Finite Group Databasebeta

← #206991

Subgroups — #206991

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2C4×#432C211yesmaximal, characteristic
3#14688C211yesmaximal, characteristic
4#9889C214yesmaximal
5#9884C214yesmaximal
6#8018C214yesmaximal
7C2×#533—21no—
8C2×#432C2212yes—
9#2868C411yescharacteristic, complemented
10#2868C411yescharacteristic, complemented
11#2862—21no—
12#2859—21no—
13#2859C412yescomplemented
14#2857—21no—
15#2857—21no—
16#2857C2212yes—
17#2857C2212yes—
18C4×C4⋊Q8—21no—
19C4×C4⋊Q8—21no—
20C4×C4⋊Q8C412yescomplemented
21C4×C4⋊Q8C412yescomplemented
22#2285—21no—
23#2285C2212yes—
24C2×#383C2212yes—
25C2×C43C2211yescharacteristic
26#1767—21no—
27C4×C4⋊C8—21no—
28#1749C2212yes—
29C4×C22⋊C8C2212yes—
30#1745—21no—
31C4×#97—21no—
32C4⋊#95—21no—
33C4⋊#95—21no—
34C4⋊#95C2212yes—
35#1434—21no—
36#1434—21no—
37#1434C2212yes—
38#1294C22116yes—
39#533—48no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×#125—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×#117—22no—
52C22×C4⋊C4—22no—
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60#432—24no—
61#432Q818yescomplemented
62C2×C4⋊C8—22no—
63C2×C22⋊C8C2314yes—
64C2×#97—22no—
65C2×#97—22no—
66C2×C4×C8—22no—
67#391—22no—
68#391—22no—
69#391—22no—
70#391—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389C2×C414yes—
78#389C2×C414yes—
79#384—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384C2×C412yes—
84#384C2×C412yes—
85#384C2×C412yes—
86#384C2×C412yes—
87#383—28no—
88#382—22no—
89#382—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380C2314yes—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2×C414yes—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95C2314yes—
104C43C2×C411yescharacteristic
105C43C2×C411yescharacteristic
106C43C2×C411yescharacteristic
107C43C2×C411yescharacteristic
108C43D411yescharacteristic
109C43D411yescharacteristic
110C43D411yescharacteristic
111C43D411yescharacteristic
112#336—22no—
113#336—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C23×C4—24no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—24no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131#125—48no—
132C22⋊Q8—48no—
133C22⋊Q8—48no—
134C4×Q8—48no—
135C4×Q8—48no—
136#117—48no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
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—22no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C414yes—
154C2×C22⋊C4—24no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q814yes—
191C2×C42#4114yes—
192C4⋊C8—28no—
193C22⋊C8—24no—
194C22⋊C8C2×Q818yes—
195#97—28no—
196#97—44no—
197C4×C8—44no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—28no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208C24C22×D411yescharacteristic
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8—44no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4#12412yes—
253C22×C4#12412yes—
254C22×C4C4⋊Q812yes—
255C22×C4C4⋊Q812yes—
256C22×C4AGL(1,ℤ/8)14yes—
257C22×C4#13714yes—
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×Q812yes—
261C22×C4C2×#4112yes—
262C2×C8—28no—
263C2×C8—44no—
264C2×C8—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—48no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—88no—
280C22⋊C4—48no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—41no—
298C42—41no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—48no—
306C42C4×Q814yes—
307C42C4×Q814yes—
308C42C22⋊Q814yes—
309C42C22⋊Q814yes—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23C22×C22⋊C411yescharacteristic
315C23C2×#12411yescharacteristic
316C23C2×C4⋊Q811yescharacteristic
317C23C2×AGL(1,ℤ/8)12yes—
318C23C2×#13712yes—
319Q8—88no—
320Q8—88no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—24no—
345C2×C4—28no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4#39112yes—
373C2×C4#39112yes—
374C2×C4#39112yes—
375C2×C4#39112yes—
376C2×C4#41012yes—
377C2×C4#41012yes—
378C2×C4#41812yes—
379C2×C4#41812yes—
380C2×C4#41912yes—
381C2×C4#41912yes—
382C2×C4#42112yes—
383C2×C4#42112yes—
384C2×C4C2×C4×Q812yes—
385C2×C4C2×C22⋊Q812yes—
386C8—48no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22C2×#39111yescharacteristic
396C22C2×#41011yescharacteristic
397C22C2×#41011yescharacteristic
398C22C2×#41811yescharacteristic
399C22C2×#41911yescharacteristic
400C22C2×#42111yescharacteristic
401C22C2×#42111yescharacteristic
402C22#314112yes—
403C22#316512yes—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—22no—
415C4—41no—
416C4—41no—
417C4—84no—
418C4—84no—
419C4—164no—
420C4#151514yes—
421C2—22no—
422C2—22no—
423C2C2×#151511yescharacteristic
424C2#1756111yescharacteristic
425C2#1758411yescharacteristic
426C2#1759511yescharacteristic
427C2#2130611yescharacteristic
428C2#2132411yescharacteristic
429C2#2133211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2868C4semidirect
10. #2868C4semidirect
13. #2859C4semidirect
20. C4×C4⋊Q8C4semidirect
21. C4×C4⋊Q8C4semidirect
61. #432Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
C4×#4322
#146882
#98892
#98842
#80182