Finite Group Databasebeta

← #224771

Subgroups — #224771

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#559C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic, complemented
3#21339C211yesmaximal, characteristic, complemented
4#9676C212yesmaximal, complemented
5#9674C211yesmaximal, characteristic
6#9645C212yesmaximal, complemented
7#9638C211yesmaximal, characteristic
8#8565C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10#8174C212yesmaximal, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#559C411yescharacteristic
13#2374—22no—
14#2367—21no—
15#2367C412yes—
16#2366C411yescharacteristic
17#2357—21no—
18#2357C412yes—
19#2355C411yescharacteristic
20#2355C411yescharacteristic
21C4×#127C2211yescharacteristic
22C4×#124—21no—
23C4×#124C2212yes—
24C4×#123—22no—
25C42×Q8C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27C42×D4—21no—
28C42×D4C2212yes—
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1916—22no—
34#1842C2211yescharacteristic
35#1831C2212yes—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41#1744C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50#1279C2212yes—
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53#559—24no—
54#559—24no—
55C2×#127C2×C411yescharacteristic
56C2×#124—21no—
57C2×#124C2×C412yes—
58C2×#123—22no—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C412yes—
70C2×C4×D4C2×C412yes—
71C2×C4×D4C2312yes—
72C22×C42—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×#97—22no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2312yes—
86C2×C4×C8C2311yescharacteristic
87#391—21no—
88#391C2×C412yes—
89#390—21no—
90#390—21no—
91#390—21no—
92#390C2×C412yes—
93#390C2×C412yes—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#386—22no—
103#385—22no—
104#384C2×C411yescharacteristic
105#382—21no—
106#382C2×C412yes—
107#381—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C43—21no—
132C43C2311yescharacteristic
133C43C2312yes—
134#336—22no—
135#336—22no—
136#336—21no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—21no—
141#336—22no—
142Q8⋊C8—22no—
143Q8⋊C8—22no—
144Q8⋊C8—42no—
145D4⋊C8—24no—
146D4⋊C8—24no—
147D4⋊C8—42no—
148D4⋊C8—42no—
149C22×Q8C22⋊C411yescharacteristic
150C22×Q8C22×C411yescharacteristic
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22×C412yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163#127—22no—
164#127M1614yes—
165#124—28no—
166#124—24no—
167#124—24no—
168#123—216no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—41no—
180C4×Q8—42no—
181C4×Q8M1614yes—
182C4×D4—28no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—42no—
191C4×D4—44no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—81no—
195C4×D4—81no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—82no—
229C2×C42—82no—
230C2×C42C22×C412yes—
231C2×C42C22×C412yes—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2411yescharacteristic
243C4⋊C8—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—22no—
247C4⋊C8—21no—
248C4⋊C8—42no—
249C4⋊C8—44no—
250C4⋊C8—42no—
251C4⋊C8—81no—
252C4⋊C8#4112yes—
253#97—82no—
254C4×C8—22no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—22no—
258C4×C8—21no—
259C4×C8—42no—
260C4×C8—42no—
261C4×C8—42no—
262C4×C8#4112yes—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274#95C22⋊C412yes—
275C24—22no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8C2×M1612yes—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—81no—
295C2×D4—81no—
296C2×D4C2×M1614yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—28no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—41no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—81no—
361C4⋊C4C2×M1612yes—
362C4⋊C4#13112yes—
363C22⋊C4—28no—
364C22⋊C4—28no—
365C22⋊C4—28no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—44no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—21no—
377C42—22no—
378C42—21no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—21no—
383C42—22no—
384C42—21no—
385C42—24no—
386C42—22no—
387C42—41no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—42no—
396C42—81no—
397C42—82no—
398C42—82no—
399C42C2×M1612yes—
400C42C2×M1612yes—
401C42#13512yes—
402C42#13512yes—
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C42H(5,𝔽2)11yescharacteristic
406C422−1+411yescharacteristic
407C23—24no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23C22×C22⋊C411yescharacteristic
415Q8—22no—
416Q8—22no—
417Q8—41no—
418Q8—41no—
419Q8—44no—
420D4—24no—
421D4—24no—
422D4—42no—
423D4—42no—
424D4—48no—
425D4—84no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—88no—
463C2×C4—88no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4#40712yes—
469C2×C4#40712yes—
470C2×C4#41612yes—
471C2×C4#41612yes—
472C2×C4#42112yes—
473C2×C4C4×#4111yescharacteristic
474C2×C4#52011yescharacteristic
475C2×C4#54811yescharacteristic
476C2×C4#54811yescharacteristic
477C2×C4#55011yescharacteristic
478C2×C4#55011yescharacteristic
479C2×C4C22×M1611yescharacteristic
480C2×C4C22×M1611yescharacteristic
481C2×C4C2×#13111yescharacteristic
482C2×C4#56811yescharacteristic
483C2×C4C2×#13511yescharacteristic
484C2×C4C2×#13511yescharacteristic
485C2×C4#57611yescharacteristic
486C2×C4#57711yescharacteristic
487C8—42no—
488C8—42no—
489C8—42no—
490C8—82no—
491C8—81no—
492C8—82no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22#234111yescharacteristic
501C22C2×#40711yescharacteristic
502C22#288011yescharacteristic
503C22C2×#41611yescharacteristic
504C22#289711yescharacteristic
505C22C2×#42111yescharacteristic
506C22#290111yescharacteristic
507C4—21no—
508C4—21no—
509C4—22no—
510C4—22no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—82no—
520C4—81no—
521C4#148814yes—
522C4#296511yescharacteristic
523C4#297011yescharacteristic
524C4#298911yescharacteristic
525C4#299011yescharacteristic
526C4#336111yescharacteristic
527C4#336511yescharacteristic
528C4#339411yescharacteristic
529C4#339611yescharacteristic
530C2—84no—
531C2C2×#148811yescharacteristic
532C2#1745311yescharacteristic
533C2#1751011yescharacteristic
534C2#1752211yescharacteristic
535C2#2253211yescharacteristic
536C2#2254311yescharacteristic
537C2#2255111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22825C2semidirect
3. #21339C2semidirect
4. #9676C2semidirect
6. #9645C2semidirect
8. #8565C2semidirect
9. #8564C2semidirect
10. #8174C2semidirect
11. C4×Q8⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5592
#228252
#213392
#96762
#96742
#96452
#96382
#85652
#85642
#81742
C4×Q8⋊C82