Finite Group Databasebeta

← #10829926

Subgroups — #10829926

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×#5577C211yesmaximal, characteristic, complemented
2C3×#2538C211yesmaximal, characteristic, complemented
3C2×#5015C211yesmaximal, characteristic, complemented
4C2×C4×#213C211yesmaximal, characteristic, complemented
5#71634C211yesmaximal, characteristic, complemented
6#69658C212yesmaximal, complemented
7#69657C212yesmaximal, complemented
8#69627C212yesmaximal, complemented
9#69625C212yesmaximal, complemented
10#69584C211yesmaximal, characteristic, complemented
11#69580C211yesmaximal, characteristic, complemented
12#24184—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C22⋊Q8—21no—
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C22×#213C2211yescharacteristic
18C22×#213C2212yes—
19#5577—24no—
20#5577—24no—
21C2×#895—21no—
22C2×#895—22no—
23C2×#874C2212yes—
24C2×#874C2211yescharacteristic
25C2×C4×D12C2212yes—
26C2×C4×D12C2211yescharacteristic
27C3×#396—22no—
28C3×#390C2212yes—
29C3×#389C2212yes—
30C3×#384C2211yescharacteristic
31C3×#380—21no—
32C2×#832C2211yescharacteristic
33#5015—24no—
34C4×#213—24no—
35C2×#809C2212yes—
36C2×#809C2211yescharacteristic
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#807C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#550—31no—
47C2×#459—31no—
48C2×C4×D8—31no—
49#2538D311yescharacteristic, complemented
50#2041—32no—
51#2011—32no—
52#1984—31no—
53#1957—31no—
54#1879—32no—
55#1862—32no—
56#1802—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C2×#239—28no—
61C2×#239—44no—
62C2×#239—44no—
63C22×D12—22no—
64C22×D12—21no—
65C22×D12—22no—
66C22×D12—21no—
67C22×D12C2312yes—
68C22×D12C2311yescharacteristic
69C22×C4×D3—22no—
70C22×C4×D3—21no—
71C3×C22⋊Q8—44no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C2×#213—22no—
89C2×#213—22no—
90C2×#213—24no—
91C2×#213—24no—
92C2×#213—22no—
93C2×#213—42no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100#895—28no—
101#895—216no—
102#874—24no—
103#874—24no—
104#874—22no—
105#874D414yes—
106C4×D12—24no—
107C4×D12—22no—
108C4×D12—24no—
109C4×D12D414yes—
110C3×#95—22no—
111#832—22no—
112#809—22no—
113#809—24no—
114#809—24no—
115#809—42no—
116#809—42no—
117#807—22no—
118C4⋊(C3⋊C8)—22no—
119C4×C3⋊C8—22no—
120C22×#41—61no—
121C22×D8—31no—
122C22×D8—32no—
123#550—64no—
124#550—64no—
125C2×#127—31no—
126C2×#127—32no—
127C2×C22⋊Q8—21no—
128C2×#121—61no—
129C2×#121—62no—
130C2×C4×Q8C2×D311yescharacteristic
131C2×C4×D4—31no—
132C2×C4×D4—32no—
133C2×C4×D4C2×D311yescharacteristic
134#459—64no—
135C4×D8—64no—
136C2×#107—31no—
137C2×C4⋊C8—31no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#102—32no—
141C2×#102—31no—
142C2×#102—32no—
143C2×#102—32no—
144C2×C4×C8—31no—
145#396—22no—
146#390C2×D312yes—
147#389C2×D312yes—
148#384C2×D311yescharacteristic
149#380—21no—
150#340—32no—
151#336—32no—
152#336—32no—
153C24×C3—21no—
154C23×D3—22no—
155C23×D3—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8D814yescomplemented
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4#4112yescomplemented
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—42no—
168C22×C3×C4—42no—
169C22×C3×C4—42no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—21no—
176#239—432no—
177C2×D12—22no—
178C2×D12—21no—
179C2×D12—22no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—22no—
183C2×D12—21no—
184C2×D12—22no—
185C2×D12—28no—
186C2×D12—24no—
187C2×D12—22no—
188C2×D12—21no—
189C2×D12—28no—
190C2×D12—24no—
191C2×D12—44no—
192C2×D12—44no—
193C2×D12—42no—
194C2×D12—42no—
195C2×D12—81no—
196C2×D12—81no—
197C2×D12—82no—
198C2×D12—82no—
199C2×D12C2×D414yes—
200C2×D12C2×D412yes—
201C2×C4×D3—28no—
202C2×C4×D3—24no—
203C2×C4×D3—44no—
