Finite Group Databasebeta

← #362807

Subgroups — #362807

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1750C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#11208C212yesmaximal, complemented
5#11207C212yesmaximal, complemented
6#10881C212yesmaximal, complemented
7#10880C212yesmaximal, complemented
8#10731C212yesmaximal
9#10730C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2317—24no—
17#2317C418yescomplemented
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#385—21no—
21C2×#385—21no—
22C2×#384—21no—
23C2×#384C2211yescharacteristic
24C2×#382—21no—
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27#1750—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1441—21no—
38#1441—21no—
39#1439—21no—
40#1439—21no—
41C4⋊#95C2212yes—
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#117—22no—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117—41no—
50C2×#117—41no—
51C2×#117—41no—
52C2×#117—41no—
53C2×#117C2×C414yes—
54C2×#117C2×C414yes—
55C2×#117D414yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—41no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4—41no—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8Q818yes—
83C2×#97—22no—
84C2×#97—41no—
85C2×#97—41no—
86C2×C4×C8—22no—
87C2×C4×C8—41no—
88C2×C4×C8—41no—
89#389—24no—
90#389—24no—
91#389C2×C418yes—
92#389C2×C418yes—
93#385—28no—
94#385—28no—
95#384—28no—
96#384—41no—
97#384—41no—
98#382—28no—
99#382—41no—
100#382—41no—
101#380—24no—
102#380—24no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—42no—
115#340—42no—
116#340—42no—
117#340—42no—
118#336—24no—
119C25C2×D411yescharacteristic
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—82no—
134C23×C4—82no—
135C23×C4C22×C412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×Q814yes—
143C23×C4#4114yes—
144C23×C4C2411yescharacteristic
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—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C2×Q814yes—
158C22×C8C2×Q814yes—
159#117—216no—
160#117—216no—
161#117—44no—
162#117—44no—
163#117—44no—
164#117—44no—
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—22no—
172C2×C4⋊C4—24no—
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—28no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C22⋊C414yes—
207C2×C4⋊C4C22⋊C414yes—
208C2×C4⋊C4C22×C414yes—
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4#4114yes—
211C2×C4⋊C4#4114yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4C22⋊C414yes—
223C2×C22⋊C4C22⋊C414yes—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42C22×C412yes—
236C2×C42C22×C412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C22⋊C8—216no—
240#97—44no—
241C4×C8—44no—
242#95—28no—
243#95—28no—
244#95—42no—
245#95—42no—
246#95—44no—
247#95—82no—
248#95—82no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24C2×C22⋊C412yes—
258C24#12412yes—
259C24C4⋊Q812yes—
260C24C22×D411yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—48no—
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×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—88no—
325C22×C4—88no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C4×Q818yes—
334C22×C4C22⋊Q818yes—
335C22×C4#12412yes—
336C22×C4C4⋊Q812yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×Q812yes—
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—48no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—88no—
380C4⋊C4—88no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4#11718yes—
384C22⋊C4—216no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—88no—
388C42—216no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—41no—
419C23#39114yes—
420C23C22×C22⋊C411yescharacteristic
421C23#52511yescharacteristic
422C23#52511yescharacteristic
423C23C2×#12411yescharacteristic
424C23C2×C4⋊Q811yescharacteristic
425C23#53411yescharacteristic
426C23#53411yescharacteristic
427C23#57612yes—
428C23#57712yes—
429C23#57712yes—
430C23#57812yes—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—816no—
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—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4#39114yes—
491C2×C4C2×#11712yes—
492C2×C4C2×C4×Q812yes—
493C2×C4#51811yescharacteristic
494C2×C4#51811yescharacteristic
495C2×C4C2×C22⋊Q812yes—
496C2×C4#52512yes—
497C2×C4#53411yescharacteristic
498C2×C4#53411yescharacteristic
499C2×C4#55312yes—
500C2×C4#55312yes—
501C2×C4#55312yes—
502C2×C4#55312yes—
503C2×C4#55412yes—
504C2×C4#55412yes—
505C2×C4#55412yes—
506C2×C4#55412yes—
507C8—44no—
508C8—44no—
509C22—28no—
510C22—24no—
511C22—44no—
512C22—42no—
513C22—42no—
514C22—44no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—44no—
520C22#228011yescharacteristic
521C22C2×#39111yescharacteristic
522C22#230811yescharacteristic
523C22#258311yescharacteristic
524C22#258311yescharacteristic
525C22#259111yescharacteristic
526C22#259111yescharacteristic
527C22#289712yes—
528C22#289812yes—
529C22#311912yes—
530C22#312012yes—
531C22#312412yes—
532C22#312512yes—
533C22#313612yes—
534C22#313712yes—
535C22#316312yes—
536C22#316412yes—
537C22#319812yes—
538C22#319911yescharacteristic
539C22#320011yescharacteristic
540C22#320011yescharacteristic
541C22#320112yes—
542C22#320211yescharacteristic
543C4—24no—
544C4—48no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—84no—
550C4—162no—
551C4—162no—
552C4#234512yes—
553C4#234512yes—
554C4#257912yes—
555C4#257912yes—
556C2—44no—
557C2#1392311yescharacteristic
558C2#2074612yes—
559C2#2132612yes—
560C2#2177011yescharacteristic
561C2#2177111yescharacteristic
562C2#2930912yes—
563C2#2931011yescharacteristic
564C2#2931111yescharacteristic
565C2#2934311yescharacteristic
566C2#2934411yescharacteristic
567C2#2934512yes—
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. #11208C2semidirect
5. #11207C2semidirect
6. #10881C2semidirect
7. #10880C2semidirect
17. #2317C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#17502
#143402
#112082
#112072
#108812
#108802
#107312
#107302