Finite Group Databasebeta

← #10829922

Subgroups — #10829922

Order 768.

1462 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
1C2×#5718C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic, complemented
3C2×C4×D24C211yesmaximal, characteristic, complemented
4#74369C211yesmaximal, characteristic, complemented
5C3×#1957C211yesmaximal, characteristic, complemented
6#70209C212yesmaximal, complemented
7#70177C212yesmaximal, complemented
8#70172C212yesmaximal, complemented
9#70168C212yesmaximal, complemented
10#69788C211yesmaximal, characteristic, complemented
11#69584C211yesmaximal, characteristic, complemented
12#24184—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×#127C2211yescharacteristic
15C2×#940C2212yes—
16C2×#939—22no—
17C2×#938—21no—
18C2×D4×Dic3C2212yes—
19C22×#213C2212yes—
20C22×D24C2211yescharacteristic
21#5718—24no—
22#5718—24no—
23C2×#897—21no—
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#384C2211yescharacteristic
31C2×#832C2212yes—
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#818C2211yescharacteristic
35#5072—24no—
36#4994—22no—
37#4988C2212yes—
38#4987—21no—
39#4984C2212yes—
40C2×#809C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C4×D24—24no—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#550—31no—
47C2×#459—31no—
48C2×C4×D8—31no—
49#2538—31no—
50#2041—32no—
51#2011—32no—
52#1984—31no—
53#1957D311yescharacteristic, complemented
54#1879—32no—
55#1862—32no—
56#1802—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×#67—22no—
61C22×#67—22no—
62C23×Dic3—22no—
63C2×#237—28no—
64C2×#237—44no—
65C2×#237—44no—
66C22×D12C2311yescharacteristic
67C22×D12C2311yescharacteristic
68C22×C4×D3—21no—
69C22×C4×D3—21no—
70C22×Dic6D411yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×#127—22no—
74C3×#127D414yes—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C42C2311yescharacteristic
79C2×#217—22no—
80#940—24no—
81#940—24no—
82#939—216no—
83#938—28no—
84D4×Dic3—24no—
85D4×Dic3—24no—
86C2×#213—24no—
87C2×#213—24no—
88C2×#213—42no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3D411yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4D412yes—
98C2×C4×Dic3C2312yes—
99C22×C3⋊C8C2312yes—
100C2×D24—22no—
101C2×D24—22no—
102C2×D24—22no—
103#897—44no—
104C4×D12—22no—
105C4×D12—22no—
106C4×D12—22no—
107C4×D12D414yes—
108C4×Dic6—22no—
109C4×Dic6—22no—
110C3×#102—24no—
111C3×#102—42no—
112C3×C4×C8—22no—
113#832—24no—
114C3⋊#95—22no—
115#821—22no—
116#821—22no—
117#818—22no—
118#809—42no—
119C4⋊(C3⋊C8)—22no—
120C22×#41—61no—
121C22×D8—32no—
122C22×D8—31no—
123#550—64no—
124#550—64no—
125C2×#127—32no—
126C2×#127C2×D311yescharacteristic
127C2×C22⋊Q8—61no—
128C2×#121—61no—
129C2×#121—62no—
130C2×C4×Q8—31no—
131C2×C4×D4—32no—
132C2×C4×D4—31no—
133C2×C4×D4—31no—
134#459—64no—
135C4×D8—64no—
136C2×#107—31no—
137C2×C4⋊C8—31no—
138C2×#102—32no—
139C2×#102—31no—
140C2×#102—31no—
141C2×#102—32no—
142C2×#102C2×D312yes—
143C2×#102C2×D312yes—
144C2×C4×C8C2×D311yescharacteristic
145#396—62no—
146#390—32no—
147#389—32no—
148#384C2×D311yescharacteristic
149#380—61no—
150#340—32no—
151#336—32no—
152#336—32no—
153C24×C3—22no—
154C23×D3—21no—
155C23×D3—21no—
156C2×C3×D4—24no—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C2×C3×D4—81no—
166C2×C3×D4—81no—
167C2×C3×D4C2×D414yes—
168C22×C3×C4—22no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C2×#67—28no—
173C2×#67—28no—
174C2×#67—44no—
175C2×#67—44no—
176C2×#67—82no—
177C2×#67—82no—
178C22×Dic3—28no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—44no—
182C22×Dic3—44no—
183C22×Dic3C2×D412yes—
184C22×Dic3C2×D412yes—
185#237—432no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12C2×D412yes—
197C2×D12#4112yescomplemented
198C2×C4×D3—24no—
199C2×C4×D3—42no—
200C2×C4×D3—42no—
201C2×C4×D3—42no—
202C2×C4×D3—42no—