204C2×C4×D3—44no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4—44no—
215C3×C4⋊C4—81no—
216C3×C22⋊C4—42no—
217C3×C22⋊C4—44no—
218C3×C22⋊C4—82no—
219C3×C42—22no—
220C3×C42—22no—
221C3×C42C2×D412yes—
222C3×C42C2×D412yes—
223#213—28no—
224#213—48no—
225(C2×D3)⋊C4—28no—
226(C2×D3)⋊C4—24no—
227C4⋊Dic3—21no—
228C4⋊Dic3C2×D412yes—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8#4112yescomplemented
238C2×#41—68no—
239C2×#41—124no—
240C2×#41—124no—
241C22×Q8#6711yescharacteristic
242C22×D4—31no—
243C22×D4—32no—
244C22×D4—62no—
245C22×D4—61no—
246C22×D4—62no—
247C22×D4—61no—
248C22×D4C22×D311yescharacteristic
249C23×C4—21no—
250C23×C4—62no—
251C23×C4—61no—
252C2×D8—62no—
253C2×D8—62no—
254C2×D8—62no—
255C2×D8—64no—
256C2×D8—64no—
257C2×D8—122no—
258C22×C8—31no—
259C22×C8—32no—
260C22×C8—31no—
261#127—34no—
262#127—64no—
263#127—64no—
264#127—62no—
265C22⋊Q8—44no—
266#121—68no—
267#121—616no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—34no—
273C4×D4—64no—
274C4×D4—64no—
275C4×D4—62no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4#6711yescharacteristic
281C2×C4⋊C4#6712yes—
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D312yes—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—62no—
288C2×C22⋊C4—61no—
289C2×C42C22×D312yes—
290C2×C42C22×D311yescharacteristic
291#107—62no—
292C4⋊C8—62no—
293#102—64no—
294#102—62no—
295#102—64no—
296#102—62no—
297#102—122no—
298#102—122no—
299C4×C8—62no—
300#95—22no—
301C23×C3—22no—
302C23×C3—41no—
303C23×C3—41no—
304C23×C3—41no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3C22×D411yescharacteristic
309C22×D3—24no—
310C22×D3—22no—
311C22×D3—41no—
312C22×D3—42no—
313C22×D3—42no—
314C22×D3—41no—
315C22×D3—42no—
316C22×D3—41no—
317C22×D3—42no—
318C22×D3—41no—
319C22×D3—42no—
320C22×D3—41no—
321C22×D3—42no—
322C22×D3—41no—
323C3×Q8—28no—
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—41no—
327C3×D4—41no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—84no—
342C2×C3×C4#12012yes—
343C2×C3×C4#12012yes—
344C2×C3×C4C2×D814yes—
345C2×C3×C4C2×D812yes—
346C2×C3×C4AGL(1,ℤ/8)12yes—
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4H(5,𝔽2)11yescharacteristic
352C2×C3×C42−1+411yescharacteristic
353C2×Dic3—22no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356D12—22no—
357D12—22no—
358D12—24no—
359D12—24no—
360D12—416no—
361D12—42no—
362D12—42no—
363D12—41no—
364D12—41no—
365D12—48no—
366D12—84no—
367D12—88no—
368C4×D3—416no—
369C4×D3—48no—
370C3⋊C8—22no—
371C3⋊C8—41no—
372C3⋊C8—42no—
373C24—21no—
374C24—62no—
375C24—61no—
376#41—1232no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8#21314yescomplemented
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—32no—
386C2×D4—34no—
387C2×D4—68no—
388C2×D4—68no—
389C2×D4—64no—
390C2×D4—64no—
391C2×D4—62no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—62no—
395C2×D4—61no—
396C2×D4—62no—
397C2×D4—62no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—62no—
401C2×D4—124no—
402C2×D4—124no—
403C2×D4—122no—
404C2×D4—122no—
405C2×D4—241no—
406C2×D4—241no—
407C2×D4—242no—
408C2×D4—242no—
409C2×D4#23912yescomplemented
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—68no—
416C22×C4—61no—
417C22×C4—64no—
418C22×C4—124no—
419C22×C4—124no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4C2×#6712yes—
423C22×C4C2×#6712yes—
424C22×C4C2×#6711yescharacteristic
425C22×C4C2×#6711yescharacteristic
426C22×C4C23×D311yescharacteristic
427D8—68no—
428D8—128no—
429C2×C8—32no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—62no—
434C2×C8—62no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—62no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—32no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—61no—
448C4⋊C4—81no—
