Finite Group Databasebeta

← #355433

Subgroups — #355433

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34065C211yesmaximal, characteristic
2C2×#1871C212yesmaximal
3#20335C211yesmaximal, characteristic, complemented
4#13761C211yesmaximal, characteristic, complemented
5#10924C211yesmaximal, characteristic, complemented
6#10923C211yesmaximal, characteristic, complemented
7#10879C211yesmaximal, characteristic, complemented
8#10878C211yesmaximal, characteristic, complemented
9#10808C212yesmaximal, complemented
10#10790C212yesmaximal, complemented
11#10732C212yesmaximal, complemented
12C23×M16C2212yes—
13C2×#525C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#104C2212yes—
17C22×#104C2212yes—
18#2324C414yescomplemented
19#2323C414yescomplemented
20#2322—22no—
21#2321—22no—
22#2297C2211yescharacteristic
23#2296C2211yescharacteristic
24#2293—22no—
25#2290—22no—
26#2280C2211yescharacteristic
27C4×#117C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29C2×#376C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1871C4116yescomplemented
32C2×#339C2212yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35C2×#339C2212yes—
36C2×#339C2212yes—
37C2×#339C2212yes—
38#1729—22no—
39#1729—22no—
40#1728C2212yes—
41#1728C2211yescharacteristic
42#1728C2211yescharacteristic
43#1437—22no—
44#1436—22no—
45C4×#95C2212yes—
46C22×#41C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—24no—
50C22×M16—42no—
51C22×M16—42no—
52C22×M16—42no—
53C22×M16—42no—
54C22×M16C2×C418yescomplemented
55C22×M16C2312yes—
56C22×M16C2312yes—
57C23×C8—22no—
58#525—24no—
59#525C2×C418yes—
60C2×#123—21no—
61C2×C22⋊Q8C2×C412yes—
62C2×#121C2×C412yes—
63C2×#120—21no—
64C4×#41—42no—
65C4×#41—42no—
66C2×C4×Q8C2×C412yes—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4C2×C412yes—
70C2×#117—21no—
71C2×#117—42no—
72C2×#117—42no—
73C2×#117C2×C414yes—
74C2×#117D412yes—
75C2×#117D412yes—
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4D412yes—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42—21no—
83C22×C42—22no—
84C22×C42C2312yes—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#104—42no—
89C2×#104—42no—
90C2×#104—42no—
91C2×#104—42no—
92C2×#104C2×C418yes—
93C2×#104C2×C418yes—
94#391C2×C412yes—
95#390—24no—
96#390C2×C412yes—
97#389—24no—
98#389C2×C414yes—
99#388—24no—
100#388—24no—
101#388—24no—
102#387—24no—
103#386—22no—
104#386—24no—
105#385—22no—
106#385—24no—
107#381—21no—
108#380C2×C414yes—
109#379—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C414yes—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4C2×C418yescomplemented
117#376—21no—
118#376—42no—
119C2×#95—22no—
120C2×#95—24no—
121C2×#95—22no—
122C43—21no—
123C43—42no—
124#339—42no—
125#339—42no—
126#339—42no—
127#339—42no—
128#339C2×C418yescomplemented
129#339C2×C418yescomplemented
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134C25C2×D411yescharacteristic
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—22no—
139C2×#41—22no—
140C2×#41C22×C414yes—
141C22×Q8C22×C411yescharacteristic
142C22×D4—21no—
143C22×D4—22no—
144C22×D4C22×C411yescharacteristic
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C22×C412yes—
154C23×C4C22×C414yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4#4114yes—
160C23×C4C2411yescharacteristic
161C2×M16—28no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—48no—
173C2×M16—48no—
174C2×M16—48no—
175C2×M16—48no—
176C2×M16—44no—
177C2×M16—44no—
178C2×M16C22×C414yes—
179C2×M16C22×C414yes—
180C22×C8—28no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—44no—
184C22×C8—44no—
185#123—28no—
186C22⋊Q8—24no—
187C22⋊Q8C4218yescomplemented
188#121—24no—
189#121C4218yescomplemented
190#120—28no—
191C4×Q8—44no—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—22no—
202#117—48no—
203#117—42no—
204#117—42no—
205#117—42no—
206#117—42no—
207#117—44no—
208#117C22×C414yes—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C22⋊C414yes—
215C2×C4⋊C4C22⋊C414yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4C22⋊C414yes—
232C2×C22⋊C4C22⋊C414yes—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42—48no—
255C2×C42C22×C414yes—
256C2×C42C22×C414yes—
257C2×C42C22×C412yes—
