Finite Group Databasebeta

← #494898

Subgroups — #494898

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#361C211yesmaximal, characteristic
2C2×#2140C211yesmaximal, characteristic
3#33324C211yesmaximal, characteristic, complemented
4C2×#1978C211yesmaximal, characteristic
5#21181C211yesmaximal, characteristic, complemented
6#13144C218yesmaximal, complemented
7#12569C211yesmaximal, characteristic, complemented
8#12559C211yesmaximal, characteristic, complemented
9C23×D8C2211yescharacteristic
10C22×M32C2211yescharacteristic
11C22×M32C2211yescharacteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14#2911—21no—
15C2×#427C2211yescharacteristic
16C2×#418—21no—
17C22×#100—21no—
18C2×#405C2211yescharacteristic
19C2×#361C418yescomplemented
20C2×#361C2214yes—
21#2140—22no—
22#2140—22no—
23C2×#357C2214yes—
24#2108—22no—
25#2108C2214yescomplemented
26C4×M32—22no—
27#1978—22no—
28#1978C2214yes—
29#1952C2214yescomplemented
30#1951—22no—
31#1742—21no—
32#1387C2214yescomplemented
33#1381C2214yescomplemented
34C23×D4—21no—
35C22×D8—21no—
36C22×D8—21no—
37C22×D8—21no—
38C22×D8—21no—
39C22×D8—21no—
40C22×D8—22no—
41C22×D8—22no—
42C22×D8—22no—
43C22×D8C2×C414yes—
44C22×D8C2312yes—
45C22×M16—21no—
46C22×M16—21no—
47C22×M16D411yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117—21no—
50C2×#117D411yescharacteristic
51C2×M32—21no—
52C2×M32—22no—
53C2×M32—22no—
54C2×M32—22no—
55C2×M32C2×C414yes—
56C2×M32D412yes—
57C2×M32D412yes—
58C2×M32C2312yes—
59C22×C16—21no—
60C22×C16C2312yes—
61C2×#108—21no—
62C2×#108—22no—
63C2×#108—22no—
64C2×#108—22no—
65C2×#108—21no—
66C2×#108C2×C414yes—
67C2×#108C2312yes—
68C2×#108C2312yes—
69#428—24no—
70#428—42no—
71#427—21no—
72#427—21no—
73#427—21no—
74#427—21no—
75#427—42no—
76C2×#107C2312yes—
77C2×#106—21no—
78#418—24no—
79#418—42no—
80C2×#102—42no—
81C2×#100—24no—
82C2×#100—44no—
83#405—21no—
84#405—21no—
85#405D412yes—
86#405C2312yes—
87C2×#97—21no—
88C2×C4×C8—21no—
89#361C2×C4116yescomplemented
90#357—28no—
91#346—24no—
92C4×C16—24no—
93#343—24no—
94C25—41no—
95C22×D4—24no—
96C22×D4—22no—
97C22×D4—41no—
98C22×D4—41no—
99C22×D4—42no—
100C22×D4—41no—
101C22×D4—41no—
102C22×D4—42no—
103C22×D4—42no—
104C22×D4—41no—
105C23×C4C2×D411yescharacteristic
106C2×D8—24no—
107C2×D8—24no—
108C2×D8—24no—
109C2×D8—24no—
110C2×D8—24no—
111C2×D8—22no—
112C2×D8—22no—
113C2×D8—22no—
114C2×D8—22no—
115C2×D8—22no—
116C2×D8—41no—
117C2×D8—41no—
118C2×D8—42no—
119C2×D8—42no—
120C2×D8—42no—
121C2×D8—42no—
122C2×D8—42no—
123C2×D8C22×C414yes—
124C2×M16—21no—
125C2×M16—22no—
126C2×M16—24no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—41no—
130C2×M16—42no—
131C2×M16—41no—
132C2×M16—42no—
133C2×M16—42no—
134C2×M16SD1614yes—
135C2×M16C2×D412yes—
136C22×C8—21no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8C22×C412yes—
140C22×C8C2×D411yescharacteristic
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C22×C8#4111yescharacteristic
144C22×C8#4111yescharacteristic
145C22×C8#4112yes—
146C22×C8#4112yes—
147C22×C8#4112yes—
148C22×C8C2411yescharacteristic
149#117—21no—
150#117—21no—
151#117—41no—
152#117—41no—
153#117—41no—
154#117—41no—
155#117D812yes—
156#117C2×D412yes—
157C2×C4⋊C4—22no—
158C2×C4⋊C4D812yes—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—41no—
161C2×C42—21no—
162C2×C42—41no—
163M32—24no—
164M32—24no—
165M32—24no—
166M32—22no—
167M32C22×C414yes—
168C2×C16—24no—
169C2×C16—22no—
170C2×C16—22no—
171C2×C16—22no—
172C2×C16C2×D414yes—
173#108—24no—
174#108—22no—
175#108—22no—
176#108—22no—
177#108—42no—
178#108C22×C414yes—
179#107—22no—
180#107—22no—
181#107—44no—
182#106—41no—
183#106—41no—
184#106—44no—
185#102—48no—
186#100—48no—
187#97—22no—
188#97—22no—
