Finite Group Databasebeta

← #215990

Subgroups — #215990

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2667C211yesmaximal, characteristic
2#21906C211yesmaximal, characteristic
3#21902C211yesmaximal, characteristic
4#9883C212yesmaximal, complemented
5#9881C212yesmaximal, complemented
6#9880C212yesmaximal, complemented
7#9879C212yesmaximal, complemented
8#8704C212yesmaximal
9#8702C212yesmaximal
10#2667—24no—
11#2667—24no—
12C2×#399C411yescharacteristic
13C2×#399C411yescharacteristic
14C2×#398C411yescharacteristic
15C2×#398C411yescharacteristic
16C2×#398C411yescharacteristic
17C2×#398C411yescharacteristic
18C2×#396C411yescharacteristic
19C2×#396C411yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#387C2211yescharacteristic
23C2×#387C2211yescharacteristic
24C2×#381C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1924—24no—
28#1923—24no—
29#1797C2212yes—
30#1797C2212yes—
31#1779C2212yes—
32#1779C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1280C2214yes—
36#1280C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C24×C4C2311yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2314yes—
70C2×#97—22no—
71C2×C4×C8—22no—
72#399—24no—
73#399—24no—
74#398—24no—
75#398—24no—
76#398—24no—
77#398—24no—
78#396—24no—
79#396—24no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#381—22no—
89#381—22no—
90#380—22no—
91#380—22no—
92C4×C22⋊C4D414yes—
93C4×C22⋊C4D414yes—
94C2×#95C2×C411yescharacteristic
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2311yescharacteristic
103C2×#95C2311yescharacteristic
104C2×#95C2311yescharacteristic
105C2×#95C2311yescharacteristic
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110C25C22×C411yescharacteristic
111C23×C4—24no—
112C23×C4—81no—
113C23×C4—81no—
114C23×C4C22×C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2411yescharacteristic
130C22×C8—24no—
131C22×C8—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—81no—
151C2×C4⋊C4—81no—
152C2×C4⋊C4—81no—
153C2×C4⋊C4—81no—
154C2×C4⋊C4—81no—
155C2×C4⋊C4—81no—
156C2×C4⋊C4—81no—
157C2×C4⋊C4—81no—
158C2×C4⋊C4—81no—
159C2×C4⋊C4—81no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—81no—
167C2×C22⋊C4—81no—
168C2×C22⋊C4—81no—
169C2×C22⋊C4—81no—
170C2×C22⋊C4C22⋊C414yes—
171C2×C22⋊C4C22⋊C414yes—
172C2×C42—81no—
173C2×C42—81no—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C4⋊C8—82no—
179C4⋊C8—82no—
180C22⋊C8—28no—
181C22⋊C8—28no—
182C22⋊C8—82no—
183C22⋊C8—82no—
184C22⋊C8—82no—
185C22⋊C8—82no—
186#97—82no—
187C4×C8—82no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95C22⋊C414yes—
203#95C22⋊C414yes—
204C24—22no—
205C24—22no—
206C24—22no—
207C24—41no—
208C24—41no—
209C24—41no—
210C24—41no—
211C24C2×C22⋊C412yes—
212C24C23×C411yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—88no—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4#13114yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C42−1+414yes—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—44no—
282C2×C8—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C22⋊C4—28no—
298C22⋊C4—44no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42—42no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—82no—
322C23—82no—
323C23—82no—
324C23—82no—
325C23C22×C22⋊C411yescharacteristic
326C23#51912yes—
327C23C2×#13112yes—
328C23#56812yes—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4#40814yes—
388C2×C4#53612yes—
389C2×C4#53612yes—
390C8—84no—
391C8—84no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—88no—
406C22#231011yescharacteristic
407C22C2×#40811yescharacteristic
408C22#287911yescharacteristic
409C22#288311yescharacteristic
410C22#288311yescharacteristic
411C22#288811yescharacteristic
412C22#288911yescharacteristic
413C22#298514yes—
414C4—22no—
415C4—22no—
416C4—24no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4—81no—
429C2—24no—
430C2—82no—
431C2#1738711yescharacteristic
432C2#1738911yescharacteristic
433C2#1740611yescharacteristic
434C2#1740811yescharacteristic
435C2#2182111yescharacteristic
436C2#2182711yescharacteristic
437C2#2183211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×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
4. #9883C2semidirect
5. #9881C2semidirect
6. #9880C2semidirect
7. #9879C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26672
#219062
#219022
#98832
#98812
#98802
#98792
#87042
#87022