Finite Group Databasebeta

← #401656

Subgroups — #401656

Order 512.

1442 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#424C211yesmaximal, characteristic
2C2×#2088C212yesmaximal, complemented
3C2×#1807C211yesmaximal, characteristic
4#14264C211yesmaximal, characteristic
5#14264C211yesmaximal, characteristic
6#11231C212yesmaximal, complemented
7#11228C212yesmaximal, complemented
8#11227C212yesmaximal, complemented
9#10722C211yesmaximal, characteristic
10#10722C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#424—28no—
18C2×#424—24no—
19C22×C4⋊C8C2212yes—
20C22×C4⋊C8C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27#2088—28no—
28#1807—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37#1444—21no—
38#1444—21no—
39#1442—21no—
40#1442—21no—
41#1439—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—24no—
50C22×M16—42no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—42no—
55C23×C8D414yes—
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—24no—
62C2×#117—24no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D412yes—
78C22×C42C2312yes—
79C2×#106—22no—
80C2×#106—22no—
81C2×#106—24no—
82C2×#106—42no—
83#424—232no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8D418yes—
90#389—24no—
91#389—24no—
92#389—24no—
93#389—24no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380Q814yes—
101#380Q814yes—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112#340—22no—
113#340—22no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#340—42no—
120C25C2×D411yescharacteristic
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×Q812yes—
136C23×C4C2×Q812yes—
137C23×C4#4112yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4C2411yescharacteristic
141C2×M16—28no—
142C2×M16—48no—
143C2×M16—48no—
144C2×M16—44no—
145C2×M16—44no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—28no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—28no—
161C22×C8—44no—
162C22×C8—48no—
163C22×C8—48no—
164C22×C8C2×D412yes—
165C22×C8C2×D414yes—
166C22×C8C2×D414yes—
167C22×C8C2×D412yes—
168#117—216no—
169#117—28no—
170#117—28no—
171C2×C4⋊C4—24no—
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—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
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—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C2×Q814yes—
208C2×C4⋊C4C2×Q814yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—81no—
217C2×C22⋊C4—81no—
218C2×C22⋊C4—81no—
219C2×C22⋊C4—81no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42C2×D414yes—
227C2×C42#4114yes—
228#106—44no—
229#106—44no—
230C4⋊C8—216no—
231C4⋊C8—216no—
232#95—24no—
233#95—24no—
234#95—42no—
235#95—42no—
236#95—44no—
237#95—44no—
238#95—82no—
239#95—82no—
240#95—82no—
241#95—82no—
242C24—22no—
243C24—22no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—22no—
249C24—21no—
250C24#12112yes—
251C24C22⋊Q812yes—
252C24C22⋊Q812yes—
253C24C22×D411yescharacteristic
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
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—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—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4#12114yes—
317C22×C4#12114yes—
318C22×C4#12112yes—
319C22×C4C22⋊Q812yes—
320C22×C4C22⋊Q812yes—
321C22×C4#12314yes—
322C22×C4#12514yes—
323C22×C4#12514yes—
324C22×C4C4⋊Q814yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C22×Q811yescharacteristic
328C22×C4C22×Q811yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4112yes—
333C22×C4H(5,𝔽2)12yes—
334C22×C42−1+412yes—
335M16—416no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—416no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8#12014yes—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—88no—
377C4⋊C4—88no—
378C22⋊C4—216no—
379C22⋊C4—84no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—24no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—24no—
392C23—24no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—24no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23#39814yes—
410C23C2×#12111yescharacteristic
411C23C2×#12112yes—
412C23C2×C22⋊Q811yescharacteristic
413C23C2×C22⋊Q811yescharacteristic
414C23#53712yes—
415C23#57611yescharacteristic
416C23#57711yescharacteristic
417C23#57711yescharacteristic
418C23#57712yes—
419C23#57712yes—
420C23#57811yescharacteristic
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
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—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
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—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4#39814yes—
480C2×C4C2×#12011yescharacteristic
481C2×C4C2×#12311yescharacteristic
482C2×C4#52511yescharacteristic
483C2×C4#52512yes—
484C2×C4C2×#12511yescharacteristic
485C2×C4C2×#12511yescharacteristic
486C2×C4C2×C4⋊Q811yescharacteristic
487C2×C4#53411yescharacteristic
488C2×C4#53611yescharacteristic
489C2×C4#53912yes—
490C2×C4#54612yes—
491C2×C4#54612yes—
492C2×C4D4×Q812yes—
493C2×C4D4×Q812yes—
494C2×C4#55212yes—
495C2×C4#55212yes—
496C2×C4#55412yes—
497C2×C4#55412yes—
498C8—24no—
499C8—42no—
500C8—44no—
501C22—24no—
502C22—24no—
503C22—24no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22C2×#39811yescharacteristic
513C22#241712yes—
514C22#241712yes—
515C22#261811yescharacteristic
516C22#284711yescharacteristic
517C22#305712yes—
518C22#305711yescharacteristic
519C22#305711yescharacteristic
520C22#306012yes—
521C22#306011yescharacteristic
522C22#306111yescharacteristic
523C22#306212yes—
524C22#306312yes—
525C22#308111yescharacteristic
526C22#308111yescharacteristic
527C22#308211yescharacteristic
528C22#308211yescharacteristic
529C22#308412yes—
530C22#308412yes—
531C22#325712yes—
532C22#325812yes—
533C22#326012yes—
534C22#326312yes—
535C4—24no—
536C4—44no—
537C4—44no—
538C4—84no—
539C4—84no—
540C4—161no—
541C4—161no—
542C4—161no—
543C4—161no—
544C4#240511yescharacteristic
545C4#243411yescharacteristic
546C4#247312yes—
547C4#247312yes—
548C4#248411yescharacteristic
549C4#258311yescharacteristic
550C2—44no—
551C2#1482111yescharacteristic
552C2#2366811yescharacteristic
553C2#2366911yescharacteristic
554C2#2485112yes—
555C2#2485212yes—
556C2#2485312yes—
557C2#2485412yes—
558C2#2978011yescharacteristic
559C2#2978111yescharacteristic
560C2#3293211yescharacteristic
561C2#3293811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
2. C2×#2088C2semidirect
6. #11231C2semidirect
7. #11228C2semidirect
8. #11227C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4242
C2×#20882
C2×#18072
#142642
#112312
#112282
#112272
#107222
#107212