449C22⋊C4—44no—
450C22⋊C4—42no—
451C22⋊C4—68no—
452C22⋊C4—64no—
453C22⋊C4—82no—
454C42—22no—
455C42—22no—
456C42D3×D412yes—
457C42D3×D412yes—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3—81no—
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—44no—
474C2×D3—44no—
475C2×D3—42no—
476C2×D3—44no—
477C2×D3—42no—
478C2×D3—81no—
479C2×D3—82no—
480C2×D3—82no—
481C2×D3—81no—
482C2×D3—82no—
483C2×D3—81no—
484C2×D3—82no—
485C2×D3—81no—
486C3×C4—24no—
487C3×C4—22no—
488C3×C4—42no—
489C3×C4—82no—
490C3×C4—81no—
491C3×C4#44714yes—
492C3×C4#52511yescharacteristic
493C3×C4#53711yescharacteristic
494C3×C4#54611yescharacteristic
495C3×C4#54711yescharacteristic
496Dic3—42no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—64no—
505C23—62no—
506C23—122no—
507C23—121no—
508C23—121no—
509C23—122no—
510C23—122no—
511C23—121no—
512C23—122no—
513C23—121no—
514C23—122no—
515C23—121no—
516C23—122no—
517C23—121no—
518C23C22×#6711yescharacteristic
519Q8—28no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524D4—62no—
525D4—62no—
526D4—64no—
527D4—64no—
528D4—1216no—
529D4—128no—
530D4—122no—
531D4—122no—
532D4—121no—
533D4—121no—
534D4—244no—
535D4—248no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—62no—
550C2×C4—84no—
551C2×C4—1216no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—128no—
555C2×C4C2×#21314yes—
556C2×C4C2×#21312yes—
557C2×C4#93712yes—
558C2×C4#93712yes—
559C2×C4#94912yes—
560C2×C4C2×D3×D411yescharacteristic
561C2×C4C2×D3×D411yescharacteristic
562C2×C4C2×#23911yescharacteristic
563C2×C4#100711yescharacteristic
564C2×C4D3×#4111yescharacteristic
565C2×C4#100911yescharacteristic
566C8—62no—
567C8—121no—
568C8—122no—
569C2×C3—82no—
570C2×C3#240711yescharacteristic
571C2×C3C2×#44711yescharacteristic
572C2×C3#301211yescharacteristic
573C2×C3#335811yescharacteristic
574C2×C3#336611yescharacteristic
575C2×C3#339111yescharacteristic
576C2×C3#339711yescharacteristic
577D3—82no—
578D3—84no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—122no—
587C22—124no—
588C22—124no—
589C22—122no—
590C22—124no—
591C22—122no—
592C22—241no—
593C22—242no—
594C22—242no—
595C22—241no—
596C22—242no—
597C22—241no—
598C22—242no—
599C22—241no—
600C22C22×#21311yescharacteristic
601C22C2×#93711yescharacteristic
602C22#581711yescharacteristic
603C22C2×#94911yescharacteristic
604C22#583411yescharacteristic
605C22#583511yescharacteristic
606C22#583711yescharacteristic
607C4—24no—
608C4—22no—
609C4—42no—
610C4—82no—
611C4—81no—
612C4—122no—
613C4#503614yes—
614C4#562711yescharacteristic
615C4#562911yescharacteristic
616C4#563011yescharacteristic
617C4#563111yescharacteristic
618C3#2418411yescharacteristic, complemented, Sylow
619C2—82no—
620C2—242no—
621C2—244no—
622C2C2×#503611yescharacteristic
623C2#8281211yescharacteristic
624C2#8342411yescharacteristic
625C2#8356211yescharacteristic
626C2#8358211yescharacteristic
627C2#8363811yescharacteristic
628C2#8367111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2538C2
socleC23×C3C22×D4
hypercenter#2538D3
hypocenterC3#24184
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#5577C2semidirect
2. C3×#2538C2semidirect
3. C2×#5015C2semidirect
4. C2×C4×#213C2semidirect
5. #71634C2semidirect
6. #69658C2semidirect
7. #69657C2semidirect
8. #69627C2semidirect
9. #69625C2semidirect
10. #69584C2semidirect
11. #69580C2semidirect
49. #2538D3semidirect
158. C2×C3×Q8D8semidirect
164. C2×C3×D4#41semidirect
237. C2×C3⋊C8#41semidirect
379. C2×Q8#213semidirect
409. C2×D4#239semidirect
618. C3#24184semidirect

Maximal subgroups

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

subgroupindex
C2×#55772
C3×#25382
C2×#50152
C2×C4×#2132
#716342
#696582
#696572
#696272
#696252
#695842
#695802
#241843