Finite Group Databasebeta

← #370180

Subgroups — #370180

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C211yesmaximal, characteristic
2C2×#1851C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4#13881C211yesmaximal, characteristic
5#13801C211yesmaximal, characteristic
6#11268C212yesmaximal, complemented
7#11231C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10850C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#382C2212yes—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1854—24no—
25#1851C418yescomplemented
26#1736—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36#1442—21no—
37#1442—21no—
38#1437—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41#1434—21no—
42C22⋊#95—22no—
43C4×#95—21no—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—42no—
49C22×M16—42no—
50C22×M16C2×C414yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—42no—
57C2×#117D414yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—42no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D412yes—
73C22×C42—21no—
74C22×C42—22no—
75C22×C42C2312yes—
76C2×#107—22no—
77C2×#107—24no—
78C2×#106—22no—
79C2×#106—42no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8—42no—
82C2×C4⋊C8C2×C418yescomplemented
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8C2×C414yes—
86#382—24no—
87#382—42no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380Q814yes—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—42no—
94C4×C22⋊C4C2×C414yes—
95C4×C22⋊C4Q814yes—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D411yescharacteristic
106C2×#95D411yescharacteristic
107#340—22no—
108#340—22no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—42no—
113#340—42no—
114#340—42no—
115#336C2×C418yescomplemented
116C25C2×D411yescharacteristic
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—44no—
128C23×C4—42no—
129C23×C4—44no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4C22×C412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×Q812yes—
138C23×C4C2×Q812yes—
139C23×C4#4112yes—
140C23×C4#4112yes—
141C23×C4C2411yescharacteristic
142C2×M16—28no—
143C2×M16—48no—
144C2×M16—44no—
145C2×M16—44no—
146C2×M16—48no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C22×C414yes—
162C22×C8C22×C412yes—
163C22×C8C22×C412yes—
164#117—28no—
165#117—44no—
166#117—48no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
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—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×Q814yes—
198C2×C4⋊C4#4114yes—
199C2×C4⋊C4#4114yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—81no—
208C2×C22⋊C4—81no—
209C2×C22⋊C4C4⋊C418yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—44no—
216C2×C42—44no—
217C2×C42—44no—
218C2×C42—82no—
219C2×C42C22×C414yes—
220C2×C42C2×Q814yes—
221#107—44no—
222#106—44no—
223C4⋊C8—48no—
224C22⋊C8—28no—
225#95—24no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—42no—
231#95—82no—
232#95—82no—
233#95—82no—
234#95C4⋊C414yes—
235#95C4⋊C414yes—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24C2×C4⋊C412yes—
245C24#12112yes—
246C24C22⋊Q812yes—
247C24C22×D411yescharacteristic
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
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—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—48no—
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—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—88no—
312C22×C4—88no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4C2×C4⋊C414yes—
316C22×C4C2×C4⋊C414yes—
317C22×C4C2×C4⋊C412yes—
318C22×C4C4×D414yes—
319C22×C4C4×Q814yes—
320C22×C4#12112yes—
321C22×C4C22⋊Q812yes—
322C22×C4#12514yes—
323C22×C4C4⋊Q814yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×Q811yescharacteristic
327C22×C4C22×Q811yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4112yes—
332C22×C4H(5,𝔽2)12yes—
333C22×C4H(5,𝔽2)12yes—
334M16—416no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—48no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4#12314yes—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—84no—
376C22⋊C4—84no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—24no—
381C42—48no—
382C42—88no—
383C23—24no—
384C23—22no—
385C23—22no—
386C23—24no—
387C23—24no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—24no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23#38914yes—
406C23C22×C4⋊C411yescharacteristic
407C23C2×#12111yescharacteristic
408C23C2×C22⋊Q811yescharacteristic
409C23#54612yes—
410C23#54612yes—
411C23#57512yes—
412C23#57512yes—
413C23#57611yescharacteristic
414C23#57711yescharacteristic
415C23#57711yescharacteristic
416C23#57811yescharacteristic
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—28no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
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—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—816no—
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—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4#38914yes—
477C2×C4C2×C4×D411yescharacteristic
478C2×C4C2×C4×Q811yescharacteristic
479C2×C4C4×#4112yes—
480C2×C4#51812yes—
481C2×C4C2×#12311yescharacteristic
482C2×C4#52511yescharacteristic
483C2×C4C2×#12511yescharacteristic
484C2×C4#52912yes—
485C2×C4C2×C4⋊Q811yescharacteristic
486C2×C4#53411yescharacteristic
487C2×C4#53511yescharacteristic
488C2×C4#53812yes—
489C2×C4#55112yes—
490C2×C4#55112yes—
491C2×C4#55412yes—
492C2×C4#55412yes—
493C2×C4#55412yes—
494C2×C4#55412yes—
495C8—42no—
496C8—44no—
497C8—42no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—44no—
502C22—44no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22C2×#38911yescharacteristic
510C22#234912yes—
511C22#241111yescharacteristic
512C22#241211yescharacteristic
513C22#245012yes—
514C22#291112yes—
515C22#291211yescharacteristic
516C22#291311yescharacteristic
517C22#304011yescharacteristic
518C22#304011yescharacteristic
519C22#304511yescharacteristic
520C22#304611yescharacteristic
521C22#308012yes—
522C22#308111yescharacteristic
523C22#308211yescharacteristic
524C22#330712yes—
525C22#331112yes—
526C22#331712yes—
527C22#332112yes—
528C22#332212yes—
529C22#332212yes—
530C22#332712yes—
531C22#332712yes—
532C4—24no—
533C4—44no—
534C4—44no—
535C4—44no—
536C4—84no—
537C4—84no—
538C4—82no—
539C4—161no—
540C4—161no—
541C4C4×#12311yescharacteristic
542C4#232111yescharacteristic
543C4#236812yes—
544C4#250211yescharacteristic
545C4#250512yes—
546C4#258311yescharacteristic
547C2—44no—
548C2#1404411yescharacteristic
549C2#2117211yescharacteristic
550C2#2117511yescharacteristic
551C2#2271012yes—
552C2#2271612yes—
553C2#2449311yescharacteristic
554C2#2449911yescharacteristic
555C2#2454311yescharacteristic
556C2#2454911yescharacteristic
557C2#2643312yes—
558C2#2643912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11268C2semidirect
7. #11231C2semidirect
8. #10907C2semidirect
9. #10850C2semidirect
25. #1851C4semidirect
82. C2×C4⋊C8C2×C4semidirect
115. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
C2×#18512
C2×#17362
#138812
#138012
#112682
#112312
#109072
#108502
#107302
#107222