Finite Group Databasebeta

← #222053

Subgroups — #222053

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36122C211yesmaximal, characteristic
2#22648C212yesmaximal, complemented
3C2×#1560C211yesmaximal, characteristic, complemented
4C22×#333C211yesmaximal, characteristic, complemented
5#9707C212yesmaximal, complemented
6#9135C212yesmaximal, complemented
7#9133C212yesmaximal, complemented
8#8587C211yesmaximal, characteristic, complemented
9#8344C211yesmaximal, characteristic, complemented
10#8342C211yesmaximal, characteristic, complemented
11#8232C211yesmaximal, characteristic, complemented
12C2×#548C412yes—
13C2×#532C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C2×#435—22no—
16C2×#423C2212yes—
17C22×C4⋊C8C2212yes—
18#2858C2211yescharacteristic
19#2857C2211yescharacteristic
20#2856C2211yescharacteristic
21#2545C412yes—
22#2491—22no—
23#2452—22no—
24#2436C412yes—
25#2432C412yes—
26#2416—22no—
27#2361—22no—
28C4×C4⋊Q8C2211yescharacteristic
29C42×D4C2211yescharacteristic
30#1940—22no—
31#1851—22no—
32#1850—22no—
33#1842C2212yes—
34#1797C2212yes—
35#1797C2212yes—
36#1767C2212yes—
37#1560—24no—
38C2×#333—24no—
39C2×#333C2211yescharacteristic
40C2×#333C2212yes—
41C2×#333C2211yescharacteristic
42C2×#333C2211yescharacteristic
43C2×#333C2212yes—
44C2×#333C2211yescharacteristic
45C2×#331C2212yes—
46C2×#331C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×#41C2×C412yes—
50C23×Q8C2×C412yes—
51C22×M16—22no—
52C23×C8—22no—
53#548—48no—
54#532—22no—
55#532—22no—
56#532—42no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×#127C2311yescharacteristic
65C2×#124—22no—
66C2×#124C2×C412yes—
67C2×#124C2×C412yes—
68C2×#124C2×C412yes—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8C2×C412yes—
71C2×#121—22no—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121C2×C412yes—
75C2×C4×Q8C2×C412yes—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4D412yes—
80C22×C4⋊C4—22no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#435—48no—
84#423—24no—
85#423—42no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—42no—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×#97—22no—
99#396—24no—
100#394—24no—
101#393—24no—
102#391C2×C412yes—
103#391C2×C412yes—
104#390—22no—
105#390—22no—
106#390—22no—
107#390C2×C412yes—
108#389—22no—
109#389—22no—
110#388—24no—
111#386—24no—
112#385—24no—
113#384C2311yescharacteristic
114#380—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4D412yes—
118C4×C22⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C411yescharacteristic
120C4×C22⋊C4D412yes—
121C4×C22⋊C4D412yes—
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—22no—
126#333—22no—
127#333—22no—
128#333—22no—
129#333—24no—
130#333—42no—
131#333D414yescomplemented
132#331—24no—
133#331—42no—
134C2×#41—24no—
135C2×#41—42no—
136C2×#41—44no—
137C2×#41—82no—
138C22×Q8—22no—
139C22×Q8—24no—
140C22×Q8—42no—
141C22×Q8—44no—
142C22×Q8—82no—
143C22×Q8C22×C412yes—
144C22×Q8C22×C412yes—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C22⋊C411yescharacteristic
149C22×D4C22×C412yes—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C2×M16—44no—
157C2×M16—44no—
158C2×M16—82no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—44no—
164C22×C8—44no—
165C22×C8—82no—
166C4⋊Q8—24no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173C4⋊Q8—42no—
174C4⋊Q8C2×D412yes—
175C4⋊Q8#4112yes—
176#127—21no—
177#127—21no—
178#127—22no—
179#124—24no—
180#124—42no—
181#124—44no—
182#124—48no—
183#124—82no—
184C22⋊Q8—48no—
185C22⋊Q8—44no—
186#121—44no—
187#121—44no—
188#121—44no—
189#121—48no—
190#121—44no—
191#121—44no—
192C4×Q8—44no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—44no—
197C4×D4—44no—
