Finite Group Databasebeta

← #505568

Subgroups — #505568

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1970C211yesmaximal, characteristic
2#20818C211yesmaximal, characteristic, complemented
3#13528C212yesmaximal, complemented
4#13482C212yesmaximal, complemented
5#13131C212yesmaximal, complemented
6#13130C212yesmaximal, complemented
7#12920C211yesmaximal, characteristic, complemented
8#12919C211yesmaximal, characteristic, complemented
9#12844C211yesmaximal, characteristic, complemented
10#12731C211yesmaximal, characteristic
11#12731C211yesmaximal, characteristic
12C23×D8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#102—21no—
15C22×C4×C8C2211yescharacteristic
16C2×#390—21no—
17#2283—21no—
18C2×#367C2212yescomplemented
19C2×#366C2212yescomplemented
20C2×#357C2212yes—
21C2×#357C2212yes—
22C2×#357C2212yes—
23C2×#357C2212yes—
24C2×C22⋊C16C2211yescharacteristic
25C2×C22⋊C16C2211yescharacteristic
26#1970—21no—
27#1970—21no—
28#1970C2212yes—
29#1970C2212yes—
30#1846C2211yescharacteristic, complemented
31#1846C2211yescharacteristic, complemented
32#1835—21no—
33#1808C2211yescharacteristic
34#1808C2211yescharacteristic
35#1807—21no—
36C4×#107C2211yescharacteristic, complemented
37C4×#107C2211yescharacteristic, complemented
38C4×#106—21no—
39#1380C2212yescomplemented
40#1380C2212yescomplemented
41#1374C2212yescomplemented
42#1374C2212yescomplemented
43C23×D4—21no—
44C22×D8—24no—
45C22×D8—21no—
46C22×D8—22no—
47C22×D8—22no—
48C22×D8—42no—
49C22×D8C2312yes—
50C23×C8D411yescharacteristic
51C23×C8C2311yescharacteristic
52C22×C4⋊C4—21no—
53C22×C22⋊C4—41no—
54C22×C22⋊C4—41no—
55C22×C42D411yescharacteristic
56C22×C16C2312yes—
57C22×C16C2312yes—
58C2×#107—21no—
59C2×#107—42no—
60C2×#107C2311yescharacteristic
61C2×#107C2311yescharacteristic
62C2×#107C2311yescharacteristic
63C2×#107C2311yescharacteristic
64C2×#107C2312yes—
65C2×#106—21no—
66C2×#106—21no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102—44no—
70C2×C4×C8—22no—
71C2×C4×C8—22no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#390—22no—
77#390—22no—
78#390—41no—
79#390—41no—
80#383—41no—
81#382—21no—
82#382—21no—
83#381—41no—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C22⋊C4—41no—
89#376—41no—
90#367—42no—
91#366—42no—
92#357—24no—
93#357—24no—
94#357—42no—
95#357—42no—
96C22⋊C16—22no—
97C22⋊C16—22no—
98#340—21no—
99#340—21no—
100#340—21no—
101#340—21no—
102C25—41no—
103C22×D4—24no—
104C22×D4—22no—
105C22×D4—42no—
106C22×D4—42no—
107C22×D4—42no—
108C22×D4—42no—
109C22×D4—41no—
110C22×D4—42no—
111C23×C4—41no—
112C23×C4D811yescharacteristic
113C23×C4D811yescharacteristic
114C23×C4C2×D411yescharacteristic
115C2×D8—24no—
116C2×D8—22no—
117C2×D8—22no—
118C2×D8—22no—
119C2×D8—22no—
120C2×D8—22no—
121C2×D8—48no—
122C2×D8—48no—
123C2×D8—42no—
124C2×D8—42no—
125C2×D8#4114yes—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—41no—
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8#4112yes—
135C22×C8#4112yes—
136C22×C8C2411yescharacteristic
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—84no—
152C2×C22⋊C4—84no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42D811yescharacteristic
158C2×C42D811yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C16—22no—
162C2×C16—22no—
163C2×C16—22no—
164C2×C16—22no—
165C2×C16—22no—
166C2×C16—22no—
167#107—22no—
168#107—22no—
169#107—22no—
170#107—22no—
171#107—22no—
172#107—22no—
173#107—42no—
174#106—42no—
175#106—42no—
176#102—44no—
177#102—44no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8—21no—
