Finite Group Databasebeta

← #134475

Subgroups — #134475

Order 512.

418 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 86 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#21529C211yesmaximal, characteristic
2#12789C211yesmaximal, characteristic, complemented
3#11711C211yesmaximal, characteristic
4#11705C211yesmaximal, characteristic
5#10817C211yesmaximal, characteristic, complemented
6#7263C211yesmaximal, characteristic, complemented
7#7262C211yesmaximal, characteristic, complemented
8C2×#466C2211yescharacteristic
9C2×#442C411yescharacteristic, complemented
10C2×#405C2211yescharacteristic
11#2324—21no—
12C2×#364—21no—
13#2112C2211yescharacteristic
14#2112C2211yescharacteristic
15#1972C411yescharacteristic, complemented
16#1943C411yescharacteristic, complemented
17#1934—21no—
18#1895—21no—
19#1807C2211yescharacteristic
20#1776C411yescharacteristic, complemented
21C4×#102—21no—
22C2×#344—21no—
23C2×#341C2211yescharacteristic
24#1735C2211yescharacteristic
25C2×H(3,ℤ/4)—21no—
26#1376—21no—
27#1349—22no—
28#1347—22no—
29C22×D8C2×C411yescharacteristic
30C22×M16—21no—
31C22×M16D411yescharacteristic
32C23×C8C2311yescharacteristic
33C2×#121—21no—
34C2×C4×D4—21no—
35C2×#117—21no—
36C2×#117D411yescharacteristic
37C2×M32—21no—
38C2×M32—21no—
39#466—21no—
40#466—21no—
41#466D412yes—
42#466Q812yes—
43#442—42no—
44C2×#107C2×C411yescharacteristic
45C2×#106C2×C411yescharacteristic
46C2×#106C2×C411yescharacteristic
47C2×#102—21no—
48C2×#102—21no—
49C2×#102—21no—
50#405—42no—
51C2×#97C2×C411yescharacteristic
52C2×#97C2×C411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8D411yescharacteristic
55#390—21no—
56#390—41no—
57#389—21no—
58#388—41no—
59#386—41no—
60#385—41no—
61#380—21no—
62C4×C4⋊C4—21no—
63#364—42no—
64C22⋊C16—21no—
65C22⋊C16—21no—
66C22⋊C16—21no—
67C22⋊C16—21no—
68#344—42no—
69#341—21no—
70#341—21no—
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#340—21no—
76H(3,ℤ/4)—24no—
77C22×D4—21no—
78C22×D4—41no—
79C23×C4—41no—
80C23×C4C2×D411yescharacteristic
81C2×D8—21no—
82C2×D8—41no—
83C2×D8—41no—
84C2×D8C4⋊C412yes—
85C2×M16—21no—
86C2×M16—21no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8C22⋊C411yescharacteristic
99C22×C8C22⋊C411yescharacteristic
100C22×C8C22×C411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×Q811yescharacteristic
104C22×C8#4111yescharacteristic
105C22×C8#4111yescharacteristic
106#121—22no—
107#121—22no—
108#121—41no—
109#121—41no—
110C4×D4—42no—
111C4×D4—42no—
112#117—22no—
113#117—22no—
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—21no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42C22⋊C411yescharacteristic
134C2×C42C22⋊C411yescharacteristic
135M32—41no—
136M32—41no—
137M32—41no—
138M32—41no—
139C2×C16—41no—
140C2×C16—41no—
141#107—21no—
142#107C4⋊C412yes—
143#106—41no—
144#106—41no—
145#102—22no—
146#102—22no—
147#102—42no—
148#102—42no—
149#97—22no—
150#97—41no—
151C4×C8—22no—
152C4×C8—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156C24—41no—
157C24AGL(1,ℤ/8)11yescharacteristic
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—161no—
167C2×D4—161no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—82no—
178C22×C4—81no—
179C22×C4—161no—
180C22×C4—161no—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4#13511yescharacteristic
185C22×C4#13511yescharacteristic
186C22×C4AGL(1,ℤ/8)11yescharacteristic
187D8—42no—
188D8—42no—
189M16—42no—
190M16—42no—
191M16—42no—
192M16—42no—
193M16—81no—
194M16—81no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—81no—
210C2×C8—81no—
211C2×C8#9912yes—
212C2×C8C2×C4⋊C411yescharacteristic
213C2×C8#11711yescharacteristic
214C2×C8#12111yescharacteristic
215C2×C8C22⋊Q811yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—81no—
225C4⋊C4—161no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—82no—
231C22⋊C4—81no—
232C22⋊C4—161no—
233C22⋊C4—161no—
234C22⋊C4—161no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—42no—
239C42—81no—
240C42—161no—
241C42#10412yes—
242C16—81no—
243C16—81no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—81no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23#41811yescharacteristic
256C23#46311yescharacteristic
257C23#47811yescharacteristic
258D4—42no—
259D4—42no—
260D4—81no—
261D4—81no—
262D4—162no—
263D4—162no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—164no—
291C2×C4—161no—
292C2×C4—161no—
293C2×C4#38011yescharacteristic
294C2×C4C2×#9911yescharacteristic
295C2×C4#41011yescharacteristic
296C2×C4#41611yescharacteristic
297C2×C4C2×#10411yescharacteristic
298C2×C4#42111yescharacteristic
299C2×C4#46311yescharacteristic
300C2×C4#47811yescharacteristic
301C8—21no—
302C8—21no—
303C8—21no—
304C8—21no—
305C8—22no—
306C8—41no—
307C8—82no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—81no—
317C22—81no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—161no—
322C22—161no—
323C22—161no—
324C22—161no—
325C22#150711yescharacteristic
326C22#151411yescharacteristic
327C22#180411yescharacteristic
328C4—21no—
329C4—21no—
330C4—42no—
331C4—81no—
332C4—82no—
333C4—161no—
334C4—161no—
335C4—162no—
336C4#150011yescharacteristic
337C4#151611yescharacteristic
338C4#179911yescharacteristic
339C4#180611yescharacteristic
340C2—21no—
341C2—21no—
342C2—41no—
343C2—41no—
344C2—162no—
345C2#857311yescharacteristic
346C2#1282611yescharacteristic
347C2#1282811yescharacteristic
subgroupG/H
center, socleC22#1804
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12789C2semidirect
5. #10817C2semidirect
6. #7263C2semidirect
7. #7262C2semidirect
9. C2×#442C4semidirect
15. #1972C4semidirect
16. #1943C4semidirect
20. #1776C4semidirect

Maximal subgroups

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

subgroupindex
#215292
#127892
#117112
#117052
#108172
#72632
#72622