Finite Group Databasebeta

← #128802

Subgroups — #128802

Order 512.

438 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 100 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#21491C211yesmaximal, characteristic
2#11697C211yesmaximal, characteristic
3#11691C211yesmaximal, characteristic
4#10803C211yesmaximal, characteristic, complemented
5#8157C211yesmaximal, characteristic, complemented
6#7253C211yesmaximal, characteristic, complemented
7#7252C211yesmaximal, characteristic, complemented
8C2×C8×D4—21no—
9C2×C8×D4C411yescharacteristic, complemented
10C2×#405C2211yescharacteristic
11#2392C2211yescharacteristic
12#1955C411yescharacteristic, complemented
13#1940C411yescharacteristic, complemented
14#1924—21no—
15#1870—21no—
16#1798C2211yescharacteristic
17C8×C4⋊C4C411yescharacteristic, complemented
18C8×C22⋊C4—21no—
19C2×#344—21no—
20#1742C2211yescharacteristic
21#1740C2211yescharacteristic
22#1729—21no—
23#1458C2211yescharacteristic
24#1458C2211yescharacteristic
25#1452—21no—
26#1346C412yescomplemented
27#1345C412yescomplemented
28#1344C412yes—
29#1343C412yes—
30#1281—21no—
31C22×M16—21no—
32C22×M16D411yescharacteristic
33C23×C8—21no—
34C23×C8D411yescharacteristic
35C2×C4×D4—21no—
36C2×C4×D4C2×C411yescharacteristic
37C2×#117—21no—
38C2×#117C2311yescharacteristic
39C8×D4—42no—
40C8×D4—42no—
41C8×D4—42no—
42#431—21no—
43#431—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C4⋊C8C2×C411yescharacteristic
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51#405—42no—
52C2×#97C2×C411yescharacteristic
53C2×#97D411yescharacteristic
54C2×C4×C8—21no—
55C2×C4×C8C2×C411yescharacteristic
56C2×C4×C8D411yescharacteristic
57#398C2×C411yescharacteristic
58#397C2×C411yescharacteristic
59#396—21no—
60#394—21no—
61#394C2×C411yescharacteristic
62#392C2×C411yescharacteristic
63C4×C4⋊C4C2×C411yescharacteristic
64C4×C22⋊C4—21no—
65#344—42no—
66#343—21no—
67#343C2×C412yes—
68#342—21no—
69#342C2×C412yes—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—42no—
74#324—21no—
75#324C2×C412yes—
76#323—21no—
77#323C2×C412yes—
78#322—42no—
79C22×D4—21no—
80C22×D4C4211yescharacteristic
81C23×C4—21no—
82C23×C4C2×D411yescharacteristic
83C2×M16—21no—
84C2×M16—21no—
85C2×M16—22no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—22no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—41no—
99C22×C8—42no—
100C22×C8—42no—
101C22×C8—41no—
102C22×C8—42no—
103C22×C8C22⋊C411yescharacteristic
104C22×C8C22⋊C411yescharacteristic
105C22×C8C22⋊C411yescharacteristic
106C22×C8C22⋊C411yescharacteristic
107C4×D4—41no—
108C4×D4—41no—
109C4×D4—41no—
110C4×D4—41no—
111C4×D4—41no—
112C4×D4—41no—
113#117—41no—
114#117—41no—
115#117—41no—
116#117—41no—
117#117—41no—
118#117—41no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4C4211yescharacteristic
123C2×C4⋊C4C22×C411yescharacteristic
124C2×C4⋊C4#4111yescharacteristic
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4C22×C411yescharacteristic
132C2×C22⋊C4#4111yescharacteristic
133C2×C42—21no—
134C2×C42—41no—
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—42no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—42no—
145C22⋊C8—42no—
146C22⋊C8—42no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149#97—41no—
150#97—41no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8—42no—
156#95—21no—
157#95—21no—
158#95C4211yescharacteristic
159#95C4211yescharacteristic
160C24—21no—
161C24C4×D411yescharacteristic
162C2×D4—41no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—42no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4C2×C4211yescharacteristic
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4#11711yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4C4×D411yescharacteristic
187C22×C4C4×D411yescharacteristic
188C22×C4AGL(1,ℤ/8)11yescharacteristic
189C22×C4AGL(1,ℤ/8)11yescharacteristic
190C22×C4AGL(1,ℤ/8)11yescharacteristic
191C22×C4#13711yescharacteristic
192M16—42no—
193M16—42no—
194M16—42no—
195M16—42no—
196M16—81no—
197M16—81no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—84no—
221C2×C8—81no—
222C2×C8—81no—
223C2×C8#9912yes—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—41no—
248C42—41no—
249C42—81no—
250C42—81no—
251C23—21no—
252C23—21no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C4×C22⋊C411yescharacteristic
263C23#44011yescharacteristic
264C23#44211yescharacteristic
265D4—42no—
266D4—44no—
267D4—44no—
268D4—42no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
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—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4C2×#9911yescharacteristic
304C2×C4#41011yescharacteristic
305C2×C4#41711yescharacteristic
306C2×C4#41811yescharacteristic
307C2×C4#42111yescharacteristic
308C2×C4#42111yescharacteristic
309C2×C4#44011yescharacteristic
310C2×C4#44211yescharacteristic
311C8—22no—
312C8—41no—
313C8—41no—
314C8—41no—
315C8—82no—
316C8—82no—
317C8—82no—
318C8—81no—
319C8—81no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—42no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22C4×#9911yescharacteristic
337C22#175911yescharacteristic
338C22#176211yescharacteristic
339C4—21no—
340C4—21no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—162no—
349C4#151011yescharacteristic
350C4#151211yescharacteristic
351C4#151311yescharacteristic
352C4#151411yescharacteristic
353C2—21no—
354C2—21no—
355C2—41no—
356C2—41no—
357C2—82no—
358C2#821611yescharacteristic
359C2#1262311yescharacteristic
360C2#1263311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
socleC22C4×#99
prime radical#392C2×C4
LCSG ▷ C22×C4 ▷ 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
4. #10803C2semidirect
5. #8157C2semidirect
6. #7253C2semidirect
7. #7252C2semidirect
9. C2×C8×D4C4semidirect
12. #1955C4semidirect
13. #1940C4semidirect
17. C8×C4⋊C4C4semidirect
26. #1346C4semidirect
27. #1345C4semidirect

Maximal subgroups

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

subgroupindex
#214912
#116972
#116912
#108032
#81572
#72532
#72522