Finite Group Databasebeta

← #192394

Subgroups — #192394

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34359C211yesmaximal, characteristic
2#21941C211yesmaximal, characteristic
3#21765C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic
5#14675C211yesmaximal, characteristic
6#8872C212yesmaximal, complemented
7#8871C212yesmaximal, complemented
8#8868C212yesmaximal, complemented
9#8867C212yesmaximal, complemented
10#8016C211yesmaximal, characteristic
11#7913C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2378—21no—
16#2378—21no—
17#2368—22no—
18#2344—22no—
19#2342—22no—
20#2332—22no—
21#2329C2212yes—
22#2329C2212yes—
23#2313—21no—
24#2310—21no—
25#2280C2211yescharacteristic
26C2×#381C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29#1894—22no—
30#1872C2212yes—
31#1872C2212yes—
32#1854—22no—
33#1853C2212yes—
34#1852C2212yes—
35#1823—21no—
36#1808—21no—
37#1807—21no—
38#1787—21no—
39#1520C418yescomplemented
40C2×#328C2212yes—
41C2×#328C2212yes—
42#1444C2212yes—
43#1442C2212yes—
44#1441—22no—
45#1439—22no—
46#1294C2212yes—
47#1294C2212yes—
48#1280C2211yescharacteristic
49#1280C2211yescharacteristic
50#1280C2211yescharacteristic
51#1280C2211yescharacteristic
52C23×Q8D411yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8C2×C414yes—
61C2×C22⋊Q8D412yes—
62C2×C22⋊Q8Q814yes—
63C2×C22⋊Q8C2312yes—
64C2×#117—22no—
65C2×#117D412yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D412yes—
75C2×#107—22no—
76C2×#106—22no—
77C2×#103—22no—
78C2×#103—22no—
79#407—22no—
80#407—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8C2×C414yes—
85C2×C22⋊C8C2311yescharacteristic
86C2×C22⋊C8C2311yescharacteristic
87C2×C22⋊C8C2311yescharacteristic
88C2×C22⋊C8C2311yescharacteristic
89#391—22no—
90#391—22no—
91#391—22no—
92#391—22no—
93#389—22no—
94#389—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#385—22no—
108#385—22no—
109#381—22no—
110#381—21no—
111#381—21no—
112#381—21no—
113#381—21no—
114#381—21no—
115#380—22no—
116#380—22no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380C2×C414yes—
122#380D411yescharacteristic
123#380D411yescharacteristic
124#380C2312yes—
125#379—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95D412yes—
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#328C2×C418yes—
146#328C2×C418yes—
147C25C2×D411yescharacteristic
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—82no—
151C22×Q8C4⋊C414yescomplemented
152C22×Q8C2×D412yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—82no—
158C23×C4C22×C412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×Q812yes—
167C23×C4#4112yes—
168C23×C4#4112yes—
169C23×C4C2411yescharacteristic
170C2×M16—42no—
171C2×M16—82no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—24no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—82no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8C4⋊C418yescomplemented
186C22⋊Q8C22×C414yes—
187C22⋊Q8C2×Q814yes—
188#117—84no—
189#117—82no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C4⋊C414yescomplemented
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—81no—
228C2×C22⋊C4—81no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234#107—82no—
235#106—82no—
236#103—82no—
237#103—82no—
238C22⋊C8—24no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—42no—
246C22⋊C8C22×C412yes—
247C22⋊C8C22×C412yes—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—22no—
266#95—41no—
267#95—41no—
268#95—82no—
269#95—82no—
270#95—82no—
271#95C22×C414yes—
272C24—22no—
273C24—22no—
274C24—41no—
275C24—42no—
276C24—41no—
277C24C4×D412yes—
278C24#12011yescharacteristic
279C24#12011yescharacteristic
280C24C22⋊Q812yes—
281C24C22×D411yescharacteristic
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—82no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×Q8C2×C4⋊C414yes—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4C2×C4⋊C412yes—
334C22×C4C2×C4⋊C412yes—
335C22×C4C4×D412yes—
336C22×C4C4×D412yes—
337C22×C4C4×D412yes—
338C22×C4#12011yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4C22⋊Q812yes—
341C22×C4C22⋊Q812yes—
342C22×C4C22⋊Q812yes—
343C22×C4#12312yes—
344C22×C4#12312yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×Q811yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351M16—84no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—84no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—28no—
365C4⋊C4—24no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—84no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—88no—
378C4⋊C4C2×C4⋊C414yes—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—28no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—81no—
387C22⋊C4—81no—
388C22⋊C4—82no—
389C22⋊C4—84no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C42—82no—
393C42—82no—
394C42—82no—
395C42—82no—
396C42—82no—
397C23—21no—
398C23—21no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—22no—
410C23—22no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23#38512yes—
420C23#38512yes—
421C23C2×C4×D411yescharacteristic
422C23C2×#12011yescharacteristic
423C23C2×C22⋊Q811yescharacteristic
424C23#57712yes—
425C23#57812yes—
426Q8—44no—
427Q8—82no—
428Q8—84no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—48no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—84no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#38512yes—
486C2×C4#38512yes—
487C2×C4C22×C4⋊C411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×C22⋊Q811yescharacteristic
490C2×C4C2×#12311yescharacteristic
491C8—42no—
492C8—42no—
493C8—81no—
494C8—81no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—82no—
514C22—84no—
515C22—82no—
516C22C2×#38511yescharacteristic
517C22#294911yescharacteristic
518C22#295011yescharacteristic
519C22#301312yes—
520C22#302111yescharacteristic
521C22#302211yescharacteristic
522C22#302612yes—
523C22#308212yes—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—22no—
529C4—22no—
530C4—44no—
531C4—44no—
532C4—84no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—162no—
538C2—21no—
539C2—22no—
540C2—21no—
541C2—42no—
542C2—42no—
543C2#1844812yes—
544C2#1844912yes—
545C2#2086511yescharacteristic
546C2#2088011yescharacteristic
547C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8872C2semidirect
7. #8871C2semidirect
8. #8868C2semidirect
9. #8867C2semidirect
39. #1520C4semidirect
151. C22×Q8C4⋊C4semidirect
185. C22⋊Q8C4⋊C4semidirect
208. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#343592
#219412
#217652
C2×#15202
#146752
#88722
#88712
#88682
#88672
#80162
#79132