Finite Group Databasebeta

← #208429

Subgroups — #208429

Order 512.

1136 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3C4×#382C211yesmaximal, characteristic, complemented
4#9908C212yesmaximal, complemented
5#9712C212yesmaximal
6#9156C214yesmaximal, complemented
7#8216C214yesmaximal, complemented
8C2×#529—21no—
9#2861—21no—
10#2860—21no—
11#2860C412yescomplemented
12#2859C411yescharacteristic, complemented
13#2858C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2856C2211yescharacteristic
16#2854—21no—
17#2854C2212yes—
18#2850—21no—
19C4×C4⋊Q8C411yescharacteristic, complemented
20C4×#127C411yescharacteristic, complemented
21C4×#124—21no—
22C4×#121—21no—
23C4×#121C2212yes—
24#2293—21no—
25#2293C412yescomplemented
26C2×#382C2212yes—
27C2×C4×C4⋊C4—21no—
28C2×C43C2211yescharacteristic
29#1839—21no—
30#1831—21no—
31#1817C2212yes—
32#1797C2212yes—
33#1767—21no—
34#1749C2212yes—
35#1745—21no—
36C2×#327C2218yes—
37C4⋊#95—21no—
38#1434—21no—
39#1434C2212yes—
40C4×#95—21no—
41C4×#95C2212yes—
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×C4⋊C4—22no—
62C22×C42—22no—
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×C4⋊C8—22no—
71C2×C22⋊C8C2314yes—
72C2×#97—22no—
73#391—21no—
74#391—21no—
75#390—21no—
76#390—21no—
77#390—21no—
78#390—21no—
79#390—21no—
80#390—21no—
81#390C2×C412yes—
82#390C2×C412yes—
83#390C2×C412yes—
84#390C2×C412yes—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#386—22no—
102#386—22no—
103#384C2×C411yescharacteristic
104#384C2×C411yescharacteristic
105#382—21no—
106#382—28no—
107#380—21no—
108#380C2312yes—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—44no—
111C4×C4⋊C4C2×C412yes—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95C2314yes—
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#327—216no—
132C22×Q8—21no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C22×C412yes—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—24no—
147C4⋊Q8—42no—
148#127—42no—
149#126—48no—
150#125—44no—
151#124—44no—
152#124—44no—
153#123—48no—
154C22⋊Q8—44no—
155#121—28no—
156#121—44no—
157#121—42no—
158#121—42no—
159C4×Q8—44no—
160C4×D4—44no—
161C4×D4—44no—
162C4×D4—44no—
163C4×D4—48no—
164#117—48no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
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×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—44no—
201C2×C42—44no—
202C2×C42—44no—
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×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C4⋊C8—44no—
232C22⋊C8—28no—
233#97—44no—
234#95—28no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—24no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95#4118yes—
246C24—22no—
247C24C22×D411yescharacteristic
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×D4—24no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—28no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4AGL(1,ℤ/8)12yes—
310C22×C4AGL(1,ℤ/8)14yes—
311C22×C4#13712yes—
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4114yes—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C4⋊C4—24no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—416no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—42no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—44no—
371C42—88no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23C22×C22⋊C411yescharacteristic
384C23#52512yes—
385C23C2×AGL(1,ℤ/8)11yescharacteristic
386C23C2×AGL(1,ℤ/8)12yes—
387C23C2×#13711yescharacteristic
388Q8—84no—
389D4—84no—
390D4—84no—
391D4—84no—
392D4—84no—
393D4—84no—
394C2×C4—24no—
395C2×C4—24no—
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—24no—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—88no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4#41012yes—
454C2×C4#41012yes—
455C2×C4#41712yes—
456C2×C4#41712yes—
457C2×C4#41812yes—
458C2×C4#41812yes—
459C2×C4#42112yes—
460C2×C4#42112yes—
461C2×C4C4×#4112yes—
462C2×C4#53212yes—
463C8—84no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22#229711yescharacteristic
480C22C2×#41011yescharacteristic
481C22C2×#41011yescharacteristic
482C22C2×#41811yescharacteristic
483C22C2×#41711yescharacteristic
484C22C2×#42111yescharacteristic
485C22C2×#42111yescharacteristic
486C22#310912yes—
487C22#311212yes—
488C4—22no—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—22no—
493C4—22no—
494C4—22no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—88no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4#151214yes—
506C2—22no—
507C2—22no—
508C2—162no—
509C2C2×#151211yescharacteristic
510C2#1756311yescharacteristic
511C2#1758311yescharacteristic
512C2#1758911yescharacteristic
513C2#2139511yescharacteristic
514C2#2141811yescharacteristic
515C2#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. #20778C2semidirect
3. C4×#382C2semidirect
4. #9908C2semidirect
6. #9156C2semidirect
7. #8216C2semidirect
11. #2860C4semidirect
12. #2859C4semidirect
13. #2858C4semidirect
19. C4×C4⋊Q8C4semidirect
20. C4×#127C4semidirect
25. #2293C4semidirect

Maximal subgroups

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

subgroupindex
#366312
#207782
C4×#3822
#99082
#97122
#91562
#82162