Finite Group Databasebeta

← #129508

Subgroups — #129508

Order 512.

857 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 857 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#443C211yesmaximal, characteristic
2#12788C211yesmaximal, characteristic, complemented
3C2×#1335C214yesmaximal, complemented
4C8×C22⋊C8C211yesmaximal, characteristic, complemented
5C22×#135C411yescharacteristic
6C22×#131—21no—
7C2×#443—28no—
8C2×#443—21no—
9C2×#443—21no—
10C2×#443C412yes—
11C2×#443C2212yes—
12C22×#108C411yescharacteristic
13C22×#104—21no—
14C22×C4×C8C2211yescharacteristic
15C2×#364C2212yescomplemented
16#2112—21no—
17C4×C22⋊C8—21no—
18C2×C82C2212yescomplemented
19#1376—21no—
20#1335—28no—
21#1273—21no—
22C22×#41—21no—
23C2×#135—24no—
24C2×#135—24no—
25C2×#135—21no—
26C2×#135—21no—
27C2×#135C812yescomplemented
28C2×#135C2×C412yes—
29C22×Q16C811yescharacteristic, complemented
30C22×SD16—21no—
31C22×D8C811yescharacteristic, complemented
32C2×#131—28no—
33C2×#131—22no—
34C2×#131—22no—
35C2×#131—44no—
36C2×#131—41no—
37C2×#131—41no—
38C22×M16—21no—
39C22×M16—41no—
40C23×C8—41no—
41C23×C8C2×C411yescharacteristic
42C23×C8D411yescharacteristic
43C22×C42D411yescharacteristic
44C2×M32—22no—
45#443—28no—
46#443—28no—
47#443—24no—
48#443—24no—
49#443—24no—
50#443D418yes—
51C2×#108—21no—
52C2×#108—24no—
53C2×#108C2×C412yes—
54C2×#104—28no—
55C2×#104—22no—
56C2×#104—22no—
57C2×C22⋊C8—22no—
58C2×C4×C8—24no—
59C2×C4×C8—22no—
60C2×C4×C8C2×C411yescharacteristic
61C2×C4×C8D411yescharacteristic
62C2×C4×C8D411yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#364—24no—
65C22⋊C16—41no—
66C22⋊C16—41no—
67#336—22no—
68C82—24no—
69C2×#41—24no—
70C2×#41—24no—
71C2×#41—22no—
72C2×#41—22no—
73C2×#41—42no—
74C2×#41—42no—
75C2×#41—41no—
76C2×#41—41no—
77C22×Q8—21no—
78C22×D4—21no—
79C22×D4—41no—
80C23×C4—21no—
81C23×C4—41no—
82C23×C4C22⋊C411yescharacteristic
83#135—24no—
84#135—28no—
85#135—24no—
86#135—24no—
87#135—24no—
88#135—22no—
89#135—22no—
90#135C22⋊C414yes—
91#135M1614yes—
92C2×Q16—24no—
93C2×Q16—21no—
94C2×Q16C2×C812yes—
95C2×SD16—28no—
96C2×SD16—22no—
97C2×SD16—22no—
98C2×D8—24no—
99C2×D8—21no—
100C2×D8C2×C812yes—
101#131—28no—
102#131—28no—
103#131—44no—
104#131—48no—
105#131—48no—
106#131—44no—
107C2×M16—24no—
108C2×M16—22no—
109C2×M16—48no—
110C2×M16—42no—
111C2×M16—41no—
112C2×M16—42no—
113C2×M16—42no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—24no—
119C22×C8—48no—
120C22×C8—41no—
121C22×C8—42no—
122C22×C8—42no—
123C22×C8C22⋊C411yescharacteristic
124C22×C8C22⋊C411yescharacteristic
125C22×C8C2×C811yescharacteristic
126C22×C8C2×C811yescharacteristic
127C22×C8M1611yescharacteristic
128C22×C8M1611yescharacteristic
129C22×C8D811yescharacteristic
130C22×C8D811yescharacteristic
131C22×C8SD1611yescharacteristic
132C22×C8SD1611yescharacteristic
133C22×C8C22×C411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C2×C42—22no—
136C2×C42—21no—
137C2×C42—41no—
138C2×C42C22⋊C411yescharacteristic
139C2×C42C2×D411yescharacteristic
140M32—42no—
141M32—42no—
142C2×C16—42no—
143#108—24no—
144#108—22no—
145#108C22⋊C414yes—
146#104—28no—
147#104—28no—
148C22⋊C8—28no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—42no—
153C4×C8—42no—
154C4×C8C2×D414yes—
155C24—41no—
156C24#10411yescharacteristic
157#41—28no—
158#41—24no—
159#41—24no—
