Finite Group Databasebeta

← #379513

Subgroups — #379513

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2098C211yesmaximal, characteristic
2#20435C211yesmaximal, characteristic, complemented
3#20422C211yesmaximal, characteristic
4#11369C211yesmaximal, characteristic, complemented
5#11282C211yesmaximal, characteristic, complemented
6#11085C211yesmaximal, characteristic
7#10944C211yesmaximal, characteristic
8#10929C211yesmaximal, characteristic
9#10920C211yesmaximal, characteristic, complemented
10#10901C211yesmaximal, characteristic, complemented
11#10898C211yesmaximal, characteristic, complemented
12#10882C211yesmaximal, characteristic, complemented
13#10817C211yesmaximal, characteristic, complemented
14#10780C211yesmaximal, characteristic
15#10748C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C2×#421C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#410C2211yescharacteristic
25C2×#408C2211yescharacteristic
26#2873C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2296C2211yescharacteristic
31#2098C418yescomplemented
32#1771—22no—
33C8×C4⋊C4—22no—
34#1765—22no—
35#1764—22no—
36C2×#344C2211yescharacteristic
37C2×#344C2211yescharacteristic
38C2×#344C2211yescharacteristic
39C2×#344C2211yescharacteristic
40#1740C2211yescharacteristic
41#1738C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44#1735C2211yescharacteristic
45#1733C2211yescharacteristic
46#1733C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1728C2211yescharacteristic
56C22×#41C2311yescharacteristic
57#568—22no—
58#568—22no—
59#568—22no—
60#568—22no—
61C2×#131—22no—
62C2×#131—41no—
63C22×M16—41no—
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#430—41no—
79#430C2×C414yes—
80C2×#108—41no—
81#424—22no—
82#424—41no—
83#424C2×C414yes—
84#424C2×C414yes—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—41no—
87#421—22no—
88#421—41no—
89#421C2×C414yes—
90#421C2×C414yes—
91C2×#104—22no—
92C2×#104—22no—
93C2×#104—22no—
94C2×#104—41no—
95C2×#104—41no—
96#416—21no—
97#416—21no—
98#416—22no—
99C2×#103Q812yes—
100C2×#102Q812yes—
101#410—22no—
102#410—41no—
103C2×#99—22no—
104#408—41no—
105#408C2×C414yes—
106C2×C22⋊C8—41no—
107#405—22no—
108#405—22no—
109#405—22no—
110#405—41no—
111#405—41no—
112#405—41no—
113C4×M16—41no—
114C2×#97—21no—
115C2×#97D412yes—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8—41no—
120C2×C4×C8—41no—
121C2×C4×C8D412yes—
122#389—22no—
123#385—22no—
124C4×C4⋊C4—22no—
125#344—22no—
126#344—22no—
127#344—22no—
128#344—41no—
129#342—22no—
130#341—21no—
131#341—21no—
132#341—22no—
133#341—41no—
134#340—22no—
135#339—22no—
136#339—22no—
137#339—22no—
138#339—41no—
139#339—41no—
140H(3,ℤ/4)—22no—
141H(3,ℤ/4)C2×C414yes—
142#336—22no—
143#336—22no—
144#336—41no—
145#336—41no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41C22×C412yes—
157C2×#41#4112yes—
158C22×Q8C2×Q811yescharacteristic
159C22×D4—21no—
160C22×D4C2×Q811yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2411yescharacteristic
164#131—28no—
165#131—44no—
166#131—44no—
167#131—44no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—42no—
199C2×M16—44no—
200C2×M16—44no—
201C2×M16C22×C412yes—
202C2×M16C22×C412yes—
203C2×M16C22×C412yes—
204C2×M16#4112yes—
205C2×M16#4112yes—
206C2×M16#4112yes—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—42no—
223C22×C8C22×C412yes—
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C22×C8C2×D412yes—
227C22×C8C2×Q812yes—
228C22×C8#4112yes—
229#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117C22×C412yes—
247#117C22×C412yes—
248#117#4112yes—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C2×Q812yes—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4#4112yes—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42C2×D412yes—
270C2×C42#4112yes—
271#108—44no—
272C4⋊C8—28no—
273C4⋊C8—44no—
274#104—28no—
275#104—44no—
