Finite Group Databasebeta

← #98942

Subgroups — #98942

Order 512.

505 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 110 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#21487C211yesmaximal, characteristic
2#10820C211yesmaximal, characteristic, complemented
3#8595C211yesmaximal, characteristic, complemented
4#8571C212yesmaximal
5#7039C212yesmaximal, complemented
6C2×#436C411yescharacteristic, complemented
7C2×#435—21no—
8C22×C4×C8C2211yescharacteristic
9C4×#121C2211yescharacteristic
10#1954C411yescharacteristic, complemented
11#1939C411yescharacteristic, complemented
12#1923—21no—
13#1869—21no—
14#1797C2212yes—
15#1797C2211yescharacteristic
16#1770C411yescharacteristic, complemented
17#1753—21no—
18C2×#340—21no—
19C2×#340—21no—
20C2×#334—21no—
21C2×#327—22no—
22C4⋊#95C2212yes—
23#1294—22no—
24#1277—21no—
25C22×M16—21no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#121D411yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2×C411yescharacteristic
31C22×C4⋊C4—22no—
32C22×C42C2311yescharacteristic
33#436—42no—
34#435—44no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—22no—
44C2×#97—21no—
45C2×C4×C8—42no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#390Q811yescharacteristic
51#389Q811yescharacteristic
52#388—21no—
53#386—21no—
54#380D411yescharacteristic
55C4×C4⋊C4C2×C411yescharacteristic
56C4×C22⋊C4—21no—
57C2×#95—22no—
58C2×#95—22no—
59#340—24no—
60#340—42no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—22no—
65#334—42no—
66#327—44no—
67C22×D4—21no—
68C22×D4C4⋊C411yescharacteristic
69C23×C4—21no—
70C23×C4—42no—
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4#4111yescharacteristic
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—42no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90#121—22no—
91#121—22no—
92C4×D4—41no—
93C4×D4—41no—
94C4×D4—42no—
95C4×D4—42no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4C4⋊C411yescharacteristic
104C2×C4⋊C4C4⋊C411yescharacteristic
105C2×C4⋊C4C4⋊C411yescharacteristic
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C42—21no—
112C2×C42—41no—
113C2×C42—41no—
114C2×C42C22×C411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×Q811yescharacteristic
117C2×C42#4111yescharacteristic
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#97—42no—
130C4×C8—22no—
131C4×C8—22no—
132C4×C8—41no—
133C4×C8—41no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—42no—
138#95—42no—
139#95—82no—
140C24—21no—
141C24#12111yescharacteristic
142C2×D4—22no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—42no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—42no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—44no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C2×C4⋊C411yescharacteristic
176C22×C4#11711yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179C22×C4C22⋊Q811yescharacteristic
180C22×C4#13512yes—
181C22×C4AGL(1,ℤ/8)12yes—
182C22×C4AGL(1,ℤ/8)12yes—
183C22×C4#13712yes—
184M16—84no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—44no—
196C2×C8—44no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—82no—
204C2×C8—82no—
205C2×C8—82no—
206C2×C8#9912yes—
207C2×C8#10412yes—
208C4⋊C4—22no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42—41no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23#38011yescharacteristic
241C23#46312yes—
242C23#46911yescharacteristic
243C23#47011yescharacteristic
244C23#47112yes—
245D4—82no—
246D4—82no—
247D4—82no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4—84no—
283C2×C4—162no—
284C2×C4—162no—
285C2×C4—162no—
286C2×C4—162no—
287C2×C4C2×#9911yescharacteristic
288C2×C4#41011yescharacteristic
289C2×C4#41611yescharacteristic
290C2×C4#41811yescharacteristic
291C2×C4C2×#10411yescharacteristic
292C2×C4#42111yescharacteristic
293C2×C4#42812yes—
294C2×C4#46812yes—
295C2×C4#47812yes—
296C8—22no—
297C8—22no—
298C8—41no—
299C8—41no—
300C8—81no—
301C8—81no—
302C8—162no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—22no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22#170112yes—
319C22#170312yes—
320C22#179911yescharacteristic
321C22#180411yescharacteristic
322C22#180611yescharacteristic
323C22#180911yescharacteristic
324C22#181211yescharacteristic
325C22#181411yescharacteristic
326C22#181511yescharacteristic
327C4—22no—
328C4—22no—
329C4—22no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—164no—
336C4#150011yescharacteristic
337C4#150711yescharacteristic
338C4#151411yescharacteristic
339C4#151611yescharacteristic
340C2—22no—
341C2—22no—
342C2—161no—
343C2#857311yescharacteristic
344C2#861011yescharacteristic
345C2#862611yescharacteristic
346C2#865611yescharacteristic
347C2#866811yescharacteristic
348C2#868011yescharacteristic
349C2#869611yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #10820C2semidirect
3. #8595C2semidirect
5. #7039C2semidirect
6. C2×#436C4semidirect
10. #1954C4semidirect
11. #1939C4semidirect
16. #1770C4semidirect

Maximal subgroups

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

subgroupindex
#214872
#108202
#85952
#85712
#70392