Finite Group Databasebeta

← #204182

Subgroups — #204182

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2323C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic, complemented
3#20536C211yesmaximal, characteristic, complemented
4C2×#1461C212yesmaximal, complemented
5C2×#1458C212yesmaximal
6#9654C214yesmaximal, complemented
7#9652C214yesmaximal
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C2×#431C2212yes—
12C2×#431C2212yes—
13#2323—24no—
14#2323—24no—
15C2×#391C411yescharacteristic
16C2×#389C411yescharacteristic
17C2×#389C2212yes—
18C2×#388C412yes—
19C2×#388C2212yes—
20C2×#387C412yes—
21C2×#380C411yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1939—24no—
24#1797C2214yes—
25#1770—24no—
26#1749C2214yes—
27#1461—28no—
28#1458—28no—
29C2×#324C2218yes—
30C2×#324C2214yes—
31C2×#323C2214yes—
32C23×Q8C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C2×C22⋊Q8—24no—
35C2×C22⋊Q8—24no—
36C2×C22⋊Q8—24no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—24no—
40C2×C4×Q8—24no—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—24no—
44C2×#117—24no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2311yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C412yes—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61#431—24no—
62#431—24no—
63#431—24no—
64#431D418yes—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—24no—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2314yes—
70C2×#97—24no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73#391—24no—
74#389—24no—
75#389—24no—
76#389—24no—
77#388—24no—
78#388—24no—
79#388—28no—
80#387—28no—
81#380—24no—
82C4×C4⋊C4D414yes—
83C4×C4⋊C4D414yes—
84C2×#95C2×C412yes—
85C2×#95C2×C412yes—
86C2×#95C2×C412yes—
87C2×#95C2×C412yes—
88C2×#95C2×C412yes—
89#336—24no—
90#336—24no—
91#324—28no—
92#324—216no—
93#323—28no—
94C25C22×C411yescharacteristic
95C22×Q8—22no—
96C22×Q8—24no—
97C22×Q8—22no—
98C22×Q8—24no—
99C22×Q8M1614yes—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—24no—
104C23×C4M1614yes—
105C23×C4C22×C412yes—
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C411yescharacteristic
108C23×C4C22×C412yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C412yes—
111C23×C4C22×C412yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C2411yescharacteristic
115C22×C8—24no—
116C22×C8—24no—
117C22⋊Q8—232no—
118C4×Q8—44no—
119C4×Q8—44no—
120C4×Q8—44no—
121C4×Q8—44no—
122#117—44no—
123#117—44no—
124#117—44no—
125#117—44no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—82no—
146C2×C4⋊C4C22⋊C414yes—
147C2×C4⋊C4C22⋊C414yes—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—82no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—28no—
163C2×C42—82no—
164C2×C42C2×D414yes—
165C2×C42C2×D414yes—
166C2×C42C2×D412yes—
167C2×C42C2×D414yes—
168C2×C42C2×D412yes—
169C2×C42#4114yes—
170C4⋊C8—28no—
171C4⋊C8—44no—
172C4⋊C8—84no—
173C22⋊C8—24no—
174C22⋊C8—24no—
175C22⋊C8—24no—
176C22⋊C8C2×D418yes—
177#97—84no—
178C4×C8—28no—
179C4×C8—44no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95C22⋊C418yes—
185C24—21no—
186C24—21no—
187C24—22no—
188C24—22no—
189C24—41no—
190C24—41no—
191C24—41no—
192C24—41no—
193C24C2×M1612yes—
194C24C23×C411yescharacteristic
195C2×Q8—28no—
196C2×Q8—216no—
197C2×Q8—44no—
198C2×Q8—44no—
199C2×Q8—48no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—216no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—48no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4C2×C22⋊C414yes—
237C22×C4C2×C22⋊C414yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C4×D414yes—
241C22×C4C4×D414yes—
242C22×C4C2×M1614yes—
243C22×C4C2×M1612yes—
244C22×C4C2×M1612yes—
245C22×C4C2×M1612yes—
246C22×C4#13114yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D412yes—
250C22×C4C2×#4112yes—
251C2×C8—28no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—44no—
257C4⋊C4—216no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—84no—
269C22⋊C4—216no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—44no—
274C22⋊C4—84no—
275C42—44no—
276C42—44no—
277C42—44no—
278C42—48no—
279C42#12114yes—
280C42#12114yes—
281C42#12414yes—
282C42#12714yes—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—24no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—42no—
299C23—41no—
300C23—82no—
301C23—82no—
302C23—82no—
303C23—82no—
304C23C22×C22⋊C411yescharacteristic
305C23C2×C4×D412yes—
306C23C22×M1611yescharacteristic
307C23C22×M1611yescharacteristic
308C23C2×#13112yes—
309Q8—416no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—28no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—416no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—84no—
348C2×C4—82no—
349C2×C4—84no—
350C2×C4—82no—
351C2×C4#39014yes—
352C2×C4#39014yes—
353C2×C4#40714yes—
354C2×C4#40814yes—
355C2×C4#43518yes—
356C2×C4#43518yes—
357C2×C4C2×#12112yes—
358C2×C4C2×#12411yescharacteristic
359C2×C4C2×#12711yescharacteristic
360C8—48no—
361C8—84no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—84no—
374C22—84no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22C2×#39011yescharacteristic
380C22C2×#40711yescharacteristic
381C22C2×#40811yescharacteristic
382C22#288311yescharacteristic
383C22#288411yescharacteristic
384C22#288911yescharacteristic
385C22#288911yescharacteristic
386C22C2×#43512yes—
387C22C2×#43512yes—
388C4—22no—
389C4—21no—
390C4—21no—
391C4—24no—
392C4—42no—
393C4—44no—
394C4—44no—
395C4—44no—
396C4—82no—
397C4—81no—
398C4—81no—
399C4#195518yes—
400C2—22no—
401C2—22no—
402C2—82no—
403C2#1737011yescharacteristic
404C2#1737111yescharacteristic
405C2#1738811yescharacteristic
406C2#1739111yescharacteristic
407C2C2×#195511yescharacteristic
408C2#2121411yescharacteristic
409C2#2122011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21951C2semidirect
3. #20536C2semidirect
4. C2×#1461C2semidirect
6. #9654C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23232
#219512
#205362
C2×#14612
C2×#14582
#96542
#96522