Finite Group Databasebeta

← #128496

Subgroups — #128496

Order 512.

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

normal subgroups only

This group has 451 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#22262C211yesmaximal, characteristic
2C4×#353C211yesmaximal, characteristic
3#11475C211yesmaximal, characteristic
4#8160C211yesmaximal, characteristic, complemented
5C8×#99C211yesmaximal, characteristic, complemented
6#7205C211yesmaximal, characteristic, complemented
7#7205C211yesmaximal, characteristic, complemented
8C2×#436C411yescharacteristic, complemented
9C2×C8×D4C2211yescharacteristic
10C4×#127C2211yescharacteristic
11C2×#353C412yes—
12#1954C411yescharacteristic, complemented
13#1954C2211yescharacteristic
14#1870—21no—
15#1870—21no—
16#1869—21no—
17#1869—21no—
18#1831C411yescharacteristic, complemented
19C4×#99C2211yescharacteristic
20C4×#99C2211yescharacteristic
21#1752—21no—
22C8×C22⋊C4—21no—
23#1744C411yescharacteristic, complemented
24#1459C2211yescharacteristic
25#1459C2211yescharacteristic
26#1393C412yes—
27#1325C412yescomplemented
28#1318C412yescomplemented
29#1282—21no—
30#1282—21no—
31C22×M16—21no—
32C23×C8D411yescharacteristic
33C23×C8D411yescharacteristic
34C2×#127C2×C411yescharacteristic
35C2×C4×D4—21no—
36C2×C4×D4—21no—
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2311yescharacteristic
39#436—41no—
40#436—41no—
41C8×D4—41no—
42C8×D4—41no—
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C4⋊C8C2×C411yescharacteristic
45C2×#99C2×C412yes—
46C2×#99C2×C412yes—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8D411yescharacteristic
55C2×C22⋊C8D411yescharacteristic
56C2×#97—41no—
57C2×C4×C8—41no—
58C2×C4×C8C2×C411yescharacteristic
59C2×C4×C8C2×C411yescharacteristic
60#390—21no—
61#390—21no—
62#390C2×C411yescharacteristic
63#390C2×C411yescharacteristic
64#384C2×C411yescharacteristic
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C43C2×C411yescharacteristic
68#353—24no—
69#342—21no—
70#342—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—41no—
76#336—41no—
77#324C2×C412yes—
78#324C2×C412yes—
79#323—21no—
80#323—21no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4C4211yescharacteristic
84C22×D4C22×C411yescharacteristic
85C23×C4—21no—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C2×M16—42no—
89C2×M16—81no—
90C2×M16—81no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—81no—
100C22×C8—81no—
101C22×C8C22⋊C411yescharacteristic
102C22×C8C22⋊C411yescharacteristic
103C22×C8C22⋊C411yescharacteristic
104C22×C8C22⋊C411yescharacteristic
105#127—22no—
106#127—22no—
107C4×D4—41no—
108C4×D4—41no—
109C4×D4—41no—
110C4×D4—41no—
111C4×D4—81no—
112C4×D4—81no—
113C4×D4—81no—
114C4×D4—81no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4C4211yescharacteristic
118C2×C4⋊C4C22×C411yescharacteristic
119C2×C22⋊C4—22no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4#4111yescharacteristic
127C2×C22⋊C4#4111yescharacteristic
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42C4211yescharacteristic
133C2×C42C4211yescharacteristic
134C2×C42C22×C411yescharacteristic
135C4⋊C8—41no—
136C4⋊C8—41no—
137#99—22no—
138#99—22no—
139#99—22no—
140#99M1614yescomplemented
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—81no—
150C22⋊C8—81no—
151C22⋊C8—81no—
152C22⋊C8—81no—
153#97—82no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—41no—
157C4×C8—82no—
158#95—41no—
159#95—41no—
160C24—21no—
161C24C4×D411yescharacteristic
162C24C4×D411yescharacteristic
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4C2×M1612yes—
176C2×D4#13112yes—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4C2×C4211yescharacteristic
197C22×C4C2×C22⋊C411yescharacteristic
198C22×C4#11711yescharacteristic
199C22×C4C4×D411yescharacteristic
200C22×C4C4×D411yescharacteristic
201M16—84no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—41no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—81no—
220C2×C8—81no—
221C2×C8—81no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—81no—
225C2×C8#9911yescharacteristic
226C2×C8#9911yescharacteristic
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C22⋊C4—24no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—41no—
247C42—42no—
248C42—41no—
249C42—41no—
250C42—81no—
251C42—81no—
252C42—81no—
253C42—81no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23C4×C22⋊C411yescharacteristic
270C23#43512yes—
271C23#43612yes—
272D4—42no—
273D4—42no—
274D4—81no—
275D4—81no—
276D4—84no—
277D4—84no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4C4×M1611yescharacteristic
314C2×C4#40511yescharacteristic
315C2×C4C2×#9911yescharacteristic
316C2×C4#41011yescharacteristic
317C8—21no—
318C8—21no—
319C8—41no—
320C8—41no—
321C8—41no—
322C8—81no—
323C8—82no—
324C8—82no—
325C8—81no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22#175311yescharacteristic
346C22C4×#9911yescharacteristic
347C22#175711yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—84no—
357C4—81no—
358C4—162no—
359C4#212711yescharacteristic
360C4#212811yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—42no—
365C2—161no—
366C2#816111yescharacteristic
367C2#1262811yescharacteristic
368C2#1263311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8160C2semidirect
5. C8×#99C2semidirect
6. #7205C2semidirect
7. #7205C2semidirect
8. C2×#436C4semidirect
12. #1954C4semidirect
18. #1831C4semidirect
23. #1744C4semidirect
27. #1325C4semidirect
28. #1318C4semidirect
140. #99M16semidirect

Maximal subgroups

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

subgroupindex
#222622
C4×#3532
#114752
#81602
C8×#992
#72052