Finite Group Databasebeta

← #224137

Subgroups — #224137

Order 512.

925 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic
3#13812C211yesmaximal, characteristic
4#9909C212yesmaximal
5#9908C211yesmaximal, characteristic
6#9888C212yesmaximal
7#9884C211yesmaximal, characteristic
8#8576C212yesmaximal
9#8572C211yesmaximal, characteristic
10#8219C212yesmaximal
11C4×#328C211yesmaximal, characteristic
12#2857C2211yescharacteristic
13#2857C2212yes—
14#2586—22no—
15#2567C411yescharacteristic, complemented
16#2551C411yescharacteristic, complemented
17#2549—22no—
18#2507—21no—
19#2507C412yescomplemented
20#2451C412yescomplemented
21#2440—21no—
22#2369—21no—
23#2369C411yescharacteristic, complemented
24Q8×C4⋊C4C411yescharacteristic, complemented
25#2345—21no—
26C4×C22⋊Q8C2211yescharacteristic
27C4×C22⋊Q8C2212yes—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1939—21no—
31#1917—22no—
32#1797C2211yescharacteristic
33#1797C2212yes—
34#1770—21no—
35#1749C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C2×#328C2211yescharacteristic
38C2×#328C2212yes—
39C2×#328C2211yescharacteristic
40C4⋊#95—21no—
41C4⋊#95C2211yescharacteristic
42#1434—22no—
43#1434C2212yes—
44C4×#95C2211yescharacteristic
45C4×#95C2212yes—
46#1294C2211yescharacteristic
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C412yes—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8C2311yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C412yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42—22no—
66C22×C42C2311yescharacteristic
67C22×C42C2312yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2311yescharacteristic
71C2×C22⋊C8C2312yes—
72C2×C22⋊C8C2311yescharacteristic
73C2×#97—22no—
74C2×C4×C8—21no—
75#398—22no—
76#398—22no—
77#398—22no—
78#395—22no—
79#395—22no—
80#395—22no—
81#391—21no—
82#391—22no—
83#391—22no—
84#391—22no—
85#391C2×C412yes—
86#391C2×C411yescharacteristic
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389—21no—
93#389—22no—
94#389—22no—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C411yescharacteristic
99#389C2×C412yes—
100#387—22no—
101#387—22no—
102#387—21no—
103#387—22no—
104#385—22no—
105#385—21no—
106#384—21no—
107#384C2×C412yes—
108#384C2×C411yescharacteristic
109#380—21no—
110#380—22no—
111#380C2312yes—
112#380C2311yescharacteristic
113C4×C4⋊C4—28no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—22no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95D412yes—
134C2×#95C2311yescharacteristic
135C2×#95C2312yes—
136C2×#95C2311yescharacteristic
137#336—21no—
138#336—22no—
139#336—21no—
140#336—22no—
141#328—22no—
142#328—24no—
143#328Q814yescomplemented
144C22×Q8—22no—
145C22×Q8—21no—
146C22×Q8C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C4⋊Q8—42no—
159C4⋊Q8—44no—
160C4⋊Q8—82no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C22⋊Q8—82no—
164C22⋊Q8—81no—
165C22⋊Q8—82no—
166C22⋊Q8—81no—
167C22⋊Q8C2×Q812yes—
168C22⋊Q8#4112yes—
169C4×Q8—42no—
170C4×Q8—82no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C412yes—
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×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C412yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C42—24no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—28no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—82no—
215C2×C42—82no—
216C2×C42—82no—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C412yes—
219C2×C42C22×C412yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C4⋊C8—42no—
226C4⋊C8—82no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—22no—
230C22⋊C8—21no—
231C22⋊C8—22no—
232C22⋊C8C2×Q812yes—
233#97—82no—
234C4×C8—42no—
235#95—28no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—22no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—21no—
253#95—22no—
254#95—42no—
255#95—44no—
256#95—81no—
257#95—82no—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262#95C2×Q812yes—
263C24C22×D411yescharacteristic
264C2×Q8—21no—
265C2×Q8—42no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—44no—
269C2×Q8—82no—
270C2×Q8—81no—
271C2×Q8—82no—
272C2×Q8—81no—
273C2×Q8C4×Q812yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4#13712yes—
326C22×C4#13712yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×Q811yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C42−1+411yescharacteristic
332C22×C42−1+411yescharacteristic
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—48no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—81no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—81no—
367C4⋊C4—82no—
368C4⋊C4#10414yes—
369C4⋊C4C4×Q812yes—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—41no—
373C22⋊C4—42no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C42—22no—
377C42—22no—
378C42—42no—
379C42—42no—
380C42—41no—
381C42—41no—
382C42—42no—
383C42—42no—
384C42—48no—
385C42—42no—
386C42—42no—
387C42—82no—
388C42—82no—
389C42—88no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23C22×C22⋊C411yescharacteristic
395C23#54811yescharacteristic
396C23D4×Q811yescharacteristic
397C23C2×#13711yescharacteristic
398C23C2×#13711yescharacteristic
399C23#57611yescharacteristic
400C23#57711yescharacteristic
401Q8—42no—
402Q8—42no—
403Q8—164no—
404Q8—162no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4#41012yes—
480C2×C4#41912yes—
481C2×C4#41912yes—
482C2×C4C2×#10414yes—
483C2×C4C2×#10412yes—
484C2×C4C2×C4×Q811yescharacteristic
485C2×C4#51911yescharacteristic
486C2×C4#54811yescharacteristic
487C2×C4D4×Q811yescharacteristic
488C8—42no—
489C8—81no—
490C8—82no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22Q8×C22⋊C411yescharacteristic
504C22C2×#41011yescharacteristic
505C22#288311yescharacteristic
506C22C2×#41911yescharacteristic
507C22#289711yescharacteristic
508C22C22×#10411yescharacteristic
509C22#290111yescharacteristic
510C22#336911yescharacteristic
511C22#337511yescharacteristic
512C22#338511yescharacteristic
513C22#338711yescharacteristic
514C4—22no—
515C4—24no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—41no—
524C4—41no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—84no—
529C4—82no—
530C4—82no—
531C4—81no—
532C4—81no—
533C4—82no—
534C4#150614yes—
535C2—21no—
536C2—21no—
537C2—21no—
538C2—21no—
539C2C2×#150611yescharacteristic
540C2#1754711yescharacteristic
541C2#1760711yescharacteristic
542C2#1762711yescharacteristic
543C2#2251311yescharacteristic
544C2#2252511yescharacteristic
545C2Q8×#10411yescharacteristic
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
15. #2567C4semidirect
16. #2551C4semidirect
19. #2507C4semidirect
20. #2451C4semidirect
23. #2369C4semidirect
24. Q8×C4⋊C4C4semidirect
143. #328Q8semidirect

Maximal subgroups

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

subgroupindex
#365642
#226292
#138122
#99092
#99082
#98882
#98842
#85762
#85722
#82192
C4×#3282