203C2×C4×D3—42no—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6D814yescomplemented
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8#4112yescomplemented
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—81no—
217C3×C42C2×D412yes—
218C3×C42C2×D412yes—
219#217—28no—
220#213—48no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—42no—
223(C2×D3)⋊C4—44no—
224(C2×D3)⋊C4—82no—
225C4⋊Dic3—22no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—44no—
230C4⋊Dic3C2×D412yes—
231Dic3⋊C4—24no—
232C4×Dic3—22no—
233C4×Dic3—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237D24—28no—
238C2×#41—68no—
239C2×#41—124no—
240C2×#41—124no—
241C22×Q8—31no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—31no—
245C22×D4—31no—
246C22×D4—62no—
247C22×D4—62no—
248C22×D4C22×D312yes—
249C23×C4—62no—
250C23×C4—61no—
251C23×C4—61no—
252C2×D8—62no—
253C2×D8—62no—
254C2×D8—64no—
255C2×D8—64no—
256C2×D8—62no—
257C2×D8—122no—
258C22×C8—32no—
259C22×C8C22×D311yescharacteristic
260C22×C8C22×D311yescharacteristic
261#127—22no—
262#127—64no—
263#127—64no—
264#127D1214yes—
265C22⋊Q8—124no—
266#121—68no—
267#121—616no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×D4—34no—
271C4×D4—64no—
272C4×D4—64no—
273C4×D4—62no—
274C4×D4—62no—
275C4×D4—62no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—32no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4C22×D312yes—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—62no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—62no—
289C2×C42—32no—
290C2×C42C22×D311yescharacteristic
291#107—62no—
292C4⋊C8—62no—
293#102—24no—
294#102—42no—
295#102—62no—
296#102—62no—
297#102—64no—
298#102—122no—
299C4×C8—22no—
300#95—62no—
301C23×C3—24no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3C22×D411yescharacteristic
309C22×D3—22no—
310C22×D3—22no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C3×D4—24no—
324C3×D4—24no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—48no—
328C3×D4—84no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4#12012yes—
335C2×C3×C4#12012yes—
336C2×C3×C4C2×D812yes—
337C2×C3×C4AGL(1,ℤ/8)12yes—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4H(5,𝔽2)11yescharacteristic
343C2×C3×C42−1+411yescharacteristic
344#67—416no—
345#67—88no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—416no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3C2×D814yes—
359D12—22no—
360D12—22no—
361D12—22no—
362D12—22no—
363D12—41no—
364D12—41no—
365D12—41no—
366D12—41no—
367C4×D3—48no—
368C4×D3—84no—
369Dic6—28no—
370C3×C8—22no—
371C3×C8—41no—
372C3⋊C8—42no—
373C24—22no—
374C24—61no—
375C24—61no—
376#41—1232no—
377C2×Q8—34no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—32no—
388C2×D4—32no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—68no—
392C2×D4—68no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—81no—
404C2×D4—81no—
405C2×D4—124no—
406C2×D4—124no—
407C2×D4—242no—
408C2×D4—242no—
409C2×D4C2×D1214yes—
410C22×C4—22no—
411C22×C4—32no—
412C22×C4—32no—
413C22×C4—68no—
414C22×C4—61no—
415C22×C4—64no—
416C22×C4—61no—
417C22×C4—124no—
418C22×C4—124no—
419C22×C4—122no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4—122no—
424C22×C4D3×D411yescharacteristic
425C22×C4D3×D411yescharacteristic
426C22×C4C23×D311yescharacteristic
427D8—68no—
428D8—128no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—62no—
435C2×C8—62no—
436C2×C8—62no—
437C2×C8#23712yescomplemented
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—32no—
441C4⋊C4—42no—
442C4⋊C4—64no—
443C4⋊C4—62no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—81no—
448C4⋊C4—124no—
449C22⋊C4—68no—
450C22⋊C4—64no—
451C22⋊C4—124no—
452C22⋊C4—122no—
453C22⋊C4—242no—
454C42—62no—
455C42—62no—
456C42C2×D1212yes—
457C42D3×D412yes—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3—82no—
464C22×C3—82no—
