Finite Group Databasebeta

← #370583

Subgroups — #370583

Order 512.

1236 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2C2×#1986C212yesmaximal
3C4⋊C4×M16C211yesmaximal, characteristic
4#13848C211yesmaximal, characteristic
5#11386C212yesmaximal
6#11226C212yesmaximal
7#10886C212yesmaximal
8#10852C211yesmaximal, characteristic
9#10851C211yesmaximal, characteristic
10#10840C211yesmaximal, characteristic
11#10701C211yesmaximal, characteristic
12C2×#533C2212yes—
13C2×#423C2211yescharacteristic
14C2×C4×M16C2212yes—
15C2×C4×M16C2211yescharacteristic
16#2552C412yescomplemented
17#2551C412yescomplemented
18#2550C412yescomplemented
19#2550C412yescomplemented
20#2549—21no—
21#2549—21no—
22#2440—22no—
23#2399—21no—
24#2399—21no—
25C4×C4⋊Q8—22no—
26#2296C2212yes—
27#2285C2211yescharacteristic
28C4×#117C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1986—28no—
34#1771—22no—
35#1770—21no—
36C8×C4⋊C4—21no—
37C2×#339C2212yes—
38C2×#339C2212yes—
39C2×#339C2212yes—
40C2×#339C2212yes—
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47#1728C2212yes—
48#1444—21no—
49#1442—21no—
50#1441—22no—
51C4×#95C2212yes—
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55#533—24no—
56#533—24no—
57#533—42no—
58#533—42no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8C2×C412yes—
62C2×#125C2×C412yes—
63C2×#125C2×C412yes—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×#117C2312yes—
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#423—22no—
80#423Q814yescomplemented
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C4×M16—22no—
84C4×M16—24no—
85C4×M16—42no—
86C4×M16Q814yescomplemented
87C2×#97—21no—
88C2×#97—22no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91#398—22no—
92#398—22no—
93#398—22no—
94#398—22no—
95#395—22no—
96#395—22no—
97#395—22no—
98#395—22no—
99#391—22no—
100#391—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C412yes—
109#385—22no—
110#385—22no—
111#384—22no—
112#384C2×C412yes—
113#382C2×C411yescharacteristic
114#382C2×C411yescharacteristic
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—42no—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4D412yes—
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—21no—
136#376—22no—
137#376—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—22no—
143C2×#95—21no—
144C2×#95D411yescharacteristic
145C2×#95D411yescharacteristic
146C43—21no—
147#339—24no—
148#339—24no—
149#339—22no—
150#339—22no—
151#339—24no—
152#339—24no—
153#339—42no—
154#339—42no—
155#339D414yes—
156#339Q814yescomplemented
157#336—22no—
158#336—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—22no—
178C2×M16—21no—
179C2×M16—22no—
180C2×M16—21no—
181C2×M16—22no—
182C2×M16C2×D412yes—
183C2×M16C2×Q812yes—
184C2×M16C2×Q814yes—
185C2×M16C2×Q812yes—
186C2×M16C2×Q812yes—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C4⋊Q8—24no—
191C4⋊Q8—24no—
192C4⋊Q8—42no—
193C4⋊Q8—42no—
194C4⋊Q8—48no—
195#125—24no—
196#125—24no—
197#125—42no—
198#125—42no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201C22⋊Q8—44no—
202C22⋊Q8—44no—
203C4×Q8—44no—
204C4×Q8—44no—
205C4×Q8—44no—
206C4×Q8—44no—
207#117—21no—
208#117—21no—
209#117—21no—
210#117—21no—
211#117—22no—
212#117—22no—
213#117—22no—
214#117—22no—
215#117C2×D412yes—
216#117C2×Q812yes—
217#117#4112yes—
218#117#4112yes—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C22⋊C412yes—
238C2×C4⋊C4C22⋊C412yes—
239C2×C4⋊C4C22⋊C412yes—
240C2×C4⋊C4C22⋊C412yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C412yes—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42C22×C412yes—
274C2×C42C22×C412yes—
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×Q812yes—
283C2×C42C2×Q812yes—
284C2×C42C2×Q812yes—
285C2×C42#4112yes—
286C2×C42#4112yes—
287C4⋊C8—24no—
288C4⋊C8—24no—
289#97—24no—
290#97—44no—
291C4×C8—24no—
292C4×C8—44no—
293#95—22no—
294#95—22no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—42no—
309#95C22⋊C412yes—
310#95C22⋊C412yes—
311#95C22⋊C411yescharacteristic
312#95C22⋊C411yescharacteristic
313#95C22⋊C411yescharacteristic
314#95C22⋊C411yescharacteristic
315C24C22×D411yescharacteristic
