Finite Group Databasebeta

← #99723

Subgroups — #99723

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#336C211yesmaximal, characteristic
2#14702C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1272C212yesmaximal, complemented
5#7025C212yesmaximal, complemented
6C24×C8—21no—
7C22×C22⋊Q8C411yescharacteristic
8C23×C42C2211yescharacteristic
9C22×C22⋊C8—21no—
10C2×#391C411yescharacteristic
11C2×#391C2211yescharacteristic
12C2×#380C411yescharacteristic
13C2×#336—21no—
14C2×#336—24no—
15C2×#336—21no—
16C2×#336C2212yes—
17C2×#336C2212yes—
18#1436—21no—
19#1436—21no—
20C4⋊#95C411yescharacteristic
21C4⋊#95C2211yescharacteristic
22C22⋊#95—21no—
23C22⋊#95—21no—
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4D411yescharacteristic
28C24×C4D411yescharacteristic
29C23×C8—24no—
30C23×C8—24no—
31C23×C8—24no—
32C23×C8—22no—
33C23×C8—22no—
34C23×C8—42no—
35C23×C8—42no—
36C23×C8—42no—
37C23×C8—42no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—42no—
40C2×C22⋊Q8—42no—
41C2×C22⋊Q8C814yescomplemented
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C22⋊C4—21no—
47C22×C22⋊C4—21no—
48C22×C42—22no—
49C22×C42—22no—
50C22×C42—42no—
51C22×C42—42no—
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42C2311yescharacteristic
56C2×C22⋊C8—24no—
57C2×C22⋊C8—42no—
58C2×C22⋊C8—42no—
59#391—22no—
60#391—22no—
61#391—42no—
62#380—42no—
63C2×#95—21no—
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
79C22×Q8—21no—
80C22×Q8—22no—
81C22×Q8—22no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—41no—
86C22×Q8C2×C812yescomplemented
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C2×C812yes—
115C23×C4M1612yes—
116C23×C4SD1612yes—
117C23×C4SD1612yes—
118C23×C4Q1612yes—
119C23×C4Q1612yes—
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C22×C8—216no—
124C22×C8—24no—
125C22×C8—24no—
126C22×C8—24no—
127C22×C8—24no—
128C22×C8—48no—
129C22×C8—48no—
130C22×C8—48no—
131C22×C8—48no—
132C22×C8—42no—
133C22×C8—42no—
134C22×C8—48no—
135C22×C8—48no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8—42no—
144C22⋊Q8—48no—
145C22⋊Q8—48no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
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—42no—
183#95—42no—
184#95—42no—
185#95—42no—
186#95—42no—
187#95—41no—
188#95—42no—
189#95—41no—
190#95—42no—
191#95—41no—
192#95—41no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24C22⋊C812yes—
202C24#9912yes—
203C24#10012yes—
204C24#10312yes—
205C24#10312yes—
206C24#10412yes—
207C24#10412yes—
208C24C2×C22⋊C411yescharacteristic
209C2×Q8—24no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—82no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8—82no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
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—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
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×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4C22⋊C812yes—
312C22×C4#9912yes—
313C22×C4#10312yes—
314C22×C4#10312yes—
315C22×C4#10412yes—
316C22×C4C4×D412yes—
317C22×C4#12011yescharacteristic
318C22×C4#12112yes—
319C22×C4#12311yescharacteristic
320C22×C4C22×C811yescharacteristic
321C22×C4C2×M1611yescharacteristic
322C22×C4#13112yes—
323C22×C4C2×SD1611yescharacteristic
324C22×C4C2×SD1611yescharacteristic
325C22×C4C2×Q1611yescharacteristic
326C22×C4C2×Q1611yescharacteristic
327C22×C4#13512yes—
328C22×C4AGL(1,ℤ/8)11yescharacteristic
329C22×C4#13711yescharacteristic
330C2×C8—28no—
331C2×C8—432no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—44no—
340C2×C8—44no—
341C2×C8—82no—
342C2×C8—82no—
343C2×C8—82no—
