Finite Group Databasebeta

← #474922

Subgroups — #474922

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2087C211yesmaximal, characteristic, complemented
2C2×#2054C211yesmaximal, characteristic
3C2×#1934C212yesmaximal, complemented
4#15222C212yesmaximal, complemented
5#11372C212yesmaximal, complemented
6#11123C212yesmaximal, complemented
7#11051C212yesmaximal, complemented
8#11022C212yesmaximal
9#10834C211yesmaximal, characteristic, complemented
10C23×D8C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#102C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#390C2212yes—
18C2×#390C2212yes—
19C2×#390C2212yes—
20C2×#389C2212yes—
21C2×#382C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23#2087—24no—
24#2054—24no—
25#1934—28no—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1439—24no—
34#1439—24no—
35C4⋊#95C2212yes—
36C23×D4C2312yes—
37C22×D8—24no—
38C22×D8—22no—
39C22×D8—42no—
40C22×D8—41no—
41C22×D8—41no—
42C23×C8C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—24no—
52C22×C22⋊C4—24no—
53C22×C22⋊C4—22no—
54C22×C42C2312yes—
55C22×C42C2311yescharacteristic
56C2×#107—22no—
57C2×#107—41no—
58C2×#107—41no—
59C2×C4⋊C8—41no—
60C2×C4⋊C8—41no—
61C2×C4⋊C8D414yes—
62C2×#102—24no—
63C2×#102—24no—
64C2×#102—24no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#102—24no—
68C2×#102—42no—
69C2×#102—42no—
70C2×#102—42no—
71C2×C4×C8—41no—
72C2×C4×C8—41no—
73C2×C4×C8D414yes—
74#390—24no—
75#390—24no—
76#390—24no—
77#390—42no—
78#390—42no—
79#390—42no—
80#389—24no—
81#389—24no—
82#382—24no—
83#382—42no—
84C4×C4⋊C4—41no—
85C4×C4⋊C4—41no—
86C2×#95—24no—
87C2×#95—22no—
88C2×#95—24no—
89C2×#95D412yes—
90#340—22no—
91#340—22no—
92#340—24no—
93#340—24no—
94#340—24no—
95#336—42no—
96#336—42no—
97C25—22no—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4#4114yes—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D414yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C2×D8—48no—
120C2×D8—44no—
121C2×D8—44no—
122C2×D8—48no—
123C2×D8—44no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8C2×D412yes—
139C22×C8C2×D414yes—
140C22×C8C2×D412yes—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
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—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—84no—
163C2×C4⋊C4#4114yes—
164C2×C22⋊C4—48no—
165C2×C22⋊C4—48no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—48no—
170C2×C22⋊C4—84no—
171C2×C22⋊C4—84no—
172C2×C22⋊C4—84no—
173C2×C22⋊C4—84no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186#107—44no—
187C4⋊C8—44no—
188#102—216no—
189#102—48no—
190#102—48no—
191C4×C8—44no—
192#95—42no—
193#95—48no—
194#95—44no—
195#95—84no—
196C24—24no—
197C24—42no—
198C24—42no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24—42no—
203C24—44no—
204C24—44no—
205C24—44no—
206C24—44no—
207C24C22×D411yescharacteristic
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4#12712yes—
265C22×C4#12712yes—
266C22×C4#13514yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271C22×C4H(5,𝔽2)11yescharacteristic
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C4H(5,𝔽2)11yescharacteristic
274C22×C4H(5,𝔽2)12yes—
275C22×C42−1+411yescharacteristic
276C22×C42−1+412yes—
277D8—416no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—48no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—88no—
309C22⋊C4—88no—
310C22⋊C4—816no—
311C22⋊C4—816no—
312C22⋊C4—88no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—42no—
321C23—48no—
322C23—48no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—84no—
327C23—84no—
328C23—82no—
329C23—82no—
330C23—82no—
331C23—82no—
332C23—82no—
333C23—84no—
334C23—84no—
335C23—82no—
336C23—82no—
337C23—84no—
338C23—84no—
339C23—84no—
340C23—84no—
341C23—82no—
342C23—84no—
343C23—84no—
344C23C2×#12711yescharacteristic
345C23#53411yescharacteristic
346C23#53711yescharacteristic
347C23#54612yes—
348C23#54812yes—
349C23C2×#13512yes—
350C23#57512yes—
351C23#57614yes—
352D4—44no—
353D4—44no—
354D4—44no—
355D4—44no—
356D4—82no—
357D4—82no—
358D4—82no—
359D4—82no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4—44no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—82no—
390C2×C4#49514yes—
391C2×C4#52512yes—
392C2×C4#53411yescharacteristic
393C2×C4#53411yescharacteristic
394C2×C4#53711yescharacteristic
395C2×C4#53712yes—
396C2×C4#53711yescharacteristic
397C2×C4#53812yes—
398C2×C4#53912yes—
399C2×C4#54012yes—
400C2×C4#54212yes—
401C2×C4#54612yes—
402C2×C4#54812yes—
403C2×C4#55111yescharacteristic
404C2×C4#55212yes—
405C2×C4#55511yescharacteristic
406C2×C4#56011yescharacteristic
407C2×C4#56312yes—
408C2×C4#56311yescharacteristic
409C8—44no—
410C8—42no—
411C8—42no—
412C22—88no—
413C22—88no—
414C22—84no—
415C22—84no—
416C22—84no—
417C22—162no—
418C22—162no—
419C22—164no—
420C22—164no—
421C22—162no—
422C22—162no—
423C22#254012yes—
424C22#265611yescharacteristic
425C22#269911yescharacteristic
426C22#270212yes—
427C22#272811yescharacteristic
428C22C2×#49511yescharacteristic
429C22#315011yescharacteristic
430C22#315512yes—
431C22#319112yes—
432C22#319912yes—
433C22#325012yes—
434C22#325912yes—
435C22#330514yes—
436C22#332214yes—
437C22#336114yes—
438C22#337014yes—
439C4—24no—
440C4—42no—
441C4—84no—
442C4—84no—
443C4—162no—
444C4—162no—
445C4#260911yescharacteristic
446C4#263012yes—
447C4#269711yescharacteristic
448C4#270712yes—
449C4#273111yescharacteristic
450C4#282311yescharacteristic
451C2—164no—
452C2#1701511yescharacteristic
453C2#2839912yes—
454C2#2840912yes—
455C2#3024111yescharacteristic
456C2#3025311yescharacteristic
457C2#3171912yes—
458C2#3180314yes—
459C2#3254512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2087C2semidirect
3. C2×#1934C2semidirect
4. #15222C2semidirect
5. #11372C2semidirect
6. #11123C2semidirect
7. #11051C2semidirect
9. #10834C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20872
C2×#20542
C2×#19342
#152222
#113722
#111232
#110512
#110222
#108342