160#41—44no—
161#41—44no—
162#41—44no—
163#41—44no—
164#41—44no—
165#41—42no—
166#41—42no—
167C2×Q8—24no—
168C2×Q8—22no—
169C2×Q8—42no—
170C2×Q8—41no—
171C2×D4—24no—
172C2×D4—22no—
173C2×D4—42no—
174C2×D4—44no—
175C2×D4—44no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—41no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4C22⋊C811yescharacteristic
194C22×C4C22⋊C811yescharacteristic
195C22×C4#10211yescharacteristic
196C22×C4#10211yescharacteristic
197C22×C4#10411yescharacteristic
198C22×C4C2×C22⋊C411yescharacteristic
199Q16—24no—
200Q16—22no—
201Q16C22⋊C814yescomplemented
202SD16—28no—
203SD16—24no—
204SD16—24no—
205D8—24no—
206D8—22no—
207D8C22⋊C814yescomplemented
208M16—24no—
209M16—48no—
210M16—44no—
211M16—42no—
212M16—48no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—28no—
226C2×C8—48no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—44no—
230C2×C8—42no—
231C2×C8—48no—
232C2×C8C22⋊C811yescharacteristic
233C2×C8C22⋊C811yescharacteristic
234C2×C8#10211yescharacteristic
235C2×C8#10211yescharacteristic
236C2×C8C2×C22⋊C412yes—
237C2×C8#12012yes—
238C2×C8C22×C811yescharacteristic
239C2×C8C2×M1612yes—
240C2×C8C2×M1611yescharacteristic
241C2×C8C2×D811yescharacteristic
242C2×C8C2×SD1611yescharacteristic
243C2×C8AGL(1,ℤ/8)12yes—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—42no—
248C42#12012yes—
249C16—82no—
250C23—22no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—44no—
255C23—42no—
256C23—82no—
257C23—82no—
258C23D4⋊C811yescharacteristic
259C23D4⋊C811yescharacteristic
260C23C2×#10411yescharacteristic
261Q8—24no—
262Q8—44no—
263Q8—42no—
264D4—24no—
265D4—44no—
266D4—48no—
267D4—44no—
268D4—44no—
269D4—42no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4D4⋊C811yescharacteristic
293C2×C4D4⋊C811yescharacteristic
294C2×C4#37911yescharacteristic
295C2×C4C2×C22⋊C811yescharacteristic
296C2×C4#40711yescharacteristic
297C2×C4C2×#10211yescharacteristic
298C2×C4#41811yescharacteristic
299C2×C4#42111yescharacteristic
300C8—21no—
301C8—21no—
302C8—21no—
303C8—21no—
304C8—48no—
305C8—44no—
306C8C2×C22⋊C812yes—
307C8#40712yes—
308C8#44712yes—
309C8#45012yes—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—42no—
318C22—42no—
319C22—44no—
320C22—88no—
321C22—82no—
322C22—82no—
323C22#133512yescomplemented
324C22C2×D4⋊C811yescharacteristic
325C22#149011yescharacteristic
326C22#178911yescharacteristic
327C4—24no—
328C4—21no—
329C4—21no—
330C4—42no—
331C4—44no—
332C4#147811yescharacteristic
333C4#148611yescharacteristic
334C4#177911yescharacteristic
335C4#178611yescharacteristic
336C2—21no—
337C2—22no—
338C2—21no—
339C2—41no—
340C2—84no—
341C2#831011yescharacteristic
342C2C2×#133511yescharacteristic
343C2#1160211yescharacteristic
subgroupG/H
centerC2×C8C2×D8
commutatorC2×C8C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22C2×D4⋊C8
prime radicalC22×#135C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12788C2semidirect
3. C2×#1335C2semidirect
4. C8×C22⋊C8C2semidirect
15. C2×#364C22semidirect
18. C2×C82C22semidirect
27. C2×#135C8semidirect
29. C22×Q16C8semidirect
31. C22×D8C8semidirect
201. Q16C22⋊C8semidirect
207. D8C22⋊C8semidirect
323. C22#1335semidirect

Maximal subgroups

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

subgroupindex
C22×#4432
#127882
C2×#13352
C8×C22⋊C82