Finite Group Databasebeta

← #358530

Subgroups — #358530

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2320C211yesmaximal, characteristic
2C2×C4×#107C211yesmaximal, characteristic
3C2×#1763C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13783C211yesmaximal, characteristic
6#11328C212yesmaximal
7#11292C212yesmaximal
8#11124C212yesmaximal
9#11067C212yesmaximal
10#10703C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2320—24no—
20#2320—24no—
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#387—22no—
25C2×#383—21no—
26C2×#382—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376C2211yescharacteristic
31C4×#107—24no—
32#1763—24no—
33C4×#103C418yescomplemented
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1436—22no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45#1434—21no—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C2×C4×Q8D414yes—
55C2×C4×Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42—21no—
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70C2×#107—24no—
71C2×#107—24no—
72C2×#107—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103C2×C414yes—
79C2×#103C2×C414yes—
80C2×#97—22no—
81C2×#97—22no—
82C2×#97—22no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8C2×C414yes—
86#391—24no—
87#391—24no—
88#389—24no—
89#389—24no—
90#389—24no—
91#389—24no—
92#387—216no—
93#383—28no—
94#382—28no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C414yes—
100C4×C4⋊C4D414yes—
101#376—22no—
102#376—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114C2×#95D412yes—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340C2×C414yes—
120#340C2×C414yes—
121#336—24no—
122#336—24no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8C22×C412yes—
131C22×Q8C2×D412yes—
132C22×Q8C2×Q812yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C22×C412yes—
158C22×C8C22×C412yes—
159C22×C8#4114yes—
160C22×C8#4114yes—
161C4×Q8—28no—
162C4×Q8—28no—
163C4×Q8—28no—
164C4×Q8C4⋊C4116yescomplemented
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×Q812yes—
198C2×C4⋊C4#4114yes—
199C2×C4⋊C4#4114yes—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—42no—
212C2×C42—81no—
213C2×C42—81no—
214C2×C42—82no—
215C2×C42—81no—
216C2×C42—81no—
217C2×C42C22×C412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×Q812yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223#107—216no—
224#103—216no—
225#103—28no—
226#103—28no—
227#97—44no—
228C4×C8—28no—
229#95—28no—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—24no—
234#95—24no—
235#95—22no—
236#95—22no—
237#95—24no—
238#95—42no—
239#95—82no—
240#95—82no—
241C24C22×D411yescharacteristic
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8C2×C4⋊C414yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4C4×D412yes—
317C22×C4C4×D412yes—
318C22×C4#12012yes—
319C22×C4#12012yes—
320C22×C4C22⋊Q812yes—
321C22×C4C22⋊Q812yes—
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×Q811yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4112yes—
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4112yes—
331C22×C4C2×#4111yescharacteristic
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8#11714yes—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—216no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—88no—
367C4⋊C4—88no—
368C4⋊C4C2×C4⋊C414yes—
369C4⋊C4#11714yes—
370C42—216no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—84no—
376C42—84no—
377C42—84no—
378C42C2×C4⋊C414yes—
379C42C4×D414yes—
380C42C22⋊Q814yes—
381C42#12314yes—
382C23C2×C4×D411yescharacteristic
383C23C2×#12011yescharacteristic
384C23C2×C22⋊Q811yescharacteristic
385C23#52512yes—
386C23#53212yes—
387C23#57611yescharacteristic
388C23#57611yescharacteristic
389C23#57712yes—
390C23#57812yes—
391C23#57811yescharacteristic
392C23#57811yescharacteristic
393Q8—24no—
394Q8—24no—
395Q8—44no—
396Q8—44no—
397Q8—42no—
398Q8—42no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4#38514yes—
460C2×C4#38514yes—
461C2×C4C22×C4⋊C411yescharacteristic
462C2×C4C2×#11711yescharacteristic
463C2×C4C2×#11711yescharacteristic
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C4×#4111yescharacteristic
466C2×C4C4×#4111yescharacteristic
467C2×C4C2×C22⋊Q811yescharacteristic
468C2×C4C2×#12311yescharacteristic
469C2×C4#52512yes—
470C2×C4#52512yes—
471C2×C4#52912yes—
472C2×C4#52912yes—
473C2×C4#52912yes—
474C2×C4#52912yes—
475C2×C4#53212yes—
476C2×C4#53212yes—
477C2×C4#53312yes—
478C2×C4#53312yes—
479C8—24no—
480C8—44no—
481C8—42no—
482C22C2×#38511yescharacteristic
483C22#229011yescharacteristic
484C22#229311yescharacteristic
485C22#238812yes—
486C22#239612yes—
487C22#294911yescharacteristic
488C22#295011yescharacteristic
489C22#295111yescharacteristic
490C22#295111yescharacteristic
491C22#300811yescharacteristic
492C22#301111yescharacteristic
493C22#301312yes—
494C22#308212yes—
495C22#308311yescharacteristic
496C22#308311yescharacteristic
497C22#311812yes—
498C22#312012yes—
499C22#312412yes—
500C22#312712yes—
501C22#317512yes—
502C22#317712yes—
503C22#317812yes—
504C22#317912yes—
505C4—24no—
506C4—24no—
507C4—44no—
508C4—44no—
509C4—44no—
510C4—82no—
511C4—84no—
512C4—84no—
513C4—82no—
514C4—82no—
515C4#228511yescharacteristic
516C4#229011yescharacteristic
517C4#229311yescharacteristic
518C4#229611yescharacteristic
519C4#239512yes—
520C4#239612yes—
521C2#1384411yescharacteristic
522C2#2088011yescharacteristic
523C2#2088111yescharacteristic
524C2#2113011yescharacteristic
525C2#2113511yescharacteristic
526C2#2128111yescharacteristic
527C2#2128511yescharacteristic
528C2#2357912yes—
529C2#2358612yes—
530C2#2378812yes—
531C2#2379212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
33. C4×#103C4semidirect
164. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23202
C2×C4×#1072
C2×#17632
C2×C4×#1032
#137832
#113282
#112922
#111242
#110672
#107032
#106942