Finite Group Databasebeta

← #500431

Subgroups — #500431

Order 512.

872 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
1C2×#3338C211yesmaximal, characteristic, complemented
2C2×#2226C211yesmaximal, characteristic, complemented
3C2×C4×SD32C211yesmaximal, characteristic, complemented
4#28227C211yesmaximal, characteristic
5#13382C212yesmaximal, complemented
6#13305C212yesmaximal, complemented
7#13282C211yesmaximal, characteristic, complemented
8#13189C211yesmaximal, characteristic, complemented
9#12991C212yesmaximal, complemented
10#12920C212yesmaximal, complemented
11#12812C211yesmaximal, characteristic, complemented
12C2×#554—21no—
13C22×SD32C2211yescharacteristic
14#3338—24no—
15#3338—24no—
16C2×#500C2211yescharacteristic
17C2×#484—22no—
18C2×#477—21no—
19C2×#467C2212yes—
20C2×#466C2212yes—
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C22×#107C2212yes—
25#2537—21no—
26#2226—24no—
27C4×SD32—24no—
28C2×#367C2212yes—
29C2×#367C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2212yes—
33C2×#357C2212yes—
34C2×#357C2211yescharacteristic
35C2×C22⋊C16C2212yes—
36C2×C22⋊C16C2212yes—
37C2×C4×C16C2211yescharacteristic
38#2033—22no—
39#1845C2211yescharacteristic
40#1808C2212yes—
41#1802—21no—
42#1380C2212yes—
43#1380C2212yes—
44#1380C2212yes—
45C22×Q16C2311yescharacteristic
46C22×D8C2311yescharacteristic
47C23×C8C2312yes—
48#554—28no—
49#554—44no—
50C2×C4⋊Q8—21no—
51C2×#125—41no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—42no—
54C2×#121—22no—
55C2×#121—41no—
56C2×C4×Q8—21no—
57C2×C4×D4—21no—
58C2×C4×D4—41no—
59C2×C4×D4D411yescharacteristic
60C22×C4⋊C4—22no—
61C2×SD32—22no—
62C2×SD32—22no—
63C2×SD32—22no—
64C22×C16C2311yescharacteristic
65C22×C16C2312yes—
66C22×C16C2311yescharacteristic
67#500—22no—
68#500D414yescomplemented
69#484—216no—
70#477—28no—
71#467—24no—
72#467—24no—
73#466—24no—
74#466—42no—
75C4×Q16—22no—
76C4×Q16D414yescomplemented
77C4×D8—22no—
78C4×D8—22no—
79C8×D4—22no—
80C8×D4—22no—
81C2×#107—22no—
82C2×#107—21no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—21no—
86C2×#107C2311yescharacteristic
87C2×#107C2312yes—
88C2×#107C2312yes—
89C2×#107C2311yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×#103—22no—
92C2×#103—21no—
93C2×#102—22no—
94C2×#102—21no—
95C2×C22⋊C8D412yes—
96C2×C4×C8C2311yescharacteristic
97#394—42no—
98#384—21no—
99#380—22no—
100#380—41no—
101#367—22no—
102#367—24no—
103C4⋊C16—22no—
104#358—22no—
105#358—42no—
106#357—22no—
107#357—24no—
108C22⋊C16—24no—
109C22⋊C16—42no—
110C4×C16—22no—
111#340—22no—
112C22×Q8—21no—
113C22×Q8—41no—
114C22×D4—21no—
115C22×D4—42no—
116C22×D4D811yescharacteristic
117C23×C4—42no—
118C23×C4—41no—
119C23×C4C2×D412yes—
120C2×Q16—21no—
121C2×Q16—21no—
122C2×Q16—21no—
123C2×Q16—21no—
124C2×Q16—21no—
125C2×Q16C2×D412yes—
126C2×D8—21no—
127C2×D8—21no—
128C2×D8—21no—
129C2×D8—21no—
130C2×D8—21no—
131C2×D8#4112yes—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8C2×D412yes—
139C22×C8C2×D411yescharacteristic
140C22×C8C2411yescharacteristic
141C4⋊Q8—24no—
142C4⋊Q8—42no—
143#125—48no—
144C22⋊Q8—28no—
145C22⋊Q8—416no—
146C22⋊Q8—44no—
147#121—44no—
148#121—82no—
149#121—84no—
150C4×Q8—24no—
151C4×Q8—42no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—48no—
155C4×D4—42no—
156C4×D4—42no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4D811yescharacteristic
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4D812yes—
175C2×C42—41no—
176C2×C42C2×D411yescharacteristic
177SD32—28no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—21no—
184C2×C16—22no—
185C2×C16—21no—
186C2×C16#4112yes—
187#107—28no—
188#107—24no—
189#107—21no—
190#107—22no—
191#107—21no—
192#107—21no—
