Finite Group Databasebeta

← #219829

Subgroups — #219829

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34929C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic
3C4×#436C211yesmaximal, characteristic
4C2×#1485C212yesmaximal, complemented
5#9673C211yesmaximal, characteristic, complemented
6#9637C211yesmaximal, characteristic, complemented
7#9077C212yesmaximal, complemented
8#8938C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10#8314C212yesmaximal, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#533C2212yes—
13C2×#436—21no—
14C2×#431C2212yes—
15C2×C4×M16C2212yes—
16#2556—22no—
17#2551C411yescharacteristic
18#2546C411yescharacteristic
19#2542C411yescharacteristic
20#2538C411yescharacteristic
21#2506—21no—
22#2492C412yes—
23#2452—21no—
24#2429C412yes—
25#2396—22no—
26C4×C4⋊Q8—21no—
27C4×C4⋊Q8C2211yescharacteristic
28C4×C22⋊Q8C2212yes—
29#2293—21no—
30C42×Q8C2211yescharacteristic
31C42×D4C2211yescharacteristic
32#1954—21no—
33#1923—22no—
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1770—21no—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1753—22no—
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43#1485—28no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2211yescharacteristic
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50C2×Q8⋊C8C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53#1279C2211yescharacteristic
54C22×M16—22no—
55#533—24no—
56#533—24no—
57#533—42no—
58#533—42no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C412yes—
63C2×#125—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8C2×C412yes—
66C2×C22⋊Q8C2×C412yes—
67C2×C22⋊Q8C2×C412yes—
68C2×C4×Q8—22no—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2312yes—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×#117—22no—
77C22×C42C2312yes—
78#436—28no—
79#431—24no—
80#431—42no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C4×M16—24no—
91C4×M16—42no—
92C2×#97—22no—
93C2×C4×C8—21no—
94C2×C4×C8—22no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2312yes—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#400—22no—
101#399—22no—
102#398—22no—
103#398—22no—
104#396—22no—
105#396—22no—
106#395—22no—
107#393—22no—
108#391—22no—
109#391C2×C412yes—
110#391C2×C412yes—
111#390—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—22no—
120#389C2×C412yes—
121#388—22no—
122#388—22no—
123#387—22no—
124#385—22no—
125#384—21no—
126#384C2×C411yescharacteristic
127#380C2×C412yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—22no—
141C4×C22⋊C4C2×C412yes—
142C4×C22⋊C4D412yes—
143C4×C22⋊C4D412yes—
144C43—21no—
145C43C2311yescharacteristic
146#336—22no—
147#336—21no—
148#336—22no—
149#336—21no—
150Q8⋊C8—24no—
151Q8⋊C8—22no—
152Q8⋊C8—24no—
153Q8⋊C8—24no—
154Q8⋊C8—42no—
155Q8⋊C8—42no—
156Q8⋊C8D414yescomplemented
157C22×Q8—22no—
158C22×Q8C22×C412yes—
159C22×Q8C22×C411yescharacteristic
160C22×D4C22⋊C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C2×M16—28no—
165C2×M16—44no—
166C2×M16—44no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8—44no—
180#125—44no—
181#125—44no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—44no—
200C4×Q8—44no—
201C4×Q8—41no—
202C4×Q8C2×D412yes—
203C4×Q8#4112yes—
204C4×D4—24no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209#117—44no—
210#117—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4C22⋊C412yes—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C412yes—
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2411yescharacteristic
261C4⋊C8—24no—
262C4⋊C8—22no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—22no—
267C4⋊C8—42no—
268C4⋊C8—81no—
269C4⋊C8C2×D412yes—
270C22⋊C8—28no—
271C22⋊C8—44no—
272C22⋊C8—82no—
273#97—44no—
274C4×C8—24no—
275C4×C8—22no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—22no—
280C4×C8—44no—
281C4×C8—42no—
282C4×C8—42no—
283C4×C8C2×D412yes—
284#95—22no—
285#95—22no—
286#95—21no—
287#95—21no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—21no—
293#95C22⋊C412yes—
294#95C22⋊C412yes—
295#95C22⋊C412yes—
296C24C2×C22⋊C412yes—
297C2×Q8—21no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—21no—
307C2×Q8—22no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—82no—
311C2×Q8C4×D412yes—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347M16—416no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—42no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—81no—
391C4⋊C4#10414yes—
392C4⋊C4C4×D412yes—
393C4⋊C4#13112yes—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—82no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—22no—
414C42—41no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—41no—
419C42—41no—
420C42—42no—
421C42—42no—
422C42—81no—
423C42#13112yes—
424C42#13112yes—
425C42#13712yes—
426C42#13712yes—
427C42C22×D411yescharacteristic
428C42C2×#4111yescharacteristic
429C42H(5,𝔽2)11yescharacteristic
430C42H(5,𝔽2)11yescharacteristic
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23C22×C22⋊C411yescharacteristic
439Q8—22no—
440Q8—22no—
441Q8—41no—
442Q8—41no—
443Q8—42no—
444Q8—42no—
445Q8—42no—
446Q8—42no—
447Q8—88no—
448D4—44no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4#40812yes—
495C2×C4#40812yes—
496C2×C4#41912yes—
497C2×C4#41912yes—
498C2×C4C2×#10414yes—
499C2×C4C2×#10412yes—
500C2×C4C2×C4×D411yescharacteristic
501C2×C4#51811yescharacteristic
502C2×C4D4211yescharacteristic
503C2×C4D4211yescharacteristic
504C2×C4#54611yescharacteristic
505C2×C4#54611yescharacteristic
506C2×C4C2×#13111yescharacteristic
507C2×C4C2×#13111yescharacteristic
508C2×C4C2×#13111yescharacteristic
509C2×C4#56811yescharacteristic
510C2×C4C2×#13711yescharacteristic
511C2×C4C2×#13711yescharacteristic
512C2×C4#57711yescharacteristic
513C2×C4#57811yescharacteristic
514C8—24no—
515C8—42no—
516C8—44no—
517C8—42no—
518C8—81no—
519C8—82no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22D4×C22⋊C411yescharacteristic
528C22C2×#40811yescharacteristic
529C22#288011yescharacteristic
530C22C2×#41911yescharacteristic
531C22#289811yescharacteristic
532C22C22×#10411yescharacteristic
533C22#290111yescharacteristic
534C4—22no—
535C4—22no—
536C4—21no—
537C4—21no—
538C4—24no—
539C4—41no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—81no—
546C4—82no—
547C4—82no—
548C4#148514yescomplemented
549C4#293111yescharacteristic
550C4#293511yescharacteristic
551C4#297911yescharacteristic
552C4#298111yescharacteristic
553C4#327611yescharacteristic
554C4#327811yescharacteristic
555C4#332811yescharacteristic
556C4#332911yescharacteristic
557C2—82no—
558C2C2×#148511yescharacteristic
559C2#1746911yescharacteristic
560C2#1751411yescharacteristic
561C2#1751811yescharacteristic
562C2#2240211yescharacteristic
563C2#2243011yescharacteristic
564C2D4×#10411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1485C2semidirect
5. #9673C2semidirect
6. #9637C2semidirect
7. #9077C2semidirect
8. #8938C2semidirect
9. #8564C2semidirect
10. #8314C2semidirect
11. C4×Q8⋊C8C2semidirect
156. Q8⋊C8D4semidirect
548. C4#1485semidirect

Maximal subgroups

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

subgroupindex
#349292
#227952
C4×#4362
C2×#14852
#96732
#96372
#90772
#89382
#85642
#83142
C4×Q8⋊C82