Finite Group Databasebeta

← #219734

Subgroups — #219734

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2361C211yesmaximal, characteristic
2#22599C212yesmaximal, complemented
3C2×#1460C212yesmaximal, complemented
4#9649C212yesmaximal, complemented
5#9080C214yesmaximal, complemented
6#8557C211yesmaximal, characteristic, complemented
7#8302C212yesmaximal, complemented
8C4×#323C211yesmaximal, characteristic, complemented
9C22×#121C411yescharacteristic
10C22×C4×D4C411yescharacteristic
11C22×C4×D4C2212yes—
12C23×C4⋊C4C411yescharacteristic
13C2×#431C2214yes—
14#2361—24no—
15#2361—22no—
16#2361—22no—
17C2×#390C2211yescharacteristic
18C2×#389C411yescharacteristic
19C2×#388C2212yes—
20C2×#386C412yes—
21C2×#385C412yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1939—22no—
25#1916—24no—
26#1817C2214yes—
27#1797C2212yes—
28C8×C4⋊C4—22no—
29C4×C22⋊C8C2212yes—
30#1460—28no—
31C2×#323C2212yes—
32C2×#323C2211yescharacteristic
33C2×#323C2214yes—
34C2×#323C2214yes—
35C2×#323C2211yescharacteristic
36#1280C2212yes—
37#1280C2212yes—
38C23×D4C2×C412yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C24×C4C2×C412yes—
43C2×#121—22no—
44C2×#121—22no—
45C2×#121—24no—
46C2×#121C818yescomplemented
47C2×C4×D4—22no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×C4×D4—44no—
55C2×C4×D4—44no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2311yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#431—28no—
75#431—44no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—24no—
78C2×C22⋊C8C2314yes—
79C2×C22⋊C8C2312yes—
80C2×C22⋊C8C2312yes—
81C2×C4×C8—24no—
82C2×C4×C8—22no—
83#390—22no—
84#390—22no—
85#389—22no—
86#389—22no—
87#388—24no—
88#388—24no—
89#386—24no—
90#386—24no—
91#385—24no—
92#385—24no—
93C4×C4⋊C4D414yes—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4D412yes—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2311yescharacteristic
102#336—22no—
103#336—24no—
104#336—24no—
105#336—22no—
106#323—24no—
107#323—22no—
108#323—28no—
109#323—44no—
110#323D414yescomplemented
111C25C22×C412yes—
112C25C22×C411yescharacteristic
113C22×D4—24no—
114C22×D4—24no—
115C22×D4—24no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—24no—
119C22×D4—22no—
120C22×D4—42no—
121C22×D4—42no—
122C22×D4—42no—
123C22×D4—42no—
124C22×D4C2×C818yes—
125C22×D4C2×C814yes—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—82no—
142C23×C4—82no—
143C23×C4C2×C814yes—
144C23×C4C22×C412yes—
145C23×C4C22×C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C412yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—24no—
156#121—216no—
157#121—216no—
158C4×D4—48no—
159C4×D4—48no—
160C4×D4—48no—
161C4×D4—48no—
162C4×D4—48no—
163C4×D4—44no—
164C4×D4—44no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—48no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C2×C814yes—
184C2×C4⋊C4M1614yes—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4C2×C818yes—
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4#4112yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C2×D414yes—
212C2×C42C2×D414yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C4⋊C8—48no—
216C4⋊C8—44no—
217C22⋊C8—22no—
218C22⋊C8—24no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—24no—
222C22⋊C8C2×D414yes—
223C4×C8—48no—
224C4×C8—44no—
225#95—22no—
226#95—24no—
227#95—22no—
228#95C22⋊C414yes—
229#95C22⋊C414yes—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24C4×D412yes—
249C24C22×C814yes—
250C24C22×C812yes—
251C24#13112yes—
252C24C23×C411yescharacteristic
253C2×D4—216no—
254C2×D4—216no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—48no—
258C2×D4—48no—
259C2×D4—48no—
260C2×D4—48no—
261C2×D4—48no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—48no—
265C2×D4—48no—
266C2×D4—42no—
267C2×D4—42no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—88no—
323C22×C4—88no—
324C22×C4—88no—
325C22×C4—88no—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C414yes—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C4×D412yes—
333C22×C4C22×C814yes—
334C22×C4C22×C812yes—
335C22×C4C22×C812yes—
336C22×C4C2×M1614yes—
337C22×C4C2×M1612yes—
338C22×C4C22×D412yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)12yes—
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—44no—
347C2×C8—42no—
348C2×C8—44no—
349C2×C8—44no—
350C2×C8—42no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—48no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4C22⋊C818yes—
359C4⋊C4C22⋊C818yes—
360C4⋊C4#10014yes—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—216no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—84no—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—42no—
376C42—42no—
377C42—42no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23C8×D414yes—
428C23C22×C22⋊C411yescharacteristic
429C23C2×C4×D411yescharacteristic
430C23#51811yescharacteristic
431C23C23×C811yescharacteristic
432C23C22×M1611yescharacteristic
433C23C2×#13111yescharacteristic
434C23#56811yescharacteristic
435D4—432no—
436D4—48no—
437D4—48no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—88no—
482C2×C4—88no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—88no—
486C2×C4—88no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4C2×C22⋊C818yes—
492C2×C4C2×C22⋊C812yes—
493C2×C4C2×C22⋊C812yes—
494C2×C4#41012yes—
495C2×C4C2×#10014yes—
496C2×C4C2×#10012yes—
497C2×C4D4212yes—
498C2×C4#54612yes—
499C8—44no—
500C8—84no—
501C8—82no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—24no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—84no—
529C22—84no—
530C22—84no—
531C22—84no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22D4×C22⋊C411yescharacteristic
537C22C22×C22⋊C811yescharacteristic
538C22#288011yescharacteristic
539C22C2×#41011yescharacteristic
540C22#288411yescharacteristic
541C22C22×#10011yescharacteristic
542C22#288811yescharacteristic
543C22C2×C8×D411yescharacteristic
544C22D4×M1611yescharacteristic
545C22#297211yescharacteristic
546C22#297411yescharacteristic
547C4—28no—
548C4—22no—
549C4—22no—
550C4—24no—
551C4—22no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—81no—
557C4—82no—
558C4#146014yescomplemented
559C2—21no—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—44no—
564C2—84no—
565C2C2×#146011yescharacteristic
566C2#1735311yescharacteristic
567C2#1741511yescharacteristic
568C2#1742511yescharacteristic
569C2D4×C22⋊C811yescharacteristic
570C2#2240611yescharacteristic
571C2D4×#10011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C23 ▷ C2 ▷ 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. #22599C2semidirect
3. C2×#1460C2semidirect
4. #9649C2semidirect
5. #9080C2semidirect
6. #8557C2semidirect
7. #8302C2semidirect
8. C4×#323C2semidirect
46. C2×#121C8semidirect
110. #323D4semidirect
558. C4#1460semidirect

Maximal subgroups

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

subgroupindex
C2×#23612
#225992
C2×#14602
#96492
#90802
#85572
#83022
C4×#3232