Finite Group Databasebeta

← #223067

Subgroups — #223067

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic
2#22603C211yesmaximal, characteristic, complemented
3#13921C211yesmaximal, characteristic, complemented
4#9909C211yesmaximal, characteristic, complemented
5#9885C211yesmaximal, characteristic, complemented
6#9157C212yesmaximal
7#9118C212yesmaximal
8#8717C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic
10C4×#327C211yesmaximal, characteristic
11#8018C212yesmaximal, complemented
12C2×#552—21no—
13C2×#432C2212yes—
14#2857C2211yescharacteristic
15#2854C2212yes—
16#2552C411yescharacteristic, complemented
17#2542C411yescharacteristic, complemented
18#2519—22no—
19#2512C412yescomplemented
20#2506C412yescomplemented
21#2498—22no—
22#2489—21no—
23#2417—21no—
24#2365C411yescharacteristic, complemented
25#2362—21no—
26#2354C411yescharacteristic, complemented
27C4×#121C2211yescharacteristic
28#2285C2212yes—
29C2×#383C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1939—21no—
32#1917—22no—
33#1817C2212yes—
34#1797C2211yescharacteristic
35#1770—21no—
36C4×C22⋊C8C2211yescharacteristic
37C2×#327C2211yescharacteristic
38C2×#327C2211yescharacteristic
39C2×#327C2212yes—
40#1443—22no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95C2211yescharacteristic
44#1434C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2211yescharacteristic
51#1294C2212yes—
52#1294C2212yes—
53#552—44no—
54#552—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—22no—
57C2×#125—21no—
58C2×#125—22no—
59C2×#123—22no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×#121C2311yescharacteristic
64C2×C4×Q8—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C411yescharacteristic
67C2×#117—22no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#432—24no—
76#432—42no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C22⋊C8C2312yes—
80C2×C22⋊C8C2311yescharacteristic
81C2×C22⋊C8C2311yescharacteristic
82C2×#97—22no—
83C2×C4×C8—21no—
84#400—22no—
85#399—22no—
86#398—22no—
87#398—22no—
88#397—22no—
89#396—22no—
90#391—21no—
91#391—21no—
92#390C2×C412yes—
93#390C2×C412yes—
94#390C2×C411yescharacteristic
95#389—21no—
96#389—21no—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C411yescharacteristic
101#388—22no—
102#388—22no—
103#388—21no—
104#388—22no—
105#388—22no—
106#387—22no—
107#387—22no—
108#386—22no—
109#386—21no—
110#384C2×C411yescharacteristic
111#384C2×C412yes—
112#384C2×C412yes—
113#383—24no—
114#383—42no—
115#382—22no—
116#380—21no—
117#380—22no—
118#380—21no—
119#380C2312yes—
120#380C2311yescharacteristic
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D411yescharacteristic
131C4×C22⋊C4—21no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95D412yes—
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142#336—21no—
143#336—22no—
144#336—22no—
145#336—21no—
146#327—22no—
147#327—22no—
148#327—42no—
149C22×Q8—21no—
150C22×D4—21no—
151C22×D4C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C4⋊Q8—42no—
165#126—48no—
166#125—48no—
167#125—44no—
168#123—44no—
169#123—44no—
170C22⋊Q8—44no—
171C22⋊Q8—44no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174#121—21no—
175#121—21no—
176#121—21no—
177#121—21no—
178C4×Q8—44no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—42no—
182#117—44no—
183#117—44no—
184#117—44no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C412yes—
236C2×C42C22×C412yes—
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C4⋊C8—44no—
243C4⋊C8—42no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—21no—
249C22⋊C8#4112yes—
250#97—44no—
251C4×C8—42no—
252#95—21no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—22no—
261#95—21no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—21no—
266#95—44no—
267#95—44no—
268#95—42no—
269#95—44no—
270#95—82no—
271#95—81no—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274#95C22⋊C412yes—
275#95C22⋊C412yes—
276#95#4112yes—
277C24—21no—
278C24C22×D411yescharacteristic
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4AGL(1,ℤ/8)12yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C42−1+411yescharacteristic
355C2×C8—22no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—41no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—41no—
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—44no—
385C4⋊C4—82no—
386C4⋊C4—81no—
387C4⋊C4—84no—
388C22⋊C4—44no—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—42no—
399C22⋊C4—81no—
400C22⋊C4—82no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—41no—
412C42—41no—
413C42—44no—
414C42—44no—
415C42—42no—
416C42—84no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23C22×C22⋊C411yescharacteristic
429C23#54611yescharacteristic
430C23#54711yescharacteristic
431C23C2×AGL(1,ℤ/8)11yescharacteristic
432C23C2×AGL(1,ℤ/8)11yescharacteristic
433C23#57711yescharacteristic
434C23#57811yescharacteristic
435Q8—84no—
436D4—81no—
437D4—81no—
438D4—81no—
439D4—81no—
440D4—81no—
441D4—81no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
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—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—84no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4#41012yes—
518C2×C4#41812yes—
519C2×C4#41812yes—
520C2×C4#42112yes—
521C2×C4C4×#4111yescharacteristic
522C2×C4#52011yescharacteristic
523C2×C4#54611yescharacteristic
524C2×C4#54711yescharacteristic
525C8—42no—
526C8—82no—
527C8—81no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22#233911yescharacteristic
548C22C2×#41011yescharacteristic
549C22#288411yescharacteristic
550C22C2×#41811yescharacteristic
551C22#289811yescharacteristic
552C22C2×#42111yescharacteristic
553C22#290111yescharacteristic
554C22#333011yescharacteristic
555C22#333211yescharacteristic
556C22#335311yescharacteristic
557C22#335411yescharacteristic
558C4—22no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—82no—
571C4—81no—
572C4—82no—
573C4—81no—
574C4—82no—
575C4—81no—
576C4—162no—
577C4#151414yes—
578C2—21no—
579C2—21no—
580C2—21no—
581C2—21no—
582C2—161no—
583C2C2×#151411yescharacteristic
584C2#1760111yescharacteristic
585C2#1761211yescharacteristic
586C2#1763111yescharacteristic
587C2#2247811yescharacteristic
588C2#2250011yescharacteristic
589C2#2250711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
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
2. #22603C2semidirect
3. #13921C2semidirect
4. #9909C2semidirect
5. #9885C2semidirect
8. #8717C2semidirect
11. #8018C2semidirect
16. #2552C4semidirect
17. #2542C4semidirect
19. #2512C4semidirect
20. #2506C4semidirect
24. #2365C4semidirect
26. #2354C4semidirect

Maximal subgroups

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

subgroupindex
#361862
#226032
#139212
#99092
#98852
#91572
#91182
#87172
#85712
C4×#3272
#80182