Finite Group Databasebeta

← #230284

Subgroups — #230284

Order 512.

662 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23156C211yesmaximal, characteristic, complemented
2#23094C211yesmaximal, characteristic
3#22969C211yesmaximal, characteristic
4#22715C211yesmaximal, characteristic
5C2×#1691C211yesmaximal, characteristic
6#9828C211yesmaximal, characteristic, complemented
7#9616C211yesmaximal, characteristic, complemented
8#9569C211yesmaximal, characteristic, complemented
9#9567C211yesmaximal, characteristic, complemented
10#8985C211yesmaximal, characteristic
11#8976C211yesmaximal, characteristic
12#8927C211yesmaximal, characteristic, complemented
13#8926C211yesmaximal, characteristic, complemented
14#8289C211yesmaximal, characteristic, complemented
15#8212C211yesmaximal, characteristic
16C2×#498C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#461C2211yescharacteristic
19C2×#436C2211yescharacteristic
20#2369C2211yescharacteristic
21#2365C2211yescharacteristic
22#2354C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1988C2211yescharacteristic
27#1987C2211yescharacteristic
28#1968—21no—
29#1956C2211yescharacteristic
30#1956C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1944—21no—
36#1941—21no—
37#1927—21no—
38#1920—21no—
39#1920—21no—
40#1919—21no—
41#1880—21no—
42#1862—21no—
43#1854—21no—
44#1853—21no—
45#1834C2211yescharacteristic
46#1809—21no—
47C4×#106—21no—
48C4×#106C2211yescharacteristic
49#1770C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1762—21no—
53C4×#103—21no—
54C4×#102C2211yescharacteristic
55#1736—21no—
56#1691C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×M16—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76C22×C4⋊C4—21no—
77#498—22no—
78#498C2×C414yescomplemented
79#476—22no—
80#476C2×C414yes—
81#461—22no—
82#461C2×C414yes—
83#436—22no—
84#436C2×C414yes—
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106D411yescharacteristic
93C2×#106D411yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—41no—
102C2×#102—21no—
103C2×#102D411yescharacteristic
104C2×#102D411yescharacteristic
105C2×C22⋊C8—21no—
106C2×#97—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391—21no—
113#391—41no—
114#390—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#388—41no—
120#387—41no—
121#385—41no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#380—21no—
125#380—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—41no—
146#340D411yescharacteristic
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#335C2×C414yescomplemented
154#333C2×C414yescomplemented
155#332C2×C414yescomplemented
156D4⋊C8—22no—
157#322—22no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×D4C2×D411yescharacteristic
161C23×C4—21no—
162C23×C4—41no—
163C2×SD16—24no—
164C2×SD16—81no—
165C2×SD16—81no—
166C2×M16—24no—
167C2×M16—81no—
168C2×M16—81no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8C22×C412yes—
185C4⋊Q8C22×C412yes—
186C4×Q8—42no—
187C4×Q8—81no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—42no—
191C4×D4—81no—
192C4×D4C22×C412yes—
193C4×D4#4112yes—
194#117—42no—
195#117—81no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42C2411yescharacteristic
229#107—81no—
230#107—81no—
231#106—24no—
232#106—24no—
233#106—24no—
234#106—22no—
235#106—41no—
236#106—81no—
237#106—81no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8C22×C412yes—
244C4⋊C8C22×C412yes—
245C4⋊C8C22×C412yes—
246#103—24no—
247#103—42no—
248#103—81no—
249#103—82no—
250#102—24no—
251#102—22no—
252#102—41no—
253C22⋊C8—24no—
254#97—81no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—42no—
260C4×C8—81no—
261C4×C8C22×C412yes—
262C4×C8#4112yes—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272C24—21no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—81no—
282C2×Q8—81no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4C4×D412yes—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—161no—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4H(5,𝔽2)11yescharacteristic
312SD16—84no—
313M16—84no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8C4×D412yes—
340C2×C8C4×D412yes—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—82no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—82no—
372C4⋊C4—161no—
373C4⋊C4C4×D412yes—
374C4⋊C4#13712yes—
375C4⋊C4#13712yes—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—161no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—82no—
388C42—82no—
389C42—161no—
390C42C23×C411yescharacteristic
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C422−1+411yescharacteristic
394C23—22no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23D4211yescharacteristic
402Q8—42no—
403Q8—42no—
404Q8—82no—
405Q8—81no—
406Q8—81no—
407Q8—161no—
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—164no—
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×C4×D411yescharacteristic
454C2×C4#52011yescharacteristic
455C2×C4#54711yescharacteristic
456C2×C4#54811yescharacteristic
457C2×C4#55311yescharacteristic
458C2×C4C2×#13711yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57511yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57611yescharacteristic
463C2×C4#57711yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57711yescharacteristic
466C8—22no—
467C8—22no—
468C8—42no—
469C8—41no—
470C8—41no—
471C8—42no—
472C8—42no—
473C8—42no—
474C8—81no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22#236111yescharacteristic
483C22#326611yescharacteristic
484C22#326611yescharacteristic
485C22#326711yescharacteristic
486C22#326811yescharacteristic
487C22#329011yescharacteristic
488C22#329111yescharacteristic
489C4—21no—
490C4—21no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—82no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—82no—
500C4—161no—
501C4—161no—
502C4C4×#13711yescharacteristic
503C4C4×#13711yescharacteristic
504C4#294611yescharacteristic
505C4#294911yescharacteristic
506C4#335111yescharacteristic
507C4#335711yescharacteristic
508C4#337211yescharacteristic
509C4#337611yescharacteristic
510C2—82no—
511C2#2004211yescharacteristic
512C2#2004611yescharacteristic
513C2#2010911yescharacteristic
514C2#2013311yescharacteristic
515C2#2301211yescharacteristic
516C2#2303511yescharacteristic
517C2#2304411yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23156C2semidirect
6. #9828C2semidirect
7. #9616C2semidirect
8. #9569C2semidirect
9. #9567C2semidirect
12. #8927C2semidirect
13. #8926C2semidirect
14. #8289C2semidirect
56. #1691C4semidirect
78. #498C2×C4semidirect
153. #335C2×C4semidirect
154. #333C2×C4semidirect
155. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#231562
#230942
#229692
#227152
C2×#16912
#98282
#96162
#95692
#95672
#89852
#89762
#89272
#89262
#82892
#82122