189C4×C8—22no—
190C4×C8—22no—
191C24—42no—
192C24—44no—
193C24—81no—
194C24—81no—
195C24—82no—
196C24—164no—
197C24#13511yescharacteristic
198C2×D4—24no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—82no—
210C2×D4—82no—
211C2×D4—81no—
212C2×D4—81no—
213C2×D4—82no—
214C2×D4—82no—
215C2×D4—82no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4C4×D412yes—
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12412yes—
222C22×C4C2×D811yescharacteristic
223C22×C4C2×D811yescharacteristic
224C22×C4C2×SD1612yes—
225C22×C4#13512yes—
226C22×C4#13511yescharacteristic
227C22×C4C22×D411yescharacteristic
228D8—24no—
229D8—24no—
230D8—42no—
231D8—42no—
232D8—44no—
233D8—44no—
234M16—22no—
235M16—24no—
236M16—24no—
237M16—42no—
238M16—44no—
239M16—42no—
240M16—42no—
241M16—82no—
242M16C2×SD1614yes—
243C2×C8—22no—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—44no—
247C2×C8—42no—
248C2×C8—82no—
249C2×C8#11712yes—
250C2×C8#11712yes—
251C2×C8C4×D412yes—
252C2×C8C4×D412yes—
253C2×C8C4×D412yes—
254C2×C8#12111yescharacteristic
255C2×C8#12111yescharacteristic
256C2×C8#12312yes—
257C2×C8#12312yes—
258C2×C8#12412yes—
259C2×C8#12612yes—
260C2×C8#12612yes—
261C2×C8C23×C411yescharacteristic
262C2×C8C22×D411yescharacteristic
263C2×C8C2×#4111yescharacteristic
264C2×C8C2×#4111yescharacteristic
265C2×C8C2×#4111yescharacteristic
266C2×C8C2×#4111yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4C2×D814yes—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C42—22no—
276C42—22no—
277C42—81no—
278C42—81no—
279C16—24no—
280C16—24no—
281C23—42no—
282C23—44no—
283C23—82no—
284C23—84no—
285C23—84no—
286C23—82no—
287C23—164no—
288C23—164no—
289C23—164no—
290C23—168no—
291C23—168no—
292C23C4×SD1612yes—
293C23#46611yescharacteristic
294C23#46611yescharacteristic
295C23#48812yes—
296C23C2×#13511yescharacteristic
297D4—44no—
298D4—44no—
299D4—82no—
300D4—84no—
301D4—84no—
302D4—82no—
303C2×C4—24no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—162no—
307C2×C4#38812yes—
308C2×C4#38812yes—
309C2×C4C4×D812yes—
310C2×C4C4×D812yes—
311C2×C4C4×SD1612yes—
312C2×C4#46512yes—
313C2×C4#46512yes—
314C2×C4#46611yescharacteristic
315C2×C4#46611yescharacteristic
316C2×C4#48612yes—
317C2×C4#48612yes—
318C2×C4#48812yes—
319C2×C4C2×C4×D411yescharacteristic
320C2×C4C2×#12111yescharacteristic
321C2×C4C2×#12411yescharacteristic
322C2×C4C22×D811yescharacteristic
323C2×C4C22×SD1611yescharacteristic
324C2×C4C2×#13511yescharacteristic
325C8—24no—
326C8—44no—
327C8—82no—
328C8#38812yes—
329C8#38812yes—
330C8C2×#11711yescharacteristic
331C8C2×C4×D411yescharacteristic
332C8C2×#12311yescharacteristic
333C8C2×#12611yescharacteristic
334C22—22no—
335C22—22no—
336C22—22no—
337C22—42no—
338C22—42no—
339C22—84no—
340C22—164no—
341C22—168no—
342C22—164no—
343C22—168no—
344C22—164no—
345C22#193412yes—
346C22#193412yes—
347C22C2×C4×SD1611yescharacteristic
348C22C2×#46611yescharacteristic
349C22C2×#48811yescharacteristic
350C4—44no—
351C4—162no—
352C4#193412yes—
353C4#193412yes—
354C4C2×#38811yescharacteristic
355C4C2×C4×D811yescharacteristic
356C4C2×#46511yescharacteristic
357C4C2×#48611yescharacteristic
358C2—21no—
359C2—21no—
360C2—22no—
361C2—22no—
362C2—164no—
363C2C2×#193411yescharacteristic
364C2#3361812yes—
subgroupG/H
center, socleC22C2×#466
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33324C2semidirect
5. #21181C2semidirect
6. #13144C2semidirect
7. #12569C2semidirect
8. #12559C2semidirect
19. C2×#361C4semidirect
25. #2108C22semidirect
29. #1952C22semidirect
32. #1387C22semidirect
33. #1381C22semidirect
89. #361C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3612
C2×#21402
#333242
C2×#19782
#211812
#131442
#125692
#125592