344C2×C8—82no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—48no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C22⋊C4—48no—
351C22⋊C4—84no—
352C42—42no—
353C42—42no—
354C42—44no—
355C42—44no—
356C42—42no—
357C42—42no—
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—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23#32314yes—
387C23Q8⋊C814yes—
388C23Q8⋊C814yes—
389C23#32814yes—
390C23#32814yes—
391C23#32914yes—
392C23#33314yes—
393C23#38611yescharacteristic
394C23C2×C22⋊C811yescharacteristic
395C23#40811yescharacteristic
396C23C2×#9911yescharacteristic
397C23C2×#9911yescharacteristic
398C23C2×#10011yescharacteristic
399C23#41311yescharacteristic
400C23C2×#10311yescharacteristic
401C23C2×#10311yescharacteristic
402C23#41611yescharacteristic
403C23#41711yescharacteristic
404C23C2×#10411yescharacteristic
405C23C2×#10411yescharacteristic
406C23C2×#10411yescharacteristic
407C23#42111yescharacteristic
408Q8—84no—
409Q8—84no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—48no—
455C2×C4—48no—
456C2×C4—48no—
457C2×C4—48no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
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—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4#32814yes—
495C2×C4C8×D412yes—
496C2×C4#43512yes—
497C2×C4#45011yescharacteristic
498C2×C4#45111yescharacteristic
499C2×C4#46512yes—
500C2×C4#46712yes—
501C2×C4#48311yescharacteristic
502C2×C4#48411yescharacteristic
503C8—48no—
504C8—84no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—24no—
519C22#127218yescomplemented
520C22C2×#32311yescharacteristic
521C22#145911yescharacteristic
522C22#146811yescharacteristic
523C22#147111yescharacteristic
524C22C2×Q8⋊C811yescharacteristic
525C22C2×Q8⋊C811yescharacteristic
526C22#147611yescharacteristic
527C22#147711yescharacteristic
528C22#148111yescharacteristic
529C22#148311yescharacteristic
530C22#149311yescharacteristic
531C22#149411yescharacteristic
532C22C2×#32811yescharacteristic
533C22C2×#32811yescharacteristic
534C22C2×#32811yescharacteristic
535C22#150111yescharacteristic
536C22#150411yescharacteristic
537C22#150411yescharacteristic
538C22#151611yescharacteristic
539C22#152011yescharacteristic
540C22C2×#32911yescharacteristic
541C22#152411yescharacteristic
542C22#153011yescharacteristic
543C22#153211yescharacteristic
544C22C2×#33311yescharacteristic
545C22#154211yescharacteristic
546C22#155011yescharacteristic
547C22#155111yescharacteristic
548C22#186911yescharacteristic
549C22#187111yescharacteristic
550C22#187211yescharacteristic
551C22#189211yescharacteristic
552C22#189611yescharacteristic
553C22#189811yescharacteristic
554C22#190011yescharacteristic
555C4—22no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—21no—
560C4—24no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—42no—
565C4—42no—
566C4—82no—
567C4—82no—
568C4—161no—
569C4—161no—
570C2—28no—
571C2C2×#127211yescharacteristic
572C2#756611yescharacteristic
573C2#757811yescharacteristic
574C2#758911yescharacteristic
575C2#765511yescharacteristic
576C2#766711yescharacteristic
577C2#768111yescharacteristic
578C2#769011yescharacteristic
579C2#893711yescharacteristic
580C2#907111yescharacteristic
581C2#907911yescharacteristic
582C2#911111yescharacteristic
583C2#911911yescharacteristic
584C2#912711yescharacteristic
585C2#914311yescharacteristic
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×#1272C2semidirect
5. #7025C2semidirect
41. C2×C22⋊Q8C8semidirect
86. C22×Q8C2×C8semidirect
158. C2×C4⋊C4C2×C8semidirect
519. C22#1272semidirect

Maximal subgroups

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

subgroupindex
C22×#3362
#147022
#107122
C2×#12722
#70252