Finite Group Databasebeta

← #134783

Subgroups — #134783

Order 512.

542 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 82 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
2#12793C212yesmaximal
3#8614C211yesmaximal, characteristic, complemented
4#8609C211yesmaximal, characteristic, complemented
5#7236C212yesmaximal, complemented
6C23×M16C411yescharacteristic, complemented
7C2×#435—21no—
8C2×C8×D4C2211yescharacteristic
9C4×#120C2211yescharacteristic
10C2×AΣL(1,𝔽16)—22no—
11#1870—21no—
12#1869C2211yescharacteristic
13#1851—21no—
14#1850C411yescharacteristic, complemented
15#1808—21no—
16#1800C2212yes—
17#1799C2212yes—
18#1779C411yescharacteristic, complemented
19#1753—21no—
20#1752C411yescharacteristic, complemented
21#1566—21no—
22#1390—22no—
23#1292—21no—
24#1289—21no—
25C24×C4C2×C411yescharacteristic
26C22×M16—21no—
27C22×M16—21no—
28C22×M16—41no—
29C22×M16—41no—
30C22×M16—44no—
31C23×C8C2×C411yescharacteristic
32C23×C8C2×C411yescharacteristic
33C23×C8D411yescharacteristic
34C2×#120D411yescharacteristic
35C2×C4×D4—21no—
36C2×C4×D4C2311yescharacteristic
37#435—44no—
38C8×D4—41no—
39C8×D4—41no—
40C2×#107—41no—
41#423—22no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44C2×#100—22no—
45C2×#99—22no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8C2×C411yescharacteristic
50C2×C22⋊C8C2×C411yescharacteristic
51C2×C22⋊C8D411yescharacteristic
52C2×#97—21no—
53C2×C4×C8D411yescharacteristic
54#386—21no—
55#386D411yescharacteristic
56#385—21no—
57#385Q811yescharacteristic
58#380—22no—
59#379Q811yescharacteristic
60C4×C22⋊C4—21no—
61C4×C22⋊C4C2×C411yescharacteristic
62AΣL(1,𝔽16)—44no—
63#343—22no—
64#342—22no—
65#340—41no—
66#336—21no—
67#336—21no—
68#336—21no—
69#334—41no—
70#334—41no—
71C25C4⋊C411yescharacteristic
72C22×D4—21no—
73C22×D4C2×D411yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—42no—
77C23×C4—41no—
78C23×C4—42no—
79C23×C4—41no—
80C23×C4C4⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—24no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—44no—
89C2×M16—44no—
90C2×M16—44no—
91C2×M16—44no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—42no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—42no—
102C22×C8—42no—
103C22×C8—42no—
104C22×C8—41no—
105C22×C8C22⋊C411yescharacteristic
106C22×C8C22⋊C411yescharacteristic
107C22×C8C22⋊C411yescharacteristic
108C22×C8C22⋊C411yescharacteristic
109#120—22no—
110#120—22no—
111C4×D4—41no—
112C4×D4—41no—
113C4×D4—42no—
114C4×D4—42no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4#4111yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4C4⋊C411yescharacteristic
124C2×C22⋊C4C4⋊C411yescharacteristic
125C2×C22⋊C4C2×Q811yescharacteristic
126C2×C22⋊C4#4111yescharacteristic
127C2×C42—21no—
128C2×C42C2×D411yescharacteristic
129#107—82no—
130C4⋊C8—41no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134#100—42no—
135#100—42no—
136#99—42no—
137#99—42no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—42no—
141C22⋊C8—42no—
142C22⋊C8—42no—
143C22⋊C8—41no—
144#97—42no—
145C4×C8—21no—
146C4×C8—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95—42no—
151C24—21no—
152C24—22no—
153C24—41no—
154C24—42no—
155C24—41no—
156C24—81no—
157C24C2×C4⋊C411yescharacteristic
158C24#12111yescharacteristic
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—81no—
165C2×D4AGL(1,ℤ/8)12yes—
166C2×D4#13712yes—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—82no—
196C22×C4C2×C22⋊C411yescharacteristic
197C22×C4#11711yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4C22⋊Q811yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201M16—44no—
202M16—48no—
203M16—44no—
204M16—82no—
205M16—82no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—42no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—81no—
225C2×C8—82no—
226C2×C8#9911yescharacteristic
227C2×C8#9911yescharacteristic
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C22⋊C4—22no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—41no—
239C22⋊C4—81no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C42—21no—
243C42—21no—
244C42—41no—
245C42—41no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—82no—
264C23—84no—
265C23—84no—
266C23—82no—
267C23#38011yescharacteristic
268C23#42812yes—
269C23#46911yescharacteristic
270C23#47011yescharacteristic
271D4—42no—
272D4—42no—
273D4—82no—
274D4—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
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—81no—
308C2×C4—162no—
309C2×C4—162no—
310C2×C4—162no—
311C2×C4C2×#9911yescharacteristic
312C2×C4#41011yescharacteristic
313C2×C4#46812yes—
314C8—21no—
315C8—42no—
316C8—41no—
317C8—41no—
318C8—41no—
319C8—81no—
320C8—162no—
321C8AΣL(1,𝔽16)12yescomplemented
322C22—21no—
323C22—21no—
324C22—21no—
325C22—22no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—42no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—84no—
342C22—84no—
343C22—81no—
344C22—81no—
345C22#179911yescharacteristic
346C22#180911yescharacteristic
347C22#181211yescharacteristic
348C4—22no—
349C4—21no—
350C4—41no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—322no—
357C4C2×AΣL(1,𝔽16)11yescharacteristic
358C4#212011yescharacteristic
359C2—21no—
360C2—21no—
361C2—41no—
362C2—42no—
363C2—41no—
364C2—82no—
365C2#861011yescharacteristic
366C2#1279311yescharacteristic
367C2#1280611yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC2×#120D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ 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
3. #8614C2semidirect
4. #8609C2semidirect
5. #7236C2semidirect
6. C23×M16C4semidirect
14. #1850C4semidirect
18. #1779C4semidirect
20. #1752C4semidirect
321. 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
#127932
#86142
#86092
#72362