Finite Group Databasebeta

← #134435

Subgroups — #134435

Order 512.

426 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 98 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#21493C211yesmaximal, characteristic
2#11697C212yesmaximal
3#10867C211yesmaximal, characteristic, complemented
4#8601C211yesmaximal, characteristic, complemented
5#7253C212yesmaximal, complemented
6C2×#436C411yescharacteristic
7C2×#435—21no—
8C22×#108—21no—
9C2×#405C2211yescharacteristic
10#2392C2211yescharacteristic
11#1955C411yescharacteristic
12#1940C411yescharacteristic
13#1924—21no—
14#1870—21no—
15#1798C2211yescharacteristic
16#1770C411yescharacteristic
17#1753—21no—
18#1742C2212yes—
19C2×#338—21no—
20#1564—21no—
21#1458C2212yes—
22#1346—22no—
23#1345—22no—
24#1275—21no—
25C22×M16—21no—
26C22×M16—22no—
27C22×M16D411yescharacteristic
28C23×C8D411yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2×C411yescharacteristic
31C2×#117C2311yescharacteristic
32#436—41no—
33#436—41no—
34#435—42no—
35#435—42no—
36#431—22no—
37C2×#108—24no—
38C2×#108—41no—
39C2×#108—41no—
40C2×#108—42no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48#405—41no—
49#405—41no—
50C2×#97—21no—
51C2×#97C2×C411yescharacteristic
52C2×#97D411yescharacteristic
53C2×C4×C8C2×C411yescharacteristic
54C2×C4×C8D411yescharacteristic
55#398D411yescharacteristic
56#397Q811yescharacteristic
57#396—21no—
58#394—21no—
59#394D411yescharacteristic
60#392Q811yescharacteristic
61C4×C4⋊C4C2×C411yescharacteristic
62C4×C22⋊C4—21no—
63#343—22no—
64#343—22no—
65#338—42no—
66#336—21no—
67#336—21no—
68#336—21no—
69#335—41no—
70#334—41no—
71#324—22no—
72#323—22no—
73C22×D4—21no—
74C22×D4C4⋊C411yescharacteristic
75C23×C4—21no—
76C23×C4C2×D411yescharacteristic
77C2×M16—21no—
78C2×M16—21no—
79C2×M16—24no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—42no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—42no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—42no—
97C22×C8C22⋊C411yescharacteristic
98C22×C8C22⋊C411yescharacteristic
99C22×C8C22⋊C411yescharacteristic
100C22×C8C22⋊C411yescharacteristic
101C22×C8D812yes—
102C22×C8SD1612yes—
103C4×D4—41no—
104C4×D4—41no—
105C4×D4—41no—
106C4×D4—41no—
107C4×D4—41no—
108C4×D4—41no—
109#117—41no—
110#117—41no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4C4⋊C411yescharacteristic
114C2×C4⋊C4C2×D411yescharacteristic
115C2×C4⋊C4#4111yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4C2×Q811yescharacteristic
122C2×C22⋊C4#4111yescharacteristic
123C2×C42—21no—
124C2×C42C22×C411yescharacteristic
125C2×C42C2×D411yescharacteristic
126#108—44no—
127#108—44no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#97—41no—
139#97—41no—
140#97—42no—
141C4×C8—21no—
142C4×C8—21no—
143C4×C8—21no—
144C4×C8—21no—
145C4×C8—42no—
146#95—21no—
147#95—21no—
148#95C4⋊C411yescharacteristic
149#95C4⋊C411yescharacteristic
150C24—21no—
151C24C22⋊Q811yescharacteristic
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—81no—
156C2×D4—81no—
157C2×D4—81no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C2×C4⋊C411yescharacteristic
173C22×C4#11711yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C2×D811yescharacteristic
178C22×C4C2×SD1611yescharacteristic
179C22×C4AGL(1,ℤ/8)11yescharacteristic
180C22×C4#13711yescharacteristic
181M16—42no—
182M16—42no—
183M16—44no—
184M16—44no—
185M16—82no—
186M16—82no—
187M16—82no—
188M16—82no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—44no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—82no—
209C2×C8—82no—
210C2×C8—82no—
211C2×C8#9911yescharacteristic
212C2×C8#9911yescharacteristic
213C2×C8#10212yes—
214C2×C8#10212yes—
215C2×C8#10412yes—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—81no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—42no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—41no—
236C42—41no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23#38011yescharacteristic
249C23#47411yescharacteristic
250C23#47611yescharacteristic
251D4—82no—
252D4—82no—
253D4—82no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
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—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4C2×#9911yescharacteristic
286C2×C4#41011yescharacteristic
287C2×C4C2×#10211yescharacteristic
288C2×C4#41711yescharacteristic
289C2×C4C2×#10411yescharacteristic
290C2×C4#42111yescharacteristic
291C2×C4#45911yescharacteristic
292C2×C4#46111yescharacteristic
293C8—22no—
294C8—41no—
295C8—41no—
296C8—84no—
297C8—81no—
298C8—81no—
299C8—162no—
300C8#32712yes—
301C8#32712yes—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22#179911yescharacteristic
319C22#180211yescharacteristic
320C22#180611yescharacteristic
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4C2×#32711yescharacteristic
331C4#150911yescharacteristic
332C4#151311yescharacteristic
333C4#151711yescharacteristic
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—161no—
339C2#857111yescharacteristic
340C2#1280611yescharacteristic
341C2#1281011yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
socleC22#1799
prime radical#392Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10867C2semidirect
4. #8601C2semidirect
5. #7253C2semidirect

Maximal subgroups

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

subgroupindex
#214932
#116972
#108672
#86012
#72532