Finite Group Databasebeta

← #276385

Subgroups — #276385

Order 512.

705 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33197C211yesmaximal, characteristic, complemented
2#29139C211yesmaximal, characteristic, complemented
3#22795C211yesmaximal, characteristic, complemented
4C4×#459C211yesmaximal, characteristic
5#10621C212yesmaximal, complemented
6#10472C212yesmaximal
7#9898C211yesmaximal, characteristic, complemented
8#9616C212yesmaximal, complemented
9#9601C211yesmaximal, characteristic, complemented
10#9564C212yesmaximal, complemented
11#8585C211yesmaximal, characteristic, complemented
12C2×#476—21no—
13C2×#459C2211yescharacteristic
14C2×C4×D8—21no—
15C2×#431—21no—
16#2868C2211yescharacteristic
17C4×#127C2211yescharacteristic
18C42×D4C2211yescharacteristic
19#2023—22no—
20#1987—21no—
21#1982C2211yescharacteristic
22#1957C2212yes—
23#1954C2212yes—
24#1948—21no—
25#1948C2212yes—
26#1947C2211yescharacteristic
27#1925—21no—
28#1923—22no—
29#1879—21no—
30#1842C2211yescharacteristic
31#1842C2211yescharacteristic
32#1834C2212yes—
33#1802—21no—
34#1802—21no—
35#1802—21no—
36#1802C2211yescharacteristic
37#1802C2212yes—
38#1797—21no—
39C8×C4⋊C4C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#102—21no—
43C2×#335C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C22×D8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×#127C2311yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C22×C42—21no—
61#476—44no—
62#459—22no—
63#459—22no—
64C4×D8—44no—
65#431—44no—
66C2×#107—21no—
67C2×#106—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8D411yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×#102—21no—
79C2×#102—22no—
80C2×#102—21no—
81C2×#102—21no—
82C2×#102—21no—
83C2×#102—22no—
84C2×#102—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8—21no—
89C2×C4×C8C2312yes—
90#390—21no—
91#390—21no—
92#390—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384C2311yescharacteristic
97#384C2312yes—
98C4×C4⋊C4D411yescharacteristic
99C4×C4⋊C4D411yescharacteristic
100C4×C4⋊C4C2311yescharacteristic
101C4×C22⋊C4—21no—
102C4×C22⋊C4—21no—
103C4×C22⋊C4—21no—
104C43C2311yescharacteristic
105#340—22no—
106#340—22no—
107#340—22no—
108#340—21no—
109#340—22no—
110#340—21no—
111#336—22no—
112#336—21no—
113#336D412yes—
114#335—24no—
115#331—24no—
116C22×Q8—41no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4C2×D411yescharacteristic
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—41no—
125C2×D8—24no—
126C2×D8—42no—
127C2×D8—42no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8C2×D412yes—
133C22×C8C2×D412yes—
134C4⋊Q8—42no—
135C4⋊Q8—42no—
136C4⋊Q8—81no—
137C4⋊Q8—81no—
138C4⋊Q8—82no—
139#127—21no—
140#127—21no—
141#127—21no—
142#127#4112yes—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—41no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—82no—
155C4×D4D814yes—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42C2411yescharacteristic
186#107—42no—
187#106—44no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C4⋊C8—42no—
200C4⋊C8—81no—
201C4⋊C8#4112yes—
202#102—24no—
203#102—42no—
204#102—44no—
205#102—42no—
206#102—42no—
207#102—82no—
208#102—82no—
209C22⋊C8—44no—
210C22⋊C8—82no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—44no—
214C4×C8—42no—
215#95—21no—
216#95—21no—
217#95—21no—
218C24—21no—
219C24—41no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×D4—21no—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—81no—
243C2×D4—81no—
244C2×D4C2×D812yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—81no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4111yescharacteristic
272D8—48no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—81no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4C2×D812yes—
309C4⋊C4#13712yes—
310C22⋊C4—24no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—22no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—42no—
326C42—42no—
327C42C2×D812yes—
328C42#13512yes—
329C42#13512yes—
330C42AGL(1,ℤ/8)12yes—
331C42C2×#4111yescharacteristic
332C42C2×#4111yescharacteristic
333C422−1+411yescharacteristic
334C422−1+411yescharacteristic
335C23—22no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23C2×#12111yescharacteristic
350Q8—164no—
351D4—22no—
352D4—22no—
353D4—42no—
354D4—42no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—41no—
359D4—82no—
360D4—82no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—81no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—48no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—84no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—164no—
396C2×C4#46312yes—
397C2×C4#46312yes—
398C2×C4#46612yes—
399C2×C4#46612yes—
400C2×C4#47112yes—
401C2×C4#52511yescharacteristic
402C2×C4#53611yescharacteristic
403C2×C4#54711yescharacteristic
404C2×C4#54811yescharacteristic
405C2×C4#55211yescharacteristic
406C2×C4#55611yescharacteristic
407C2×C4C22×D811yescharacteristic
408C2×C4C2×#13511yescharacteristic
409C2×C4C2×#13511yescharacteristic
410C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
411C2×C4C2×#13711yescharacteristic
412C2×C4#57511yescharacteristic
413C2×C4#57611yescharacteristic
414C2×C4#57711yescharacteristic
415C8—42no—
416C8—42no—
417C8—42no—
418C8—82no—
419C8—82no—
420C8—81no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22#241511yescharacteristic
436C22C2×#46311yescharacteristic
437C22#304511yescharacteristic
438C22C2×#46611yescharacteristic
439C22#305611yescharacteristic
440C22C2×#47111yescharacteristic
441C22#306811yescharacteristic
442C4—21no—
443C4—21no—
444C4—22no—
445C4—22no—
446C4—41no—
447C4—42no—
448C4—42no—
449C4—81no—
450C4—161no—
451C4—162no—
452C4#168114yes—
453C4#310011yescharacteristic
454C4#310711yescharacteristic
455C4#322611yescharacteristic
456C4#323311yescharacteristic
457C4#334111yescharacteristic
458C4#335411yescharacteristic
459C4#336011yescharacteristic
460C4#336611yescharacteristic
461C2—82no—
462C2—162no—
463C2C2×#168111yescharacteristic
464C2#1916811yescharacteristic
465C2#1941011yescharacteristic
466C2#1949411yescharacteristic
467C2#2467711yescharacteristic
468C2#2470011yescharacteristic
469C2#2472111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33197C2semidirect
2. #29139C2semidirect
3. #22795C2semidirect
5. #10621C2semidirect
7. #9898C2semidirect
8. #9616C2semidirect
9. #9601C2semidirect
10. #9564C2semidirect
11. #8585C2semidirect

Maximal subgroups

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

subgroupindex
#331972
#291392
#227952
C4×#4592
#106212
#104722
#98982
#96162
#96012
#95642
#85852