Finite Group Databasebeta

← #216721

Subgroups — #216721

Order 512.

1062 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3C4×#384C211yesmaximal, characteristic
4#9884C212yesmaximal
5#9679C212yesmaximal
6#9144C214yesmaximal
7#8578C214yesmaximal
8#2864—21no—
9#2863—21no—
10#2863C412yescomplemented
11#2862C411yescharacteristic, complemented
12#2859C411yescharacteristic, complemented
13#2857C2211yescharacteristic
14#2857C2211yescharacteristic
15#2855—21no—
16#2855C2212yes—
17#2851—21no—
18C4×C4⋊Q8C411yescharacteristic, complemented
19C4×C4⋊Q8C411yescharacteristic, complemented
20C4×#125—21no—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8C2212yes—
23#2295—21no—
24#2295C412yescomplemented
25C2×#384—21no—
26#2283—21no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C43C2211yescharacteristic
31#1839—21no—
32#1831—21no—
33#1817C2212yes—
34#1797C2212yes—
35C4×C4⋊C8—21no—
36C4×C22⋊C8C2212yes—
37#1744—21no—
38C2×#328C2218yes—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42#1294C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2312yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C42—22no—
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×C4⋊C8—22no—
62C2×C22⋊C8C2314yes—
63C2×C4×C8—22no—
64#391—21no—
65#391—21no—
66#391—21no—
67#391—21no—
68#391C2×C412yes—
69#391C2×C412yes—
70#391C2×C412yes—
71#391C2×C412yes—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389C2×C412yes—
81#389C2×C412yes—
82#389C2×C412yes—
83#389C2×C412yes—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#385—22no—
93#385—22no—
94#384—28no—
95#384C2×C411yescharacteristic
96#384C2×C411yescharacteristic
97#383—22no—
98#382—21no—
99#382—21no—
100#382—21no—
101#381—22no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380C2312yes—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—44no—
112C4×C4⋊C4—44no—
113C4×C4⋊C4—44no—
114C4×C4⋊C4C2×C412yes—
115C4×C22⋊C4—22no—
116#376—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95C2314yes—
120C43C2×C411yescharacteristic
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123C43C2×C411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#328—48no—
133C22×Q8—21no—
134C22×Q8C22×C412yes—
135C23×C4—24no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—24no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146#125—44no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C4×Q8—44no—
152C4×Q8—44no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—24no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42—44no—
191C2×C42—44no—
192C2×C42—44no—
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×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42SD1614yes—
210C2×C42Q1614yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C4⋊C8—44no—
224C22⋊C8—24no—
225C22⋊C8—24no—
226C4×C8—44no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—28no—
242#95—44no—
243#95—44no—
244C24C22×D411yescharacteristic
245C2×Q8—24no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—28no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×SD1612yes—
295C22×C4C2×SD1612yes—
296C22×C4C2×SD1612yes—
297C22×C4C2×Q1612yes—
298C22×C4C2×Q1612yes—
299C22×C4C2×Q1612yes—
300C22×C4AGL(1,ℤ/8)12yes—
301C22×C4#13712yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4H(5,𝔽2)12yes—
305C22×C42−1+412yes—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C4⋊C4—24no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—416no—
313C4⋊C4—416no—
314C4⋊C4—416no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—44no—
356C42—44no—
357C42—88no—
358C42#10314yes—
359C42#10314yes—
360C42#10314yes—
361C42#10314yes—
362C42#10414yes—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23C22×C22⋊C411yescharacteristic
368C23#53511yescharacteristic
369C23#53611yescharacteristic
370C23C22×SD1611yescharacteristic
371C23C22×Q1611yescharacteristic
372C23C2×AGL(1,ℤ/8)11yescharacteristic
373C23C2×#13711yescharacteristic
374Q8—84no—
375Q8—84no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—48no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4#41012yes—
438C2×C4#41012yes—
439C2×C4C2×#10312yes—
440C2×C4C2×#10312yes—
441C2×C4C2×#10312yes—
442C2×C4C2×#10312yes—
443C2×C4C2×#10312yes—
444C2×C4C2×#10312yes—
445C2×C4#41712yes—
446C2×C4#41712yes—
447C2×C4C2×#10412yes—
448C2×C4C2×#10412yes—
449C2×C4C2×#10412yes—
450C2×C4#42112yes—
451C2×C4#52012yes—
452C2×C4#53712yes—
453C8—84no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22#231111yescharacteristic
467C22C2×#41011yescharacteristic
468C22C2×#41011yescharacteristic
469C22C22×#10311yescharacteristic
470C22C2×#41711yescharacteristic
471C22C22×#10411yescharacteristic
472C22C2×#42111yescharacteristic
473C22#320811yescharacteristic
474C22#320911yescharacteristic
475C22#322911yescharacteristic
476C22#323011yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—22no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—22no—
489C4—22no—
490C4—88no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4#150314yes—
496C2—21no—
497C2—21no—
498C2—21no—
499C2—21no—
500C2C2×#150311yescharacteristic
501C2#1754611yescharacteristic
502C2#1757511yescharacteristic
503C2#1758111yescharacteristic
504C2#2186011yescharacteristic
505C2#2187511yescharacteristic
506C2#2189911yescharacteristic
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
10. #2863C4semidirect
11. #2862C4semidirect
12. #2859C4semidirect
18. C4×C4⋊Q8C4semidirect
19. C4×C4⋊Q8C4semidirect
24. #2295C4semidirect

Maximal subgroups

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

subgroupindex
#366352
#207772
C4×#3842
#98842
#96792
#91442
#85782