181C4×C8—21no—
182C4×C8—21no—
183C4×C8—21no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—41no—
192C24—44no—
193C24—42no—
194C24—82no—
195C24—82no—
196C24—82no—
197C24—82no—
198C24—82no—
199C24—82no—
200C24C2×D811yescharacteristic
201C2×D4—24no—
202C2×D4—48no—
203C2×D4—48no—
204C2×D4—44no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—82no—
211C2×D4—84no—
212C2×D4—84no—
213C2×D4—82no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4#12312yes—
233C22×C4C2×D811yescharacteristic
234C22×C4C2×D811yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4#13712yes—
239C22×C4C22×D411yescharacteristic
240D8—24no—
241D8—24no—
242D8—48no—
243D8—42no—
244D8—42no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—44no—
255C2×C8#12312yes—
256C2×C8AGL(1,ℤ/8)12yes—
257C2×C8C2×#4111yescharacteristic
258C2×C8C2×#4111yescharacteristic
259C2×C8C2×#4112yes—
260C2×C8H(5,𝔽2)12yes—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—82no—
272C22⋊C4—84no—
273C22⋊C4—84no—
274C22⋊C4—84no—
275C22⋊C4—82no—
276C22⋊C4—82no—
277C22⋊C4—84no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—22no—
283C42—22no—
284C42—82no—
285C42—82no—
286C16—42no—
287C16—42no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—44no—
293C23—88no—
294C23—88no—
295C23—88no—
296C23—82no—
297C23—82no—
298C23—82no—
299C23—82no—
300C23—82no—
301C23—82no—
302C23—164no—
303C23—164no—
304C23—164no—
305C23—164no—
306C23#48012yes—
307C23#50812yes—
308C23#50912yes—
309C23C22×D811yescharacteristic
310D4—44no—
311D4—44no—
312D4—88no—
313D4—82no—
314D4—82no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—164no—
350C2×C4#48012yes—
351C2×C4#50912yes—
352C2×C4C2×#12311yescharacteristic
353C2×C4#52511yescharacteristic
354C2×C4#53511yescharacteristic
355C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
356C2×C4C2×#13711yescharacteristic
357C2×C4#57511yescharacteristic
358C8—22no—
359C8—22no—
360C8—41no—
361C8#55112yes—
362C8#55112yes—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—84no—
370C22—84no—
371C22—84no—
372C22—1616no—
373C22—162no—
374C22—162no—
375C22—162no—
376C22—162no—
377C22#223414yes—
378C22C2×#48011yescharacteristic
379C22#310011yescharacteristic
380C22#320511yescharacteristic
381C22C2×#50811yescharacteristic
382C22C2×#50911yescharacteristic
383C22C2×#50911yescharacteristic
384C22#341411yescharacteristic
385C4—22no—
386C4—22no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—161no—
391C4—161no—
392C4—161no—
393C4—161no—
394C4—162no—
395C4#248211yescharacteristic
396C4#309411yescharacteristic
397C4#311011yescharacteristic
398C4#321511yescharacteristic
399C2—24no—
400C2—164no—
401C2#2758211yescharacteristic
402C2C2×#223411yescharacteristic
403C2#3364811yescharacteristic
404C2#3366511yescharacteristic
405C2#3367411yescharacteristic
406C2#3377711yescharacteristic
407C2#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
2. #20818C2semidirect
3. #13528C2semidirect
4. #13482C2semidirect
5. #13131C2semidirect
6. #13130C2semidirect
7. #12920C2semidirect
8. #12919C2semidirect
9. #12844C2semidirect
18. C2×#367C22semidirect
19. C2×#366C22semidirect
30. #1846C22semidirect
31. #1846C22semidirect
36. C4×#107C22semidirect
37. C4×#107C22semidirect
39. #1380C22semidirect
40. #1380C22semidirect
41. #1374C22semidirect
42. #1374C22semidirect

Maximal subgroups

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

subgroupindex
C2×#19702
#208182
#135282
#134822
#131312
#131302
#129202
#129192
#128442
#127312