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—82no—
321C2×Q8—82no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
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—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C412yes—
361C22×C4#12412yes—
362C22×C4#12412yes—
363C22×C4C4⋊Q812yes—
364C22×C4C4⋊Q812yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×Q811yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C22×Q811yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C4H(5,𝔽2)11yescharacteristic
377C22×C42−1+411yescharacteristic
378C22×C42−1+411yescharacteristic
379M16—22no—
380M16—22no—
381M16—22no—
382M16—22no—
383M16—22no—
384M16—24no—
385M16—24no—
386M16—42no—
387M16—42no—
388M16—41no—
389M16—41no—
390M16C4⋊Q814yescomplemented
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—42no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—416no—
420C4⋊C4—44no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4C4×D414yes—
425C4⋊C4C4×Q814yes—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—44no—
449C42—41no—
450C42—41no—
451C42—42no—
452C42—42no—
453C42—41no—
454C42—41no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—82no—
460C42C4×Q812yes—
461C42C4×Q812yes—
462C42C4×Q812yes—
463C42C4×Q812yes—
464C42C22⋊Q812yes—
465C42C22⋊Q812yes—
466C42C22⋊Q812yes—
467C42C22⋊Q812yes—
468C23—22no—
469C23—22no—
470C23C22×C22⋊C411yescharacteristic
471C23C2×#12411yescharacteristic
472C23C2×C4⋊Q811yescharacteristic
473C23D4211yescharacteristic
474C23#54611yescharacteristic
475C23#54811yescharacteristic
476C23D4×Q811yescharacteristic
477Q8—88no—
478Q8—88no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4#39112yes—
532C2×C4#39112yes—
533C2×C4#39112yes—
534C2×C4#39112yes—
535C2×C4#42114yes—
536C2×C4C2×C4×D411yescharacteristic
537C2×C4C2×C4×Q811yescharacteristic
538C2×C4C2×C4×Q811yescharacteristic
539C2×C4C2×C4×Q811yescharacteristic
540C2×C4#51811yescharacteristic
541C2×C4#51911yescharacteristic
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4C2×C22⋊Q811yescharacteristic
544C2×C4C2×C4⋊Q811yescharacteristic
545C2×C4#53211yescharacteristic
546C2×C4#53311yescharacteristic
547C2×C4#53311yescharacteristic
548C2×C4#54211yescharacteristic
549C2×C4#54211yescharacteristic
550C2×C4#54411yescharacteristic
551C2×C4#54411yescharacteristic
552C2×C4D4211yescharacteristic
553C2×C4#54611yescharacteristic
554C2×C4#54711yescharacteristic
555C2×C4#54811yescharacteristic
556C2×C4D4×Q811yescharacteristic
557C2×C4D4×Q811yescharacteristic
558C2×C4#55411yescharacteristic
559C2×C4#55411yescharacteristic
560C2×C4#55511yescharacteristic
561C2×C4#55611yescharacteristic
562C2×C4#55711yescharacteristic
563C2×C4Q8211yescharacteristic
564C8—24no—
565C8—42no—
566C8—44no—
567C22—22no—
568C22—22no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—22no—
575C22C2×#39111yescharacteristic
576C22D4×C22⋊C411yescharacteristic
577C22Q8×C22⋊C411yescharacteristic
578C22#249111yescharacteristic
579C22#249211yescharacteristic
580C22#255011yescharacteristic
581C22#255111yescharacteristic
582C22C2×#42112yes—
583C22#290112yes—
584C4—22no—
585C4—22no—
586C4—22no—
587C4—22no—
588C4—24no—
589C4—44no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—44no—
594C4—84no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4C4×C4⋊Q811yescharacteristic
600C4#233111yescharacteristic
601C4#236511yescharacteristic
602C4#236911yescharacteristic
603C4#243911yescharacteristic
604C4#244011yescharacteristic
605C4#245111yescharacteristic
606C4#245211yescharacteristic
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2#1405311yescharacteristic
612C2#2130611yescharacteristic
613C2#2130711yescharacteristic
614C2#2243412yes—
615C2#2252912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2552C4semidirect
17. #2551C4semidirect
18. #2550C4semidirect
19. #2550C4semidirect
80. #423Q8semidirect
86. C4×M16Q8semidirect
156. #339Q8semidirect
390. M16C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#349382
C2×#19862
C4⋊C4×M162
#138482
#113862
#112262
#108862
#108522
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