Finite Group Databasebeta

← #207281

Subgroups — #207281

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35239C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic
3#13750C211yesmaximal, characteristic
4#9910C212yesmaximal
5#9910C212yesmaximal
6#9909C212yesmaximal
7#9908C212yesmaximal
8#8022C211yesmaximal, characteristic
9#8021C211yesmaximal, characteristic
10#8014C211yesmaximal, characteristic
11#8013C211yesmaximal, characteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2851C2212yes—
16#2666—22no—
17#2571C411yescharacteristic, complemented
18#2570C412yescomplemented
19#2552C411yescharacteristic, complemented
20#2552C411yescharacteristic, complemented
21#2551C411yescharacteristic, complemented
22#2550C412yescomplemented
23#2442—21no—
24#2440—21no—
25#2438—22no—
26#2331—21no—
27#2330—21no—
28#2285C2211yescharacteristic
29#2283C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×#376C2211yescharacteristic
35#1940—22no—
36#1939—21no—
37#1939—21no—
38#1797C2212yes—
39#1797C2212yes—
40C4×#95—22no—
41C4×#95C2212yes—
42C4×#95C2212yes—
43#1294C2211yescharacteristic
44#1294C2211yescharacteristic
45#1294C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C412yes—
55C2×#125—21no—
56C2×#125C2×C412yes—
57C2×C22⋊Q8—22no—
58C2×#117—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C42—21no—
63C22×C42—21no—
64C22×C42—22no—
65C22×C42—22no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68#432—22no—
69#432—22no—
70#431—22no—
71#431—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C22⋊C8C2311yescharacteristic
77C2×C22⋊C8C2311yescharacteristic
78C2×C22⋊C8C2312yes—
79C2×#97—21no—
80C2×C4×C8—21no—
81#400—22no—
82#400—22no—
83#398—22no—
84#398—22no—
85#398—22no—
86#398—22no—
87#395—22no—
88#395—22no—
89#391—22no—
90#391—22no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—22no—
96#389—22no—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#384—21no—
108#384C2×C412yes—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#383—21no—
112#382—21no—
113#382—21no—
114#382C2×C411yescharacteristic
115#382C2×C412yes—
116#382C2×C411yescharacteristic
117#381—21no—
118#380—22no—
119#380—21no—
120#380C2311yescharacteristic
121#380C2312yes—
122#380C2311yescharacteristic
123C4×C4⋊C4—28no—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D412yes—
136C4×C4⋊C4Q814yescomplemented
137C4×C22⋊C4—21no—
138#376—21no—
139#376—22no—
140#376Q814yescomplemented
141C2×#95—21no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95D412yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148#336—22no—
149#336—22no—
150C22×Q8—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C4⋊Q8—44no—
163C4⋊Q8—82no—
164#125—44no—
165#125—82no—
166C22⋊Q8—82no—
167C22⋊Q8—82no—
168#117—81no—
169#117—81no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—28no—
209C2×C42—28no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—82no—
213C2×C42—81no—
214C2×C42—81no—
215C2×C42—82no—
216C2×C42—82no—
217C2×C42—82no—
218C2×C42—82no—
219C2×C42C22×C412yes—
220C2×C42C22×C412yes—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×Q812yes—
228C2×C42C2×Q812yes—
229C2×C42#4112yes—
230C2×C42#4112yes—
231C4⋊C8—42no—
232C4⋊C8—42no—
233C4⋊C8—82no—
234C4⋊C8—82no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241#97—42no—
242C4×C8—42no—
243#95—28no—
244#95—28no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—21no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—24no—
259#95—21no—
260#95—21no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95C22⋊C412yes—
266#95C22⋊C412yes—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269#95C2×Q812yes—
270#95C2×Q812yes—
271#95C2×Q814yes—
272C24C22×D411yescharacteristic
273C2×Q8—44no—
274C2×Q8—82no—
275C2×Q8—82no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C22⋊Q812yes—
328C22×C4C22⋊Q812yes—
329C22×C4C22⋊Q812yes—
330C22×C4C22⋊Q812yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×Q811yescharacteristic
334C22×C4C22×Q811yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C4⋊C4—216no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4#10414yes—
371C4⋊C4#10414yes—
372C22⋊C4—42no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C42—216no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—84no—
386C42—82no—
387C42—88no—
388C42—88no—
389C42C4×Q812yes—
390C42C4×Q812yes—
391C42C4×Q812yes—
392C42C4×Q812yes—
393C42#12412yes—
394C42#12412yes—
395C42C4⋊Q812yes—
396C42C4⋊Q812yes—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23C22×C22⋊C411yescharacteristic
402C23C2×C22⋊Q811yescharacteristic
403C23C2×C22⋊Q811yescharacteristic
404C23#57611yescharacteristic
405C23#57711yescharacteristic
406C23#57711yescharacteristic
407C23#57811yescharacteristic
408Q8—164no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—28no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4#39112yes—
483C2×C4#39112yes—
484C2×C4#39112yes—
485C2×C4#39112yes—
486C2×C4C2×#10414yes—
487C2×C4C2×#10414yes—
488C2×C4C2×#10412yes—
489C2×C4C2×#10412yes—
490C2×C4C2×C4×Q811yescharacteristic
491C2×C4C2×C4×Q811yescharacteristic
492C2×C4C2×#12411yescharacteristic
493C2×C4C2×C4⋊Q811yescharacteristic
494C8—81no—
495C8—81no—
496C8—82no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22C2×#39111yescharacteristic
510C22#288311yescharacteristic
511C22#288411yescharacteristic
512C22#289711yescharacteristic
513C22#289811yescharacteristic
514C22C22×#10411yescharacteristic
515C22C22×#10411yescharacteristic
516C22#308111yescharacteristic
517C22#308211yescharacteristic
518C22#308311yescharacteristic
519C22#308411yescharacteristic
520C4—22no—
521C4—22no—
522C4—22no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—24no—
528C4—24no—
529C4—48no—
530C4—44no—
531C4—44no—
532C4—42no—
533C4—42no—
534C4—82no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—82no—
539C4—81no—
540C4#199414yes—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—21no—
545C2#1754711yescharacteristic
546C2#1754811yescharacteristic
547C2#1754911yescharacteristic
548C2#1755011yescharacteristic
549C2#2131511yescharacteristic
550C2C2×#199411yescharacteristic
551C2#2133411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #2571C4semidirect
18. #2570C4semidirect
19. #2552C4semidirect
20. #2552C4semidirect
21. #2551C4semidirect
22. #2550C4semidirect
136. C4×C4⋊C4Q8semidirect
140. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#352392
#219532
#137502
#99102
#99092
#99082
#80222
#80212
#80142
#80132