Finite Group Databasebeta

← #99731

Subgroups — #99731

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#336C211yesmaximal, characteristic
2#14697C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1271C212yesmaximal, complemented
5#7025C212yesmaximal, complemented
6C24×C8—21no—
7C23×C42C2211yescharacteristic
8C22×C22⋊C8—21no—
9C2×#389C411yescharacteristic
10C2×#389C2211yescharacteristic
11C2×#385—21no—
12C2×#380C411yescharacteristic
13C2×#380C411yescharacteristic
14C2×#336—21no—
15C2×#336—24no—
16C2×#336—21no—
17C2×#336C2212yes—
18C2×#336C2212yes—
19#1436—21no—
20#1436—21no—
21C4⋊#95C411yescharacteristic
22C4⋊#95C2211yescharacteristic
23C22⋊#95—21no—
24#1271—24no—
25C24×C4C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C24×C4D411yescharacteristic
28C23×C8—24no—
29C23×C8—24no—
30C23×C8—24no—
31C23×C8—22no—
32C23×C8—22no—
33C23×C8—42no—
34C23×C8—42no—
35C23×C8—42no—
36C23×C8—42no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C42—22no—
46C22×C42—22no—
47C22×C42—42no—
48C22×C42—42no—
49C22×C42C2×C411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42C2311yescharacteristic
53C2×C22⋊C8—24no—
54C2×C22⋊C8—42no—
55C2×C22⋊C8—42no—
56#389—22no—
57#389—22no—
58#389—42no—
59#385—42no—
60#385—42no—
61#380—22no—
62#380—42no—
63#380C814yescomplemented
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72#336—24no—
73#336—22no—
74#336—22no—
75#336—22no—
76#336—42no—
77#336D414yes—
78C25C22⋊C411yescharacteristic
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4—41no—
102C23×C4—41no—
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C2×C812yes—
107C23×C4M1612yes—
108C23×C4D812yes—
109C23×C4SD1612yes—
110C23×C4SD1612yes—
111C23×C4Q1612yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C22×C8—216no—
116C22×C8—24no—
117C22×C8—24no—
118C22×C8—24no—
119C22×C8—24no—
120C22×C8—48no—
121C22×C8—48no—
122C22×C8—48no—
123C22×C8—48no—
124C22×C8—42no—
125C22×C8—42no—
126C22×C8—48no—
127C22×C8—48no—
128C22×C8—42no—
129C22×C8—42no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8—42no—
133C22×C8—42no—
134C22×C8—42no—
135C22×C8—42no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4C2×C812yescomplemented
158C2×C4⋊C4C2×C812yescomplemented
159C2×C22⋊C4—24no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—42no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42—41no—
176C2×C42—41no—
177C2×C42—48no—
178C2×C42—48no—
179C2×C42C2×D412yes—
180C2×C42#4112yes—
181C22⋊C8—48no—
182#95—24no—
183#95—42no—
184#95—42no—
185#95—42no—
186#95—42no—
187#95—42no—
188#95—41no—
189#95—41no—
190#95—42no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24C22⋊C812yes—
200C24#9912yes—
201C24#10112yes—
202C24#10212yes—
203C24#10312yes—
204C24#10412yes—
205C24#10412yes—
206C24C2×C22⋊C411yescharacteristic
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4C22⋊C812yes—
300C22×C4#9912yes—
301C22×C4#10212yes—
302C22×C4#10312yes—
303C22×C4#10412yes—
304C22×C4C4×D412yes—
305C22×C4#12011yescharacteristic
306C22×C4#12112yes—
307C22×C4#12311yescharacteristic
308C22×C4C22×C811yescharacteristic
309C22×C4C2×M1611yescharacteristic
310C22×C4#13112yes—
311C22×C4C2×D811yescharacteristic
312C22×C4C2×SD1611yescharacteristic
313C22×C4C2×SD1611yescharacteristic
314C22×C4C2×Q1611yescharacteristic
315C22×C4#13512yes—
316C22×C4AGL(1,ℤ/8)11yescharacteristic
317C22×C4#13711yescharacteristic
318C2×C8—28no—
319C2×C8—432no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—44no—
328C2×C8—44no—
