Finite Group Databasebeta

← #481820

Subgroups — #481820

Order 512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34633C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic, complemented
3C2×#1464C211yesmaximal, characteristic, complemented
4#11801C212yesmaximal, complemented
5#11800C212yesmaximal, complemented
6#11798C214yesmaximal, complemented
7#11692C214yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525—21no—
10C22×#117C2211yescharacteristic
11C2×#432—21no—
12#2621C411yescharacteristic, complemented
13#2621C2211yescharacteristic
14#2620—21no—
15#2620—21no—
16#2619C411yescharacteristic, complemented
17#2619C2211yescharacteristic
18#2618—21no—
19#2618C412yescomplemented
20#2617—21no—
21#2617C2212yes—
22#2392—21no—
23#2392—21no—
24#2392C412yescomplemented
25#2392C2212yes—
26#2297—21no—
27#2285C411yescharacteristic, complemented
28C2×#381—21no—
29#1820—21no—
30#1820—21no—
31#1820C2212yes—
32#1820C2212yes—
33C2×#342—21no—
34C2×#342C2212yes—
35#1464—21no—
36#1464—21no—
37#1464C2212yes—
38#1464C2212yes—
39C2×#324—21no—
40C2×#323C2212yes—
41#1344C2218yes—
42#1343C2218yes—
43C22×#41—21no—
44C24×C4D411yescharacteristic
45#532—41no—
46#532—41no—
47#532—41no—
48#532—41no—
49C2×C4⋊Q8C2×C411yescharacteristic, complemented
50C2×#127C2×C411yescharacteristic, complemented
51C2×#124—21no—
52#525—48no—
53C2×#123—22no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121C2×C412yes—
60C2×#120—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yescomplemented
65C2×#117—21no—
66C2×#117—41no—
67C2×#117—41no—
68C2×#117—41no—
69C2×#117—41no—
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C22⋊C4—22no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4C2311yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42D411yescharacteristic
85#432—22no—
86#432—22no—
87#432—41no—
88#432—41no—
89C2×C4⋊C8—41no—
90C2×C22⋊C8—22no—
91C2×#97—41no—
92#398—21no—
93#398—21no—
94#398C2×C412yescomplemented
95#398C2×C412yescomplemented
96#397—21no—
97#397—21no—
98#397C2×C412yes—
99#397C2312yes—
100#396—22no—
101#396—22no—
102#396—22no—
103#396—22no—
104#394—21no—
105#394—21no—
106#394—22no—
107#394—22no—
108#394—22no—
109#394—22no—
110#394C2×C412yescomplemented
111#394C2×C412yescomplemented
112#392—21no—
113#392—21no—
114#392C2×C412yes—
115#392C2312yes—
116#391—21no—
117#391—21no—
118#390—21no—
119#390—21no—
120#386—22no—
121#386—22no—
122#384—21no—
123#382C2×C411yescharacteristic, complemented
124#382C2×C411yescharacteristic, complemented
125#381—22no—
126#381—22no—
127#381—41no—
128#381—41no—
129#380—21no—
130#380C2×C412yes—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2×C412yescomplemented
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C2×#95—41no—
136C2×#95—41no—
137#342—22no—
138#342—22no—
139#342—22no—
140#342C2314yes—
141#324—22no—
142#324—22no—
143#323—22no—
144#323C2314yes—
145C25C2×D411yescharacteristic
146C2×#41—44no—
147C2×#41—44no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—41no—
151C2×#41—41no—
152C22×Q8—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4C22×C412yes—
159C23×C4—24no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C22×C8—42no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180#127—41no—
181#127—41no—
182#124—42no—
183#124—42no—
184#123—48no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187#121—42no—
188#121—42no—
189#121—44no—
190#121—44no—
191#121—42no—
192#121—42no—
193#120—48no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—42no—
198C4×D4—48no—
199C4×D4—48no—
200#117—41no—
201#117—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—41no—
207#117—41no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—41no—
212#117—41no—
213#117—41no—
214#117—41no—
215#117—41no—
216#117—44no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4C22⋊C411yescharacteristic
257C2×C22⋊C4C22⋊C411yescharacteristic
258C2×C22⋊C4C22×C411yescharacteristic
259C2×C22⋊C4C2×D411yescharacteristic
260C2×C22⋊C4C2×D411yescharacteristic
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2411yescharacteristic
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C4⋊C8—42no—
282C4⋊C8—42no—
283C22⋊C8—24no—
284C22⋊C8—42no—
285#97—42no—
286#97—42no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—21no—
291#95—21no—
292#95—22no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—42no—
297#95C22×C412yes—
298#95C22×C412yes—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24—81no—
304C24—81no—
305C24C2×C22⋊C411yescharacteristic
306C24C2×C22⋊C411yescharacteristic
307C24AGL(1,ℤ/8)14yes—
308C24C22×D411yescharacteristic
309#41—84no—
310#41—84no—
311#41—84no—
312#41—84no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—44no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—82no—
330C2×D4—82no—
331C2×D4—82no—
332C2×D4—82no—
333C2×D4—82no—
334C2×D4—82no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4#9911yescharacteristic
385C22×C4#9911yescharacteristic
386C22×C4#9911yescharacteristic
387C22×C4#9911yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C2×C8—44no—
401C2×C8—84no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C22⋊C4—84no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—44no—
457C23—21no—
458C23—21no—
459C23—24no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—81no—
479C23—81no—
480C23—84no—
481C23—84no—
482C23C2×#9911yescharacteristic
483C23C2×#9911yescharacteristic
484C23#41812yes—
485C23#41812yes—
486C23#42112yes—
487C23C22×C22⋊C411yescharacteristic
488C23C2×AGL(1,ℤ/8)12yes—
489Q8—82no—
490Q8—82no—
491D4—82no—
492D4—82no—
493D4—84no—
494D4—84no—
495D4—88no—
496D4—88no—
497D4—84no—
498D4—84no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—168no—
562C2×C4C2×#9911yescharacteristic
563C2×C4C2×#9911yescharacteristic
564C2×C4C2×#9911yescharacteristic
565C2×C4C2×#9911yescharacteristic
566C8—84no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—22no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—48no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—81no—
585C22—81no—
586C22—84no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22#151012yes—
595C22#151012yes—
596C22C22×#9911yescharacteristic
597C22C2×#41811yescharacteristic
598C22C2×#42111yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—164no—
617C2—21no—
618C2—22no—
619C2—21no—
620C2—22no—
621C2—42no—
622C2—42no—
623C2—162no—
624C2C2×#151011yescharacteristic
625C2#3334711yescharacteristic
626C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#34633C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1820C2semidirect
3. C2×#1464C2semidirect
4. #11801C2semidirect
5. #11800C2semidirect
6. #11798C2semidirect
7. #11692C2semidirect
8. C2×#532C4semidirect
12. #2621C4semidirect
16. #2619C4semidirect
19. #2618C4semidirect
24. #2392C4semidirect
27. #2285C4semidirect
49. C2×C4⋊Q8C2×C4semidirect
50. C2×#127C2×C4semidirect
64. C2×C4×D4C2×C4semidirect
94. #398C2×C4semidirect
95. #398C2×C4semidirect
110. #394C2×C4semidirect
111. #394C2×C4semidirect
123. #382C2×C4semidirect
124. #382C2×C4semidirect
132. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#346332
C2×#18202
C2×#14642
#118012
#118002
#117982
#116922