Finite Group Databasebeta

← #498955

Subgroups — #498955

Order 512.

950 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic
2C2×#2141C211yesmaximal, characteristic
3C22×#358C211yesmaximal, characteristic
4#21232C211yesmaximal, characteristic
5#13366C214yesmaximal, complemented
6#13364C214yesmaximal, complemented
7#12557C211yesmaximal, characteristic
8#12556C211yesmaximal, characteristic
9#12545C211yesmaximal, characteristic
10C23×Q16C2211yescharacteristic
11C2×#544—21no—
12C22×M32C2211yescharacteristic
13C23×C16C2211yescharacteristic
14#3164—24no—
15#3164—24no—
16C2×#501—22no—
17C2×#500C2212yes—
18C2×#499C2212yes—
19C2×#428—21no—
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C2×#419—21no—
24C2×#405C2211yescharacteristic
25#2310—21no—
26#2141—24no—
27C2×#358—24no—
28C2×#358C2212yes—
29C2×#358C2214yes—
30C2×#358C2212yes—
31#1971—22no—
32#1969C2214yes—
33#1380C2212yes—
34#1380C2212yes—
35#1380C2214yes—
36#1374C2214yes—
37C23×Q8—21no—
38C22×Q16—22no—
39C22×Q16—22no—
40C22×Q16—21no—
41C22×Q16—22no—
42C22×Q16—22no—
43C22×Q16—22no—
44C22×Q16Q814yescomplemented
45C22×Q16C2312yes—
46C22×M16D411yescharacteristic
47C23×C8C2311yescharacteristic
48#544—28no—
49#544—44no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—41no—
52C2×#125—22no—
53C2×#125—41no—
54C2×C22⋊Q8—42no—
55C2×#117—21no—
56C2×#117—21no—
57C2×#117D411yescharacteristic
58C22×C4⋊C4—21no—
59C2×M32—22no—
60C2×M32—22no—
61C2×M32—22no—
62C22×C16—22no—
63C22×C16—22no—
64C22×C16—22no—
65C22×C16C2312yes—
66C22×C16C2312yes—
67#501—216no—
68#500—24no—
69#500—24no—
70#499—24no—
71#499—24no—
72#428—28no—
73#427—22no—
74#427—22no—
75#427—22no—
76#427D414yes—
77C2×#107—22no—
78C2×#107—21no—
79C2×#107—22no—
80C2×#107D414yes—
81C2×#107C2312yes—
82C2×#107C2311yescharacteristic
83C2×#107C2312yes—
84C2×#107C2311yescharacteristic
85C2×#106—22no—
86C2×#106—21no—
87C2×#106—21no—
88#419—44no—
89C2×#103—24no—
90#405—22no—
91#405—22no—
92C2×#97D412yes—
93C2×C4×C8C2312yes—
94#391—24no—
95#391—41no—
96#391—41no—
97#381—42no—
98#358—28no—
99#358—24no—
100#358—24no—
101C22×Q8—24no—
102C22×Q8—22no—
103C22×Q8—41no—
104C22×Q8—42no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—42no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4C2×D411yescharacteristic
113C2×Q16—28no—
114C2×Q16—28no—
115C2×Q16—24no—
116C2×Q16—22no—
117C2×Q16—22no—
118C2×Q16—22no—
119C2×Q16—22no—
120C2×Q16—22no—
121C2×Q16—42no—
122C2×Q16—44no—
123C2×Q16—44no—
124C2×Q16—42no—
125C2×Q16C2×Q814yes—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16SD1614yes—
129C22×C8C2×D412yes—
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D412yes—
132C22×C8C2×D412yes—
133C22×C8C2×Q812yes—
134C22×C8#4112yes—
135C22×C8#4112yes—
136C22×C8#4112yes—
137C22×C8C2411yescharacteristic
138C4⋊Q8—28no—
139C4⋊Q8—44no—
140C4⋊Q8—48no—
141#125—28no—
142#125—44no—
143#125—48no—
144C22⋊Q8—416no—
145#117—22no—
146#117—22no—
147#117—24no—
148#117—42no—
149#117—42no—
150#117—42no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4D811yescharacteristic
162C2×C4⋊C4D811yescharacteristic
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4D812yes—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42C2×D412yes—
169M32—44no—
170C2×C16—22no—
171C2×C16—22no—
172C2×C16—22no—
173C2×C16—22no—
174C2×C16—22no—
175C2×C16—22no—
176C2×C16—44no—
177#107—28no—
178#107—24no—
