Finite Group Databasebeta

← #410418

Subgroups — #410418

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C211yesmaximal, characteristic
2C2×C4×#107C211yesmaximal, characteristic, complemented
3#13993C212yesmaximal
4#11328C211yesmaximal, characteristic, complemented
5#11256C211yesmaximal, characteristic, complemented
6#11228C212yesmaximal, complemented
7#11200C212yesmaximal
8#11096C211yesmaximal, characteristic
9#11096C211yesmaximal, characteristic
10#10820C211yesmaximal, characteristic, complemented
11#10807C212yesmaximal, complemented
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1923—24no—
24C4×#107—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1443—22no—
36#1439—22no—
37#1437—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C24×C4C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D411yescharacteristic
54C22×C22⋊C4D411yescharacteristic
55C22×C42—22no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—24no—
61C2×#107—24no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—42no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—42no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8D414yes—
72#389—24no—
73#389—42no—
74#380—24no—
75#380—24no—
76#380—24no—
77#380—42no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—24no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D411yescharacteristic
91C2×#95D411yescharacteristic
92#340—24no—
93#340—22no—
94#340—22no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#336—22no—
101#336—22no—
102C25C2×D411yescharacteristic
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141C22×C8#4112yes—
142C22×C8#4112yes—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4#4114yes—
168C2×C4⋊C4#4114yes—
169C2×C4⋊C4#4114yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—82no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4Q1614yes—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—44no—
190C2×C42—82no—
191C2×C42—82no—
192C2×C42C2×D412yes—
193C2×C42#4112yes—
194#107—28no—
195#107—28no—
196C4⋊C8—44no—
197C22⋊C8—44no—
198C22⋊C8—44no—
199C4×C8—28no—
200#95—28no—
201#95—22no—
202#95—22no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—82no—
208#95—82no—
209#95—82no—
210C24—21no—
211C24—22no—
212C24—22no—
213C24—21no—
214C24—21no—
215C24—21no—
216C24—22no—
217C24—21no—
218C24—21no—
219C24—22no—
220C24C2×Q1612yes—
221C24C22×D411yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4#12112yes—
291C22×C4#12112yes—
292C22×C4C2×Q1612yes—
293C22×C4C2×Q1612yes—
294C22×C4#13512yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4112yes—
299C22×C4C2×#4112yes—
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4112yes—
302C22×C4H(5,𝔽2)12yes—
303C22×C42−1+411yescharacteristic
304C22×C42−1+411yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8#12314yes—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—88no—
342C22⋊C4—28no—
343C22⋊C4—44no—
344C22⋊C4—88no—
345C22⋊C4—88no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—41no—
351C42—41no—
352C42—42no—
353C42—88no—
354C23—22no—
355C23—22no—
356C23—24no—
357C23—24no—
358C23—22no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23C2×#12111yescharacteristic
378C23#52512yes—
379C23#54612yes—
380C23#54711yescharacteristic
381C23#54711yescharacteristic
382C23C22×Q1611yescharacteristic
383C23C2×#13511yescharacteristic
384C23#57512yes—
385C23#57611yescharacteristic
386C23#57611yescharacteristic
387C23#57712yes—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—41no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4#46714yes—
449C2×C4C2×#12311yescharacteristic
450C2×C4#52511yescharacteristic
451C2×C4#52511yescharacteristic
452C2×C4#52912yes—
453C2×C4#52912yes—
454C2×C4#52912yes—
455C2×C4#53211yescharacteristic
456C2×C4#53511yescharacteristic
457C2×C4#53611yescharacteristic
458C2×C4#54212yes—
459C2×C4#54212yes—
460C2×C4#55212yes—
461C2×C4#55211yescharacteristic
462C2×C4#55211yescharacteristic
463C2×C4#55312yes—
464C2×C4#55312yes—
465C2×C4#55512yes—
466C2×C4#55612yes—
467C8—24no—
468C8—42no—
469C8—81no—
470C8—81no—
471C22—42no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—42no—
476C22—81no—
477C22—81no—
478C22—82no—
479C22—82no—
480C22—81no—
481C22—81no—
482C22#239211yescharacteristic
483C22#246812yes—
484C22#247412yes—
485C22#248611yescharacteristic
486C22#253511yescharacteristic
487C22C2×#46711yescharacteristic
488C22#305411yescharacteristic
489C22#305811yescharacteristic
490C22#306011yescharacteristic
491C22#310212yes—
492C22#310412yes—
493C22#311812yes—
494C22#311912yes—
495C22#329912yes—
496C22#330412yes—
497C22#332212yes—
498C22#332212yes—
499C22#333811yescharacteristic
500C22#333811yescharacteristic
501C22#333911yescharacteristic
502C22#333911yescharacteristic
503C22#334812yes—
504C22#335012yes—
505C4—24no—
506C4—42no—
507C4—42no—
508C4—84no—
509C4—84no—
510C4—81no—
511C4—81no—
512C4—84no—
513C4—162no—
514C4#250011yescharacteristic
515C4#252512yes—
516C4#252912yes—
517C4#260411yescharacteristic
518C4#263211yescharacteristic
519C4#285311yescharacteristic
520C2—82no—
521C2#1507811yescharacteristic
522C2#2369811yescharacteristic
523C2#2370211yescharacteristic
524C2#2710012yes—
525C2#2710812yes—
526C2#2733212yes—
527C2#2734012yes—
528C2#2766211yescharacteristic
529C2#2766511yescharacteristic
530C2#2814211yescharacteristic
531C2#2814611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#107C2semidirect
4. #11328C2semidirect
5. #11256C2semidirect
6. #11228C2semidirect
10. #10820C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×C4×#1072
#139932
#113282
#112562
#112282
#112002
#110962
#108202
#108072