Finite Group Databasebeta

← #204204

Subgroups — #204204

Order 512.

1185 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2663C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic, complemented
3#20526C211yesmaximal, characteristic, complemented
4C2×#1472C212yesmaximal, complemented
5C2×#1468C212yesmaximal
6#9654C214yesmaximal, complemented
7#9652C214yesmaximal, complemented
8C22×#123C2211yescharacteristic
9C22×C22⋊Q8C2211yescharacteristic
10C2×#407C2212yes—
11C22×C22⋊C8C2212yes—
12#2663—24no—
13#2663—24no—
14C2×#400C411yescharacteristic
15C2×#399C411yescharacteristic
16C2×#397C412yes—
17C2×#395C411yescharacteristic
18C2×#394C412yes—
19C2×#393C411yescharacteristic
20C2×#389C2212yes—
21C2×#388C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1851—24no—
24#1797C2214yes—
25#1752—24no—
26#1749C2214yes—
27#1472—28no—
28#1468—28no—
29C2×#324C2218yes—
30C2×#324C2214yes—
31C2×#323C2214yes—
32C23×Q8C2×C411yescharacteristic
33C23×D4C2×C411yescharacteristic
34C24×C4C2311yescharacteristic
35C22×M16—22no—
36C23×C8—22no—
37C2×#123—22no—
38C2×#123—22no—
39C2×#123—24no—
40C2×#123—24no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—24no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2311yescharacteristic
58C22×C42C2312yes—
59C2×C4⋊C8—24no—
60#407—24no—
61#407—24no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8D418yes—
66C2×C22⋊C8C2314yes—
67C2×C22⋊C8C2314yes—
68C2×#97—24no—
69#400—24no—
70#399—24no—
71#397—28no—
72#395—24no—
73#394—28no—
74#393—24no—
75#389—24no—
76#389—24no—
77#388—24no—
78#388—24no—
79C4×C22⋊C4D414yes—
80C4×C22⋊C4D414yes—
81C2×#95C2×C412yes—
82C2×#95C2×C412yes—
83C2×#95C2×C412yes—
84C2×#95C2×C412yes—
85C2×#95C2×C411yescharacteristic
86C2×#95C2×C411yescharacteristic
87#336—24no—
88#336—24no—
89#324—28no—
90#324—216no—
91#323—28no—
92C25C22×C411yescharacteristic
93C25C22×C411yescharacteristic
94C22×Q8—22no—
95C22×Q8—24no—
96C22×Q8—82no—
97C22×D4—22no—
98C22×D4—24no—
99C22×D4—82no—
100C23×C4—82no—
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C412yes—
103C23×C4C22×C412yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C412yes—
107C23×C4C22×C412yes—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D414yes—
112C23×C4#4114yes—
113C23×C4C2411yescharacteristic
114C2×M16—44no—
115C2×M16—84no—
116C22×C8—24no—
117C22×C8—28no—
118C22×C8—24no—
119C22×C8—84no—
120#123—44no—
121#123—44no—
122#123—44no—
123#123—44no—
124C22⋊Q8—44no—
125C22⋊Q8—44no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—82no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—44no—
157C2×C22⋊C4—44no—
158C2×C22⋊C4—82no—
159C2×C22⋊C4C22⋊C418yes—
160C2×C42—82no—
161C2×C42C2×D414yes—
162C2×C42C2×D414yes—
163C4⋊C8—84no—
164C22⋊C8—28no—
165C22⋊C8—24no—
166C22⋊C8—24no—
167C22⋊C8—24no—
168C22⋊C8—44no—
169C22⋊C8C2×D418yes—
170#97—84no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95C22⋊C414yes—
176#95C22⋊C414yes—
177C24—24no—
178C24—21no—
179C24—22no—
180C24—24no—
181C24—42no—
182C24—41no—
183C24—41no—
184C24—42no—
185C24—41no—
186C24—41no—
187C24C2×C22⋊C412yes—
188C24C4×D414yes—
189C24C23×C411yescharacteristic
190C2×Q8—44no—
191C2×Q8—84no—
192C2×Q8—88no—
193C2×D4—44no—
194C2×D4—84no—
195C2×D4—88no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4C2×C22⋊C414yes—
227C22×C4C2×C22⋊C414yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C4×D414yes—
230C22×C4#12114yes—
231C22×C4#12114yes—
232C22×C4#12414yes—
233C22×C4#12714yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D412yes—
237C22×C4C2×#4112yes—
238M16—88no—
239C2×C8—28no—
240C2×C8—48no—
241C2×C8—48no—
242C2×C8—44no—
243C2×C8—88no—
244C4⋊C4—48no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—84no—
254C22⋊C4—48no—
255C22⋊C4—48no—
256C22⋊C4—48no—
257C22⋊C4—44no—
258C22⋊C4—42no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—44no—
262C22⋊C4—84no—
263C42—44no—
264C42—44no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—24no—
269C23—24no—
270C23—22no—
271C23—22no—
272C23—28no—
273C23—44no—
274C23—44no—
275C23—42no—
276C23—42no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—44no—
281C23—84no—
282C23—84no—
283C23—82no—
284C23—82no—
285C23—84no—
286C23—84no—
287C23#39014yes—
288C23C22×C22⋊C411yescharacteristic
289C23C2×C4×D412yes—
290C23#56814yes—
291Q8—88no—
292D4—88no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—48no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—84no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—84no—
329C2×C4#39014yes—
330C2×C4C2×#12112yes—
331C2×C4C2×#12411yescharacteristic
332C2×C4C2×#12711yescharacteristic
333C8—48no—
334C8—84no—
335C22—22no—
336C22—22no—
337C22—24no—
338C22—24no—
339C22—24no—
340C22—24no—
341C22—24no—
342C22—44no—
343C22—44no—
344C22—88no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—82no—
349C22—82no—
350C22—164no—
351C22C2×#39011yescharacteristic
352C22#287912yes—
353C22#288312yes—
354C22#288911yescharacteristic
355C22#288911yescharacteristic
356C22#293214yes—
357C4—22no—
358C4—21no—
359C4—21no—
360C4—24no—
361C4—44no—
362C4—44no—
363C4—82no—
364C4—81no—
365C4—81no—
366C4—82no—
367C4—81no—
368C2—24no—
369C2—44no—
370C2—161no—
371C2#1740612yes—
372C2#1740712yes—
373C2#2118811yescharacteristic
374C2#2121311yescharacteristic
375C2#2122011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21934C2semidirect
3. #20526C2semidirect
4. C2×#1472C2semidirect
6. #9654C2semidirect
7. #9652C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26632
#219342
#205262
C2×#14722
C2×#14682
#96542
#96522