Finite Group Databasebeta

← #128808

Subgroups — #128808

Order 512.

456 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 102 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#21526C211yesmaximal, characteristic
2#12605C211yesmaximal, characteristic, complemented
3#11711C211yesmaximal, characteristic
4#11705C211yesmaximal, characteristic
5#10819C211yesmaximal, characteristic, complemented
6#7263C211yesmaximal, characteristic, complemented
7#7262C211yesmaximal, characteristic, complemented
8C2×#466C2211yescharacteristic
9C2×#442C411yescharacteristic, complemented
10C22×#97C2211yescharacteristic
11C4×#121—21no—
12#2112C2211yescharacteristic
13#2112C2211yescharacteristic
14C2×#347—21no—
15#1972C411yescharacteristic, complemented
16#1943C411yescharacteristic, complemented
17#1936—21no—
18#1893—21no—
19#1807C2211yescharacteristic
20#1776C411yescharacteristic, complemented
21#1762—21no—
22C2×#341C2211yescharacteristic
23#1735C2211yescharacteristic
24C2×#339—21no—
25#1729—21no—
26#1375—21no—
27#1349C414yescomplemented
28#1347C414yescomplemented
29C22×D8C2×C411yescharacteristic
30C22×M16—21no—
31C23×C8D411yescharacteristic
32C23×C8C2311yescharacteristic
33C2×#121—21no—
34C2×C4×D4—21no—
35C2×C4×D4—41no—
36C2×#117—21no—
37C22×C42D411yescharacteristic
38C2×M32—21no—
39C2×M32—21no—
40#466—21no—
41#466—21no—
42#466C2×C412yes—
43#466C2×C412yes—
44#442—42no—
45C2×#107C2×C411yescharacteristic
46C2×#106C2×C411yescharacteristic
47C2×#106C2×C411yescharacteristic
48C2×#102—21no—
49C2×#102—21no—
50C2×#102—21no—
51C2×#97—42no—
52C2×#97C2×C411yescharacteristic
53C2×#97C2×C411yescharacteristic
54C2×#97D411yescharacteristic
55C2×#97D411yescharacteristic
56#390—21no—
57#389—21no—
58#388—41no—
59#386—41no—
60#380—21no—
61C4×C4⋊C4—21no—
62C4×C22⋊C4—41no—
63C22⋊C16—21no—
64C22⋊C16—21no—
65C22⋊C16C2×C412yescomplemented
66C22⋊C16C2×C412yescomplemented
67#347—42no—
68#341—21no—
69#341C2×C412yescomplemented
70#340—21no—
71#340—21no—
72#340—21no—
73#340—21no—
74#340C2×C412yescomplemented
75#339—42no—
76#336—42no—
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—22no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8C22⋊C411yescharacteristic
100C22×C8C22⋊C411yescharacteristic
101C22×C8C22×C411yescharacteristic
102C22×C8C22×C411yescharacteristic
103C22×C8C22×C411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8#4111yescharacteristic
106C22×C8#4111yescharacteristic
107#121—22no—
108#121—22no—
109#121—41no—
110#121—41no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—84no—
114#117—41no—
115#117—41no—
116#117—41no—
117#117—41no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
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—41no—
137M32—41no—
138M32—41no—
139M32—41no—
140C2×C16—22no—
141C2×C16—22no—
142#107—21no—
143#107C4212yes—
144#106—41no—
145#106—41no—
146#102—22no—
147#102—22no—
148#102—42no—
149#102—42no—
150#97—22no—
151#97—22no—
152#97—41no—
153#97—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—42no—
181C22×C4—82no—
182C22×C4—81no—
183C22×C4—82no—
184C22×C4—82no—
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—42no—
194M16—42no—
195M16—81no—
196M16—81no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
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—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—82no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—82no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—42no—
241C42—81no—
242C42—81no—
243C42—82no—
244C42#9912yes—
245C16—42no—
246C16—42no—
247C23—22no—
248C23—21no—
249C23—21no—
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—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—84no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—164no—
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—21no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—42no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—161no—
325C22—161no—
326C22—161no—
327C22—161no—
328C22#151011yescharacteristic
329C22#151411yescharacteristic
330C22#176211yescharacteristic
331C22AGL(1,ℤ/16)12yes—
332C4—21no—
333C4—21no—
334C4—42no—
335C4—81no—
336C4—82no—
337C4—161no—
338C4—162no—
339C4—161no—
340C4#151211yescharacteristic
341C4#151311yescharacteristic
342C4C4×#9911yescharacteristic
343C4#175911yescharacteristic
344C2—21no—
345C2—21no—
346C2—22no—
347C2—41no—
348C2—162no—
349C2#821611yescharacteristic
350C2#1264711yescharacteristic
351C2#1264911yescharacteristic
subgroupG/H
center, socleC22#1762
commutatorC2×C8C2×C42
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12605C2semidirect
5. #10819C2semidirect
6. #7263C2semidirect
7. #7262C2semidirect
9. C2×#442C4semidirect
15. #1972C4semidirect
16. #1943C4semidirect
20. #1776C4semidirect
27. #1349C4semidirect
28. #1347C4semidirect
65. C22⋊C16C2×C4semidirect
66. C22⋊C16C2×C4semidirect
69. #341C2×C4semidirect
74. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
#215262
#126052
#117112
#117052
#108192
#72632
#72622