Finite Group Databasebeta

← #207754

Subgroups — #207754

Order 512.

861 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2650C211yesmaximal, characteristic
2#21912C211yesmaximal, characteristic, complemented
3#20536C211yesmaximal, characteristic, complemented
4#9882C212yesmaximal, complemented
5#9652C212yesmaximal
6#9082C212yesmaximal, complemented
7#9080C212yesmaximal
8#8155C212yesmaximal
9#8154C212yesmaximal, complemented
10C2×#431C2212yes—
11#2650—24no—
12#2650—24no—
13C2×#400C411yescharacteristic
14C2×#400C411yescharacteristic
15C2×#400C411yescharacteristic
16C2×#397C411yescharacteristic
17C2×#397C411yescharacteristic
18C2×#396C411yescharacteristic
19C2×#395C411yescharacteristic
20C2×#393C411yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#388C2211yescharacteristic
23C2×#388C2211yescharacteristic
24C2×#388C2211yescharacteristic
25C2×#387C2211yescharacteristic
26C2×#387C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1916—24no—
29#1817C2212yes—
30#1797C2212yes—
31#1770—24no—
32#1749C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×Q8C2×C411yescharacteristic
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⋊C4C2311yescharacteristic
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#431—28no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8C2314yes—
62C2×C22⋊C8C2314yes—
63C2×#97—22no—
64C2×#97—22no—
65C2×C4×C8—22no—
66C2×C4×C8—22no—
67#400—24no—
68#400—24no—
69#400—24no—
70#397—24no—
71#397—24no—
72#396—24no—
73#395—24no—
74#393—24no—
75#391—22no—
76#391—22no—
77#388—22no—
78#388—22no—
79#388—22no—
80#388—22no—
81#388—22no—
82#388—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87C4×C4⋊C4D414yes—
88C4×C4⋊C4D414yes—
89C2×#95C2×C411yescharacteristic
90C2×#95C2×C411yescharacteristic
91C2×#95C2×C411yescharacteristic
92C2×#95C2×C411yescharacteristic
93C2×#95C2×C411yescharacteristic
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×#95C2×C411yescharacteristic
103C2×#95C2311yescharacteristic
104C2×#95C2311yescharacteristic
105#336—22no—
106#336—24no—
107#336—22no—
108#336—22no—
109#336—22no—
110#324—28no—
111#323—28no—
112C25C22×C411yescharacteristic
113C22×Q8—22no—
114C22×Q8—42no—
115C22×Q8—81no—
116C22×Q8—81no—
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×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—24no—
132C22×C8—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—81no—
145C2×C4⋊C4—81no—
146C2×C4⋊C4—81no—
147C2×C4⋊C4—81no—
148C2×C4⋊C4C22⋊C414yes—
149C2×C4⋊C4C22⋊C414yes—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—81no—
162C2×C22⋊C4—81no—
163C2×C22⋊C4—81no—
164C2×C22⋊C4—81no—
165C2×C42—24no—
166C2×C42—81no—
167C2×C42—81no—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C4⋊C8—28no—
175C4⋊C8—82no—
176C22⋊C8—24no—
177C22⋊C8—24no—
178C22⋊C8—24no—
179C22⋊C8#4118yes—
180#97—82no—
181#97—82no—
182C4×C8—28no—
183C4×C8—82no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95C22⋊C414yes—
201#95C22⋊C414yes—
202C24—21no—
203C24—22no—
204C24—22no—
205C24—22no—
206C24—41no—
207C24—41no—
208C24—41no—
209C24—41no—
210C24C23×C411yescharacteristic
211C2×Q8—44no—
212C2×Q8—82no—
213C2×Q8—82no—
214C2×Q8—88no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
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×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4#13114yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4114yes—
272C2×C8—28no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—44no—
278C4⋊C4—28no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C22⋊C4—44no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—82no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C42—22no—
296C42—22no—
297C42—24no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42—42no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—44no—
314C23—82no—
315C23—82no—
316C23—82no—
317C23—82no—
318C23—82no—
319C23—82no—
320C23C22×C22⋊C411yescharacteristic
321C23C4×#4112yes—
322C23C2×#13112yes—
323C23#56812yes—
324Q8—88no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—28no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
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—41no—
350C2×C4—41no—
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—44no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4#40814yes—
382C2×C4#52512yes—
383C2×C4#53212yes—
384C8—48no—
385C8—84no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—164no—
400C22#229711yescharacteristic
401C22C2×#40811yescharacteristic
402C22#287911yescharacteristic
403C22#288311yescharacteristic
404C22#288411yescharacteristic
405C22#288811yescharacteristic
406C22#288911yescharacteristic
407C22#296814yes—
408C4—24no—
409C4—22no—
410C4—22no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—81no—
423C2—24no—
424C2—161no—
425C2#1738811yescharacteristic
426C2#1739011yescharacteristic
427C2#1740511yescharacteristic
428C2#1741011yescharacteristic
429C2#2137711yescharacteristic
430C2#2142311yescharacteristic
431C2#2142711yescharacteristic
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. #21912C2semidirect
3. #20536C2semidirect
4. #9882C2semidirect
6. #9082C2semidirect
9. #8154C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26502
#219122
#205362
#98822
#96522
#90822
#90802
#81552
#81542