Finite Group Databasebeta

← #128987

Subgroups — #128987

Order 512.

524 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 92 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#21443C211yesmaximal, characteristic
2#11733C211yesmaximal, characteristic
3#11725C211yesmaximal, characteristic
4C8×#100C211yesmaximal, characteristic, complemented
5#8160C211yesmaximal, characteristic, complemented
6#7285C211yesmaximal, characteristic, complemented
7#7282C211yesmaximal, characteristic, complemented
8C22×#131C411yescharacteristic, complemented
9C2×#435—21no—
10C2×C8×D4C2211yescharacteristic
11#2388C2211yescharacteristic
12#1870C2211yescharacteristic
13#1869—21no—
14#1851C411yescharacteristic, complemented
15#1850—21no—
16#1780C411yescharacteristic, complemented
17C4×#100C2211yescharacteristic
18C4×#99C2211yescharacteristic
19#1753—21no—
20#1752—21no—
21#1752C411yescharacteristic, complemented
22#1460C2211yescharacteristic
23#1459C2211yescharacteristic
24C8×M16—21no—
25#1354C412yescomplemented
26#1351C412yes—
27#1345C412yescomplemented
28#1343C412yes—
29#1283—21no—
30#1282—21no—
31C22×#41C2×C411yescharacteristic
32C2×#131—41no—
33C2×#131—41no—
34C2×#131—44no—
35C22×M16—21no—
36C22×M16C2×C411yescharacteristic
37C23×C8—21no—
38C23×C8C2×C411yescharacteristic
39C23×C8D411yescharacteristic
40C2×C4×D4—21no—
41C2×C4×D4C2311yescharacteristic
42#435—44no—
43C8×D4—41no—
44C8×D4—41no—
45#431—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8D411yescharacteristic
48C2×#100—21no—
49C2×#99—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8C2×C411yescharacteristic
55C2×C22⋊C8C2×C411yescharacteristic
56C2×C22⋊C8D411yescharacteristic
57C4×M16—21no—
58C2×#97—21no—
59C2×#97—41no—
60C2×C4×C8D411yescharacteristic
61#395C2×C411yescharacteristic
62#394—21no—
63#394C2×C411yescharacteristic
64#393C2×C411yescharacteristic
65#392—21no—
66#392C2×C411yescharacteristic
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4C2×C411yescharacteristic
70#343C2×C412yes—
71#342C2×C412yes—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—41no—
76#324C2×C412yes—
77#323—21no—
78#323—21no—
79#323C2×C412yes—
80#322—41no—
81C82—41no—
82C2×#41—42no—
83C2×#41—42no—
84C2×#41—41no—
85C2×#41—41no—
86C22×Q8C4211yescharacteristic
87C22×D4—21no—
88C22×D4—21no—
89C22×D4C4211yescharacteristic
90C22×D4#4111yescharacteristic
91C23×C4—21no—
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D411yescharacteristic
94#131—44no—
95#131—44no—
96#131—44no—
97#131—44no—
98C2×M16—24no—
99C2×M16—22no—
100C2×M16—42no—
101C2×M16—42no—
102C2×M16—41no—
103C2×M16—41no—
104C2×M16—41no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8—42no—
114C22×C8—42no—
115C22×C8—42no—
116C22×C8—41no—
117C22×C8C22⋊C411yescharacteristic
118C22×C8C22⋊C411yescharacteristic
119C22×C8C22⋊C411yescharacteristic
120C22×C8C22⋊C411yescharacteristic
121C4×D4—41no—
122C4×D4—41no—
123C4×D4—42no—
124C4×D4—42no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4C22×C411yescharacteristic
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4C22×C411yescharacteristic
134C2×C22⋊C4#4111yescharacteristic
135C2×C42—21no—
136C2×C42—41no—
137C2×C42C2×D411yescharacteristic
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—42no—
141#100—24no—
142#99—42no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—42no—
148C22⋊C8—42no—
149C22⋊C8—41no—
150C22⋊C8—42no—
151C22⋊C8—41no—
152#97—41no—
153#97—42no—
154#97—82no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—41no—
158C4×C8—41no—
159#95—21no—
160#95—21no—
161#95—41no—
162#95C4211yescharacteristic
163#95C4211yescharacteristic
164C24—21no—
165C24#11711yescharacteristic
166C24C4×D411yescharacteristic
167#41—42no—
168#41—44no—
169#41—44no—
170#41—42no—
171#41—42no—
172#41—42no—
173C2×Q8—42no—
174C2×Q8—41no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—81no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4C2×C4211yescharacteristic
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C4×D411yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4C4×D411yescharacteristic
209M16—24no—
210M16—44no—
211M16—42no—
212M16—42no—
213M16—44no—
214M16—42no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—41no—
223C2×C8—48no—
224C2×C8—41no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—42no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—81no—
236C2×C8—81no—
237C2×C8—81no—
238C2×C8#9911yescharacteristic
239C2×C8#9911yescharacteristic
240C4⋊C4—41no—
241C4⋊C4—42no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C42—21no—
252C42—21no—
253C42—41no—
254C42—41no—
255C42—81no—
256C42—81no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23C4×C22⋊C411yescharacteristic
274C23#44312yes—
275Q8—42no—
276Q8—42no—
277D4—44no—
278D4—42no—
279D4—44no—
280D4—44no—
281D4—42no—
282D4—84no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4C2×#9911yescharacteristic
316C2×C4#41011yescharacteristic
317C2×C4#44412yes—
318C8—21no—
319C8—44no—
320C8—41no—
321C8—42no—
322C8—41no—
323C8—41no—
324C8—81no—
325C8—82no—
326C8—82no—
327C8AΣL(1,𝔽16)12yes—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—42no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22C4×#9911yescharacteristic
348C22#176411yescharacteristic
349C22#176511yescharacteristic
350C4—22no—
351C4—21no—
352C4—41no—
353C4—42no—
354C4—41no—
355C4—41no—
356C4—81no—
357C4—81no—
358C4—162no—
359C4C2×AΣL(1,𝔽16)11yescharacteristic
360C4#212011yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—81no—
365C2—82no—
366C2#824011yescharacteristic
367C2C4×AΣL(1,𝔽16)11yescharacteristic
368C2#1263411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radical#2388C22
LCSG ▷ C22×C4 ▷ C23 ▷ 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. C8×#100C2semidirect
5. #8160C2semidirect
6. #7285C2semidirect
7. #7282C2semidirect
8. C22×#131C4semidirect
14. #1851C4semidirect
16. #1780C4semidirect
21. #1752C4semidirect
25. #1354C4semidirect
27. #1345C4semidirect

Maximal subgroups

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

subgroupindex
#214432
#117332
#117252
C8×#1002
#81602
#72852
#72822