Finite Group Databasebeta

← #128485

Subgroups — #128485

Order 512.

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

normal subgroups only

This group has 570 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#21439C211yesmaximal, characteristic
2C4×AΣL(1,𝔽16)C211yesmaximal, characteristic
3#11471C211yesmaximal, characteristic
4#8164C211yesmaximal, characteristic, complemented
5C8×#99C211yesmaximal, characteristic, complemented
6#7204C211yesmaximal, characteristic, complemented
7#7203C211yesmaximal, characteristic, complemented
8C23×M16C411yescharacteristic, complemented
9C2×#435—21no—
10C2×C8×D4C2211yescharacteristic
11C4×#120C2211yescharacteristic
12C2×AΣL(1,𝔽16)C412yes—
13#1870—21no—
14#1869C2211yescharacteristic
15#1851—21no—
16#1850C411yescharacteristic, complemented
17#1779C411yescharacteristic, complemented
18C4×#100C2211yescharacteristic
19C4×#99C2211yescharacteristic
20#1753—21no—
21#1752C411yescharacteristic, complemented
22C8×C22⋊C4—21no—
23#1460C2211yescharacteristic
24#1459C2211yescharacteristic
25#1451—21no—
26#1390C412yes—
27#1326C412yescomplemented
28#1316C412yescomplemented
29#1283—21no—
30#1282—21no—
31C24×C4C2×C411yescharacteristic
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—41no—
35C22×M16—41no—
36C22×M16—44no—
37C23×C8C2×C411yescharacteristic
38C23×C8C2×C411yescharacteristic
39C23×C8D411yescharacteristic
40C2×#120C2×C411yescharacteristic
41C2×C4×D4—21no—
42C2×C4×D4C2311yescharacteristic
43#435—44no—
44C8×D4—41no—
45C8×D4—41no—
46#431—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D411yescharacteristic
49C2×#100C2×C412yes—
50C2×#99C2×C412yes—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8C2×C411yescharacteristic
56C2×C22⋊C8C2×C411yescharacteristic
57C2×C22⋊C8D411yescharacteristic
58C4×M16—21no—
59C2×#97—21no—
60C2×C4×C8—41no—
61C2×C4×C8D411yescharacteristic
62#386—21no—
63#386C2×C411yescharacteristic
64#385—21no—
65#385C2×C411yescharacteristic
66#379C2×C411yescharacteristic
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4C2×C411yescharacteristic
70AΣL(1,𝔽16)—24no—
71#343—21no—
72#342—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—41no—
77#324—21no—
78#323—21no—
79#323C2×C412yes—
80#323C2×C412yes—
81#322—41no—
82#322—41no—
83C25C4211yescharacteristic
84C22×D4—21no—
85C22×D4C22×C411yescharacteristic
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—42no—
89C23×C4—41no—
90C23×C4—42no—
91C23×C4—41no—
92C23×C4C4211yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C2×M16—24no—
96C2×M16—22no—
97C2×M16—42no—
98C2×M16—42no—
99C2×M16—42no—
100C2×M16—42no—
101C2×M16—44no—
102C2×M16—44no—
103C2×M16—44no—
104C2×M16—44no—
105C2×M16—41no—
106C2×M16—41no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—42no—
114C22×C8—42no—
115C22×C8—42no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8C22⋊C411yescharacteristic
119C22×C8C22⋊C411yescharacteristic
120C22×C8C22⋊C411yescharacteristic
121C22×C8C22⋊C411yescharacteristic
122#120—22no—
123#120—22no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—42no—
127C4×D4—42no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4#4111yescharacteristic
130C2×C22⋊C4—22no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4C4211yescharacteristic
136C2×C22⋊C4C4211yescharacteristic
137C2×C22⋊C4C22×C411yescharacteristic
138C2×C22⋊C4#4111yescharacteristic
139C2×C42—21no—
140C2×C42—41no—
141C2×C42C2×D411yescharacteristic
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—42no—
145#100—22no—
146#100—22no—
147#99—22no—
148#99M1614yescomplemented
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—41no—
156C22⋊C8—42no—
157C22⋊C8—41no—
158#97—41no—
159#97—42no—
160C4×C8—21no—
161C4×C8—21no—
162C4×C8—41no—
163C4×C8—41no—
164C4×C8—82no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—41no—
169C24—21no—
170C24—21no—
171C24—21no—
172C24—41no—
173C24—42no—
174C24—41no—
175C24—81no—
176C24C2×C4211yescharacteristic
177C24C4×D411yescharacteristic
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—81no—
183C2×D4C2×M1612yes—
184C2×D4#13112yes—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4#11711yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4C4×D411yescharacteristic
219M16—44no—
220M16—48no—
221M16—44no—
222M16—42no—
223M16—42no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—42no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—81no—
246C2×C8—81no—
247C2×C8—81no—
248C2×C8#9911yescharacteristic
249C2×C8#9911yescharacteristic
250C4⋊C4—41no—
251C4⋊C4—42no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—24no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—41no—
262C22⋊C4—81no—
263C42—21no—
264C42—21no—
265C42—41no—
266C42—41no—
267C42—81no—
268C42—81no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—22no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—42no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—81no—
287C23—81no—
288C23—82no—
289C23—82no—
290C23—84no—
291C23—81no—
292C23—81no—
293C23C4×C22⋊C411yescharacteristic
294C23C4×M1611yescharacteristic
295C23#40511yescharacteristic
296C23#43512yes—
297D4—42no—
298D4—42no—
299D4—84no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4C2×#9911yescharacteristic
337C2×C4#41011yescharacteristic
338C2×C4#43612yes—
339C8—21no—
340C8—42no—
341C8—41no—
342C8—41no—
343C8—41no—
344C8—81no—
345C8—82no—
346C8—82no—
347C8—82no—
348C8AΣL(1,𝔽16)12yescomplemented
349C22—21no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—42no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—81no—
373C22—81no—
374C22#175311yescharacteristic
375C22C4×#9911yescharacteristic
376C22#175711yescharacteristic
377C4—21no—
378C4—22no—
379C4—41no—
380C4—42no—
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C4—84no—
386C4C2×AΣL(1,𝔽16)11yescharacteristic
387C4#212011yescharacteristic
388C2—21no—
389C2—21no—
390C2—42no—
391C2—41no—
392C2—41no—
393C2—82no—
394C2#816111yescharacteristic
395C2C4×AΣL(1,𝔽16)11yescharacteristic
396C2#1262411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radicalC2×#120C2×C4
LCSG ▷ C24 ▷ 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. #8164C2semidirect
5. C8×#99C2semidirect
6. #7204C2semidirect
7. #7203C2semidirect
8. C23×M16C4semidirect
16. #1850C4semidirect
17. #1779C4semidirect
21. #1752C4semidirect
27. #1326C4semidirect
28. #1316C4semidirect
148. #99M16semidirect
348. C8AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
#214392
C4×AΣL(1,𝔽16)2
#114712
#81642
C8×#992
#72042
#72032