Finite Group Databasebeta

← #368290

Subgroups — #368290

Order 512.

1290 conjugacy classes of subgroups.

The subgroup lattice has 613 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×#384C211yesmaximal, characteristic
6#11340C212yesmaximal
7#11276C212yesmaximal
8#10899C212yesmaximal
9#10891C212yesmaximal
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×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#2274C2212yes—
29C2×C4×C4⋊C4—22no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C43C2211yescharacteristic
33#1994C418yescomplemented
34#1989—22no—
35#1988—21no—
36#1987—21no—
37#1949—22no—
38#1948—21no—
39#1947—21no—
40C2×#343C2212yes—
41C2×#343C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C4⋊#95—22no—
50C4⋊#95C2211yescharacteristic
51C4⋊#95C2211yescharacteristic
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⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42D412yes—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#428—24no—
82#428—42no—
83C2×#107—22no—
84C2×#106—22no—
85#423—22no—
86#423—22no—
87#423—22no—
88#423C2×C414yes—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93#391—24no—
94#391—22no—
95#389—24no—
96#389—22no—
97#389—22no—
98#389—22no—
99#387—24no—
100#387—24no—
101#387—24no—
102#385—24no—
103#384—22no—
104#384—22no—
105#384—22no—
106#384—22no—
107#384D411yescharacteristic
108#384D411yescharacteristic
109#384Q814yescomplemented
110#382—22no—
111#382D412yes—
112#380—21no—
113#380—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
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×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—48no—
131C4×C22⋊C4—24no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95D412yes—
136C43—22no—
137C43C2×C414yes—
138#343—24no—
139#343—42no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#339—24no—
147#339—42no—
148#339C2×C418yes—
149#339C2×C418yes—
150C22×Q8—21no—
151C22×Q8—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
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×M16—22no—
172C2×M16C22×C414yes—
173C22×C8—22no—
174C22×C8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8C4⋊C414yescomplemented
178C4⋊Q8C4⋊C414yescomplemented
179#125—24no—
180#125C4⋊C418yescomplemented
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—84no—
190#117—22no—
191#117—22no—
192#117—22no—
193#117—22no—
194#117—22no—
195#117—22no—
196#117C22×C414yes—
197#117C2×Q814yes—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
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—28no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×Q812yes—
239C2×C4⋊C4C2×Q812yes—
240C2×C4⋊C4#4112yes—
241C2×C4⋊C4#4112yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—41no—
267C2×C42—41no—
268C2×C42C22×C412yes—
269C2×C42C22×C412yes—
270C2×C42C22×C412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×Q812yes—
276C2×C42C2×Q812yes—
277C2×C42#4112yes—
278C2×C42#4112yes—
279C2×C42#4112yes—
280#107—44no—
281#106—44no—
282C4⋊C8—24no—
283C4⋊C8—24no—
284C4⋊C8—42no—
285C4⋊C8—42no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—44no—
295#95—44no—
296#95—41no—
297#95—41no—
298C24C22×D411yescharacteristic
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—81no—
304C2×Q8—81no—
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—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4C4×D412yes—
347C22×C4C4×D412yes—
348C22×C4#12011yescharacteristic
349C22×C4#12011yescharacteristic
350C22×C4#12011yescharacteristic
351C22×C4#12011yescharacteristic
352C22×C4C22⋊Q812yes—
353C22×C4C22⋊Q814yes—
354C22×C4C22⋊Q812yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×Q811yescharacteristic