179#107—22no—
180#107—21no—
181#107—21no—
182#107—22no—
183#107—22no—
184#107C2×D414yes—
185#107C2×D412yes—
186#107C2×D412yes—
187#106—28no—
188#106—24no—
189#106—42no—
190#103—48no—
191#97—24no—
192C4×C8—22no—
193C4×C8—22no—
194#95—44no—
195#95—44no—
196C24C2×D811yescharacteristic
197C2×Q8—24no—
198C2×Q8—48no—
199C2×Q8—42no—
200C2×Q8—44no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—48no—
206C2×Q8—48no—
207C2×Q8—82no—
208C2×Q8—84no—
209C2×Q8—82no—
210C2×Q8—84no—
211C2×Q8—164no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—162no—
222C22×C4#12012yes—
223C22×C4C22⋊Q812yes—
224C22×C4#12412yes—
225C22×C4C2×D812yes—
226C22×C4C2×D812yes—
227C22×C4C2×D811yescharacteristic
228C22×C4C2×SD1612yes—
229C22×C4AGL(1,ℤ/8)12yes—
230C22×C4#13712yes—
231C22×C4C22×D411yescharacteristic
232Q16—24no—
233Q16—24no—
234Q16—42no—
235Q16—42no—
236Q16—48no—
237M16—28no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8#12012yes—
241C2×C8C22⋊Q814yes—
242C2×C8C22⋊Q814yes—
243C2×C8C22⋊Q812yes—
244C2×C8#12414yes—
245C2×C8#12414yes—
246C2×C8#12412yes—
247C2×C8C2×SD1614yes—
248C2×C8C22×D411yescharacteristic
249C2×C8C22×D411yescharacteristic
250C2×C8C22×Q811yescharacteristic
251C2×C8C2×#4111yescharacteristic
252C2×C8C2×#4111yescharacteristic
253C2×C8C2×#4111yescharacteristic
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—48no—
260C4⋊C4—48no—
261C4⋊C4—48no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—44no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C22⋊C4—24no—
269C22⋊C4—44no—
270C22⋊C4—82no—
271C42—22no—
272C42—22no—
273C42—44no—
274C42—82no—
275C16—42no—
276C16—42no—
277C23—22no—
278C23—22no—
279C23#44712yes—
280C23#47412yes—
281C23#48612yes—
282C23C22×D811yescharacteristic
283Q8—44no—
284Q8—44no—
285Q8—82no—
286Q8—82no—
287Q8—88no—
288Q8—168no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—82no—
298C2×C4—84no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—88no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—164no—
307C2×C4#39314yes—
308C2×C4#44712yes—
309C2×C4#44814yes—
310C2×C4#47412yes—
311C2×C4#47614yes—
312C2×C4#48612yes—
313C2×C4#48914yes—
314C2×C4C2×#12011yescharacteristic
315C2×C4C2×C22⋊Q811yescharacteristic
316C2×C4C2×#12411yescharacteristic
317C2×C4C22×SD1611yescharacteristic
318C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
319C2×C4C2×#13711yescharacteristic
320C8—24no—
321C8#39314yes—
322C8C2×C22⋊Q811yescharacteristic
323C8C2×C22⋊Q811yescharacteristic
324C8C2×#12411yescharacteristic
325C8C2×#12411yescharacteristic
326C22—24no—
327C22—24no—
328C22—22no—
329C22—22no—
330C22#202314yes—
331C22C2×#44711yescharacteristic
332C22C2×#47411yescharacteristic
333C22C2×#48611yescharacteristic
334C22#341612yes—
335C22#341712yes—
336C4—42no—
337C4—42no—
338C4—84no—
339C4—84no—
340C4—82no—
341C4—82no—
342C4—162no—
343C4#202314yes—
344C4C2×#39311yescharacteristic
345C4C2×#44811yescharacteristic
346C4C2×#47611yescharacteristic
347C4C2×#48911yescharacteristic
348C2—24no—
349C2C2×#202311yescharacteristic
350C2#3352311yescharacteristic
351C2#3352511yescharacteristic
352C2#3363212yes—
353C2#3370312yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13366C2semidirect
6. #13364C2semidirect
44. C22×Q16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
C2×#21412
C22×#3582
#212322
#133662
#133642
#125572
#125562
#125452