Finite Group Databasebeta

← #183174

Subgroups — #183174

Order 512.

990 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2994C211yesmaximal, characteristic
2#21403C211yesmaximal, characteristic
3#20538C211yesmaximal, characteristic, complemented
4C2×#1482C211yesmaximal, characteristic
5C2×#1453C211yesmaximal, characteristic, complemented
6#8212C211yesmaximal, characteristic, complemented
7#8211C211yesmaximal, characteristic, complemented
8#8209C211yesmaximal, characteristic, complemented
9#8206C211yesmaximal, characteristic, complemented
10#8205C211yesmaximal, characteristic, complemented
11#8202C211yesmaximal, characteristic, complemented
12#8200C211yesmaximal, characteristic, complemented
13C8×#103C211yesmaximal, characteristic, complemented
14#7893C211yesmaximal, characteristic, complemented
15#7873C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2994—24no—
18#2994C418yescomplemented
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#435—21no—
23C2×C8×D4—21no—
24C2×C8×D4C2211yescharacteristic
25C2×#433—21no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#424—21no—
30C22×C4⋊C8C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2285C2211yescharacteristic
33#1992C412yes—
34#1959C412yes—
35#1895C412yes—
36#1894C412yes—
37#1770—21no—
38#1770C2211yescharacteristic
39#1770C2211yescharacteristic
40C8×C4⋊C4—21no—
41C8×C4⋊C4C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C4×#103C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47#1750—21no—
48#1750—21no—
49#1482C418yescomplemented
50C2×Q8⋊C8C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×D4⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54#1453—24no—
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C22×M16C2×C412yes—
69C23×C8—21no—
70C23×C8—21no—
71#529—22no—
72#529—22no—
73#529C2×C414yes—
74#529C2×C414yes—
75C2×#126—21no—
76C2×#125C2×C411yescharacteristic
77C2×#124C2×C411yescharacteristic
78C2×#123—21no—
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×#121C2×C411yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4C2311yescharacteristic
84C2×#117—21no—
85C2×#117—21no—
86C22×C42C2311yescharacteristic
87#446—22no—
88#446C2×C414yes—
89C8×Q8—22no—
90C8×Q8C2×C414yes—
91#436—22no—
92#436C2×C414yes—
93#435—28no—
94C8×D4—28no—
95C8×D4—22no—
96C8×D4C2×C414yes—
97#433—28no—
98#431—22no—
99#431—22no—
100#431C2×C414yes—
101#431C2×C414yes—
102#431C2×C414yes—
103#431D414yes—
104#424—28no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8C2×C414yes—
111C2×C4⋊C8D411yescharacteristic
112C2×C4⋊C8D411yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×#103C2×C412yes—
118C2×#103C2×C412yes—
119C2×#102C2×C412yes—
120C2×#102C2×C412yes—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8D411yescharacteristic
126C2×C22⋊C8D411yescharacteristic
127C4×M16—22no—
128C4×M16—22no—
129C2×#97—21no—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97C2×C412yes—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143#384C2×C412yes—
144#382—21no—
145#380—21no—
146#380C2×C412yes—
147C4×C4⋊C4—21no—
148C4×C4⋊C4C2311yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
150#340C2×C412yes—
151#340C2×C412yes—
152#340C2×C412yes—
153#340C2×C412yes—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158Q8⋊C8C2×C414yes—
159Q8⋊C8C2×C414yes—
160D4⋊C8C2×C414yes—
161D4⋊C8C2×C414yes—
162#322—22no—
163#322—22no—
164#322—22no—
165C82—22no—
166C22×Q8C22×C411yescharacteristic
167C22×D4—21no—
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4#4111yescharacteristic
173C2×M16—24no—
174C2×M16—24no—
175C2×M16—24no—
176C2×M16—42no—
177C2×M16—81no—
178C2×M16—81no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—24no—
186C22×C8—24no—
187C22×C8—81no—
188C22×C8—81no—
189C22×C8—81no—
190C22×C8—81no—
191C22×C8C22×C412yes—
192C22×C8C22×C412yes—
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197#126—24no—
198#126—24no—
199#125—22no—
200#125C4214yes—
201#124—22no—
202#124C4214yes—
203#123—24no—
204#123—24no—
205C22⋊Q8—22no—
206C22⋊Q8C4214yes—
207#121—22no—
208#121C4214yes—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8C22×C412yes—
212C4×Q8C22×C412yes—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4C22×C412yes—
218C4×D4C22×C412yes—
219#117—24no—
220#117—24no—
221#117—81no—
