Finite Group Databasebeta

← #402466

Subgroups — #402466

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic
3C2×#1863C211yesmaximal, characteristic
4#15200C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11293C212yesmaximal
8#11219C212yesmaximal
9#10789C211yesmaximal, characteristic
10#10789C212yesmaximal
11#10789C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×#107C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18Q8×C4⋊C4—24no—
19Q8×C4⋊C4—24no—
20C2×#389—22no—
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#384—21no—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#2100—24no—
28#1863—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1445—21no—
40#1442—22no—
41#1442—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×Q8—24no—
50C2×C4×Q8—24no—
51C2×C4×Q8—24no—
52C2×C4×Q8—24no—
53C2×C4×Q8—22no—
54C2×C4×Q8—24no—
55C2×C4×Q8—24no—
56C2×C4×Q8—42no—
57C2×C4×Q8—42no—
58C2×C4×Q8D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—21no—
73C22×C42—22no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#107—22no—
77C2×#107—24no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103Q814yes—
86#389—216no—
87#389—24no—
88#389—24no—
89#389—22no—
90#389—22no—
91#384—28no—
92#384—22no—
93#384D414yes—
94C4×C4⋊C4—28no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D412yes—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—22no—
114#336—42no—
115#336—42no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8C2×D412yes—
122C22×Q8C2×Q812yes—
123C22×Q8#4112yes—
124C22×Q8#4114yes—
125C22×Q8#4114yes—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8C2×Q812yes—
154C4×Q8—216no—
155C4×Q8—28no—
156C4×Q8—28no—
157C4×Q8—48no—
158C4×Q8—48no—
159C4×Q8—48no—
160C4×Q8—48no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4Q1614yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×Q812yes—
206C2×C4⋊C4#4114yes—
207C2×C42—28no—
208C2×C42—24no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—28no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—44no—
219C2×C42—44no—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222#107—44no—
223#103—28no—
224#103—44no—
225#103—48no—
226#95—28no—
227#95—24no—
228#95—44no—
229#95—42no—
230#95—42no—
231#95—81no—
232#95—82no—
233#95—82no—
234#95—82no—
235C24C22×D411yescharacteristic
236C2×Q8—22no—
237C2×Q8—24no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8C22⋊Q814yes—
250C2×Q8#12314yes—
251C2×Q8#12414yes—
252C2×Q8#12518yes—
253C2×Q8#12618yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4#12012yes—
308C22×C4#12012yes—
309C22×C4C2×Q1614yes—
310C22×C4C2×Q1612yes—
311C22×C4#13712yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×Q811yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4112yes—
318C22×C4C2×#4112yes—
319C22×C4H(5,𝔽2)12yes—
320C22×C42−1+411yescharacteristic
321C22×C42−1+411yescharacteristic
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C4⋊C4—216no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—28no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—416no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C4⋊C4—88no—
364C4⋊C4#12314yes—
365C42—216no—
366C42—22no—
367C42—22no—
368C42—24no—
369C42—416no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—42no—
374C23C2×#12011yescharacteristic
375C23#52512yes—
376C23#54612yes—
377C23#54811yescharacteristic
378C23D4×Q811yescharacteristic
379C23C22×Q1611yescharacteristic
380C23C2×#13711yescharacteristic
381C23#57512yes—
382C23#57611yescharacteristic
383C23#57712yes—
384C23#57711yescharacteristic
385Q8—24no—
386Q8—28no—
387Q8—28no—
388Q8—24no—
389Q8—24no—
390Q8#40018yes—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4#45114yes—
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4C2×#12311yescharacteristic
451C2×C4C2×#12311yescharacteristic
452C2×C4#52512yes—
453C2×C4C2×#12411yescharacteristic
454C2×C4C2×#12512yes—
455C2×C4C2×#12612yes—
456C2×C4#53511yescharacteristic
457C2×C4#53611yescharacteristic
458C2×C4#53912yes—
459C2×C4#53912yes—
460C2×C4#54612yes—
461C2×C4#54811yescharacteristic
462C2×C4D4×Q811yescharacteristic
463C2×C4#55212yes—
464C2×C4#55212yes—
465C2×C4#55312yes—
466C2×C4#55412yes—
467C8—42no—
468C8—81no—
469C8—82no—
470C22#240911yescharacteristic
471C22#243511yescharacteristic
472C22#247312yes—
473C22#248112yes—
474C22#284911yescharacteristic
475C22C2×#45111yescharacteristic
476C22#300511yescharacteristic
477C22#301011yescharacteristic
478C22#301211yescharacteristic
479C22#310412yes—
480C22#311712yes—
481C22#311912yes—
482C22#312512yes—
483C22#329912yes—
484C22#330912yes—
485C22#331712yes—
486C22#332912yes—
487C22#336311yescharacteristic
488C22#337112yes—
489C22#337511yescharacteristic
490C22Q8×Q1611yescharacteristic
491C22#338711yescharacteristic
492C22#338812yes—
493C4—24no—
494C4—28no—
495C4—44no—
496C4—42no—
497C4—44no—
498C4—82no—
499C4—82no—
500C4—84no—
501C4—162no—
502C4—161no—
503C4C2×#40011yescharacteristic
504C4#247312yes—
505C4#248112yes—
506C4#248711yescharacteristic
507C4#250311yescharacteristic
508C4#262311yescharacteristic
509C2#1484311yescharacteristic
510C2#2433311yescharacteristic
511C2#2434011yescharacteristic
512C2#2588811yescharacteristic
513C2#2589311yescharacteristic
514C2#2730012yes—
515C2#2731412yes—
516C2#2755612yes—
517C2#2757212yes—
518C2#3301011yescharacteristic
519C2#3301311yescharacteristic
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×Q8×C4⋊C42
C2×#21002
C2×#18632
#152002
#113572
#113462
#112932
#112192
#107892