Finite Group Databasebeta

← #188854

Subgroups — #188854

Order 512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2974C211yesmaximal, characteristic
2#21948C212yesmaximal, complemented
3#17362C211yesmaximal, characteristic
4C2×#1465C211yesmaximal, characteristic
5#8613C214yesmaximal, complemented
6#8612C212yesmaximal, complemented
7#8611C212yesmaximal, complemented
8#8009C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10#2974—22no—
11#2974—22no—
12#2974—24no—
13C2×#436C412yes—
14C2×#435C412yes—
15C2×#435C2212yes—
16C2×C8×D4C2212yes—
17C2×#432C411yescharacteristic
18C2×#431C411yescharacteristic
19C2×#407C411yescharacteristic
20C2×#407C2212yes—
21C2×#407C2212yes—
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C411yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#1855—22no—
26#1855—22no—
27#1855—24no—
28#1810C2214yescomplemented
29#1810C2214yescomplemented
30#1465—21no—
31#1465—21no—
32#1465C2211yescharacteristic
33#1465C2212yes—
34#1465C2211yescharacteristic
35#1465C2212yes—
36#1465C2212yes—
37#1463—21no—
38#1462—21no—
39#1459—21no—
40#1459—21no—
41C2×#324—21no—
42C2×#323—21no—
43#1293C2214yescomplemented
44#1293C2214yescomplemented
45C23×D4Q811yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16C2×C412yes—
51C22×M16C2×C412yes—
52C23×C8C2×C412yes—
53C23×C8C2312yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4Q811yescharacteristic
63C2×C4×D4Q811yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4Q811yescharacteristic
69C22×C42C2×C411yescharacteristic
70#436—24no—
71#436—24no—
72#435—24no—
73#435—24no—
74#435—24no—
75#435—24no—
76C8×D4—24no—
77C8×D4—24no—
78#432—22no—
79#432—22no—
80#431—22no—
81#431—21no—
82#431—21no—
83C2×#108—24no—
84C2×#108—24no—
85C2×C4⋊C8C2×C412yes—
86C2×C4⋊C8C2×C412yes—
87#407—22no—
88#407—22no—
89#407—22no—
90#407—22no—
91#407—22no—
92#407—22no—
93#407—22no—
94#407—22no—
95#407—22no—
96#407—22no—
97#407C2312yes—
98#407C2314yes—
99#407C2312yes—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8C2×C411yescharacteristic
108C2×C22⋊C8C2×C412yes—
109C2×C22⋊C8C2×C411yescharacteristic
110C2×C22⋊C8C2×C412yes—
111C2×C22⋊C8C2×C411yescharacteristic
112C2×C22⋊C8C2×C411yescharacteristic
113C2×C22⋊C8C2×C412yes—
114C2×C22⋊C8D412yes—
115C2×C22⋊C8D412yes—
116C2×C22⋊C8D414yes—
117C2×C22⋊C8C2312yes—
118C2×#97C2×C412yes—
119C2×C4×C8C2×C412yes—
120#341—24no—
121#341—24no—
122#324—22no—
123#324—22no—
124#324—21no—
125#324—21no—
126#324—22no—
127#323—22no—
128#323—22no—
129#323—21no—
130#323—21no—
131#323—22no—
132C25C4⋊C411yescharacteristic
133C25C2×Q811yescharacteristic
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4C4⋊C411yescharacteristic
139C22×D4C4⋊C411yescharacteristic
140C22×D4C2×Q811yescharacteristic
141C22×D4C2×Q811yescharacteristic
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4C4⋊C411yescharacteristic
149C23×C4C4⋊C411yescharacteristic
150C23×C4C4⋊C411yescharacteristic
151C23×C4C4⋊C411yescharacteristic
152C23×C4C4⋊C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×Q811yescharacteristic
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C2×M16—24no—
166C2×M16—24no—
167C2×M16—24no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16—82no—
173C2×M16—82no—
174C2×M16—84no—
175C2×M16—84no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—42no—
179C22×C8—42no—
180C22×C8—82no—
181C22×C8—82no—
182C22×C8C22×C412yes—
183C22×C8C22×C412yes—
184C22×C8C22×C412yes—
185C22×C8C22×C412yes—
186C22×C8C2×D414yes—
187C22×C8C2×D414yes—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4C4⋊C411yescharacteristic
207C2×C4⋊C4C4⋊C411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4C4⋊C411yescharacteristic
215C2×C22⋊C4C4⋊C411yescharacteristic
216C2×C22⋊C4C4⋊C411yescharacteristic
217C2×C22⋊C4C4⋊C411yescharacteristic
218C2×C22⋊C4C2×D411yescharacteristic
219C2×C22⋊C4C2×D411yescharacteristic
220C2×C22⋊C4C2×Q811yescharacteristic
221C2×C22⋊C4C2×Q811yescharacteristic