193#107—22no—
194#107—22no—
195#107—81no—
196#107C2×D412yes—
197#107C2×D414yes—
198C4⋊C8—22no—
199#103—28no—
200#103—24no—
201#102—44no—
202#102—42no—
203C22⋊C8—24no—
204C4×C8C2×D412yes—
205C4×C8C2×D412yes—
206#95—42no—
207C24—41no—
208C24C2×D812yes—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—44no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—161no—
217C2×Q8—161no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—84no—
227C2×D4—84no—
228C2×D4—162no—
229C2×D4SD3214yescomplemented
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—48no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—162no—
246C22×C4—162no—
247C22×C4C2×D812yes—
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C22×D411yescharacteristic
251Q16—22no—
252Q16—22no—
253Q16—41no—
254Q16—41no—
255D8—22no—
256D8—22no—
257D8—41no—
258D8—41no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8#12012yes—
269C2×C8C22×D411yescharacteristic
270C2×C8C22×D411yescharacteristic
271C2×C8C2×#4111yescharacteristic
272C2×C8C2×#4111yescharacteristic
273C2×C8H(5,𝔽2)11yescharacteristic
274C2×C82−1+411yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—48no—
279C4⋊C4—48no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—82no—
289C4⋊C4—161no—
290C22⋊C4—24no—
291C22⋊C4—48no—
292C22⋊C4—84no—
293C22⋊C4—84no—
294C22⋊C4—82no—
295C42—44no—
296C42#12012yes—
297C42AGL(1,ℤ/8)12yes—
298C16—22no—
299C16—41no—
300C16—42no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—42no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23C2×SD3214yes—
315C23C22×D811yescharacteristic
316Q8—42no—
317Q8—42no—
318Q8—81no—
319Q8—81no—
320Q8—164no—
321D4—28no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—81no—
326D4—168no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—168no—
352C2×C4—164no—
353C2×C4#44712yes—
354C2×C4#44712yes—
355C2×C4C2×SD3212yes—
356C2×C4#51012yes—
357C2×C4C2×#12011yescharacteristic
358C2×C4#54611yescharacteristic
359C2×C4#54711yescharacteristic
360C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
361C2×C4#57711yescharacteristic
362C2×C4#57811yescharacteristic
363C8—22no—
364C8—42no—
365C8#52511yescharacteristic
366C8#53711yescharacteristic
367C8#54811yescharacteristic
368C8#55011yescharacteristic
369C22—24no—
370C22—24no—
371C22—24no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22—161no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22C2×#44711yescharacteristic
384C22#329411yescharacteristic
385C22#333411yescharacteristic
386C22C22×SD3211yescharacteristic
387C22C2×#51011yescharacteristic
388C22#341511yescharacteristic
389C22#341611yescharacteristic
390C4—22no—
391C4—42no—
392C4—82no—
393C4—82no—
394C4—84no—
395C4—162no—
396C4—161no—
397C4#218514yes—
398C4#240711yescharacteristic
399C4#301311yescharacteristic
400C4#333011yescharacteristic
401C4#335411yescharacteristic
402C2—44no—
403C2—162no—
404C2#2411611yescharacteristic
405C2C2×#218511yescharacteristic
406C2#3352611yescharacteristic
407C2#3386211yescharacteristic
408C2#3389811yescharacteristic
409C2#3390211yescharacteristic
410C2#3392411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#28227C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3338C2semidirect
2. C2×#2226C2semidirect
3. C2×C4×SD32C2semidirect
5. #13382C2semidirect
6. #13305C2semidirect
7. #13282C2semidirect
8. #13189C2semidirect
9. #12991C2semidirect
10. #12920C2semidirect
11. #12812C2semidirect
68. #500D4semidirect
76. C4×Q16D4semidirect
229. C2×D4SD32semidirect

Maximal subgroups

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

subgroupindex
C2×#33382
C2×#22262
C2×C4×SD322
#282272
#133822
#133052
#132822
#131892
#129912
#129202
#128122