329C2×C8—82no—
330C2×C8—82no—
331C2×C8—82no—
332C2×C8—82no—
333C4⋊C4—24no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—84no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C22⋊C4—48no—
345C22⋊C4—84no—
346C42—42no—
347C42—42no—
348C42—44no—
349C42—44no—
350C42—42no—
351C42—42no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23#32414yes—
381C23D4⋊C814yes—
382C23Q8⋊C814yes—
383C23#32714yes—
384C23#32814yes—
385C23#33014yes—
386C23#33214yes—
387C23#38611yescharacteristic
388C23C2×C22⋊C811yescharacteristic
389C23#40811yescharacteristic
390C23C2×#9911yescharacteristic
391C23C2×#9911yescharacteristic
392C23C2×#10111yescharacteristic
393C23#41311yescharacteristic
394C23C2×#10211yescharacteristic
395C23C2×#10311yescharacteristic
396C23#41611yescharacteristic
397C23#41711yescharacteristic
398C23C2×#10411yescharacteristic
399C23C2×#10411yescharacteristic
400C23C2×#10411yescharacteristic
401C23#42111yescharacteristic
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4#32714yes—
487C2×C4C8×D412yes—
488C2×C4#43512yes—
489C2×C4#44711yescharacteristic
490C2×C4#44811yescharacteristic
491C2×C4#46512yes—
492C2×C4#46712yes—
493C2×C4#48011yescharacteristic
494C2×C4#48111yescharacteristic
495C8—48no—
496C8—84no—
497C22—24no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—24no—
503C22—24no—
504C22—24no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22#127118yescomplemented
512C22C2×#32411yescharacteristic
513C22#145911yescharacteristic
514C22#146811yescharacteristic
515C22#147211yescharacteristic
516C22C2×D4⋊C811yescharacteristic
517C22C2×Q8⋊C811yescharacteristic
518C22#147611yescharacteristic
519C22#147711yescharacteristic
520C22#148011yescharacteristic
521C22#148311yescharacteristic
522C22#149311yescharacteristic
523C22#149411yescharacteristic
524C22C2×#32711yescharacteristic
525C22C2×#32711yescharacteristic
526C22C2×#32811yescharacteristic
527C22#150111yescharacteristic
528C22#150411yescharacteristic
529C22#150411yescharacteristic
530C22#151611yescharacteristic
531C22#152011yescharacteristic
532C22C2×#33011yescharacteristic
533C22#152411yescharacteristic
534C22#153111yescharacteristic
535C22#153211yescharacteristic
536C22C2×#33211yescharacteristic
537C22#154011yescharacteristic
538C22#155011yescharacteristic
539C22#155211yescharacteristic
540C22#186911yescharacteristic
541C22#187111yescharacteristic
542C22#187411yescharacteristic
543C22#189211yescharacteristic
544C22#189611yescharacteristic
545C22#189811yescharacteristic
546C22#190111yescharacteristic
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—21no—
552C4—24no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4—82no—
560C4—161no—
561C4—161no—
562C2—28no—
563C2C2×#127111yescharacteristic
564C2#756511yescharacteristic
565C2#757911yescharacteristic
566C2#758911yescharacteristic
567C2#765411yescharacteristic
568C2#766811yescharacteristic
569C2#768111yescharacteristic
570C2#769111yescharacteristic
571C2#893911yescharacteristic
572C2#907111yescharacteristic
573C2#908111yescharacteristic
574C2#911111yescharacteristic
575C2#912111yescharacteristic
576C2#913111yescharacteristic
577C2#914711yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1271C2semidirect
5. #7025C2semidirect
63. #380C8semidirect
157. C2×C4⋊C4C2×C8semidirect
158. C2×C4⋊C4C2×C8semidirect
511. C22#1271semidirect

Maximal subgroups

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

subgroupindex
C22×#3362
#146972
#107122
C2×#12712
#70252