Finite Group Databasebeta

← #238126

Subgroups — #238126

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2#21859C211yesmaximal, characteristic
3#21386C211yesmaximal, characteristic
4#21258C211yesmaximal, characteristic
5#10350C214yesmaximal, complemented
6#10348C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic, complemented
8#7901C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8—21no—
12C22×C4⋊Q8—21no—
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#417—21no—
17C2×#417—21no—
18#2551—22no—
19#2551—22no—
20#2551—22no—
21#2551C2214yes—
22C2×#395—22no—
23C2×#389—21no—
24C2×#389—21no—
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990C2212yes—
29#1988C2212yes—
30#1983—22no—
31#1982C2212yes—
32#1982C2212yes—
33#1981C2212yes—
34#1957C2212yes—
35#1874—22no—
36#1874—21no—
37#1874—21no—
38#1802—22no—
39C4×#102—22no—
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C22×#41—21no—
44C23×Q8—22no—
45C22×M16—21no—
46#532—22no—
47#532—22no—
48#532—22no—
49#532Q814yescomplemented
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—48no—
55C2×C4⋊Q8—44no—
56C2×C4⋊Q8—44no—
57C2×C4⋊Q8C2311yescharacteristic
58C2×#127C2311yescharacteristic
59C2×#124—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D412yes—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#423—22no—
81#423—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86#417—44no—
87#417—44no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—24no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8C2312yes—
99#395—28no—
100#395—28no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389—24no—
105#384—21no—
106#384—21no—
107#384D412yes—
108#384C2312yes—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4C2312yes—
116C4×C4⋊C4C2312yes—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—21no—
120#340—24no—
121#340—22no—
122#340—22no—
123C2×#41—24no—
124C2×#41—42no—
125C2×#41—42no—
126C2×#41—42no—
127C2×#41—42no—
128C2×#41—42no—
129C22×Q8—28no—
130C22×Q8—21no—
131C22×Q8—48no—
132C22×Q8—48no—
133C22×Q8—84no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—41no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—42no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—416no—
157C4⋊Q8—48no—
158C4⋊Q8—48no—
159C4⋊Q8—416no—
160C4⋊Q8—48no—
161C4⋊Q8—48no—
162C4⋊Q8C2×Q812yescomplemented
163#127—21no—
164#127—21no—
165#127—21no—
166#127C2×Q812yescomplemented
167#124—24no—
168#124—42no—
169#121—24no—
170#121—24no—
171#121—42no—
172#121—42no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—42no—
176C4×D4—42no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4D814yes—
201C2×C4⋊C4SD1614yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D414yes—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—44no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42C2411yescharacteristic
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230#103—44no—
231#103—44no—
232#102—44no—
233#102—44no—
234#102—44no—
235#102—44no—
236#102—84no—
237C22⋊C8—42no—
238C22⋊C8—42no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—42no—
242#95—28no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247C24—41no—
248C24#12011yescharacteristic
249#41—48no—
250#41—48no—
251#41—84no—
252#41—84no—
253C2×Q8—22no—
254C2×Q8—416no—
255C2×Q8—416no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—88no—
262C2×Q8—832no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—44no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—82no—
279C2×D4—82no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—41no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4C22⋊Q812yes—
312C22×C4C22⋊Q812yes—
313C22×C4C22⋊Q812yes—
314C22×C4#12412yes—
315C22×C4#12412yes—
316C22×C4#12412yes—
317C22×C4C2×D812yes—
318C22×C4C2×SD1612yes—
319C22×C4AGL(1,ℤ/8)12yes—
320C22×C4#13712yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324M16—84no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—416no—
347C4⋊C4—416no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—82no—
355C4⋊C4C2×D814yes—
356C4⋊C4C2×SD1614yes—
357C4⋊C4AGL(1,ℤ/8)14yes—
358C4⋊C4#13714yes—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C42—44no—
362C42—42no—
363C42—42no—
364C42—44no—
365C42—44no—
366C42C22⋊Q812yes—
367C42C22⋊Q812yes—
368C42C22⋊Q812yes—
369C42#12412yes—
370C42#12412yes—
371C42#12412yes—
372C42C22×Q811yescharacteristic
373C42C2×#4111yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—42no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23#39312yes—
388C23#44712yes—
389C23#44812yes—
390C23C2×#12011yescharacteristic
391Q8—42no—
392Q8—42no—
393Q8—832no—
394Q8—81no—
395Q8—81no—
396D4—44no—
397D4—44no—
398D4—42no—
399D4—42no—
400D4—82no—
401D4—88no—
402D4—81no—
403D4—81no—
404D4—82no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—88no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—88no—
439C2×C4#39312yes—
440C2×C4#39312yes—
441C2×C4#39312yes—
442C2×C4#44712yes—
443C2×C4#44812yes—
444C2×C4#47414yes—
445C2×C4#47614yes—
446C2×C4#48614yes—
447C2×C4#48914yes—
448C2×C4C2×C22⋊Q811yescharacteristic
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4C2×C22⋊Q811yescharacteristic
451C2×C4C2×#12411yescharacteristic
452C2×C4C2×#12411yescharacteristic
453C2×C4C2×#12411yescharacteristic
454C2×C4C22×D811yescharacteristic
455C2×C4C22×SD1611yescharacteristic
456C2×C4C2×AGL(1,ℤ/8)12yes—
457C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
458C2×C4C2×#13712yes—
459C2×C4C2×#13711yescharacteristic
460C8—82no—
461C8—82no—
462C8—82no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22#202314yes—
477C22C2×#39311yescharacteristic
478C22C2×#44711yescharacteristic
479C22C2×#44811yescharacteristic
480C22#300311yescharacteristic
481C22#300411yescharacteristic
482C22#301911yescharacteristic
483C22#302011yescharacteristic
484C4—22no—
485C4—21no—
486C4—21no—
487C4—22no—
488C4—44no—
489C4—44no—
490C4—44no—
491C4—44no—
492C4—84no—
493C4—82no—
494C4#202314yes—
495C4C2×#47411yescharacteristic
496C4C2×#47611yescharacteristic
497C4#307912yes—
498C4C2×#48611yescharacteristic
499C4C2×#48911yescharacteristic
500C4#314112yes—
501C2—24no—
502C2—162no—
503C2#1874611yescharacteristic
504C2#1875811yescharacteristic
505C2#1876811yescharacteristic
506C2#1877611yescharacteristic
507C2C2×#202311yescharacteristic
508C2#2345211yescharacteristic
509C2#2346211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2551C2semidirect
5. #10350C2semidirect
6. #10348C2semidirect
7. #8043C2semidirect
8. #7901C2semidirect
9. #7892C2semidirect
49. #532Q8semidirect
162. C4⋊Q8C2×Q8semidirect
166. #127C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#218592
#213862
#212582
#103502
#103482
#80432
#79012
#78922