Finite Group Databasebeta

← #206966

Subgroups — #206966

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2C4×#432C211yesmaximal, characteristic
3#13755C211yesmaximal, characteristic
4#9887C214yesmaximal
5#9886C214yesmaximal
6#8022C212yesmaximal
7#8022C212yesmaximal
8C2×#432C2212yes—
9#2864—21no—
10#2864—21no—
11#2859C411yescharacteristic, complemented
12#2859C411yescharacteristic, complemented
13#2859C411yescharacteristic, complemented
14#2859C411yescharacteristic, complemented
15#2851—21no—
16#2851—21no—
17#2851C2212yes—
18#2851C2212yes—
19C4×#125—21no—
20C4×#125—21no—
21C4×#125C412yescomplemented
22C4×#125C412yescomplemented
23#2296—21no—
24#2296—21no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×#376C2212yes—
29C2×C43C2211yescharacteristic
30#1767—21no—
31C4×C4⋊C8—21no—
32#1749C2212yes—
33C4×C22⋊C8C2212yes—
34#1745—21no—
35C4×#97—21no—
36C4⋊#95—21no—
37C4⋊#95C2212yes—
38#1434—21no—
39#1434C2212yes—
40C4×#95—21no—
41#1294C2218yes—
42#1294C2218yes—
43C2×#125—21no—
44C2×#125—21no—
45C2×#125C2×C412yes—
46C2×#125C2×C412yes—
47C2×#117—21no—
48C2×#117—21no—
49C22×C4⋊C4—22no—
50C22×C42—22no—
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#432—24no—
59#432Q818yes—
60C2×C4⋊C8—22no—
61C2×C22⋊C8C2314yes—
62C2×#97—22no—
63C2×#97—22no—
64C2×C4×C8—22no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#385—22no—
90#385—22no—
91#385—22no—
92#385—22no—
93#384C2×C411yescharacteristic
94#384C2×C411yescharacteristic
95#384C2×C411yescharacteristic
96#384C2×C411yescharacteristic
97#382—21no—
98#382—21no—
99#382—21no—
100#382—21no—
101#382C2×C412yes—
102#382C2×C412yes—
103#380—21no—
104#380—21no—
105#380C2312yes—
106#380C2312yes—
107C4×C4⋊C4—28no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116#376—28no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95C2314yes—
122C43C2×C411yescharacteristic
123C43C2×C411yescharacteristic
124C43C2×C411yescharacteristic
125C43C2×C411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128C43D411yescharacteristic
129C43D411yescharacteristic
130#336—22no—
131#336—22no—
132C23×C4—24no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—24no—
141#125—44no—
142#125—44no—
143#125—44no—
144#125—44no—
145#117—44no—
146#117—44no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C42—24no—
173C2×C42—28no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—44no—
182C2×C42—44no—
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×Q814yes—
212C2×C42#4114yes—
213C4⋊C8—28no—
214C22⋊C8—24no—
215C22⋊C8C2×Q818yes—
216#97—28no—
217#97—44no—
218C4×C8—44no—
219#95—28no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—28no—
233#95—44no—
234#95—44no—
235#95—44no—
236C24C22×D411yescharacteristic
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—28no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4#12412yes—
279C22×C4#12412yes—
280C22×C4C4⋊Q812yes—
281C22×C4C4⋊Q812yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×Q812yes—
285C22×C4C2×#4112yes—
286C2×C8—28no—
287C2×C8—44no—
288C2×C8—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—416no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—416no—
336C42—44no—
337C42—44no—
338C42#10414yes—
339C42C4×Q814yes—
340C42C4×Q814yes—
341C42C22⋊Q814yes—
342C42C22⋊Q814yes—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23C22×C22⋊C411yescharacteristic
348C23C2×#12411yescharacteristic
349C23C2×C4⋊Q811yescharacteristic
350C23#57512yes—
351C23#57512yes—
352C2×C4—28no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4#39112yes—
412C2×C4#39112yes—
413C2×C4#39112yes—
414C2×C4#39112yes—
415C2×C4#41012yes—
416C2×C4#41012yes—
417C2×C4C2×#10412yes—
418C2×C4C2×#10412yes—
419C2×C4C2×#10412yes—
420C2×C4#42112yes—
421C2×C4C2×C4×Q812yes—
422C2×C4C2×C22⋊Q812yes—
423C8—48no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22C2×#39111yescharacteristic
433C22C2×#41011yescharacteristic
434C22C2×#41011yescharacteristic
435C22#289611yescharacteristic
436C22#289611yescharacteristic
437C22C22×#10411yescharacteristic
438C22C2×#42111yescharacteristic
439C22#314212yes—
440C22#316612yes—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—88no—
455C4—82no—
456C4—82no—
457C4—82no—
458C4—82no—
459C4#198614yes—
460C2—22no—
461C2—22no—
462C2#1754111yescharacteristic
463C2#1754311yescharacteristic
464C2#1759111yescharacteristic
465C2#1759411yescharacteristic
466C2C2×#198611yescharacteristic
467C2#2132511yescharacteristic
468C2#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
11. #2859C4semidirect
12. #2859C4semidirect
13. #2859C4semidirect
14. #2859C4semidirect
21. C4×#125C4semidirect
22. C4×#125C4semidirect

Maximal subgroups

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

subgroupindex
#366402
C4×#4322
#137552
#98872
#98862
#80222