Finite Group Databasebeta

← #400428

Subgroups — #400428

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic
3C2×#1920C211yesmaximal, characteristic
4C2×#1865C212yesmaximal
5#15194C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11190C212yesmaximal
8#11067C212yesmaximal
9#11010C211yesmaximal, characteristic
10#10806C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×#106C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19#2365—24no—
20#2365—24no—
21C2×#391C2212yes—
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#387—22no—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2043—24no—
30#1920—24no—
31#1865—28no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—21no—
42#1444—21no—
43#1441—22no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×Q8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8D414yes—
54C2×C4×Q8—24no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C2×C4×Q8—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—22no—
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#106—22no—
77C2×#106—22no—
78C2×#106D414yes—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—42no—
90C2×#103—42no—
91#391—24no—
92#391—24no—
93#389—28no—
94#389—24no—
95#389—22no—
96#389—22no—
97#389—42no—
98#387—216no—
99C4×C4⋊C4—28no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4D414yes—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D412yes—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—24no—
119#336—24no—
120#336—22no—
121#336—22no—
122#336—42no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8C2×D414yes—
135C22×Q8#4114yes—
136C22×Q8#4114yes—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8C2×D412yes—
165C4⋊Q8—28no—
166C4⋊Q8—28no—
167C4⋊Q8—216no—
168C4×Q8—48no—
169C4×Q8—48no—
170C4×Q8—48no—
171C4×Q8—48no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4#4114yes—
214C2×C4⋊C4#4112yes—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—28no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—44no—
226C2×C42—44no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231#106—28no—
232#103—216no—
233#103—48no—
234#103—48no—
235#103—48no—
236#95—28no—
237#95—28no—
238#95—24no—
239#95—24no—
240#95—44no—
241#95—42no—
242#95—42no—
243#95—81no—
244#95—82no—
245#95—82no—
246C24C22×D411yescharacteristic
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8#12318yes—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4#12012yes—
331C22×C4#12012yes—
332C22×C4#13512yes—
333C22×C4#13714yes—
334C22×C4#13712yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C42−1+412yes—
345C2×C8—22no—
346C2×C8—24no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—28no—
363C4⋊C4—216no—
364C4⋊C4—216no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—416no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—88no—
385C4⋊C4—84no—
386C4⋊C4#12114yes—
387C42—24no—
388C42—22no—
389C42—22no—
390C42—416no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—48no—
396C23C2×#12011yescharacteristic
397C23#52512yes—
398C23D4211yescharacteristic
399C23#54611yescharacteristic
400C23#54812yes—
401C23C2×#13511yescharacteristic
402C23C2×#13711yescharacteristic
403C23C2×#13712yes—
404C23#57711yescharacteristic
405C23#57712yes—
406C23#57811yescharacteristic
407Q8—28no—
408Q8—28no—
409Q8—44no—
410Q8—44no—
411Q8—44no—
412Q8—44no—
413Q8—44no—
414Q8—44no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—28no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
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—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4#45214yes—
475C2×C4C2×#12111yescharacteristic
476C2×C4C2×#12312yes—
477C2×C4#52511yescharacteristic
478C2×C4#52511yescharacteristic
479C2×C4#52512yes—
480C2×C4#52912yes—
481C2×C4#53211yescharacteristic
482C2×C4#53511yescharacteristic
483C2×C4#53611yescharacteristic
484C2×C4#53912yes—
485C2×C4#53912yes—
486C2×C4D4211yescharacteristic
487C2×C4#54611yescharacteristic
488C2×C4#54712yes—
489C2×C4#54812yes—
490C2×C4#55112yes—
491C2×C4#55212yes—
492C2×C4#55312yes—
493C8—42no—
494C8—81no—
495C8—82no—
496C22#240811yescharacteristic
497C22#243211yescharacteristic
498C22#246012yes—
499C22#246312yes—
500C22#284911yescharacteristic
501C22C2×#45211yescharacteristic
502C22#300411yescharacteristic
503C22#301011yescharacteristic
504C22#301311yescharacteristic
505C22#310812yes—
506C22#311312yes—
507C22#312112yes—
508C22#312612yes—
509C22#326612yes—
510C22#327611yescharacteristic
511C22#328411yescharacteristic
512C22#330211yescharacteristic
513C22#332012yes—
514C22#333111yescharacteristic
515C22#336012yes—
516C22#336912yes—
517C22#337212yes—
518C22#337412yes—
519C4—24no—
520C4—44no—
521C4—44no—
522C4—48no—
523C4—42no—
524C4—84no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—161no—
529C4#239211yescharacteristic
530C4#241912yes—
531C4#244312yes—
532C4#253611yescharacteristic
533C4#260311yescharacteristic
534C4#263311yescharacteristic
535C2#1480111yescharacteristic
536C2#2424111yescharacteristic
537C2#2425111yescharacteristic
538C2#2564611yescharacteristic
539C2#2565711yescharacteristic
540C2#2675912yes—
541C2#2677512yes—
542C2#2690312yes—
543C2#2691112yes—
544C2#3300811yescharacteristic
545C2#3301511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#20432
C2×#19202
C2×#18652
#151942
#113462
#111902
#110672
#110102
#108062
#107892