Finite Group Databasebeta

← #134887

Subgroups — #134887

Order 512.

498 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 74 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#21442C211yesmaximal, characteristic
2#11820C212yesmaximal
3#8622C211yesmaximal, characteristic, complemented
4#8609C211yesmaximal, characteristic, complemented
5#7320C212yesmaximal, complemented
6C22×#131C411yescharacteristic, complemented
7C2×C8×D4—21no—
8C2×C8×D4C2211yescharacteristic
9#2388C2211yescharacteristic
10#1870—21no—
11#1870C2211yescharacteristic
12#1851—21no—
13#1851C411yescharacteristic, complemented
14#1808—21no—
15#1799C2212yes—
16#1780C411yescharacteristic, complemented
17C8×C22⋊C4—21no—
18C8×C22⋊C4C411yescharacteristic, complemented
19#1568—21no—
20#1466C2212yes—
21#1351—22no—
22#1343—22no—
23#1290—21no—
24#1289—21no—
25C22×#41C2×C411yescharacteristic
26C2×#131—44no—
27C2×#131—41no—
28C2×#131—41no—
29C22×M16—21no—
30C22×M16C2×C411yescharacteristic
31C23×C8—21no—
32C23×C8C2×C411yescharacteristic
33C23×C8D411yescharacteristic
34C2×C4×D4—21no—
35C2×C4×D4C2311yescharacteristic
36C8×D4—41no—
37C8×D4—41no—
38C8×D4—44no—
39#432—22no—
40C2×#107—41no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8D411yescharacteristic
43C2×#99—22no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8C2×C411yescharacteristic
48C2×C22⋊C8C2×C411yescharacteristic
49C2×C22⋊C8D411yescharacteristic
50C2×C4×C8—21no—
51C2×C4×C8D411yescharacteristic
52#395Q811yescharacteristic
53#394—21no—
54#394Q811yescharacteristic
55#393D411yescharacteristic
56#392—21no—
57#392D411yescharacteristic
58#380—22no—
59C4×C22⋊C4—21no—
60C4×C22⋊C4C2×C411yescharacteristic
61#342—22no—
62#340—41no—
63#336—21no—
64#336—21no—
65#336—21no—
66#335—41no—
67#334—41no—
68#323—22no—
69#323—22no—
70C2×#41—42no—
71C2×#41—42no—
72C2×#41—41no—
73C2×#41—41no—
74C22×Q8C4⋊C411yescharacteristic
75C22×D4—21no—
76C22×D4—21no—
77C22×D4C4⋊C411yescharacteristic
78C22×D4#4111yescharacteristic
79C23×C4—21no—
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82#131—44no—
83#131—44no—
84#131—44no—
85#131—44no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—41no—
91C22×C8—24no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—42no—
99C22×C8—42no—
100C22×C8—42no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8C22⋊C411yescharacteristic
104C22×C8C22⋊C411yescharacteristic
105C22×C8C22⋊C411yescharacteristic
106C22×C8C22⋊C411yescharacteristic
107C4×D4—41no—
108C4×D4—41no—
109C4×D4—42no—
110C4×D4—42no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2×Q811yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4C2×D411yescharacteristic
121C2×C22⋊C4#4111yescharacteristic
122C2×C42—21no—
123C2×C42C2×D411yescharacteristic
124#107—82no—
125C4⋊C8—41no—
126C4⋊C8—42no—
127C4⋊C8—42no—
128#99—82no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134C22⋊C8—41no—
135C22⋊C8—42no—
136#97—42no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—42no—
140#95—21no—
141#95—21no—
142#95—42no—
143#95C4⋊C411yescharacteristic
144#95C4⋊C411yescharacteristic
145C24—21no—
146C24#11711yescharacteristic
147C24#12111yescharacteristic
148#41—42no—
149#41—44no—
150#41—44no—
151#41—42no—
152#41—42no—
153#41—42no—
154C2×Q8—42no—
155C2×Q8—41no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—42no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—82no—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4C2×C4⋊C411yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192M16—42no—
193M16—48no—
194M16—42no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—44no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—42no—
205C2×C8—44no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—81no—
214C2×C8—82no—
215C2×C8#9911yescharacteristic
216C2×C8#9911yescharacteristic
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C42—21no—
231C42—21no—
232C42—41no—
233C42—41no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23#38011yescharacteristic
251C23#47211yescharacteristic
252C23#47311yescharacteristic
253Q8—42no—
254Q8—42no—
255D4—42no—
256D4—42no—
257D4—42no—
258D4—44no—
259D4—44no—
260D4—44no—
261D4—44no—
262D4—42no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—162no—
294C2×C4—162no—
295C2×C4—162no—
296C2×C4C2×#9911yescharacteristic
297C2×C4#41011yescharacteristic
298C8—21no—
299C8—42no—
300C8—41no—
301C8—41no—
302C8—41no—
303C8—81no—
304C8—162no—
305C8AΣL(1,𝔽16)12yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—42no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22#179911yescharacteristic
328C22#181411yescharacteristic
329C22#181611yescharacteristic
330C4—21no—
331C4—42no—
332C4—41no—
333C4—41no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—322no—
339C4C2×AΣL(1,𝔽16)11yescharacteristic
340C4#212011yescharacteristic
341C2—21no—
342C2—21no—
343C2—41no—
344C2—41no—
345C2—81no—
346C2—82no—
347C2#862711yescharacteristic
348C2#1279311yescharacteristic
349C2#1280711yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radical#2388C22
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. #8622C2semidirect
4. #8609C2semidirect
5. #7320C2semidirect
6. C22×#131C4semidirect
13. #1851C4semidirect
16. #1780C4semidirect
18. C8×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#214422
#118202
#86222
#86092
#73202