276#104—44no—
277#103—22no—
278#103—22no—
279#102—22no—
280#102—22no—
281#99—44no—
282C22⋊C8—44no—
283#97—24no—
284#97—42no—
285#97—42no—
286C4×C8—24no—
287C4×C8—42no—
288C4×C8—42no—
289C4×C8—42no—
290C4×C8—42no—
291#95—24no—
292C24—21no—
293C24C2×#4111yescharacteristic
294#41—22no—
295#41—22no—
296#41—22no—
297#41—22no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—22no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8C4×Q812yes—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4C4×Q812yes—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—44no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358M16—22no—
359M16—22no—
360M16—22no—
361M16—22no—
362M16—22no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—48no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—48no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8C22⋊Q812yes—
417C2×C8C22⋊Q812yes—
418C2×C8#12314yes—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—48no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42C4×D414yes—
454C23—22no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—82no—
459C23C4×#4111yescharacteristic
460C23#52511yescharacteristic
461C23#52911yescharacteristic
462C23#55111yescharacteristic
463C23#55311yescharacteristic
464C23#55411yescharacteristic
465C23#55911yescharacteristic
466Q8—22no—
467Q8—22no—
468Q8—41no—
469Q8—41no—
470D4—22no—
471D4—22no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—41no—
477D4—41no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4C2×C4×D411yescharacteristic
501C2×C4C2×C4×Q811yescharacteristic
502C2×C4C4×#4111yescharacteristic
503C2×C4C4×#4111yescharacteristic
504C2×C4#51811yescharacteristic
505C2×C4#51811yescharacteristic
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×#12311yescharacteristic
508C2×C4#52511yescharacteristic
509C2×C4#52911yescharacteristic
510C2×C4#52911yescharacteristic
511C2×C4#53311yescharacteristic
512C2×C4#53511yescharacteristic
513C2×C4#53511yescharacteristic
514C2×C4#54011yescharacteristic
515C2×C4#54011yescharacteristic
516C2×C4#55111yescharacteristic
517C2×C4#55111yescharacteristic
518C2×C4#55111yescharacteristic
519C2×C4#55111yescharacteristic
520C2×C4#55111yescharacteristic
521C2×C4#55311yescharacteristic
522C2×C4#55311yescharacteristic
523C2×C4#55311yescharacteristic
524C2×C4#55411yescharacteristic
525C2×C4#55411yescharacteristic
526C2×C4#55411yescharacteristic
527C2×C4#55911yescharacteristic
528C8—24no—
529C8—22no—
530C8—22no—
531C8—42no—
532C8—42no—
533C8—42no—
534C8—42no—
535C8—42no—
536C8—81no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—42no—
545C22—82no—
546C22—82no—
547C22—161no—
548C22C4×#12311yescharacteristic
549C22#236811yescharacteristic
550C22#237411yescharacteristic
551C22#251611yescharacteristic
552C22#252211yescharacteristic
553C22#258011yescharacteristic
554C22#259811yescharacteristic
555C4—44no—
556C4—44no—
557C4—42no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4—161no—
562C4C4×C22⋊Q811yescharacteristic
563C4#232811yescharacteristic
564C4#236811yescharacteristic
565C4#237411yescharacteristic
566C4#248211yescharacteristic
567C4#248711yescharacteristic
568C4#251611yescharacteristic
569C4#252211yescharacteristic
570C2—21no—
571C2—21no—
572C2—21no—
573C2—21no—
574C2—21no—
575C2—21no—
576C2—82no—
577C2#1424011yescharacteristic
578C2#2927012yes—
subgroupG/H
centerC2×C4#535
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2580
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20435C2semidirect
4. #11369C2semidirect
5. #11282C2semidirect
9. #10920C2semidirect
10. #10901C2semidirect
11. #10898C2semidirect
12. #10882C2semidirect
13. #10817C2semidirect
31. #2098C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20982
#204352
#204222
#113692
#112822
#110852
#109442
#109292
#109202
#109012
#108982
#108822
#108172
#107802
#107482