Finite Group Databasebeta

← #508569

Subgroups — #508569

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33208C211yesmaximal, characteristic, complemented
2C2×#1970C211yesmaximal, characteristic
3#13528C212yesmaximal, complemented
4#13476C212yesmaximal, complemented
5#13474C212yesmaximal, complemented
6#13365C212yesmaximal, complemented
7#12920C211yesmaximal, characteristic, complemented
8#12919C211yesmaximal, characteristic, complemented
9#12783C211yesmaximal, characteristic, complemented
10#12731C211yesmaximal, characteristic
11#12731C211yesmaximal, characteristic
12C23×D8C2211yescharacteristic
13C2×#499C2211yescharacteristic, complemented
14C22×#107C2211yescharacteristic
15C22×#102—21no—
16C22×C4×C8C2211yescharacteristic
17#2851—21no—
18C2×#390—21no—
19C2×#367C2212yescomplemented
20C2×#366C2212yescomplemented
21C2×C4⋊C16C2212yescomplemented
22C2×#357C2212yes—
23C2×#357C2212yes—
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×C22⋊C16C2211yescharacteristic
27C2×C22⋊C16C2211yescharacteristic
28#1991—21no—
29#1970—21no—
30#1970—21no—
31#1970C2212yes—
32#1970C2212yes—
33#1942C2211yescharacteristic, complemented
34#1942C2211yescharacteristic, complemented
35#1941—21no—
36#1846C2211yescharacteristic, complemented
37#1808C2211yescharacteristic
38#1808C2211yescharacteristic
39#1807—21no—
40#1380C2212yescomplemented
41#1380C2212yescomplemented
42#1374C2212yescomplemented
43C23×D4—21no—
44C22×D8—21no—
45C22×D8—22no—
46C22×D8—42no—
47C22×D8—42no—
48C22×D8Q814yes—
49C22×D8C2312yes—
50C23×C8D411yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#125—21no—
53C22×C4⋊C4—21no—
54C22×C22⋊C4—41no—
55C22×C22⋊C4—41no—
56C22×C42D411yescharacteristic
57C22×C16C2312yes—
58C22×C16C2312yes—
59#499—42no—
60C2×#107—21no—
61C2×#107—42no—
62C2×#107C2311yescharacteristic
63C2×#107C2311yescharacteristic
64C2×#107C2311yescharacteristic
65C2×#107C2311yescharacteristic
66C2×#107C2312yes—
67C2×#106—21no—
68C2×#106—21no—
69C2×#102—22no—
70C2×#102—22no—
71C2×#102—44no—
72C2×C4×C8—42no—
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#390—22no—
78#390—22no—
79#390—41no—
80#390—41no—
81#389—21no—
82#389—21no—
83#387—41no—
84#387—41no—
85#385—41no—
86#385—41no—
87#382—21no—
88#380—21no—
89#380—21no—
90#367—42no—
91#366—42no—
92C4⋊C16—42no—
93#357—42no—
94#357—42no—
95#357—42no—
96#357—42no—
97C22⋊C16—22no—
98C22⋊C16—22no—
99#340—21no—
100#340—21no—
101#340—21no—
102#340—21no—
103C25—41no—
104C22×D4—24no—
105C22×D4—22no—
106C22×D4—42no—
107C22×D4—42no—
108C22×D4—42no—
109C22×D4—42no—
110C22×D4—41no—
111C22×D4—42no—
112C23×C4—41no—
113C23×C4D811yescharacteristic
114C23×C4D811yescharacteristic
115C23×C4C2×D411yescharacteristic
116C2×D8—24no—
117C2×D8—22no—
118C2×D8—22no—
119C2×D8—22no—
120C2×D8—22no—
121C2×D8—22no—
122C2×D8—48no—
123C2×D8—42no—
124C2×D8—42no—
125C2×D8—48no—
126C2×D8C2×Q814yes—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—41no—
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C22×C8C2×Q812yes—
136C22×C8#4112yes—
137C22×C8C2411yescharacteristic
138#125—42no—
139#125—42no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—84no—
159C2×C22⋊C4—84no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42D811yescharacteristic
163C2×C42D811yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C16—22no—
167C2×C16—22no—
168C2×C16—22no—
169C2×C16—22no—
170C2×C16—22no—
171C2×C16—22no—
172#107—22no—
173#107—22no—
174#107—22no—
175#107—22no—
176#107—22no—
177#107—22no—
178#107—42no—
