Finite Group Databasebeta

← #192192

Subgroups — #192192

Order 512.

1207 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2C2×#1870C211yesmaximal, characteristic
3#20851C212yesmaximal, complemented
4#17362C211yesmaximal, characteristic
5#8860C214yesmaximal, complemented
6#8859C214yesmaximal, complemented
7#8057C212yesmaximal, complemented
8C23×M16C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10#2975C2212yes—
11#2974—24no—
12#2973—21no—
13#2973C2212yes—
14#2972—21no—
15D4×M16—22no—
16D4×M16—22no—
17#2920—21no—
18C2×#430—22no—
19C2×#407—21no—
20C2×#407C2212yes—
21C2×#407C2212yes—
22C22×C22⋊C8—21no—
23C22×C22⋊C8C2211yescharacteristic
24#1870—21no—
25#1870—21no—
26#1870C2212yes—
27#1870C2212yes—
28#1855—24no—
29#1855C2218yescomplemented
30#1738—22no—
31C2×#336—21no—
32#1465C412yes—
33#1465C412yes—
34#1463C412yes—
35#1462C412yes—
36#1459C2212yes—
37#1459C2212yes—
38#1298C2218yescomplemented
39C23×D4Q811yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C22×M16—21no—
44C22×M16—21no—
45C22×M16—21no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—44no—
49C22×M16D412yes—
50C22×M16D412yes—
51C22×M16C2312yes—
52C23×C8—21no—
53C23×C8—21no—
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⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4Q811yescharacteristic
69C22×C42D411yescharacteristic
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—22no—
82#430—48no—
83C2×#108—28no—
84#423—21no—
85#423—21no—
86#407—22no—
87#407—22no—
88#407—22no—
89#407—22no—
90#407—22no—
91#407—22no—
92#407—22no—
93#407C2×C414yes—
94#407C2×C414yes—
95#407D414yes—
96#407C2314yes—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—44no—
108C2×C22⋊C8C2312yes—
109C2×C22⋊C8C2314yes—
110C4×M16—22no—
111#341—28no—
112#336—22no—
113#336—22no—
114#324C2×C414yes—
115#324C2×C414yes—
116#323C2×C414yes—
117#323C2×C414yes—
118C25C4⋊C411yescharacteristic
119C25C2×Q811yescharacteristic
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—42no—
123C22×D4—42no—
124C22×D4C4⋊C411yescharacteristic
125C22×D4C4⋊C411yescharacteristic
126C22×D4C2×Q811yescharacteristic
127C22×D4C2×Q811yescharacteristic
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4C4⋊C411yescharacteristic
137C23×C4C4⋊C411yescharacteristic
138C23×C4C4⋊C411yescharacteristic
139C23×C4C4⋊C411yescharacteristic
140C23×C4C4⋊C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×Q811yescharacteristic
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×M16—24no—
154C2×M16—24no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—28no—
160C2×M16—24no—
161C2×M16—44no—
162C2×M16—44no—
163C2×M16—48no—
164C2×M16—44no—
165C2×M16—44no—
166C2×M16—44no—
167C2×M16—42no—
168C2×M16—44no—
169C2×M16—44no—
170C2×M16—82no—
171C2×M16—82no—
172C2×M16C2×D414yes—
173C2×M16C2×D414yes—
174C22×C8—24no—
175C22×C8—22no—
176C22×C8—24no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—42no—
183C22×C8—41no—
184C22×C8—41no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—41no—
188C22×C8—41no—
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—
204C4×D4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C4⋊C411yescharacteristic
208C2×C4⋊C4C4⋊C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4C4⋊C411yescharacteristic
216C2×C22⋊C4C4⋊C411yescharacteristic
217C2×C22⋊C4C4⋊C411yescharacteristic
218C2×C22⋊C4C4⋊C411yescharacteristic
219C2×C22⋊C4C22×C411yescharacteristic
220C2×C22⋊C4C22×C411yescharacteristic
221C2×C22⋊C4C2×Q811yescharacteristic
222C2×C22⋊C4C2×Q811yescharacteristic
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42C4⋊C411yescharacteristic
228C2×C42C4⋊C411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231#108—416no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C22⋊C8—24no—
237C22⋊C8—24no—
238C22⋊C8—24no—
239C22⋊C8—24no—
240C22⋊C8—24no—
241C22⋊C8—24no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—24no—
245C22⋊C8—24no—
246C22⋊C8—44no—
