Finite Group Databasebeta

← #183235

Subgroups — #183235

Order 512.

1224 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34222C211yesmaximal, characteristic
2#20469C211yesmaximal, characteristic, complemented
3C2×#1514C211yesmaximal, characteristic
4C2×#1502C211yesmaximal, characteristic
5#13806C211yesmaximal, characteristic, complemented
6#8220C212yesmaximal, complemented
7#8218C212yesmaximal, complemented
8#8216C212yesmaximal, complemented
9C4×#327C212yesmaximal, complemented
10#8025C211yesmaximal, characteristic, complemented
11#8021C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2372—22no—
16#2364—22no—
17#2353—21no—
18#2353—21no—
19#2339—21no—
20#2339—21no—
21#2331C412yescomplemented
22#2328C412yescomplemented
23#2319C411yescharacteristic, complemented
24#2319C411yescharacteristic, complemented
25C4×#121C2212yes—
26C4×#121C2212yes—
27#2283C2211yescharacteristic
28C2×#383C2211yescharacteristic
29#2275C411yescharacteristic, complemented
30#2275C411yescharacteristic, complemented
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1768—22no—
34#1749C2212yes—
35C4×C22⋊C8C2212yes—
36#1746—21no—
37#1726—21no—
38#1514C418yescomplemented
39#1502C418yescomplemented
40C2×#327C2211yescharacteristic
41C2×#327C2212yes—
42C2×#327C2211yescharacteristic
43C2×#327C2211yescharacteristic
44C2×#327C2212yes—
45C2×#327C2211yescharacteristic
46#1436—22no—
47#1434—21no—
48#1434C2212yes—
49C4×#95—21no—
50C4×#95C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55C22×#41—21no—
56#532—22no—
57#532—22no—
58#532C2×C414yes—
59#532C2×C414yes—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#127C2×C411yescharacteristic
62C2×#124—21no—
63C2×#121—21no—
64C2×#121C2311yescharacteristic
65C2×#121C2311yescharacteristic
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C412yes—
72C2×C4×D4C2×C412yes—
73C22×C4⋊C4—21no—
74C22×C42—21no—
75C22×C42—22no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#432—22no—
79#432C2×C414yescomplemented
80#431—22no—
81#431C2×C414yescomplemented
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C22⋊C8C2311yescharacteristic
85C2×C22⋊C8C2311yescharacteristic
86C2×C22⋊C8C2312yes—
87C2×#97—21no—
88C2×#97—22no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91#391—22no—
92#391—22no—
93#390C2×C412yes—
94#390C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#387—22no—
102#387—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#383—21no—
110#383—22no—
111#383C2×C414yescomplemented
112#382C2×C412yes—
113#381—21no—
114#381—21no—
115#381—21no—
116#380—21no—
117#380—21no—
118#380C2312yes—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C414yescomplemented
125C4×C22⋊C4—21no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132#376—21no—
133#376C2×C411yescharacteristic
134#376C2×C412yes—
135#376C2×C411yescharacteristic
136#376D411yescharacteristic
137#376D411yescharacteristic
138#376D412yes—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95D411yescharacteristic
147C2×#95D411yescharacteristic
148C2×#95D412yes—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152#336—22no—
153#336—22no—
154#327C2×C414yescomplemented
155#327C2×C414yescomplemented
156#327C2×C414yescomplemented
157#327C2×C414yescomplemented
158#327C2×C418yescomplemented
159#327C2×C418yescomplemented
160C2×#41—24no—
161C2×#41—24no—
162C2×#41—82no—
163C2×#41—82no—
164C22×Q8—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4C22×C411yescharacteristic
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C4⋊Q8—22no—
187C4⋊Q8C4214yescomplemented
188#127—22no—
189#127C4214yescomplemented
190#124—24no—
191#124—24no—
192#121—24no—
193#121—24no—
194#121—21no—
195#121—21no—
196#121—21no—
197#121—21no—
198#121C22×C412yes—
199#121C22×C412yes—
200#121C22×C412yes—
201#121C22×C412yes—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—82no—
209C4×D4—82no—
210C4×D4—82no—
211C4×D4—82no—
212C4×D4C4214yescomplemented
213C4×D4C4214yescomplemented
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—24no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—82no—
244C2×C42—82no—
245C2×C42C22×C412yes—
246C2×C42C22×C412yes—
