Finite Group Databasebeta

← #404887

Subgroups — #404887

Order 512.

1224 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic
3C2×#1865C211yesmaximal, characteristic
4#15149C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11190C212yesmaximal
8#11067C212yesmaximal
9#10908C212yesmaximal
10#10885C212yesmaximal
11#10789C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#103C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C2×#398—21no—
18#2355—24no—
19#2355—24no—
20C2×#391C2212yes—
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#387—22no—
24C2×#384C2211yescharacteristic
25C2×#382—21no—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2099—24no—
30#1865—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1442—22no—
42#1441—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×Q8—24no—
50C2×C4×Q8—24no—
51C2×C4×Q8—22no—
52C2×C4×Q8D414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42—22no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×#106—22no—
70C2×#106—24no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—42no—
79#398—44no—
80#391—24no—
81#391—24no—
82#389—24no—
83#389—22no—
84#389—22no—
85#389—42no—
86#387—216no—
87#384—22no—
88#384D414yes—
89#382—28no—
90C4×C4⋊C4—28no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95D412yes—
105#340—24no—
106#340—24no—
107#340—22no—
108#340—24no—
109#340—24no—
110#340—24no—
111#336—24no—
112#336—22no—
113#336—22no—
114#336—42no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8C2×D412yes—
124C22×Q8#4114yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8#4112yes—
153C4×Q8—216no—
154C4×Q8—28no—
155C4×Q8—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4#4114yes—
198C2×C4⋊C4#4112yes—
199C2×C4⋊C4#4114yes—
200C2×C42—28no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—28no—
211C2×C42—82no—
212C2×C42—82no—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215#106—44no—
216#103—28no—
217#103—48no—
218#103—44no—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—24no—
223#95—44no—
224#95—44no—
225#95—42no—
226#95—42no—
227#95—44no—
228#95—81no—
229C24C22×D411yescharacteristic
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8#12314yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4#12012yes—
304C22×C4#12012yes—
305C22×C4#13512yes—
306C22×C4AGL(1,ℤ/8)12yes—
307C22×C4#13714yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4112yes—
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C42−1+412yes—
317C22×C42−1+411yescharacteristic
318C2×C8—22no—
319C2×C8—24no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—28no—
336C4⋊C4—216no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—84no—
357C4⋊C4—88no—
358C4⋊C4—84no—
359C4⋊C4#12314yes—
360C42—216no—
361C42—22no—
362C42—22no—
363C42—24no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42—88no—
369C23C2×#12011yescharacteristic
370C23#52512yes—
371C23#54611yescharacteristic
372C23#54711yescharacteristic
373C23#54812yes—
374C23C2×#13511yescharacteristic
375C23C2×AGL(1,ℤ/8)11yescharacteristic
376C23C2×#13712yes—
377C23#57711yescharacteristic
378C23#57712yes—
379C23#57811yescharacteristic
380Q8—24no—
381Q8—24no—
382Q8—42no—
383Q8—42no—
384Q8—44no—
385Q8—44no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4#44914yes—
446C2×C4C2×#12311yescharacteristic
447C2×C4C2×#12311yescharacteristic
448C2×C4#52511yescharacteristic
449C2×C4#52512yes—
450C2×C4#52912yes—
451C2×C4#52912yes—
452C2×C4#53211yescharacteristic
453C2×C4#53611yescharacteristic
454C2×C4#53711yescharacteristic
455C2×C4#54112yes—
456C2×C4#54212yes—
457C2×C4#54712yes—
458C2×C4#54811yescharacteristic
459C2×C4#55011yescharacteristic
460C2×C4#55112yes—
461C2×C4#55212yes—
462C2×C4#55212yes—
463C2×C4#55212yes—
464C8—42no—
465C8—82no—
466C8—81no—
467C22#240711yescharacteristic
468C22#243211yescharacteristic
469C22#247512yes—
470C22#247712yes—
471C22#284911yescharacteristic
472C22C2×#44911yescharacteristic
473C22#300411yescharacteristic
474C22#300911yescharacteristic
475C22#301311yescharacteristic
476C22#310812yes—
477C22#311212yes—
478C22#312012yes—
479C22#312512yes—
480C22#330211yescharacteristic
481C22#331612yes—
482C22#332411yescharacteristic
483C22#334011yescharacteristic
484C22#334412yes—
485C22#335311yescharacteristic
486C22#336512yes—
487C22#337212yes—
488C22#337312yes—
489C22#337412yes—
490C4—24no—
491C4—44no—
492C4—44no—
493C4—42no—
494C4—44no—
495C4—84no—
496C4—84no—
497C4—82no—
498C4—82no—
499C4—161no—
500C4#239511yescharacteristic
501C4#247512yes—
502C4#247912yes—
503C4#248411yescharacteristic
504C4#249811yescharacteristic
505C4#262511yescharacteristic
506C2#1490611yescharacteristic
507C2#2411911yescharacteristic
508C2#2413411yescharacteristic
509C2#2565211yescharacteristic
510C2#2565711yescharacteristic
511C2#2735912yes—
512C2#2737512yes—
513C2#2742512yes—
514C2#2743912yes—
515C2#3300311yescharacteristic
516C2#3301611yescharacteristic
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×#23552
C2×#20992
C2×#18652
#151492
#113572
#113462
#111902
#110672
#109082
#108852
#107892