Finite Group Databasebeta

← #128974

Subgroups — #128974

Order 512.

411 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 90 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#22031C211yesmaximal, characteristic
2#11733C211yesmaximal, characteristic
3#11725C211yesmaximal, characteristic
4#8164C211yesmaximal, characteristic, complemented
5#8160C211yesmaximal, characteristic, complemented
6#7285C211yesmaximal, characteristic, complemented
7#7282C211yesmaximal, characteristic, complemented
8C2×C8×D4C2211yescharacteristic
9C2×#408C411yescharacteristic, complemented
10#2388C2211yescharacteristic
11#1955—21no—
12#1924—21no—
13#1923—21no—
14#1870C2211yescharacteristic
15#1869—21no—
16#1851C411yescharacteristic, complemented
17#1780C411yescharacteristic, complemented
18C4×#100C2211yescharacteristic
19C4×#99C2211yescharacteristic
20#1752—21no—
21#1752C411yescharacteristic, complemented
22#1460C2211yescharacteristic
23#1459C2211yescharacteristic
24#1451—21no—
25#1354C412yescomplemented
26#1351C412yes—
27#1345C412yescomplemented
28#1343C412yes—
29#1283—21no—
30#1282—21no—
31C22×#41C2×C411yescharacteristic
32C22×M16C2×C411yescharacteristic
33C23×C8C2×C411yescharacteristic
34C23×C8D411yescharacteristic
35C2×C4×D4—21no—
36C2×C4×D4C2311yescharacteristic
37C8×D4—41no—
38C8×D4—41no—
39#431—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8D411yescharacteristic
42C2×#100—21no—
43C2×#99—21no—
44#408—41no—
45#408—41no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8C2×C411yescharacteristic
53C2×C22⋊C8C2×C411yescharacteristic
54C2×C22⋊C8D411yescharacteristic
55C4×M16—21no—
56C2×#97—21no—
57C2×#97—41no—
58C2×C4×C8D411yescharacteristic
59#395C2×C411yescharacteristic
60#394—21no—
61#394C2×C411yescharacteristic
62#393C2×C411yescharacteristic
63#392—21no—
64#392C2×C411yescharacteristic
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4C2×C411yescharacteristic
68#343C2×C412yes—
69#342C2×C412yes—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—41no—
74#324C2×C412yes—
75#323—21no—
76#323—21no—
77#323C2×C412yes—
78#322—41no—
79#322—41no—
80C2×#41—41no—
81C2×#41—41no—
82C2×#41—81no—
83C2×#41—81no—
84C22×Q8C4211yescharacteristic
85C22×D4—21no—
86C22×D4—21no—
87C22×D4C4211yescharacteristic
88C22×D4#4111yescharacteristic
89C23×C4—21no—
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C2×M16—22no—
93C2×M16—41no—
94C2×M16—81no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—41no—
102C22×C8—81no—
103C22×C8—81no—
104C22×C8C22⋊C411yescharacteristic
105C22×C8C22⋊C411yescharacteristic
106C22×C8C22⋊C411yescharacteristic
107C22×C8C22⋊C411yescharacteristic
108C4×D4—41no—
109C4×D4—41no—
110C4×D4—81no—
111C4×D4—81no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4C22×C411yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4C22×C411yescharacteristic
121C2×C22⋊C4#4111yescharacteristic
122C2×C42—21no—
123C2×C42—41no—
124C2×C42C2×D411yescharacteristic
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—81no—
128#100—42no—
129#99—42no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—42no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—81no—
140C22⋊C8—81no—
141C22⋊C8—81no—
142#97—41no—
143#97—81no—
144#97—82no—
145C4×C8—21no—
146C4×C8—21no—
147C4×C8—41no—
148C4×C8—41no—
149#95—21no—
150#95—21no—
151#95—41no—
152#95C4211yescharacteristic
153#95C4211yescharacteristic
154C24—21no—
155C24#11711yescharacteristic
156C24C4×D411yescharacteristic
157#41—84no—
158#41—82no—
159#41—82no—
160C2×Q8—41no—
161C2×Q8—81no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4C2×C4211yescharacteristic
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4C4×D411yescharacteristic
195C22×C4C4×D411yescharacteristic
196M16—42no—
197M16—42no—
198M16—82no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—81no—
217C2×C8—81no—
218C2×C8—81no—
219C2×C8—81no—
220C2×C8—81no—
221C2×C8#9911yescharacteristic
222C2×C8#9911yescharacteristic
223C4⋊C4—41no—
224C4⋊C4—81no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C42—21no—
235C42—21no—
236C42—41no—
237C42—41no—
238C42—81no—
239C42—81no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23C4×C22⋊C411yescharacteristic
257C23#44412yes—
258Q8—82no—
259D4—44no—
260D4—82no—
261D4—84no—
262D4—84no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
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—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4C2×#9911yescharacteristic
296C2×C4#41011yescharacteristic
297C2×C4#44412yes—
298C8—21no—
299C8—21no—
300C8—41no—
301C8—41no—
302C8—41no—
303C8—81no—
304C8—82no—
305C8—82no—
306C8—82no—
307C8—81no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22C4×#9911yescharacteristic
328C22#176511yescharacteristic
329C22#176511yescharacteristic
330C4—22no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—162no—
339C4#212111yescharacteristic
340C4#212211yescharacteristic
341C2—21no—
342C2—21no—
343C2—42no—
344C2—81no—
345C2—161no—
346C2#823911yescharacteristic
347C2#1262211yescharacteristic
348C2#1263411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C4×#99
prime radical#2388C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ 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
4. #8164C2semidirect
5. #8160C2semidirect
6. #7285C2semidirect
7. #7282C2semidirect
9. C2×#408C4semidirect
16. #1851C4semidirect
17. #1780C4semidirect
21. #1752C4semidirect
25. #1354C4semidirect
27. #1345C4semidirect

Maximal subgroups

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

subgroupindex
#220312
#117332
#117252
#81642
#81602
#72852
#72822