247C22⋊C8—44no—
248C22⋊C8—42no—
249C22⋊C8C22×C418yes—
250#97—24no—
251C4×C8—24no—
252C24—22no—
253C24—21no—
254C24—21no—
255C24—42no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24C2×C4⋊C411yescharacteristic
263C24C2×C4⋊C411yescharacteristic
264C24C2×C4⋊C411yescharacteristic
265C24C2×C4⋊C411yescharacteristic
266C24C22⋊Q812yes—
267C24C22⋊Q812yes—
268C24C22×Q811yescharacteristic
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—48no—
282C2×D4—48no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—82no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—82no—
328C22×C4C2×C4⋊C411yescharacteristic
329C22×C4C2×C4⋊C411yescharacteristic
330C22×C4C2×C4⋊C411yescharacteristic
331C22×C4C2×C4⋊C411yescharacteristic
332C22×C4C2×C4⋊C411yescharacteristic
333C22×C4C2×C4⋊C411yescharacteristic
334C22×C4C2×C4⋊C411yescharacteristic
335C22×C4C2×C4⋊C411yescharacteristic
336C22×C4C4×D412yes—
337C22×C4C4×D412yes—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4#12011yescharacteristic
341C22×C4#12012yes—
342C22×C4#12011yescharacteristic
343C22×C4C22⋊Q812yes—
344C22×C4C22⋊Q812yes—
345C22×C4#12311yescharacteristic
346C22×C4#12311yescharacteristic
347C22×C4#12312yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D412yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4112yes—
352M16—28no—
353M16—48no—
354M16—48no—
355M16—44no—
356M16—48no—
357M16—44no—
358M16—44no—
359M16—44no—
360M16—44no—
361M16—82no—
362M16—82no—
363C2×C8—24no—
364C2×C8—28no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—44no—
368C2×C8—42no—
369C2×C8—44no—
370C2×C8—44no—
371C2×C8—42no—
372C2×C8—44no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—82no—
377C2×C8—82no—
378C2×C8—82no—
379C2×C8—82no—
380C2×C8—82no—
381C2×C8—82no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C23—22no—
403C23—22no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—24no—
408C23—24no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—41no—
418C23—42no—
419C23—41no—
420C23—42no—
421C23—42no—
422C23—41no—
423C23—42no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—48no—
432C23—42no—
433C23—42no—
434C23—81no—
435C23—81no—
436C23—84no—
437C23—84no—
438C23—84no—
439C23#38512yes—
440C23#38511yescharacteristic
441C23#38511yescharacteristic
442C23#43012yescomplemented
443C23C22×C4⋊C411yescharacteristic
444C23C2×C22⋊Q812yes—
445D4—28no—
446D4—44no—
447D4—44no—
448D4—44no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4#38511yescharacteristic
500C2×C4#38512yes—
501C2×C4#38511yescharacteristic
502C2×C4#43012yescomplemented
503C2×C4B(4,𝔽2)12yes—
504C2×C4#45812yes—
505C2×C4C2×C4×D412yes—
506C2×C4C2×#12011yescharacteristic
507C2×C4C2×#12311yescharacteristic
508C8—48no—
509C8—44no—
510C8—44no—
511C8—44no—
512C8—82no—
513C22—24no—
514C22—24no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—42no—
529C22—42no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—81no—
536C22—81no—
537C22—84no—
538C22—84no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—84no—
543C22—84no—
544C22—81no—
545C22#185812yes—
546C22#186712yes—
547C22C2×#38511yescharacteristic
548C22C2×#43011yescharacteristic
549C22C2×#43011yescharacteristic
550C4—22no—
551C4—22no—
552C4—21no—
553C4—21no—
554C4—22no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4#186712yes—
561C4C2×B(4,𝔽2)11yescharacteristic
562C4C2×#45811yescharacteristic
563C2—21no—
564C2—22no—
565C2—22no—
566C2—21no—
567C2—42no—
568C2—42no—
569C2—42no—
570C2—81no—
571C2#2085111yescharacteristic
572C2C2×#185811yescharacteristic
573C2C2×#186711yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20851C2semidirect
5. #8860C2semidirect
6. #8859C2semidirect
7. #8057C2semidirect
29. #1855C22semidirect
38. #1298C22semidirect
442. C23#430semidirect
502. C2×C4#430semidirect

Maximal subgroups

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

subgroupindex
#602302
C2×#18702
#208512
#173622
#88602
#88592
#80572