179#106—42no—
180#106—42no—
181#102—44no—
182#102—44no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—41no—
186C4×C8—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—41no—
192#95—41no—
193C24—42no—
194C24—44no—
195C24—82no—
196C24—82no—
197C24—82no—
198C24—82no—
199C24—82no—
200C24—82no—
201C24C2×D811yescharacteristic
202C2×D4—24no—
203C2×D4—48no—
204C2×D4—48no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—82no—
212C2×D4—82no—
213C2×D4—84no—
214C2×D4—84no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4C22⋊Q812yes—
234C22×C4C2×D811yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4#13712yes—
240C22×C4C22×D411yescharacteristic
241D8—24no—
242D8—24no—
243D8—48no—
244D8—42no—
245D8—42no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—44no—
256C2×C8C22⋊Q812yes—
257C2×C8AGL(1,ℤ/8)12yes—
258C2×C8C22×Q811yescharacteristic
259C2×C8C2×#4111yescharacteristic
260C2×C8H(5,𝔽2)12yes—
261C2×C8H(5,𝔽2)12yes—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C22⋊C4—84no—
278C22⋊C4—84no—
279C22⋊C4—84no—
280C22⋊C4—82no—
281C22⋊C4—82no—
282C22⋊C4—84no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—41no—
288C42—41no—
289C16—42no—
290C16—42no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—44no—
296C23—88no—
297C23—88no—
298C23—82no—
299C23—82no—
300C23—82no—
301C23—82no—
302C23—82no—
303C23—82no—
304C23—88no—
305C23—164no—
306C23—164no—
307C23—164no—
308C23—164no—
309C23#47412yes—
310C23#50812yes—
311C23#50912yes—
312C23C22×D811yescharacteristic
313D4—44no—
314D4—44no—
315D4—88no—
316D4—82no—
317D4—82no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—164no—
353C2×C4#47412yes—
354C2×C4#50912yes—
355C2×C4C2×C22⋊Q811yescharacteristic
356C2×C4#53411yescharacteristic
357C2×C4#53511yescharacteristic
358C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
359C2×C4C2×#13711yescharacteristic
360C2×C4#57511yescharacteristic
361C8—22no—
362C8—22no—
363C8—41no—
364C8#53812yes—
365C8#54312yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—84no—
373C22—84no—
374C22—84no—
375C22—1616no—
376C22—162no—
377C22—162no—
378C22—162no—
379C22—162no—
380C22#222814yes—
381C22C2×#47411yescharacteristic
382C22#318611yescharacteristic
383C22#320511yescharacteristic
384C22C2×#50811yescharacteristic
385C22C2×#50911yescharacteristic
386C22C2×#50911yescharacteristic
387C22#341411yescharacteristic
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—161no—
394C4—161no—
395C4—161no—
396C4—161no—
397C4—162no—
398C4#262611yescharacteristic
399C4#307711yescharacteristic
400C4#319311yescharacteristic
401C4#321511yescharacteristic
402C2—24no—
403C2—164no—
404C2#2995011yescharacteristic
405C2C2×#222811yescharacteristic
406C2#3362711yescharacteristic
407C2#3375411yescharacteristic
408C2#3376011yescharacteristic
409C2#3377711yescharacteristic
410C2#3378311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33208C2semidirect
3. #13528C2semidirect
4. #13476C2semidirect
5. #13474C2semidirect
6. #13365C2semidirect
7. #12920C2semidirect
8. #12919C2semidirect
9. #12783C2semidirect
13. C2×#499C22semidirect
19. C2×#367C22semidirect
20. C2×#366C22semidirect
21. C2×C4⋊C16C22semidirect
33. #1942C22semidirect
34. #1942C22semidirect
36. #1846C22semidirect
40. #1380C22semidirect
41. #1380C22semidirect
42. #1374C22semidirect

Maximal subgroups

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

subgroupindex
#332082
C2×#19702
#135282
#134762
#134742
#133652
#129202
#129192
#127832
#127312