Finite Group Databasebeta

← #368291

Subgroups — #368291

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2C2×#1994C211yesmaximal, characteristic
3#21312C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic
5C4×#382C211yesmaximal, characteristic
6#11339C212yesmaximal
7#11276C212yesmaximal
8#10899C212yesmaximal
9#10892C212yesmaximal
10#10847C211yesmaximal, characteristic
11#10839C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#428C2212yes—
14C2×#423C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2370—22no—
17#2369—21no—
18#2365—21no—
19#2355—21no—
20#2355—22no—
21Q8×C4⋊C4—21no—
22#2329—22no—
23#2317C2212yes—
24C4×C22⋊Q8—22no—
25#2285C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28#2274C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1994C418yescomplemented
33#1989—22no—
34#1988—21no—
35#1987—21no—
36#1949—22no—
37#1948—21no—
38#1947—21no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48C4⋊#95C2211yescharacteristic
49C4⋊#95C2211yescharacteristic
50#1434—22no—
51C4×#95—22no—
52C22×M16C2312yes—
53C22×M16C2312yes—
54#533—22no—
55#533—22no—
56#533C2×C414yes—
57#533Q814yes—
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8D411yescharacteristic
60C2×#125D412yes—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117C2312yes—
67C2×#117C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—21no—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42D412yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#428—24no—
80#428—42no—
81C2×#107—22no—
82C2×#106—22no—
83#423—22no—
84#423—22no—
85#423—22no—
86#423C2×C414yes—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91#391—24no—
92#391—22no—
93#389—24no—
94#389—22no—
95#389—22no—
96#389—22no—
97#387—24no—
98#387—24no—
99#387—24no—
100#385—24no—
101#384—22no—
102#384D411yescharacteristic
103#384D411yescharacteristic
104#382—22no—
105#382—22no—
106#382—22no—
107#382—22no—
108#382D412yes—
109#382Q814yescomplemented
110#380—21no—
111#380—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C22⋊C4—24no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95D412yes—
135C43—22no—
136C43C2×C414yes—
137#343—24no—
138#343—42no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#339—24no—
146#339—42no—
147#339C2×C418yes—
148#339C2×C418yes—
149C22×Q8—21no—
150C22×Q8—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16C22×C414yes—
172C22×C8—22no—
173C22×C8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8C4⋊C414yes—
177C4⋊Q8C4⋊C414yes—
178#125—24no—
179#125C4⋊C418yes—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—84no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117—22no—
193#117—22no—
194#117—22no—
195#117C22×C414yes—
196#117C2×Q814yes—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×Q812yes—
232C2×C4⋊C4C2×Q812yes—
233C2×C4⋊C4#4112yes—
234C2×C4⋊C4#4112yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42C22×C412yes—
262C2×C42C22×C412yes—
263C2×C42C22×C412yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×Q812yes—
269C2×C42C2×Q812yes—
270C2×C42#4112yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273#107—44no—
274#106—44no—
275C4⋊C8—24no—
276C4⋊C8—24no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279#95—28no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—28no—
289#95—44no—
290#95—44no—
291#95—41no—
292#95—41no—
293C24C22×D411yescharacteristic
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—81no—
299C2×Q8—81no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4C4×D412yes—
342C22×C4C4×D412yes—
343C22×C4#12011yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4#12011yescharacteristic
347C22×C4C22⋊Q812yes—
348C22×C4C22⋊Q814yes—
349C22×C4C22⋊Q812yes—
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q811yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C42−1+411yescharacteristic
362C22×C42−1+411yescharacteristic
363C22×C42−1+411yescharacteristic
364M16—24no—
365M16—24no—
366M16—42no—
367M16—42no—
368M16—42no—
369M16—42no—
370M16—42no—
371M16—42no—
372C2×C8—24no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—81no—
407C4⋊C4—81no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4C2×C4⋊C414yes—
411C4⋊C4C2×C4⋊C414yes—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C42—24no—
417C42—22no—
418C42—24no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—48no—
424C42—48no—
425C42—41no—
426C42—41no—
427C42—42no—
428C42—42no—
429C42—44no—
430C42—82no—
431C42—82no—
432C42—84no—
433C42—84no—
434C42—82no—
435C42C2×C4⋊C412yes—
436C42C2×C4⋊C412yes—
437C42C4×D412yes—
438C42C4×D412yes—
439C42C4×Q814yescomplemented
440C42C22⋊Q812yes—
441C42C22⋊Q812yes—
442C42#12312yes—
443C42#12312yes—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23C2×C4×D411yescharacteristic
449C23C2×#12011yescharacteristic
450C23C2×C22⋊Q811yescharacteristic
451C23C2×C22⋊Q811yescharacteristic
452C23#52511yescharacteristic
453C23#53611yescharacteristic
454C23#53711yescharacteristic
455Q8—48no—
456Q8—84no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4#38512yes—
514C2×C4#38512yes—
515C2×C4#38512yes—
516C2×C4#38512yes—
517C2×C4#45412yes—
518C2×C4#45512yes—
519C2×C4C22×C4⋊C411yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C2×C4×Q811yescharacteristic
522C2×C4C4×#4111yescharacteristic
523C2×C4#51911yescharacteristic
524C2×C4#52011yescharacteristic
525C2×C4C2×C22⋊Q811yescharacteristic
526C2×C4C2×#12311yescharacteristic
527C2×C4#52911yescharacteristic
528C2×C4#53311yescharacteristic
529C2×C4#54111yescharacteristic
530C2×C4#54211yescharacteristic
531C2×C4#54611yescharacteristic
532C2×C4#54711yescharacteristic
533C2×C4#54811yescharacteristic
534C2×C4#54811yescharacteristic
535C2×C4#54811yescharacteristic
536C2×C4D4×Q811yescharacteristic
537C2×C4D4×Q811yescharacteristic
538C2×C4#55011yescharacteristic
539C2×C4#55611yescharacteristic
540C2×C4#55611yescharacteristic
541C2×C4#55711yescharacteristic
542C2×C4#55711yescharacteristic
543C2×C4#55711yescharacteristic
544C2×C4#55711yescharacteristic
545C2×C4Q8211yescharacteristic
546C2×C4#55911yescharacteristic
547C8—44no—
548C8—82no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22C2×#38511yescharacteristic
562C22C4×C22⋊Q811yescharacteristic
563C22#232311yescharacteristic
564C22#240711yescharacteristic
565C22#240911yescharacteristic
566C22#258111yescharacteristic
567C22#258211yescharacteristic
568C22C2×#45511yescharacteristic
569C22C2×#45411yescharacteristic
570C22#301811yescharacteristic
571C22#301811yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—44no—
576C4—44no—
577C4—44no—
578C4—44no—
579C4—44no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4#186014yes—
586C4Q8×C4⋊C411yescharacteristic
587C4#235111yescharacteristic
588C4#236511yescharacteristic
589C4#236711yescharacteristic
590C4#244611yescharacteristic
591C4#244811yescharacteristic
592C4#245111yescharacteristic
593C4#245311yescharacteristic
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2#1401711yescharacteristic
599C2C2×#186011yescharacteristic
600C2#2086311yescharacteristic
601C2#2417711yescharacteristic
602C2#2417811yescharacteristic
603C2#2431311yescharacteristic
604C2#2431411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
32. #1994C4semidirect
109. #382Q8semidirect
439. C42C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#342132
C2×#19942
#213122
#211702
C4×#3822
#113392
#112762
#108992
#108922
#108472
#108392