
616C4—241no—
617C4#582411yescharacteristic
618C4#582911yescharacteristic
619C4#583111yescharacteristic
620C4#583411yescharacteristic
621C3#2418111yescharacteristic, complemented, Sylow
622C2—21no—
623C2—21no—
624C2—82no—
625C2—242no—
626C2#8409711yescharacteristic
627C2#8411611yescharacteristic
628C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3507C2
socleC22×C3C2×#120
prime radical#83297C2
hypercenter#3507D3
hypocenterC3#24181
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
1. C3×#3507C2semidirect
2. #83350C2semidirect
4. #82455C2semidirect
5. #71867C2semidirect
6. #71863C2semidirect
7. #71840C2semidirect
8. #71816C2semidirect
9. #71667C2semidirect
10. #71664C2semidirect
11. #71637C2semidirect
52. #3507D3semidirect
621. C3#24181semidirect

Maximal subgroups

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

subgroupindex
C3×#35072
#833502
#832972
#824552
#718672
#718632
#718402
#718162
#716672
#716642
#716372
#241813