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C4⋊C411yescharacteristic
227C2×C42C4⋊C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230#108—84no—
231#108—84no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—24no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—24no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—24no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8C22⋊C412yes—
254C22⋊C8C22⋊C411yescharacteristic
255C22⋊C8C22⋊C411yescharacteristic
256C22⋊C8C22⋊C412yes—
257C22⋊C8C22⋊C412yes—
258C22⋊C8C22⋊C414yes—
259C22⋊C8C22⋊C414yes—
260#97—24no—
261C4×C8—22no—
262C4×C8—22no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—42no—
270C24—42no—
271C24—81no—
272C24—81no—
273C24C2×C4⋊C411yescharacteristic
274C24C2×C4⋊C411yescharacteristic
275C24C2×C4⋊C411yescharacteristic
276C24C2×C4⋊C411yescharacteristic
277C24C22⋊Q812yes—
278C24C22⋊Q812yes—
279C24C22×Q811yescharacteristic
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—48no—
293C2×D4—48no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C4⋊C411yescharacteristic
334C22×C4C2×C4⋊C411yescharacteristic
335C22×C4C2×C4⋊C411yescharacteristic
336C22×C4C2×C4⋊C411yescharacteristic
337C22×C4C2×C4⋊C411yescharacteristic
338C22×C4C2×C4⋊C411yescharacteristic
339C22×C4C2×C4⋊C411yescharacteristic
340C22×C4C2×C4⋊C411yescharacteristic
341C22×C4#11711yescharacteristic
342C22×C4#11711yescharacteristic
343C22×C4#11712yes—
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4C22⋊Q812yes—
349C22×C4C22⋊Q812yes—
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D412yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C2×#4112yes—
354M16—44no—
355M16—44no—
356M16—44no—
357M16—44no—
358M16—84no—
359M16—84no—
360M16—84no—
361M16—84no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—44no—
365C2×C8—44no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—44no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—82no—
374C2×C8—82no—
375C2×C8—84no—
376C2×C8—84no—
377C2×C8C2×C22⋊C412yes—
378C2×C8C2×C22⋊C412yes—
379C2×C8C2×C22⋊C414yes—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—24no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—42no—
431C23—81no—
432C23—81no—
433C23—84no—
434C23—84no—
435C23—84no—
436C23—84no—
437C23#38012yes—
438C23#38011yescharacteristic
439C23#38011yescharacteristic
440C23#43012yes—
441C23C22×C4⋊C411yescharacteristic
442C23C2×C22⋊Q812yes—
443D4—28no—
444D4—44no—
445D4—44no—
446D4—44no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
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—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4#38011yescharacteristic
493C2×C4#38011yescharacteristic
494C2×C4#38012yes—
495C2×C4#43012yes—
496C2×C4C22×C22⋊C411yescharacteristic
497C2×C4C2×#11711yescharacteristic
498C2×C4C2×#12112yes—
499C8—22no—
500C8—22no—
501C8—24no—
502C8—42no—
503C8—42no—
504C8—84no—
505C8—82no—
506C8—82no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—22no—
514C22—21no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—21no—
519C22—22no—
520C22—22no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—81no—
533C22—81no—
534C22—84no—
535C22—84no—
536C22—81no—
537C22—81no—
538C22—84no—
539C22—84no—
540C22#181312yes—
541C22C2×#38011yescharacteristic
542C22C2×#43011yescharacteristic
543C22C2×#43011yescharacteristic
544C4—22no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—22no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4#288312yes—
556C2—21no—
557C2—21no—
558C2—22no—
559C2—42no—
560C2—41no—
561C2—42no—
562C2—42no—
563C2—81no—
564C2#2063311yescharacteristic
565C2#2066711yescharacteristic
566C2C2×#181311yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
5. #8613C2semidirect
6. #8612C2semidirect
7. #8611C2semidirect
8. #8009C2semidirect
28. #1810C22semidirect
29. #1810C22semidirect
43. #1293C22semidirect
44. #1293C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29742
#219482
#173622
C2×#14652
#86132
#86122
#86112
#80092