Finite Group Databasebeta

← #134823

Subgroups — #134823

Order 512.

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

normal subgroups only

This group has 507 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#21439C211yesmaximal, characteristic
2#11525C212yesmaximal
3#8620C211yesmaximal, characteristic, complemented
4#8612C211yesmaximal, characteristic, complemented
5#7247C212yesmaximal, complemented
6C23×M16C411yescharacteristic
7C2×#435—21no—
8C2×C8×D4C2211yescharacteristic
9C4×#120C2211yescharacteristic
10#1870—21no—
11#1869C2211yescharacteristic
12#1851—21no—
13#1850C411yescharacteristic
14#1810—21no—
15#1779C411yescharacteristic
16#1753—21no—
17#1752C411yescharacteristic
18#1568—21no—
19#1466C2212yes—
20#1465C2212yes—
21#1326—22no—
22#1316—22no—
23#1293—21no—
24#1290—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×#120Q811yescharacteristic
35C2×C4×D4—21no—
36C2×C4×D4C2311yescharacteristic
37#435—44no—
38C8×D4—41no—
39C8×D4—41no—
40#432—22no—
41C2×#108—41no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44#407—22no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8C2×C411yescharacteristic
49C2×C22⋊C8C2×C411yescharacteristic
50C2×C22⋊C8D411yescharacteristic
51C2×#97—21no—
52C2×C4×C8D411yescharacteristic
53#386—21no—
54#386Q811yescharacteristic
55#385—21no—
56#385D411yescharacteristic
57#379D411yescharacteristic
58C4×C22⋊C4—21no—
59C4×C22⋊C4C2×C411yescharacteristic
60#341—41no—
61#336—21no—
62#336—21no—
63#336—21no—
64#335—41no—
65#334—41no—
66#324—22no—
67#323—22no—
68#323—22no—
69#323—22no—
70C25C4⋊C411yescharacteristic
71C22×D4—21no—
72C22×D4C2×Q811yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—42no—
76C23×C4—41no—
77C23×C4—42no—
78C23×C4—41no—
79C23×C4C4⋊C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C2×M16—24no—
83C2×M16—42no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16—44no—
88C2×M16—44no—
89C2×M16—44no—
90C2×M16—44no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—42no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8—42no—
103C22×C8—41no—
104C22×C8C22⋊C411yescharacteristic
105C22×C8C22⋊C411yescharacteristic
106C22×C8C22⋊C411yescharacteristic
107C22×C8C22⋊C411yescharacteristic
108#120—42no—
109C4×D4—41no—
110C4×D4—41no—
111C4×D4—42no—
112C4×D4—42no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4#4111yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4C4⋊C411yescharacteristic
120C2×C22⋊C4C4⋊C411yescharacteristic
121C2×C22⋊C4C2×D411yescharacteristic
122C2×C22⋊C4#4111yescharacteristic
123C2×C42—21no—
124C2×C42C2×D411yescharacteristic
125#108—82no—
126C4⋊C8—41no—
127C4⋊C8—42no—
128C4⋊C8—42no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134C22⋊C8—41no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#97—42no—
139#97—42no—
140C4×C8—21no—
141C4×C8—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—21no—
146C24—22no—
147C24—41no—
148C24—42no—
149C24—41no—
150C24—81no—
151C24C2×C4⋊C411yescharacteristic
152C24C22⋊Q811yescharacteristic
153C2×D4—22no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—81no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4#11711yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192M16—44no—
193M16—48no—
194M16—44no—
195M16—82no—
196M16—82no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—42no—
203C2×C8—41no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—81no—
216C2×C8—82no—
217C2×C8—82no—
218C2×C8#9911yescharacteristic
219C2×C8#9911yescharacteristic
220C4⋊C4—41no—
221C4⋊C4—42no—
222C22⋊C4—22no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—41no—
229C22⋊C4—81no—
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—42no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—82no—
253C23—84no—
254C23—84no—
255C23—82no—
256C23#38011yescharacteristic
257C23#43012yes—
258D4—44no—
259D4—84no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4C2×#9911yescharacteristic
295C2×C4#41011yescharacteristic
296C8—21no—
297C8—42no—
298C8—41no—
299C8—41no—
300C8—41no—
301C8—81no—
302C8—162no—
303C8—162no—
304C8AΣL(1,𝔽16)12yes—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—41no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—84no—
323C22—84no—
324C22—81no—
325C22—81no—
326C22#179911yescharacteristic
327C22#181111yescharacteristic
328C22#181311yescharacteristic
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—42no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4C2×AΣL(1,𝔽16)11yescharacteristic
338C4#212011yescharacteristic
339C2—21no—
340C2—21no—
341C2—42no—
342C2—81no—
343C2—82no—
344C2#861311yescharacteristic
345C2#1279311yescharacteristic
346C2#1280611yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC2×#120Q8
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. #8620C2semidirect
4. #8612C2semidirect
5. #7247C2semidirect

Maximal subgroups

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

subgroupindex
#214392
#115252
#86202
#86122
#72472