258C2×C42C22×C412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262#104—48no—
263#104—48no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—28no—
269#95—48no—
270#95—44no—
271C24—21no—
272C24—21no—
273C24—22no—
274C24—21no—
275C24—44no—
276C24C2×C22⋊C412yes—
277C24C4×D414yes—
278C24C22×D411yescharacteristic
279#41—24no—
280#41—24no—
281#41—24no—
282#41—24no—
283#41—42no—
284#41—42no—
285#41—42no—
286#41—42no—
287C2×Q8—21no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8C2×C4212yes—
291C2×D4—24no—
292C2×D4—28no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—84no—
300C2×D4C2×C4212yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—84no—
344C22×C4C2×C4214yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C414yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4#11714yes—
350C22×C4C4×D414yes—
351C22×C4C4×D414yes—
352C22×C4C4×D414yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C23×C412yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C2×#4112yes—
357C22×C4H(5,𝔽2)12yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C4H(5,𝔽2)12yes—
360C22×C42−1+412yes—
361M16—24no—
362M16—24no—
363M16—24no—
364M16—24no—
365M16—416no—
366M16—42no—
367M16—42no—
368M16—42no—
369M16—42no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—416no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—22no—
378C4⋊C4—28no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—84no—
385C4⋊C4C2×C4214yes—
386C22⋊C4—24no—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—416no—
392C22⋊C4—44no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—42no—
396C22⋊C4—84no—
397C22⋊C4—84no—
398C42—24no—
399C42—24no—
400C42—24no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—24no—
406C42—24no—
407C42—24no—
408C42—24no—
409C42—44no—
410C42—416no—
411C42—42no—
412C42—41no—
413C42—41no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—84no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—44no—
428C23—44no—
429C23—41no—
430C23—41no—
431C23—44no—
432C23—82no—
433C23—82no—
434C23—84no—
435C23C4×C22⋊C414yes—
436C23C22×C22⋊C411yescharacteristic
437C23C2×C4×D412yes—
438C23#53411yescharacteristic
439C23#53411yescharacteristic
440C23#53712yes—
441Q8—24no—
442Q8—42no—
443D4—24no—
444D4—44no—
445D4—44no—
446D4—42no—
447D4—88no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—48no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—88no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4C4×C22⋊C414yes—
492C2×C4C22×C4211yescharacteristic
493C2×C4C2×#11711yescharacteristic
494C2×C4C2×C4×D412yes—
495C2×C4#51812yes—
496C2×C4#51812yes—
497C2×C4#51812yes—
498C2×C4#52012yes—
499C2×C4#52012yes—
500C2×C4#52012yes—
501C2×C4#53412yes—
502C2×C4#53511yescharacteristic
503C2×C4#53611yescharacteristic
504C2×C4#53812yes—
505C2×C4#53812yes—
506C2×C4#53912yes—
507C2×C4#53912yes—
508C8—44no—
509C8—44no—
510C22—22no—
511C22—22no—
512C22—24no—
513C22—22no—
514C22—22no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—42no—
522C22—84no—
523C22—84no—
524C22—82no—
525C22—82no—
526C22C2×C4×C22⋊C411yescharacteristic
527C22#230811yescharacteristic
528C22#231111yescharacteristic
529C22#232512yes—
530C22#232512yes—
531C22#290114yes—
532C4—24no—
533C4—48no—
534C4—44no—
535C4—42no—
536C4—42no—
537C4—44no—
538C4—42no—
539C4—84no—
540C4—82no—
541C4—82no—
542C4#227512yes—
543C4#231812yes—
544C4#232112yes—
545C4#232212yes—
546C2—22no—
547C2—22no—
548C2—44no—
549C2—82no—
550C2#1378411yescharacteristic
551C2#2040512yes—
552C2#2177512yes—
553C2#2185812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20335C2semidirect
4. #13761C2semidirect
5. #10924C2semidirect
6. #10923C2semidirect
7. #10879C2semidirect
8. #10878C2semidirect
9. #10808C2semidirect
10. #10790C2semidirect
11. #10732C2semidirect
18. #2324C4semidirect
19. #2323C4semidirect
31. #1871C4semidirect
54. C22×M16C2×C4semidirect
116. C4×C22⋊C4C2×C4semidirect
128. #339C2×C4semidirect
129. #339C2×C4semidirect
187. C22⋊Q8C42semidirect
189. #121C42semidirect

Maximal subgroups

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

subgroupindex
#340652
C2×#18712
#203352
#137612
#109242
#109232
#108792
#108782
#108082
#107902
#107322