Finite Group Databasebeta

← #219841

Subgroups — #219841

Order 512.

998 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34929C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic
3#21264C211yesmaximal, characteristic
4C2×#1497C212yesmaximal, complemented
5#9673C211yesmaximal, characteristic, complemented
6#9637C211yesmaximal, characteristic, complemented
7#9073C212yesmaximal, complemented
8#8941C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10#8314C212yesmaximal, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#533C2212yes—
13C2×#432C2212yes—
14C2×#431C2212yes—
15#2556—22no—
16#2551C411yescharacteristic
17#2546C411yescharacteristic
18#2542C411yescharacteristic
19#2538C411yescharacteristic
20#2506—21no—
21#2492C412yes—
22#2452—21no—
23#2429C412yes—
24#2396—22no—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8C2211yescharacteristic
27C4×C22⋊Q8C2212yes—
28#2293—21no—
29C42×Q8C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1954—21no—
32#1954—21no—
33#1939—21no—
34#1924—22no—
35#1923—22no—
36#1842C2211yescharacteristic
37#1797C2212yes—
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43#1497—28no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2211yescharacteristic
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50C2×Q8⋊C8C2211yescharacteristic
51#1279C2211yescharacteristic
52#1279C2212yes—
53#1279C2211yescharacteristic
54#533—24no—
55#533—24no—
56#533—42no—
57#533—42no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8C2×C412yes—
62C2×#125—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2×C412yes—
65C2×C22⋊Q8C2×C412yes—
66C2×C22⋊Q8C2×C412yes—
67C2×C4×Q8—22no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2312yes—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×#117—22no—
76C22×C42C2312yes—
77#432—24no—
78#432—42no—
79#431—24no—
80#431—42no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2311yescharacteristic
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×#97—22no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#400—22no—
100#399—22no—
101#398—22no—
102#398—22no—
103#396—22no—
104#396—22no—
105#395—22no—
106#393—22no—
107#391—22no—
108#391C2×C412yes—
109#391C2×C412yes—
110#390—21no—
111#390—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—22no—
119#389C2×C412yes—
120#388—22no—
121#388—22no—
122#387—22no—
123#385—22no—
124#384—21no—
125#384C2×C411yescharacteristic
126#380C2×C412yes—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—22no—
140C4×C22⋊C4C2×C412yes—
141C4×C22⋊C4D412yes—
142C4×C22⋊C4D412yes—
143C43—21no—
144C43C2311yescharacteristic
145#336—22no—
146#336—21no—
147#336—22no—
148#336—21no—
149Q8⋊C8—24no—
150Q8⋊C8—22no—
151Q8⋊C8—24no—
152Q8⋊C8—24no—
153Q8⋊C8—42no—
154Q8⋊C8—42no—
155Q8⋊C8D414yescomplemented
156C22×Q8—22no—
157C22×Q8C22×C412yes—
158C22×Q8C22×C411yescharacteristic
159C22×D4C22⋊C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—22no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—44no—
176#125—44no—
177#125—44no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—44no—
196C4×Q8—44no—
197C4×Q8—41no—
198C4×Q8C2×D412yes—
199C4×Q8#4112yes—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—42no—
205#117—44no—
206#117—44no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C22⋊C412yes—
235C2×C22⋊C4C22⋊C412yes—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C412yes—
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2411yescharacteristic
257C4⋊C8—22no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—22no—
262C4⋊C8—44no—
263C4⋊C8—42no—
264C4⋊C8—42no—
265C4⋊C8—81no—
266C4⋊C8C2×D412yes—
267C22⋊C8—44no—
268C22⋊C8—44no—
269C22⋊C8—44no—
270C22⋊C8—82no—
271#97—44no—
272C4×C8—22no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—22no—
277C4×C8—42no—
278C4×C8—42no—
279C4×C8—42no—
280C4×C8C2×D412yes—
281#95—22no—
282#95—22no—
283#95—21no—
284#95—21no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—21no—
290#95C22⋊C412yes—
291#95C22⋊C412yes—
292#95C22⋊C412yes—
293C24C2×C22⋊C412yes—
294C2×Q8—21no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—21no—
304C2×Q8—22no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—82no—
308C2×Q8C4×D412yes—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
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—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—42no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—81no—
388C4⋊C4C4×D412yes—
389C4⋊C4#13112yes—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—22no—
410C42—41no—
411C42—42no—
412C42—42no—
413C42—42no—
414C42—41no—
415C42—41no—
416C42—42no—
417C42—42no—
418C42—81no—
419C42#13112yes—
420C42#13112yes—
421C42#13712yes—
422C42#13712yes—
423C42C22×D411yescharacteristic
424C42C2×#4111yescharacteristic
425C42H(5,𝔽2)11yescharacteristic
426C42H(5,𝔽2)11yescharacteristic
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23C22×C22⋊C411yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—41no—
438Q8—41no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442Q8—42no—
443Q8—88no—
444D4—44no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4#40812yes—
492C2×C4#40812yes—
493C2×C4#41912yes—
494C2×C4#41912yes—
495C2×C4#42112yes—
496C2×C4C2×C4×D411yescharacteristic
497C2×C4#51811yescharacteristic
498C2×C4D4211yescharacteristic
499C2×C4D4211yescharacteristic
500C2×C4#54611yescharacteristic
501C2×C4#54611yescharacteristic
502C2×C4C2×#13111yescharacteristic
503C2×C4C2×#13111yescharacteristic
504C2×C4C2×#13111yescharacteristic
505C2×C4#56811yescharacteristic
506C2×C4C2×#13711yescharacteristic
507C2×C4C2×#13711yescharacteristic
508C2×C4#57711yescharacteristic
509C2×C4#57811yescharacteristic
510C8—42no—
511C8—42no—
512C8—42no—
513C8—82no—
514C8—81no—
515C8—82no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22D4×C22⋊C411yescharacteristic
524C22C2×#40811yescharacteristic
525C22#288011yescharacteristic
526C22C2×#41911yescharacteristic
527C22#289811yescharacteristic
528C22C2×#42111yescharacteristic
529C22#290111yescharacteristic
530C4—22no—
531C4—22no—
532C4—21no—
533C4—21no—
534C4—41no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—81no—
542C4—82no—
543C4—82no—
544C4#149714yescomplemented
545C4#293111yescharacteristic
546C4#293511yescharacteristic
547C4#297911yescharacteristic
548C4#298111yescharacteristic
549C4#327611yescharacteristic
550C4#327811yescharacteristic
551C4#332811yescharacteristic
552C4#332911yescharacteristic
553C2—82no—
554C2C2×#149711yescharacteristic
555C2#1748411yescharacteristic
556C2#1751411yescharacteristic
557C2#1752311yescharacteristic
558C2#2240211yescharacteristic
559C2#2243011yescharacteristic
560C2#2243511yescharacteristic
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×#1497C2semidirect
5. #9673C2semidirect
6. #9637C2semidirect
7. #9073C2semidirect
8. #8941C2semidirect
9. #8564C2semidirect
10. #8314C2semidirect
11. C4×Q8⋊C8C2semidirect
155. Q8⋊C8D4semidirect
544. C4#1497semidirect

Maximal subgroups

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

subgroupindex
#349292
#227992
#212642
C2×#14972
#96732
#96372
#90732
#89412
#85642
#83142
C4×Q8⋊C82