Finite Group Databasebeta

← #391837

Subgroups — #391837

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#1798C211yesmaximal, characteristic
3#14343C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11358C212yesmaximal, complemented
6#11228C212yesmaximal, complemented
7#11227C212yesmaximal, complemented
8#11200C212yesmaximal
9#11199C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2628—24no—
16#2628C418yescomplemented
17C2×#400—21no—
18C2×#399—21no—
19C2×#395—21no—
20C2×#393—21no—
21C2×#389C2212yes—
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26#1798—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36#1445—21no—
37#1445—21no—
38#1440—21no—
39#1440—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×Q8—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8C2×C414yes—
52C2×#117—41no—
53C2×#117—41no—
54C2×#117—41no—
55C2×#117—41no—
56C2×#117C2×C414yes—
57C2×#117D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—41no—
76C2×C4⋊C8—41no—
77C2×C4⋊C8—41no—
78C2×C4⋊C8—41no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—42no—
82#400—28no—
83#399—28no—
84#395—28no—
85#393—28no—
86#389—24no—
87#389C2×C418yes—
88#382—22no—
89#382C2×C414yes—
90#380—24no—
91#380—41no—
92#380—41no—
93#380—41no—
94#380—41no—
95#380C2×C418yes—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106#340—42no—
107#340—42no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340—42no—
114#336—24no—
115C25C2×D411yescharacteristic
116C22×Q8—24no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—82no—
120C22×Q8—82no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C22×C412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159#117—44no—
160#117—44no—
161#117—44no—
162#117—44no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
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—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C22⋊C414yes—
196C2×C4⋊C4C22⋊C414yes—
197C2×C4⋊C4C22×C414yes—
198C2×C4⋊C4C22×C414yes—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4C22⋊C414yes—
211C2×C22⋊C4C22⋊C414yes—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C22×C412yes—
221C2×C42C22×C412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C4⋊C8—44no—
225C4⋊C8—44no—
226C22⋊C8—48no—
227#95—28no—
228#95—24no—
229#95—24no—
230#95—28no—
231#95—42no—
232#95—42no—
233#95—82no—
234#95—82no—
235#95—82no—
236#95—82no—
237C24—21no—
238C24—21no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24C2×C22⋊C412yes—
246C24C22×D411yescharacteristic
247C2×Q8—48no—
248C2×Q8—88no—
249C2×Q8—88no—
250C2×Q8—84no—
251C2×Q8—84no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
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—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4H(5,𝔽2)11yescharacteristic
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+412yes—
331C22×C42−1+412yes—
332C22×C42−1+412yes—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—48no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—88no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—88no—
370C22⋊C4—84no—
371C22⋊C4—84no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—24no—
384C23—24no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—41no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23C22×C22⋊C411yescharacteristic
402C23#53411yescharacteristic
403C23#53411yescharacteristic
404C23#53612yes—
405C23#53712yes—
406C23#57612yes—
407C23#57712yes—
408C23#57712yes—
409C23#57812yes—
410Q8—816no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—82no—
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—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4#51811yescharacteristic
471C2×C4#51811yescharacteristic
472C2×C4#51912yes—
473C2×C4#52012yes—
474C2×C4#53411yescharacteristic
475C2×C4#53411yescharacteristic
476C2×C4#53511yescharacteristic
477C2×C4#53611yescharacteristic
478C2×C4#53611yescharacteristic
479C2×C4#53611yescharacteristic
480C2×C4#56212yes—
481C2×C4#56212yes—
482C2×C4#56212yes—
483C2×C4#56212yes—
484C2×C4#56312yes—
485C2×C4#56312yes—
486C2×C4#56312yes—
487C2×C4#56312yes—
488C8—82no—
489C8—82no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—44no—
494C22—44no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22#230811yescharacteristic
500C22#231011yescharacteristic
501C22#231111yescharacteristic
502C22#281412yes—
503C22#281412yes—
504C22#289712yes—
505C22#289812yes—
506C22#319812yes—
507C22#319911yescharacteristic
508C22#320011yescharacteristic
509C22#320011yescharacteristic
510C22#320112yes—
511C22#320211yescharacteristic
512C22#324112yes—
513C22#324212yes—
514C22#324312yes—
515C22#324412yes—
516C22#325712yes—
517C22#325812yes—
518C22#326212yes—
519C22#326312yes—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—84no—
526C4—84no—
527C4—162no—
528C4—162no—
529C4#237812yes—
530C4#237812yes—
531C4#281611yescharacteristic
532C4#281611yescharacteristic
533C4#281711yescharacteristic
534C4#281711yescharacteristic
535C2—82no—
536C2#1466011yescharacteristic
537C2#2177011yescharacteristic
538C2#2177111yescharacteristic
539C2#2184312yes—
540C2#2189612yes—
541C2#3275012yes—
542C2#3275112yes—
543C2#3275212yes—
544C2#3275312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11384C2semidirect
5. #11358C2semidirect
6. #11228C2semidirect
7. #11227C2semidirect
16. #2628C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#17982
#143432
#113842
#113582
#112282
#112272
#112002
#111992