359C22×C4C22×Q811yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C42−1+411yescharacteristic
367C22×C42−1+411yescharacteristic
368C22×C42−1+411yescharacteristic
369M16—24no—
370M16—24no—
371M16—42no—
372M16—42no—
373M16—42no—
374M16—42no—
375M16—42no—
376M16—42no—
377C2×C8—24no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—416no—
402C4⋊C4—416no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—84no—
416C4⋊C4—84no—
417C4⋊C4C2×C4⋊C414yes—
418C4⋊C4C2×C4⋊C414yes—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C42—216no—
424C42—24no—
425C42—22no—
426C42—24no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—48no—
432C42—41no—
433C42—41no—
434C42—42no—
435C42—42no—
436C42—44no—
437C42—82no—
438C42—82no—
439C42—84no—
440C42—84no—
441C42—82no—
442C42C2×C4⋊C412yes—
443C42C2×C4⋊C412yes—
444C42C4×D412yes—
445C42C4×D412yes—
446C42C4×Q814yescomplemented
447C42C22⋊Q812yes—
448C42C22⋊Q812yes—
449C42#12312yes—
450C42#12312yes—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23C2×C4×D411yescharacteristic
456C23C2×#12011yescharacteristic
457C23C2×C22⋊Q811yescharacteristic
458C23C2×C22⋊Q811yescharacteristic
459C23#52511yescharacteristic
460C23#53611yescharacteristic
461C23#53711yescharacteristic
462Q8—48no—
463Q8—84no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4#38512yes—
521C2×C4#38512yes—
522C2×C4#38512yes—
523C2×C4#38512yes—
524C2×C4#45312yes—
525C2×C4#45612yes—
526C2×C4C22×C4⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×Q811yescharacteristic
529C2×C4C4×#4111yescharacteristic
530C2×C4#51911yescharacteristic
531C2×C4#52011yescharacteristic
532C2×C4C2×C22⋊Q811yescharacteristic
533C2×C4C2×#12311yescharacteristic
534C2×C4#52911yescharacteristic
535C2×C4#53311yescharacteristic
536C2×C4#54111yescharacteristic
537C2×C4#54211yescharacteristic
538C2×C4#54611yescharacteristic
539C2×C4#54711yescharacteristic
540C2×C4#54811yescharacteristic
541C2×C4#54811yescharacteristic
542C2×C4#54811yescharacteristic
543C2×C4D4×Q811yescharacteristic
544C2×C4D4×Q811yescharacteristic
545C2×C4#55011yescharacteristic
546C2×C4#55611yescharacteristic
547C2×C4#55611yescharacteristic
548C2×C4#55711yescharacteristic
549C2×C4#55711yescharacteristic
550C2×C4#55711yescharacteristic
551C2×C4#55711yescharacteristic
552C2×C4Q8211yescharacteristic
553C2×C4#55911yescharacteristic
554C8—44no—
555C8—82no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22C2×#38511yescharacteristic
569C22C4×C22⋊Q811yescharacteristic
570C22#232311yescharacteristic
571C22#240711yescharacteristic
572C22#240911yescharacteristic
573C22#258111yescharacteristic
574C22#258211yescharacteristic
575C22C2×#45311yescharacteristic
576C22C2×#45611yescharacteristic
577C22#301811yescharacteristic
578C22#301811yescharacteristic
579C4—28no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—44no—
584C4—44no—
585C4—44no—
586C4—44no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4#185914yes—
593C4Q8×C4⋊C411yescharacteristic
594C4#235111yescharacteristic
595C4#236511yescharacteristic
596C4#236711yescharacteristic
597C4#244611yescharacteristic
598C4#244811yescharacteristic
599C4#245111yescharacteristic
600C4#245311yescharacteristic
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2#1401711yescharacteristic
606C2C2×#185911yescharacteristic
607C2#2086311yescharacteristic
608C2#2417611yescharacteristic
609C2#2417911yescharacteristic
610C2#2431211yescharacteristic
611C2#2431511yescharacteristic
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
33. #1994C4semidirect
109. #384Q8semidirect
177. C4⋊Q8C4⋊C4semidirect
178. C4⋊Q8C4⋊C4semidirect
180. #125C4⋊C4semidirect
446. 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×#3842
#113402
#112762
#108992
#108912
#108472
#108392