Finite Group Databasebeta

← #360913

Subgroups — #360913

Order 512.

1086 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1921C212yesmaximal
2C2×C4×#107C211yesmaximal, characteristic
3#13954C211yesmaximal, characteristic
4#13804C211yesmaximal, characteristic
5#11292C212yesmaximal
6#11219C212yesmaximal
7#10884C212yesmaximal
8#10850C211yesmaximal, characteristic
9#10850C211yesmaximal, characteristic
10#10842C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1921—28no—
25C4×#107C418yescomplemented
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1444—22no—
37#1436—22no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41#1434—21no—
42C4×#95—21no—
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C42—21no—
56C22×C42—21no—
57C22×C42—22no—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×#107—22no—
61C2×#107—22no—
62C2×#107C2×C414yes—
63C2×#107C2×C414yes—
64C2×#107D414yes—
65C2×#107Q814yes—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8C2×C414yes—
72#389—24no—
73#389—24no—
74#389—42no—
75#389—42no—
76#382—22no—
77#382—22no—
78#382—22no—
79#382Q814yes—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—42no—
86C4×C4⋊C4C2×C414yes—
87C4×C4⋊C4C2×C414yes—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95D411yescharacteristic
100C2×#95D411yescharacteristic
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—42no—
106#340—42no—
107#340C2×C414yes—
108#340C2×C414yes—
109#336—24no—
110#336—42no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8C22×C412yes—
140C22×C8C22×C412yes—
141C22×C8C2×D412yes—
142C22×C8C2×Q812yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×Q814yes—
184C2×C4⋊C4C2×Q812yes—
185C2×C4⋊C4C2×Q812yes—
186C2×C4⋊C4#4112yes—
187C2×C4⋊C4#4112yes—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—44no—
197C2×C42—44no—
198C2×C42—82no—
199C2×C42—82no—
200C2×C42—82no—
201C2×C42C22×C412yes—
202C2×C42C2×Q812yes—
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207#107—28no—
208#107C4⋊C4116yescomplemented
209C4⋊C8—44no—
210C4×C8—28no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—22no—
216#95—22no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—42no—
221#95—42no—
222#95—44no—
223#95—82no—
224C24C22×D411yescharacteristic
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4C4×D412yes—
277C22×C4C4×D412yes—
278C22×C4C22⋊Q812yes—
279C22×C4C22⋊Q812yes—
280C22×C4#12312yes—
281C22×C4#12312yes—
282C22×C4#13514yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×Q811yescharacteristic
286C22×C4C22×Q811yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4H(5,𝔽2)11yescharacteristic
294C22×C4H(5,𝔽2)11yescharacteristic
295C22×C42−1+411yescharacteristic
296C22×C42−1+411yescharacteristic
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8C2×C4⋊C414yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—48no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—88no—
341C4⋊C4—88no—
342C4⋊C4C2×C4⋊C414yes—
343C4⋊C4C2×C4⋊C414yes—
344C4⋊C4#11714yes—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—84no—
350C42—84no—
351C42—88no—
352C42#11714yes—
353C42C4×Q814yes—
354C42#12514yes—
355C42#12614yes—
356C23C2×C4×D411yescharacteristic
357C23C2×C22⋊Q811yescharacteristic
358C23C2×#12311yescharacteristic
359C23D4211yescharacteristic
360C23#54611yescharacteristic
361C23#54811yescharacteristic
362C23D4×Q811yescharacteristic
363C23C2×#13512yes—
364C23#57512yes—
365C23#57612yes—
366C23#57812yes—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4#38714yes—
424C2×C4#38714yes—
425C2×C4C22×C4⋊C411yescharacteristic
426C2×C4C2×#11711yescharacteristic
427C2×C4C2×#11711yescharacteristic
428C2×C4C2×C4×Q811yescharacteristic
429C2×C4#52011yescharacteristic
430C2×C4#52011yescharacteristic
431C2×C4C2×#12511yescharacteristic
432C2×C4C2×#12611yescharacteristic
433C2×C4#53211yescharacteristic
434C2×C4#53211yescharacteristic
435C2×C4#53311yescharacteristic
436C2×C4#53311yescharacteristic
437C2×C4#54011yescharacteristic
438C2×C4#54111yescharacteristic
439C2×C4#54211yescharacteristic
440C2×C4#54211yescharacteristic
441C2×C4#54711yescharacteristic
442C2×C4D4×Q811yescharacteristic
443C2×C4#55011yescharacteristic
444C2×C4#55211yescharacteristic
445C2×C4#55311yescharacteristic
446C2×C4#55411yescharacteristic
447C2×C4#55411yescharacteristic
448C2×C4#55611yescharacteristic
449C2×C4#55711yescharacteristic
450C2×C4#55711yescharacteristic
451C2×C4Q8211yescharacteristic
452C2×C4#55911yescharacteristic
453C8—24no—
454C8—42no—
455C8—82no—
456C22C2×#38711yescharacteristic
457C22D4×C4⋊C411yescharacteristic
458C22#235211yescharacteristic
459C22#244511yescharacteristic
460C22#244611yescharacteristic
461C22#245111yescharacteristic
462C22#245211yescharacteristic
463C22C4×#13511yescharacteristic
464C22#294311yescharacteristic
465C22#294811yescharacteristic
466C22#295011yescharacteristic
467C22#307612yes—
468C22#308312yes—
469C22#309112yes—
470C22#309812yes—
471C22#327312yes—
472C22#328512yes—
473C22#329712yes—
474C22#332712yes—
475C22#336112yes—
476C22#337712yes—
477C22#338312yes—
478C22#338912yes—
479C4—24no—
480C4—24no—
481C4—44no—
482C4—44no—
483C4—44no—
484C4—44no—
485C4—84no—
486C4—82no—
487C4—82no—
488C4—162no—
489C4#228511yescharacteristic
490C4#231411yescharacteristic
491C4Q8×C4⋊C411yescharacteristic
492C4#235511yescharacteristic
493C4#250611yescharacteristic
494C4#250711yescharacteristic
495C4#251111yescharacteristic
496C4#251211yescharacteristic
497C2#1388411yescharacteristic
498C2#2103211yescharacteristic
499C2#2103711yescharacteristic
500C2#2267211yescharacteristic
501C2#2268511yescharacteristic
502C2#2276111yescharacteristic
503C2#2277211yescharacteristic
504C2#2623812yes—
505C2#2628412yes—
506C2#2644812yes—
507C2#2646812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. C4×#107C4semidirect
208. #107C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19212
C2×C4×#1072
#139542
#138042
#112922
#112192
#108842
#108502
#108422
#108342