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C412yes—
250C2×C42C22×C412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4112yes—
256C2×C42#4112yes—
257C4⋊C8—24no—
258C4⋊C8—24no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—22no—
262C22⋊C8C22×C412yes—
263C22⋊C8C22×C412yes—
264C22⋊C8C22×C414yes—
265#97—24no—
266#97—44no—
267C4×C8—24no—
268C4×C8—44no—
269#95—21no—
270#95—21no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—24no—
278#95—24no—
279#95—22no—
280#95—21no—
281#95—21no—
282#95—22no—
283#95—44no—
284#95—44no—
285#95—82no—
286#95C22⋊C411yescharacteristic
287#95C22⋊C411yescharacteristic
288#95C22⋊C412yes—
289#95C22⋊C411yescharacteristic
290#95C22⋊C411yescharacteristic
291#95C22⋊C412yes—
292#95C22⋊C412yes—
293#95C22⋊C412yes—
294#95C22⋊C412yes—
295#95C22⋊C412yes—
296#95C22×C412yes—
297#95C22×C412yes—
298#95C22×C414yes—
299C24—21no—
300C24—21no—
301C24C22×D411yescharacteristic
302#41—88no—
303#41—84no—
304#41—84no—
305C2×Q8—24no—
306C2×Q8—82no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—81no—
322C2×D4—81no—
323C2×D4—82no—
324C2×D4C2×C4212yes—
325C2×D4C2×C4212yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C412yes—
378C22×C4C4×D412yes—
379C22×C4C4×D412yes—
380C22×C4C4×D412yes—
381C22×C4C4×D412yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C23×C411yescharacteristic
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—24no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—48no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4C2×C4212yes—
404C4⋊C4C2×C4212yes—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C42—24no—
420C42—24no—
421C42—42no—
422C42—42no—
423C42—81no—
424C42—81no—
425C42—84no—
426C42—82no—
427C42—82no—
428C42—88no—
429C42C2×C4212yes—
430C42C2×C4212yes—
431C42#11712yes—
432C42#11712yes—
433C42C4×D412yes—
434C42C4×D412yes—
435C42C4×D412yes—
436C42C4×D412yes—
437C23—22no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23C22×C22⋊C411yescharacteristic
451C23C2×C4×D411yescharacteristic
452C23C2×C4×D411yescharacteristic
453C23#57612yes—
454C23#57712yes—
455Q8—84no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—44no—
461D4—44no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4—42no—
466D4—84no—
467D4—84no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—164no—
530C2×C4—164no—
531C2×C4C4×C22⋊C412yes—
532C2×C4C4×C22⋊C412yes—
533C2×C4C4×C22⋊C412yes—
534C2×C4C4×C22⋊C412yes—
535C2×C4#41014yes—
536C2×C4C22×C4211yescharacteristic
537C2×C4C2×#11711yescharacteristic
538C2×C4C2×C4×D411yescharacteristic
539C2×C4C2×C4×D411yescharacteristic
540C8—42no—
541C8—42no—
542C8—44no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22C2×C4×C22⋊C411yescharacteristic
562C22C2×#41012yes—
563C22#289712yes—
564C22#290111yescharacteristic
565C22#290111yescharacteristic
566C22#294612yes—
567C22#294811yescharacteristic
568C22#294911yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—24no—
578C4—44no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—82no—
585C4—84no—
586C4—82no—
587C4—82no—
588C2—21no—
589C2—22no—
590C2—21no—
591C2—82no—
592C2—82no—
593C2#1760412yes—
594C2#1760712yes—
595C2C4×#41011yescharacteristic
596C2#2040511yescharacteristic
597C2#2041611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20469C2semidirect
5. #13806C2semidirect
6. #8220C2semidirect
7. #8218C2semidirect
8. #8216C2semidirect
9. C4×#327C2semidirect
10. #8025C2semidirect
11. #8021C2semidirect
21. #2331C4semidirect
22. #2328C4semidirect
23. #2319C4semidirect
24. #2319C4semidirect
29. #2275C4semidirect
30. #2275C4semidirect
38. #1514C4semidirect
39. #1502C4semidirect
79. #432C2×C4semidirect
81. #431C2×C4semidirect
111. #383C2×C4semidirect
124. C4×C4⋊C4C2×C4semidirect
154. #327C2×C4semidirect
155. #327C2×C4semidirect
156. #327C2×C4semidirect
157. #327C2×C4semidirect
158. #327C2×C4semidirect
159. #327C2×C4semidirect
187. C4⋊Q8C42semidirect
189. #127C42semidirect
212. C4×D4C42semidirect
213. C4×D4C42semidirect

Maximal subgroups

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

subgroupindex
#342222
#204692
C2×#15142
C2×#15022
#138062
#82202
#82182
#82162
C4×#3272
#80252
#80212