Finite Group Databasebeta

← #207902

Subgroups — #207902

Order 512.

1072 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2#20463C211yesmaximal, characteristic, complemented
3#14690C211yesmaximal, characteristic, complemented
4#9889C212yesmaximal, complemented
5#9684C212yesmaximal
6#9114C214yesmaximal, complemented
7#8216C214yesmaximal, complemented
8C2×#529—21no—
9C2×#431C2212yes—
10#2861—21no—
11#2860—21no—
12#2860C412yescomplemented
13#2859C411yescharacteristic, complemented
14#2858C411yescharacteristic, complemented
15#2857C2211yescharacteristic
16#2856C2211yescharacteristic
17#2854—21no—
18#2854C2212yes—
19#2850—21no—
20C4×C4⋊Q8C411yescharacteristic, complemented
21C4×#127C411yescharacteristic, complemented
22C4×#124—21no—
23C4×#121—21no—
24C4×#121C2212yes—
25#2293—21no—
26#2293C412yescomplemented
27C2×C43C2211yescharacteristic
28#1767—21no—
29#1767—21no—
30#1749C2212yes—
31#1749C2212yes—
32#1744—21no—
33C4×#97—21no—
34C2×#327C2218yes—
35#1434—21no—
36#1434—21no—
37#1434—21no—
38#1434—21no—
39#1434C2212yes—
40#1434C2212yes—
41#1434C2212yes—
42#1294C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45#529—48no—
46C2×#126—22no—
47C2×#125—21no—
48C2×#124—21no—
49C2×#124—21no—
50C2×#123—22no—
51C2×C22⋊Q8—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121C2312yes—
55C2×C4×Q8—21no—
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yes—
60C2×#117—22no—
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#431—28no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2314yes—
71C2×#97—22no—
72C2×#97—22no—
73C2×C4×C8—22no—
74#391—21no—
75#391—21no—
76#390—21no—
77#390—21no—
78#390—21no—
79#390—21no—
80#390—21no—
81#390—21no—
82#390C2×C412yes—
83#390C2×C412yes—
84#390C2×C412yes—
85#390C2×C412yes—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#386—22no—
103#386—22no—
104#384C2×C411yescharacteristic
105#384C2×C411yescharacteristic
106#382—21no—
107#380—21no—
108#380C2312yes—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C412yes—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—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#327—216no—
131C22×Q8—21no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4C22×C412yes—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—24no—
146C4⋊Q8—42no—
147#127—42no—
148#126—48no—
149#125—44no—
150#124—44no—
151#124—44no—
152#123—48no—
153C22⋊Q8—44no—
154#121—28no—
155#121—44no—
156#121—44no—
157C4×Q8—44no—
158C4×D4—44no—
159C4×D4—44no—
160C4×D4—44no—
161C4×D4—48no—
162#117—48no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—22no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42—42no—
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×C42C22⋊C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C4⋊C8—28no—
223C22⋊C8—24no—
224C22⋊C8#4118yes—
225#97—44no—
226#97—44no—
227C4×C8—28no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—28no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241C24—22no—
242C24C22×D411yescharacteristic
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×D4—24no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—42no—
257C2×D4—42no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4AGL(1,ℤ/8)14yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4114yes—
306C2×C8—28no—
307C2×C8—44no—
308C2×C8—44no—
309C4⋊C4—24no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
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—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—44no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23C22×C22⋊C411yescharacteristic
373C23#53212yes—
374C23C2×AGL(1,ℤ/8)12yes—
375C23#57512yes—
376Q8—84no—
377D4—84no—
378D4—84no—
379D4—84no—
380D4—88no—
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—28no—
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—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
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—88no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4#41012yes—
440C2×C4#41012yes—
441C2×C4#41812yes—
442C2×C4#41812yes—
443C2×C4#42112yes—
444C2×C4#42112yes—
445C2×C4C4×#4112yes—
446C2×C4#52512yes—
447C8—48no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22#229711yescharacteristic
462C22C2×#41011yescharacteristic
463C22C2×#41011yescharacteristic
464C22C2×#41811yescharacteristic
465C22#289611yescharacteristic
466C22C2×#42111yescharacteristic
467C22C2×#42111yescharacteristic
468C22#318014yes—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—22no—
476C4—22no—
477C4—41no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4—164no—
487C2—24no—
488C2—162no—
489C2#1755811yescharacteristic
490C2#1756111yescharacteristic
491C2#1757911yescharacteristic
492C2#1758911yescharacteristic
493C2#2138311yescharacteristic
494C2#2140811yescharacteristic
495C2#2142111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
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
2. #20463C2semidirect
3. #14690C2semidirect
4. #9889C2semidirect
6. #9114C2semidirect
7. #8216C2semidirect
12. #2860C4semidirect
13. #2859C4semidirect
14. #2858C4semidirect
20. C4×C4⋊Q8C4semidirect
21. C4×#127C4semidirect
26. #2293C4semidirect

Maximal subgroups

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

subgroupindex
#366312
#204632
#146902
#98892
#96842
#91142
#82162