465C22×C3C2×#12011yescharacteristic
466C22×C3#54811yescharacteristic
467C22×C3#55011yescharacteristic
468C22×C3C22×D811yescharacteristic
469C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
470C22×C3#57611yescharacteristic
471C22×C3#57711yescharacteristic
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—42no—
475C2×D3—42no—
476C2×D3—42no—
477C2×D3—42no—
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—81no—
481C2×D3—81no—
482C2×D3—81no—
483C2×D3—81no—
484C2×D3—81no—
485C2×D3—81no—
486C3×C4—22no—
487C3×C4—82no—
488C3×C4#44714yes—
489C3×C4#52511yescharacteristic
490C3×C4#53711yescharacteristic
491C3×C4#54611yescharacteristic
492C3×C4#54711yescharacteristic
493Dic3—24no—
494Dic3—42no—
495Dic3—42no—
496Dic3—81no—
497C23—24no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—62no—
505C23—62no—
506C23—121no—
507C23—121no—
508C23—121no—
509C23—121no—
510C23—121no—
511C23—121no—
512C23—121no—
513C23—121no—
514C23—121no—
515C23—121no—
516C23—121no—
517C23—121no—
518C23C2×D3×D411yescharacteristic
519Q8—68no—
520D4—24no—
521D4—24no—
522D4—48no—
523D4—42no—
524D4—42no—
525D4—62no—
526D4—62no—
527D4—62no—
528D4—62no—
529D4—84no—
530D4—1216no—
531D4—121no—
532D4—121no—
533D4—121no—
534D4—121no—
535D4—248no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—34no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—1216no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—128no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—244no—
556C2×C4#88212yes—
557C2×C4#88212yes—
558C2×C4D3×D812yes—
559C2×C4#91412yes—
560C2×C4C22×D1211yescharacteristic
561C2×C4C2×D3×D411yescharacteristic
562C2×C4C2×#23711yescharacteristic
563C2×C4#100411yescharacteristic
564C2×C4D3×#4111yescharacteristic
565C2×C4#101011yescharacteristic
566C8—22no—
567C8—41no—
568C8—122no—
569C2×C3—84no—
570C2×C3#240711yescharacteristic
571C2×C3C2×#44711yescharacteristic
572C2×C3#301211yescharacteristic
573C2×C3#335811yescharacteristic
574C2×C3#336611yescharacteristic
575C2×C3#339111yescharacteristic
576C2×C3#339711yescharacteristic
577D3—82no—
578D3—82no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—122no—
587C22—122no—
588C22—122no—
589C22—122no—
590C22—122no—
591C22—122no—
592C22—241no—
593C22—241no—
594C22—241no—
595C22—241no—
596C22—241no—
597C22—241no—
598C22—241no—
599C22—241no—
600C22C2×#88211yescharacteristic
601C22#567711yescharacteristic
602C22#571811yescharacteristic
603C22C2×D3×D811yescharacteristic
604C22#575711yescharacteristic
605C22C2×#91411yescharacteristic
606C22#576211yescharacteristic
607C4—22no—
608C4—64no—
609C4—82no—
610C4—122no—
611C4—122no—
612C4—241no—
613C4#477314yes—
614C4#555411yescharacteristic
615C4#555511yescharacteristic
616C4#560611yescharacteristic
617C4#560811yescharacteristic
618C3#2418411yescharacteristic, complemented, Sylow
619C2—84no—
620C2—242no—
621C2—242no—
622C2#7643511yescharacteristic
623C2C2×#477311yescharacteristic
624C2#7874711yescharacteristic
625C2#8158411yescharacteristic
626C2#8158611yescharacteristic
627C2#8175711yescharacteristic
628C2#8177811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1957C2
socleC23×C3C22×D4
hypercenter#1957D3
hypocenterC3#24184
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
1. C2×#5718C2semidirect
2. C2×#5072C2semidirect
3. C2×C4×D24C2semidirect
4. #74369C2semidirect
5. C3×#1957C2semidirect
6. #70209C2semidirect
7. #70177C2semidirect
8. #70172C2semidirect
9. #70168C2semidirect
10. #69788C2semidirect
11. #69584C2semidirect
53. #1957D3semidirect
197. C2×D12#41semidirect
206. C2×Dic6D8semidirect
212. C2×C3×C8#41semidirect
437. C2×C8#237semidirect
618. C3#24184semidirect

Maximal subgroups

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

subgroupindex
C2×#57182
C2×#50722
C2×C4×D242
#743692
C3×#19572
#702092
#701772
#701722
#701682
#697882
#695842
#241843