222#117—81no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C412yes—
244C2×C42C22×C412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D412yes—
247C2×C42#4111yescharacteristic
248C2×C42#4112yes—
249C2×C42C2411yescharacteristic
250C4⋊C8—24no—
251C4⋊C8—24no—
252C4⋊C8—24no—
253C4⋊C8—24no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8C22⋊C414yes—
259C4⋊C8C22⋊C414yes—
260C4⋊C8C22×C412yes—
261C4⋊C8C22×C412yes—
262C4⋊C8C22×C412yes—
263C4⋊C8C22×C412yes—
264#103—24no—
265#103—42no—
266#102—24no—
267#102—42no—
268C22⋊C8—24no—
269C22⋊C8—24no—
270C22⋊C8—24no—
271C22⋊C8—24no—
272C22⋊C8C22⋊C414yes—
273C22⋊C8C22⋊C414yes—
274#97—24no—
275#97—24no—
276#97—81no—
277#97—81no—
278C4×C8—24no—
279C4×C8—24no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—81no—
287C4×C8—81no—
288C4×C8C22×C412yes—
289C4×C8C22×C412yes—
290C4×C8C22×C412yes—
291C4×C8C22×C412yes—
292C4×C8C2×D412yes—
293C4×C8C2×D412yes—
294#95—22no—
295#95—21no—
296#95—21no—
297C24—21no—
298C24C4×D411yescharacteristic
299C2×Q8—21no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8C2×C4212yes—
303C2×D4—24no—
304C2×D4—21no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—44no—
308C2×D4C2×C4212yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4C2×C4212yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4#11712yes—
330C22×C4C4×D412yes—
331C22×C4C4×D412yes—
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4C4×D411yescharacteristic
335C22×C4C4×D412yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4111yescharacteristic
339M16—48no—
340M16—84no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—84no—
371C2×C8—84no—
372C2×C8C2×C22⋊C412yes—
373C2×C8C2×C22⋊C412yes—
374C2×C8C2×C22⋊C412yes—
375C2×C8C2×C22⋊C412yes—
376C2×C8#13114yes—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—44no—
389C4⋊C4—82no—
390C4⋊C4C2×C4212yes—
391C4⋊C4C2×C4212yes—
392C22⋊C4—24no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—82no—
399C42—22no—
400C42—22no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42C2×C4212yes—
406C42C2×C22⋊C412yes—
407C42#11712yes—
408C42C4×D412yes—
409C42C4×D412yes—
410C42C4×D412yes—
411C42C23×C411yescharacteristic
412C42C23×C411yescharacteristic
413C42C22×D411yescharacteristic
414C42C2×#4111yescharacteristic
415C23—22no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23C4×C22⋊C412yes—
425C23C2×C4×D411yescharacteristic
426Q8—22no—
427Q8—22no—
428Q8—42no—
429D4—22no—
430D4—22no—
431D4—48no—
432D4—42no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4C4×C22⋊C412yes—
465C2×C4C4×C22⋊C412yes—
466C2×C4C4×C22⋊C412yes—
467C2×C4C22×C4211yescharacteristic
468C2×C4C22×C22⋊C411yescharacteristic
469C2×C4C2×#11711yescharacteristic
470C2×C4C2×C4×D411yescharacteristic
471C2×C4C2×C4×D411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×#13112yes—
474C2×C4#56812yes—
475C2×C4#57611yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57711yescharacteristic
478C2×C4#57811yescharacteristic
479C8—22no—
480C8—22no—
481C8—22no—
482C8—22no—
483C8—24no—
484C8—42no—
485C8—42no—
486C8—42no—
487C8—42no—
488C8—42no—
489C8—42no—
490C8—42no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—82no—
500C22—82no—
501C22C2×C4×C22⋊C411yescharacteristic
502C22#293512yes—
503C22#294611yescharacteristic
504C22#294711yescharacteristic
505C22#295512yes—
506C4—22no—
507C4—22no—
508C4—21no—
509C4—21no—
510C4—22no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—82no—
515C4—82no—
516C4C4×#13111yescharacteristic
517C4#287511yescharacteristic
518C4#289711yescharacteristic
519C4#289811yescharacteristic
520C4#293312yes—
521C4#295111yescharacteristic
522C4#295211yescharacteristic
523C2—42no—
524C2—82no—
525C2#1808212yes—
526C2#1808312yes—
527C2#2038711yescharacteristic
528C2#2041511yescharacteristic
529C2#2042011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20538C2semidirect
5. C2×#1453C2semidirect
6. #8212C2semidirect
7. #8211C2semidirect
8. #8209C2semidirect
9. #8206C2semidirect
10. #8205C2semidirect
11. #8202C2semidirect
12. #8200C2semidirect
13. C8×#103C2semidirect
14. #7893C2semidirect
15. #7873C2semidirect
18. #2994C4semidirect
49. #1482C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29942
#214032
#205382
C2×#14822
C2×#14532
#82122
#82112
#82092
#82062
#82052
#82022
#82002
C8×#1032
#78932
#78732