Finite Group Databasebeta

← #224586

Subgroups — #224586

Order 512.

919 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic
3#13812C211yesmaximal, characteristic
4#9711C211yesmaximal, characteristic
5#9711C212yesmaximal
6#9683C212yesmaximal
7#9679C211yesmaximal, characteristic
8#8576C212yesmaximal
9#8572C211yesmaximal, characteristic
10#8219C212yesmaximal
11C4×#328C211yesmaximal, characteristic
12#2857C2211yescharacteristic
13#2857C2212yes—
14#2585—22no—
15#2569—22no—
16#2552C411yescharacteristic, complemented
17#2546C411yescharacteristic, complemented
18#2511C412yescomplemented
19#2509—21no—
20#2446C412yescomplemented
21#2442—21no—
22#2370—21no—
23#2365C411yescharacteristic, complemented
24#2355C411yescharacteristic, 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×#328C2211yescharacteristic
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#328C2212yes—
42C2×#328C2212yes—
43C2×#328C2212yes—
44C4⋊#95—21no—
45C4⋊#95C2211yescharacteristic
46#1434—22no—
47#1434C2212yes—
48C4×#95C2211yescharacteristic
49C4×#95C2212yes—
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C412yes—
55C2×#125—21no—
56C2×C22⋊Q8C2312yes—
57C2×C22⋊Q8C2311yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42—22no—
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8C2311yescharacteristic
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2311yescharacteristic
72C2×#97—22no—
73C2×C4×C8—21no—
74#400—22no—
75#400—22no—
76#400—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#391C2×C412yes—
81#391C2×C412yes—
82#391C2×C412yes—
83#391C2×C411yescharacteristic
84#391C2×C412yes—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389—22no—
92#389—21no—
93#389C2×C412yes—
94#389C2×C411yescharacteristic
95#387—22no—
96#387—22no—
97#387—21no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—21no—
104#385—22no—
105#384C2×C411yescharacteristic
106#384C2×C412yes—
107#382—21no—
108#380—21no—
109#380—22no—
110#380—22no—
111#380—21no—
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×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—22no—
127C4×C22⋊C4—21no—
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#328—22no—
144#328—24no—
145#328—42no—
146#328—42no—
147#328—42no—
148C22×Q8C22×C411yescharacteristic
149C22×Q8C22×C412yes—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C4⋊Q8—42no—
162C4⋊Q8—82no—
163#125—44no—
164C22⋊Q8—22no—
165C22⋊Q8—21no—
166C22⋊Q8—22no—
167C22⋊Q8—21no—
168C22⋊Q8—22no—
169C22⋊Q8—21no—
170C22⋊Q8—22no—
171C22⋊Q8—21no—
172C4×Q8—42no—
173C4×Q8—82no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—24no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—28no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—82no—
217C2×C42—82no—
218C2×C42—82no—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C412yes—
221C2×C42C22×C412yes—
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C4⋊C8—42no—
228C4⋊C8—82no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—22no—
232C22⋊C8—21no—
233C22⋊C8—22no—
234C22⋊C8#4112yes—
235#97—82no—
236C4×C8—42no—
237#95—28no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—22no—
256#95—21no—
257#95—22no—
258#95—42no—
259#95—44no—
260#95—81no—
261#95—82no—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266#95#4112yes—
267C24C22×D411yescharacteristic
268C2×Q8—21no—
269C2×Q8—22no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—41no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
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—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4#13712yes—
328C22×C4#13712yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4H(5,𝔽2)11yescharacteristic
334C22×C42−1+411yescharacteristic
335C2×C8—22no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—48no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4#10414yes—
373C22⋊C4—42no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C42—22no—
380C42—22no—
381C42—42no—
382C42—42no—
383C42—41no—
384C42—41no—
385C42—42no—
386C42—42no—
387C42—48no—
388C42—42no—
389C42—42no—
390C42—82no—
391C42—82no—
392C42—88no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23C22×C22⋊C411yescharacteristic
398C23#54811yescharacteristic
399C23#55011yescharacteristic
400C23C2×#13711yescharacteristic
401C23C2×#13711yescharacteristic
402C23#57711yescharacteristic
403C23#57811yescharacteristic
404Q8—81no—
405Q8—82no—
406Q8—81no—
407Q8—82no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4#41012yes—
483C2×C4#41912yes—
484C2×C4#41912yes—
485C2×C4C2×#10414yes—
486C2×C4C2×#10412yes—
487C2×C4C4×#4111yescharacteristic
488C2×C4#52011yescharacteristic
489C2×C4#54811yescharacteristic
490C2×C4#55011yescharacteristic
491C8—42no—
492C8—81no—
493C8—82no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22#234111yescharacteristic
507C22C2×#41011yescharacteristic
508C22#288311yescharacteristic
509C22C2×#41911yescharacteristic
510C22#289811yescharacteristic
511C22C22×#10411yescharacteristic
512C22#290111yescharacteristic
513C22#337211yescharacteristic
514C22#337411yescharacteristic
515C22#339811yescharacteristic
516C22#340011yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—22no—
522C4—24no—
523C4—44no—
524C4—42no—
525C4—41no—
526C4—41no—
527C4—42no—
528C4—42no—
529C4—84no—
530C4—82no—
531C4—81no—
532C4—82no—
533C4—81no—
534C4—82no—
535C4—81no—
536C4—81no—
537C4#150614yes—
538C2—21no—
539C2—21no—
540C2—21no—
541C2—21no—
542C2C2×#150611yescharacteristic
543C2#1754811yescharacteristic
544C2#1761211yescharacteristic
545C2#1762711yescharacteristic
546C2#2253511yescharacteristic
547C2#2254711yescharacteristic
548C2#2255011yescharacteristic
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
16. #2552C4semidirect
17. #2546C4semidirect
18. #2511C4semidirect
20. #2446C4semidirect
23. #2365C4semidirect
24. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#365992
#226292
#138122
#97112
#96832
#96792
#85762
#85722
#82192
C4×#3282