Finite Group Databasebeta

← #404893

Subgroups — #404893

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic
3C2×#1863C211yesmaximal, characteristic
4#15194C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11190C212yesmaximal
8#11067C212yesmaximal
9#10907C212yesmaximal
10#10884C212yesmaximal
11#10789C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#103C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17#2355—24no—
18#2355—24no—
19C2×#391C2212yes—
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#387—22no—
23C2×#384C2211yescharacteristic
24C2×#382—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#2099—24no—
29#1863—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1445—21no—
41#1444—21no—
42#1441—22no—
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—22no—
57C22×C4⋊C4—21no—
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×#107—22no—
70C2×#107—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#391—24no—
80#391—24no—
81#389—24no—
82#389—22no—
83#389—22no—
84#389—42no—
85#387—216no—
86#384—22no—
87#384D414yes—
88#382—28no—
89C4×C4⋊C4—28no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D412yes—
104#340—24no—
105#340—24no—
106#340—22no—
107#340—24no—
108#340—24no—
109#340—24no—
110#336—24no—
111#336—22no—
112#336—22no—
113#336—42no—
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8C2×D412yes—
123C22×Q8#4114yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8#4112yes—
152C4×Q8—216no—
153C4×Q8—28no—
154C4×Q8—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4#4114yes—
196C2×C4⋊C4#4112yes—
197C2×C4⋊C4#4114yes—
198C2×C42—28no—
199C2×C42—24no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—28no—
209C2×C42—82no—
210C2×C42—82no—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213#107—44no—
214#103—28no—
215#103—48no—
216#103—44no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95—24no—
221#95—44no—
222#95—42no—
223#95—42no—
224#95—82no—
225#95—81no—
226#95—82no—
227C24C22×D411yescharacteristic
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8#12314yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
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—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4#12012yes—
300C22×C4#12012yes—
301C22×C4#13512yes—
302C22×C4#13714yes—
303C22×C4#13712yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4112yes—
307C22×C4C2×#4112yes—
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4H(5,𝔽2)11yescharacteristic
312C22×C42−1+412yes—
313C22×C42−1+411yescharacteristic
314C2×C8—22no—
315C2×C8—24no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—28no—
332C4⋊C4—216no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—84no—
353C4⋊C4—88no—
354C4⋊C4—84no—
355C4⋊C4#12314yes—
356C42—216no—
357C42—22no—
358C42—22no—
359C42—24no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—88no—
365C23C2×#12011yescharacteristic
366C23#52512yes—
367C23#54611yescharacteristic
368C23#54711yescharacteristic
369C23#54812yes—
370C23C2×#13511yescharacteristic
371C23C2×#13711yescharacteristic
372C23C2×#13712yes—
373C23#57612yes—
374C23#57711yescharacteristic
375C23#57811yescharacteristic
376Q8—24no—
377Q8—24no—
378Q8—42no—
379Q8—42no—
380Q8—44no—
381Q8—44no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4#45214yes—
440C2×C4C2×#12311yescharacteristic
441C2×C4C2×#12311yescharacteristic
442C2×C4#52511yescharacteristic
443C2×C4#52512yes—
444C2×C4#52912yes—
445C2×C4#52912yes—
446C2×C4#53211yescharacteristic
447C2×C4#53611yescharacteristic
448C2×C4#53711yescharacteristic
449C2×C4#54112yes—
450C2×C4#54212yes—
451C2×C4#54712yes—
452C2×C4#54811yescharacteristic
453C2×C4#55011yescharacteristic
454C2×C4#55112yes—
455C2×C4#55212yes—
456C2×C4#55212yes—
457C2×C4#55212yes—
458C8—42no—
459C8—82no—
460C8—81no—
461C22#240711yescharacteristic
462C22#243211yescharacteristic
463C22#247512yes—
464C22#247712yes—
465C22#284911yescharacteristic
466C22C2×#45211yescharacteristic
467C22#300411yescharacteristic
468C22#300811yescharacteristic
469C22#301311yescharacteristic
470C22#310812yes—
471C22#311312yes—
472C22#311912yes—
473C22#312412yes—
474C22#330411yescharacteristic
475C22#331412yes—
476C22#332811yescharacteristic
477C22#334211yescharacteristic
478C22#334612yes—
479C22#335611yescharacteristic
480C22#336112yes—
481C22#337212yes—
482C22#337412yes—
483C22#337512yes—
484C4—24no—
485C4—44no—
486C4—44no—
487C4—42no—
488C4—44no—
489C4—84no—
490C4—84no—
491C4—82no—
492C4—82no—
493C4—161no—
494C4#239511yescharacteristic
495C4#247512yes—
496C4#247912yes—
497C4#248411yescharacteristic
498C4#249811yescharacteristic
499C4#262511yescharacteristic
500C2#1490611yescharacteristic
501C2#2412311yescharacteristic
502C2#2413011yescharacteristic
503C2#2564811yescharacteristic
504C2#2565711yescharacteristic
505C2#2736112yes—
506C2#2737712yes—
507C2#2742312yes—
508C2#2744112yes—
509C2#3300811yescharacteristic
510C2#3301211yescharacteristic
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×#18632
#151942
#113572
#113462
#111902
#110672
#109072
#108842
#107892