Finite Group Databasebeta

← #416750

Subgroups — #416750

Order 512.

1164 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#1949C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic
4#13955C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11385C212yesmaximal
7#11220C212yesmaximal
8#11219C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×#125C2212yes—
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2552—24no—
18#2552—24no—
19C2×#398—22no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#384—21no—
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1949—24no—
29#1836—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1442—21no—
41#1442—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—22no—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×#125—24no—
49C2×#125—24no—
50C2×#125—24no—
51C2×#125—24no—
52C2×#125—42no—
53C2×#125—42no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#107—22no—
72C2×#107—24no—
73C2×#106—22no—
74C2×#106—24no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8Q814yes—
78C2×#97—22no—
79C2×#97—22no—
80C2×#97Q814yes—
81#398—216no—
82#389—28no—
83#389—24no—
84#389—24no—
85#384—28no—
86#384—22no—
87#384D414yes—
88#382—22no—
89#382—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C4×C4⋊C4Q814yes—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
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—22no—
110#340—24no—
111#340—24no—
112#340—42no—
113#340—42no—
114#336—42no—
115#336—42no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C2×Q814yes—
142C22×C8C2×Q812yes—
143C22×C8C2×Q812yes—
144#125—48no—
145#125—48no—
146#125—48no—
147#125—48no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
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—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—81no—
185C2×C4⋊C4—81no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×Q814yes—
192C2×C4⋊C4#4114yes—
193C2×C4⋊C4#4114yes—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—44no—
203C2×C42—81no—
204C2×C42—81no—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×Q812yes—
208C2×C42C2×Q812yes—
209C2×C42C2×Q812yes—
210C2×C42#4112yes—
211#107—44no—
212#106—44no—
213C4⋊C8—28no—
214#97—28no—
215#95—28no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—44no—
220#95—44no—
221#95—42no—
222#95—42no—
223#95—82no—
224#95—82no—
225C24C22×D411yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4#12012yes—
281C22×C4#12012yes—
282C22×C4C22⋊Q812yes—
283C22×C4C22⋊Q812yes—
284C22×C4C4⋊Q812yes—
285C22×C4C4⋊Q812yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×Q811yescharacteristic
289C22×C4C22×Q811yescharacteristic
290C22×C4C22×Q811yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4112yes—
293C22×C4C2×#4112yes—
294C22×C4H(5,𝔽2)12yes—
295C22×C42−1+412yes—
296C2×C8—24no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—416no—
325C4⋊C4—416no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—84no—
337C4⋊C4—88no—
338C4⋊C4#12414yes—
339C4⋊C4#12414yes—
340C42—48no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—84no—
346C42C22⋊Q814yes—
347C42C22⋊Q814yes—
348C42C4⋊Q814yes—
349C42C4⋊Q814yes—
350C23C2×#12011yescharacteristic
351C23C2×C22⋊Q811yescharacteristic
352C23C2×C4⋊Q811yescharacteristic
353C23#54612yes—
354C23#54812yes—
355C23#57512yes—
356C23#57512yes—
357C23#57611yescharacteristic
358C23#57711yescharacteristic
359C23#57711yescharacteristic
360C23#57811yescharacteristic
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4#39514yes—
420C2×C4#39514yes—
421C2×C4C2×C22⋊Q811yescharacteristic
422C2×C4C2×C22⋊Q811yescharacteristic
423C2×C4C2×#12411yescharacteristic
424C2×C4C2×#12411yescharacteristic
425C2×C4C2×C4⋊Q811yescharacteristic
426C2×C4C2×C4⋊Q811yescharacteristic
427C2×C4#53312yes—
428C2×C4#53312yes—
429C2×C4#53511yescharacteristic
430C2×C4#53611yescharacteristic
431C2×C4#54412yes—
432C2×C4#54412yes—
433C2×C4#54612yes—
434C2×C4#54812yes—
435C2×C4#55412yes—
436C2×C4#55512yes—
437C2×C4#55612yes—
438C2×C4#55712yes—
439C8—42no—
440C8—42no—
441C8—44no—
442C22C2×#39511yescharacteristic
443C22#243511yescharacteristic
444C22#243611yescharacteristic
445C22#243912yes—
446C22#255012yes—
447C22#300511yescharacteristic
448C22#300511yescharacteristic
449C22#301211yescharacteristic
450C22#301311yescharacteristic
451C22#308012yes—
452C22#308111yescharacteristic
453C22#308211yescharacteristic
454C22#316311yescharacteristic
455C22#316411yescharacteristic
456C22#316612yes—
457C22#330712yes—
458C22#330912yes—
459C22#331512yes—
460C22#331712yes—
461C22#337012yes—
462C22#337112yes—
463C22#337612yes—
464C22#337712yes—
465C4—24no—
466C4—24no—
467C4—44no—
468C4—44no—
469C4—82no—
470C4—82no—
471C4—84no—
472C4—84no—
473C4—161no—
474C4—161no—
475C4#243912yes—
476C4#255012yes—
477C4#256311yescharacteristic
478C4#256811yescharacteristic
479C4#266011yescharacteristic
480C4#286211yescharacteristic
481C2#1523011yescharacteristic
482C2#2355311yescharacteristic
483C2#2355711yescharacteristic
484C2#2589211yescharacteristic
485C2#2589311yescharacteristic
486C2#2594011yescharacteristic
487C2#2594111yescharacteristic
488C2#2603712yes—
489C2#2603912yes—
490C2#2880512yes—
491C2#2880712yes—
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×#25522
C2×#19492
C2×#18362
#139552
#113852
#112202
#112192
#108492
#108352
#107032