Finite Group Databasebeta

← #239785

Subgroups — #239785

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#21959C211yesmaximal, characteristic, complemented
3#21833C211yesmaximal, characteristic, complemented
4#20645C211yesmaximal, characteristic
5C22×#332C211yesmaximal, characteristic, complemented
6C2×#1477C211yesmaximal, characteristic
7#10418C214yesmaximal, complemented
8#10417C214yesmaximal, complemented
9#7893C211yesmaximal, characteristic, complemented
10C2×#536—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×#416—21no—
17#2857—22no—
18#2621—22no—
19#2547—21no—
20#2547C2212yes—
21#2546—21no—
22#2546C2212yes—
23#2486—22no—
24#2486—22no—
25#2310—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#1990C2214yes—
29#1988—22no—
30#1948—22no—
31#1948C2214yes—
32#1823—22no—
33#1807—22no—
34#1805—21no—
35#1804—21no—
36#1803C2212yes—
37#1802C2212yes—
38#1735—21no—
39C2×#332—24no—
40C2×#332C2214yes—
41#1477—24no—
42C2×Q8⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44#1279C2214yes—
45C22×#41D411yescharacteristic
46C23×Q8—21no—
47C23×C8—21no—
48C23×C8—21no—
49#536—44no—
50#536—44no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D412yes—
59C2×C4⋊Q8C2312yes—
60C2×#124—22no—
61C2×#123—24no—
62C2×C22⋊Q8—24no—
63C4×#41—22no—
64C4×#41—22no—
65C4×#41—22no—
66C4×#41—22no—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4D411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—21no—
76C2×#117D411yescharacteristic
77C22×C4⋊C4—21no—
78C22×C4⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431—22no—
83C2×#106—24no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8D414yes—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93#416—44no—
94C2×#103—22no—
95C2×#102—22no—
96C2×C22⋊C8—21no—
97C2×C4×C8C2312yes—
98#398—24no—
99#398—24no—
100#397—24no—
101#397—24no—
102#391—24no—
103#391—22no—
104#391—22no—
105#389—24no—
106#389—24no—
107#384D412yes—
108#384C2312yes—
109#382—21no—
110#382—21no—
111#381—24no—
112#380—24no—
113#380—21no—
114#380—24no—
115#380—21no—
116#380—21no—
117#380D412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2312yes—
122#340—22no—
123#340—22no—
124#340—24no—
125#340—28no—
126#332—28no—
127Q8⋊C8—24no—
128D4⋊C8—24no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—22no—
132C2×#41—22no—
133C2×#41—22no—
134C2×#41SD1614yes—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8C2×D411yescharacteristic
144C22×D4—21no—
145C22×D4C2×D411yescharacteristic
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8—22no—
154C22×C8—24no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—42no—
158C22×C8—42no—
159C22×C8—42no—
160C22×C8—82no—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173#124—44no—
174#124—44no—
175#123—416no—
176C22⋊Q8—416no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—42no—
193#117—22no—
194#117—22no—
195#117—42no—
196#117—42no—
197#117—42no—
198#117—42no—
199#117—42no—
200#117—42no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D414yes—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—41no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—41no—
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42C2411yescharacteristic
241#106—84no—
242C4⋊C8—24no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8C2×D414yes—
249#103—44no—
250#102—44no—
251C22⋊C8—42no—
252C4×C8—22no—
253C4×C8—22no—
254#95—28no—
255#95—28no—
256#95—22no—
257#95—22no—
258#95—41no—
259#95—41no—
260C24—21no—
261C24#12011yescharacteristic
262#41—28no—
263#41—28no—
264#41—44no—
265#41—44no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8C2×SD1612yes—
285C2×D4—24no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4C2×SD1612yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12112yes—
321C22×C4#12312yes—
322C22×C4#12312yes—
323C22×C4#12412yes—
324C22×C4C2×SD1612yes—
325C22×C4#13512yes—
326C22×C4AGL(1,ℤ/8)12yes—
327C22×C4AGL(1,ℤ/8)12yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C2×C8—24no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—84no—
339C2×C8—84no—
340C4⋊C4—24no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—24no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—44no—
349C4⋊C4—416no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C22⋊C4—416no—
364C22⋊C4—42no—
365C22⋊C4—44no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—42no—
369C22⋊C4—82no—
370C42—41no—
371C42—41no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—82no—
377C42#12112yes—
378C42#12112yes—
379C42#12112yes—
380C42#12312yes—
381C42#12312yes—
382C42#12412yes—
383C42C22×D411yescharacteristic
384C42C2×#4111yescharacteristic
385C42C2×#4111yescharacteristic
386C42C2×#4111yescharacteristic
387C23—22no—
388C23—22no—
389C23—22no—
390C23—41no—
391C23—41no—
392C23—42no—
393C23—42no—
394C23#39412yes—
395C23#44912yes—
396C23#45012yes—
397C23C2×#12011yescharacteristic
398Q8—22no—
399Q8—22no—
400Q8—41no—
401Q8—41no—
402Q8—82no—
403Q8—82no—
404Q8—82no—
405Q8—82no—
406Q8—88no—
407D4—22no—
408D4—22no—
409D4—48no—
410D4—41no—
411D4—41no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—48no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—84no—
435C2×C4—88no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4#39412yes—
443C2×C4#39412yes—
444C2×C4#39412yes—
445C2×C4#44912yes—
446C2×C4#45012yes—
447C2×C4#46514yes—
448C2×C4#46914yes—
449C2×C4#48114yes—
450C2×C4#48214yes—
451C2×C4C2×#12111yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4C2×#12111yescharacteristic
454C2×C4C2×#12311yescharacteristic
455C2×C4C2×#12311yescharacteristic
456C2×C4C2×#12411yescharacteristic
457C2×C4C22×SD1611yescharacteristic
458C2×C4C2×#13511yescharacteristic
459C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
460C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
461C2×C4#57712yes—
462C2×C4#57812yes—
463C8—44no—
464C8—82no—
465C8—82no—
466C22—24no—
467C22—24no—
468C22—22no—
469C22—22no—
470C22—42no—
471C22—44no—
472C22—82no—
473C22—81no—
474C22—81no—
475C22#203414yes—
476C22C2×#39411yescharacteristic
477C22C2×#45011yescharacteristic
478C22C2×#44911yescharacteristic
479C22#301312yes—
480C22#302012yes—
481C4—21no—
482C4—22no—
483C4—21no—
484C4—22no—
485C4—42no—
486C4—84no—
487C4—84no—
488C4—81no—
489C4—81no—
490C4—162no—
491C4#203414yes—
492C4#304211yescharacteristic
493C4#304411yescharacteristic
494C4C2×#46511yescharacteristic
495C4C2×#46911yescharacteristic
496C4C2×#48211yescharacteristic
497C4C2×#48111yescharacteristic
498C4#313712yes—
499C2—24no—
500C2—82no—
501C2#1888312yes—
502C2#1888612yes—
503C2C2×#203411yescharacteristic
504C2#2349411yescharacteristic
505C2#2351211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
2. #21959C2semidirect
3. #21833C2semidirect
5. C22×#332C2semidirect
7. #10418C2semidirect
8. #10417C2semidirect
9. #7893C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#219592
#218332
#206452
C22×#3322
C2×#14772
#104182
#104172
#78932