Finite Group Databasebeta

← #128804

Subgroups — #128804

Order 512.

450 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 100 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
1#21538C211yesmaximal, characteristic
2#12607C211yesmaximal, characteristic, complemented
3C2×#1341C211yesmaximal, characteristic
4C2×#1339C211yesmaximal, characteristic
5#10813C211yesmaximal, characteristic, complemented
6#7255C211yesmaximal, characteristic, complemented
7#7254C211yesmaximal, characteristic, complemented
8C2×#469C2211yescharacteristic
9C2×#442C411yescharacteristic, complemented
10C2×C4×M16C2211yescharacteristic
11C4×#121—21no—
12C2×#349C2211yescharacteristic
13C2×#349C2211yescharacteristic
14C2×#347—21no—
15#1970C411yescharacteristic, complemented
16#1941C411yescharacteristic, complemented
17#1934—21no—
18#1895—21no—
19#1809C2211yescharacteristic
20#1776C411yescharacteristic, complemented
21#1762—21no—
22C2×#344—21no—
23C2×#339—21no—
24C2×#338C2211yescharacteristic
25C2×H(3,ℤ/4)C2211yescharacteristic
26#1376—21no—
27#1341C414yescomplemented
28#1339C414yescomplemented
29C22×D8C2×C411yescharacteristic
30C22×M16—21no—
31C22×M16D411yescharacteristic
32C22×M16C2311yescharacteristic
33C2×#121—21no—
34C2×C4×D4—21no—
35C2×C4×D4—41no—
36C2×#117—21no—
37C22×C42D411yescharacteristic
38C2×M32—21no—
39C2×M32—21no—
40#469—21no—
41#469—21no—
42#469C2×C412yes—
43#469C2×C412yes—
44#442—42no—
45C2×#107C2×C411yescharacteristic
46C2×#106C2×C411yescharacteristic
47C2×#106C2×C411yescharacteristic
48C2×#102—21no—
49C2×#102—21no—
50C2×#102—21no—
51C4×M16—42no—
52C2×#97C2×C411yescharacteristic
53C2×#97D411yescharacteristic
54C2×C4×C8C2×C411yescharacteristic
55C2×C4×C8D411yescharacteristic
56#390—21no—
57#389—21no—
58#388—41no—
59#386—41no—
60#380—21no—
61C4×C4⋊C4—21no—
62C4×C22⋊C4—41no—
63#349—21no—
64#349—21no—
65#349C2×C412yescomplemented
66#349C2×C412yescomplemented
67#347—42no—
68#344—42no—
69#340—21no—
70#340—21no—
71#340—21no—
72#339—42no—
73#338—21no—
74#338C2×C412yescomplemented
75H(3,ℤ/4)—21no—
76H(3,ℤ/4)C2×C412yescomplemented
77C22×D4—21no—
78C22×D4—41no—
79C23×C4—21no—
80C23×C4—41no—
81C23×C4C2×D411yescharacteristic
82C2×D8—21no—
83C2×D8—41no—
84C2×D8—41no—
85C2×D8C4212yes—
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—22no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16C22×C411yescharacteristic
99C2×M16C22×C411yescharacteristic
100C2×M16#4111yescharacteristic
101C2×M16#4111yescharacteristic
102C22×C8—41no—
103C22×C8C22⋊C411yescharacteristic
104C22×C8C22⋊C411yescharacteristic
105C22×C8C22×C411yescharacteristic
106C22×C8C2×D411yescharacteristic
107#121—22no—
108#121—22no—
109#121—41no—
110#121—41no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—84no—
114#117—22no—
115#117—41no—
116#117—41no—
117#117—41no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C42—22no—
130C2×C42—21no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42C22⋊C411yescharacteristic
135C2×C42C22⋊C411yescharacteristic
136M32—22no—
137M32—22no—
138M32—41no—
139M32—41no—
140C2×C16—41no—
141C2×C16—41no—
142#107—41no—
143#106—21no—
144#106—41no—
145#106C4212yes—
146#102—22no—
147#102—22no—
148#102—42no—
149#102—42no—
150#97—22no—
151#97—22no—
152C4×C8—41no—
153C4×C8—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157C24—41no—
158C24AGL(1,ℤ/8)11yescharacteristic
159C2×D4—22no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—82no—
168C2×D4—82no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—82no—
181C22×C4—81no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—81no—
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4C4×D411yescharacteristic
187C22×C4C4×D411yescharacteristic
188C22×C4AGL(1,ℤ/8)11yescharacteristic
189C22×C4AGL(1,ℤ/8)11yescharacteristic
190C22×C4#13711yescharacteristic
191D8—42no—
192D8—42no—
193M16—41no—
194M16—41no—
195M16—41no—
196M16—41no—
197M16—42no—
198M16—42no—
199M16—42no—
200M16—42no—
201M16—81no—
202M16—81no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—42no—
214C2×C8—81no—
215C2×C8—81no—
216C2×C8C2×C4211yescharacteristic
217C2×C8#11711yescharacteristic
218C2×C8C4×D411yescharacteristic
219C2×C8C4×D411yescharacteristic
220C4⋊C4—22no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—81no—
228C4⋊C4—82no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C22⋊C4—81no—
236C22⋊C4—82no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—42no—
241C42—42no—
242C42—81no—
243C42—82no—
244C42#9912yes—
245C16—42no—
246C16—42no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23—81no—
256C23—81no—
257C23—81no—
258C23#41811yescharacteristic
259C23#44011yescharacteristic
260C23#44211yescharacteristic
261D4—42no—
262D4—42no—
263D4—84no—
264D4—84no—
265D4—81no—
266D4—81no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—84no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—164no—
295C2×C4—161no—
296C2×C4—161no—
297C2×C4C4×C22⋊C411yescharacteristic
298C2×C4C2×#9911yescharacteristic
299C2×C4#41011yescharacteristic
300C2×C4#41711yescharacteristic
301C2×C4#42111yescharacteristic
302C2×C4#42111yescharacteristic
303C2×C4#44011yescharacteristic
304C2×C4#44211yescharacteristic
305C8—21no—
306C8—21no—
307C8—21no—
308C8—21no—
309C8—41no—
310C8—41no—
311C8—82no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—161no—
325C22—161no—
326C22—161no—
327C22—161no—
328C22#151011yescharacteristic
329C22#151411yescharacteristic
330C22#176211yescharacteristic
331C4—21no—
332C4—21no—
333C4—42no—
334C4—42no—
335C4—82no—
336C4—161no—
337C4—162no—
338C4—161no—
339C4#151211yescharacteristic
340C4#151311yescharacteristic
341C4C4×#9911yescharacteristic
342C4#175911yescharacteristic
343C2—21no—
344C2—21no—
345C2—41no—
346C2—41no—
347C2—162no—
348C2#821611yescharacteristic
349C2#1183611yescharacteristic
350C2#1183711yescharacteristic
subgroupG/H
center, socleC22#1514
commutatorC2×C8C2×C42
Frattini, cosocleC22×M16C23
prime radicalC2×#469C22
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12607C2semidirect
5. #10813C2semidirect
6. #7255C2semidirect
7. #7254C2semidirect
9. C2×#442C4semidirect
15. #1970C4semidirect
16. #1941C4semidirect
20. #1776C4semidirect
27. #1341C4semidirect
28. #1339C4semidirect
65. #349C2×C4semidirect
66. #349C2×C4semidirect
74. #338C2×C4semidirect
76. H(3,ℤ/4)C2×C4semidirect

Maximal subgroups

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

subgroupindex
#215382
#126072
C2×#13412
C2×#13392
#108132
#72552
#72542