198C4×D4—48no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4C22⋊C411yescharacteristic
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C411yescharacteristic
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2411yescharacteristic
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—44no—
245C4⋊C8—44no—
246C4⋊C8—44no—
247C4⋊C8—82no—
248C4⋊C8C2×D414yes—
249C22⋊C8—44no—
250C22⋊C8—44no—
251#97—44no—
252#95—24no—
253#95—24no—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257C24—22no—
258C24C2×C22⋊C411yescharacteristic
259C24C2×C22⋊C411yescharacteristic
260#41—88no—
261#41—88no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—82no—
272C2×Q8—88no—
273C2×Q8—88no—
274C2×Q8C4×D414yes—
275C2×D4—22no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—44no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—48no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—82no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C412yes—
326C22×C4AGL(1,ℤ/8)14yes—
327C22×C4#13714yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D412yes—
330M16—88no—
331C2×C8—24no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—88no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—48no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C22⋊C4—24no—
357C22⋊C4—22no—
358C22⋊C4—24no—
359C22⋊C4—48no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—48no—
365C22⋊C4—84no—
366C22⋊C4#10014yes—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—41no—
381C42—44no—
382C42C22×D411yescharacteristic
383C42C2×#4111yescharacteristic
384C42H(5,𝔽2)12yes—
385C23—21no—
386C23—21no—
387C23—22no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—22no—
392C23—22no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—44no—
397C23—44no—
398C23C2×#10012yes—
399C23#41714yes—
400C23C22×C22⋊C411yescharacteristic
401Q8—44no—
402Q8—44no—
403Q8—82no—
404Q8—82no—
405Q8—88no—
406D4—44no—
407D4—82no—
408D4—82no—
409D4—88no—
410D4—88no—
411D4—82no—
412D4—82no—
413D4—82no—
414D4—82no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—24no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—88no—
454C2×C4—88no—
455C2×C4C2×#10012yes—
456C2×C4C2×#10012yes—
457C2×C4#41714yes—
458C2×C4C2×C4×D411yescharacteristic
459C2×C4#51811yescharacteristic
460C2×C4D4212yes—
461C2×C4#54612yes—
462C2×C4C2×AGL(1,ℤ/8)12yes—
463C2×C4C2×#13712yes—
464C2×C4#57712yes—
465C2×C4#57812yes—
466C8—44no—
467C8—82no—
468C8—82no—
469C8—82no—
470C22—24no—
471C22—22no—
472C22—22no—
473C22—24no—
474C22—41no—
475C22—42no—
476C22—41no—
477C22—41no—
478C22—42no—
479C22—82no—
480C22—84no—
481C22—82no—
482C22—82no—
483C22—84no—
484C22#156014yescomplemented
485C22D4×C22⋊C411yescharacteristic
486C22C22×#10011yescharacteristic
487C22#288811yescharacteristic
488C22C2×#41712yes—
489C22#289812yes—
490C4—22no—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—24no—
495C4—41no—
496C4—42no—
497C4—41no—
498C4—44no—
499C4—42no—
500C4—82no—
501C4—82no—
502C4#326512yes—
503C4#327012yes—
504C4#330812yes—
505C4#331612yes—
506C2—24no—
507C2—81no—
508C2—162no—
509C2C2×#156011yescharacteristic
510C2#1776412yes—
511C2#1777911yescharacteristic
512C2D4×#10011yescharacteristic
513C2#2242112yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36122C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22648C2semidirect
3. C2×#1560C2semidirect
4. C22×#333C2semidirect
5. #9707C2semidirect
6. #9135C2semidirect
7. #9133C2semidirect
8. #8587C2semidirect
9. #8344C2semidirect
10. #8342C2semidirect
11. #8232C2semidirect
131. #333D4semidirect
484. C22#1560semidirect

Maximal subgroups

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

subgroupindex
#361222
#226482
C2×#15602
C22×#3332
#97072
#91352
#91332
#85872
#83442
#83422
#82322