Finite Group Databasebeta

← #391821

Subgroups — #391821

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#1818C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11372C212yesmaximal, complemented
6#11208C212yesmaximal, complemented
7#11207C212yesmaximal, complemented
8#11200C212yesmaximal
9#11199C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2628—24no—
15#2628C418yescomplemented
16C2×#400—21no—
17C2×#399—21no—
18C2×#395—21no—
19C2×#393—21no—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26#1818—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1441—21no—
38#1441—21no—
39#1439—21no—
40#1439—21no—
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⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—41no—
77C2×C22⋊C8—41no—
78C2×C22⋊C8—41no—
79C2×C22⋊C8—41no—
80#400—28no—
81#399—28no—
82#395—28no—
83#393—28no—
84#389—24no—
85#389C2×C418yes—
86#384—41no—
87#384—41no—
88#382—22no—
89#382—41no—
90#382—41no—
91#382C2×C414yes—
92#380—24no—
93#380—42no—
94#380—42no—
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—
105#340—42no—
106#340—42no—
107#340—42no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#340—42no—
113#336—24no—
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—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4C22⋊C414yes—
200C2×C4⋊C4C22⋊C414yes—
201C2×C4⋊C4C22×C414yes—
202C2×C4⋊C4C22×C414yes—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4C22⋊C414yes—
214C2×C22⋊C4C22⋊C414yes—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C22×C412yes—
224C2×C42C22×C412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C22⋊C8—44no—
228C22⋊C8—44no—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—28no—
233#95—44no—
234#95—42no—
235#95—42no—
236#95—82no—
237#95—82no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24C2×C22⋊C412yes—
249C24C22×D411yescharacteristic
250C2×Q8—48no—
251C2×Q8—88no—
252C2×Q8—88no—
253C2×Q8—84no—
254C2×Q8—84no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
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×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C4H(5,𝔽2)11yescharacteristic
332C22×C4H(5,𝔽2)12yes—
333C22×C42−1+412yes—
334C22×C42−1+412yes—
335C22×C42−1+412yes—
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—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—48no—
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—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—88no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—88no—
375C22⋊C4—88no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—24no—
388C23—24no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23C22×C22⋊C411yescharacteristic
408C23#53411yescharacteristic
409C23#53411yescharacteristic
410C23#53511yescharacteristic
411C23#53611yescharacteristic
412C23#53611yescharacteristic
413C23#53611yescharacteristic
414C23#57612yes—
415C23#57712yes—
416C23#57712yes—
417C23#57812yes—
418Q8—816no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
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—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4#51811yescharacteristic
479C2×C4#51811yescharacteristic
480C2×C4#51912yes—
481C2×C4#52012yes—
482C2×C4#53411yescharacteristic
483C2×C4#53411yescharacteristic
484C2×C4#53612yes—
485C2×C4#53712yes—
486C2×C4#56212yes—
487C2×C4#56212yes—
488C2×C4#56212yes—
489C2×C4#56212yes—
490C2×C4#56312yes—
491C2×C4#56312yes—
492C2×C4#56312yes—
493C2×C4#56312yes—
494C8—82no—
495C8—82no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—81no—
502C22—81no—
503C22—82no—
504C22—81no—
505C22—82no—
506C22—81no—
507C22#230811yescharacteristic
508C22#231011yescharacteristic
509C22#231111yescharacteristic
510C22#281611yescharacteristic
511C22#281611yescharacteristic
512C22#281711yescharacteristic
513C22#281711yescharacteristic
514C22#289712yes—
515C22#289812yes—
516C22#319812yes—
517C22#319911yescharacteristic
518C22#320011yescharacteristic
519C22#320011yescharacteristic
520C22#320112yes—
521C22#320211yescharacteristic
522C22#322412yes—
523C22#322512yes—
524C22#324112yes—
525C22#324212yes—
526C22#324312yes—
527C22#324312yes—
528C22#324412yes—
529C22#324412yes—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—84no—
536C4—84no—
537C4—162no—
538C4—162no—
539C4#237812yes—
540C4#237812yes—
541C4#281412yes—
542C4#281412yes—
543C2—82no—
544C2#1466011yescharacteristic
545C2#2177011yescharacteristic
546C2#2177111yescharacteristic
547C2#2184312yes—
548C2#2187112yes—
549C2#3277911yescharacteristic
550C2#3278011yescharacteristic
551C2#3278112yes—
552C2#3279211yescharacteristic
553C2#3279312yes—
554C2#3279411yescharacteristic
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. #11372C2semidirect
6. #11208C2semidirect
7. #11207C2semidirect
15. #2628C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#18182
#143402
#113842
#113722
#112082
#112072
#112002
#111992