Finite Group Databasebeta

← #224743

Subgroups — #224743

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2363C211yesmaximal, characteristic
2#22629C212yesmaximal, complemented
3#9653C214yesmaximal, complemented
4#9649C212yesmaximal, complemented
5#8557C212yesmaximal, complemented
6#8557C211yesmaximal, characteristic, complemented
7#8152C212yesmaximal, complemented
8C4×#323C211yesmaximal, characteristic, complemented
9C22×#121C411yescharacteristic
10C22×C4×D4C411yescharacteristic
11C22×#117C411yescharacteristic
12#2363—24no—
13#2363—22no—
14#2363—22no—
15C2×#390C2211yescharacteristic
16C2×#390C2212yes—
17C2×#389C411yescharacteristic
18C2×#388C412yes—
19C2×#386C412yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23#1939—22no—
24#1917—24no—
25#1797C2212yes—
26#1797C2214yes—
27#1770—22no—
28#1749C2214yes—
29C4×C22⋊C8C2212yes—
30C2×#323C2214yes—
31C2×#323C2212yes—
32C2×#323C2214yes—
33C2×#323C2212yes—
34#1280C2211yescharacteristic
35#1280C2212yes—
36#1280C2211yescharacteristic
37C23×D4C2×C412yes—
38C23×D4C2×C411yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C412yes—
41C2×#121—24no—
42C2×#121—24no—
43C2×#121—22no—
44C2×#121—22no—
45C2×C4×D4—22no—
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4—44no—
49C2×#117—21no—
50C2×#117—22no—
51C2×#117—21no—
52C2×#117—44no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2312yes—
63C22×C22⋊C4C2311yescharacteristic
64C22×C42C2×C412yes—
65C22×C42C2×C411yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2312yes—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—24no—
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2314yes—
72C2×C22⋊C8C2312yes—
73C2×#97—24no—
74C2×C4×C8—22no—
75#390—22no—
76#390—22no—
77#390—24no—
78#390—24no—
79#389—22no—
80#389—22no—
81#388—24no—
82#388—24no—
83#386—24no—
84#386—24no—
85C4×C4⋊C4D414yes—
86C4×C4⋊C4D412yes—
87C4×C4⋊C4D412yes—
88C4×C22⋊C4—24no—
89C4×C22⋊C4—24no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C2×#95C2×C412yes—
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C412yes—
95C2×#95C2311yescharacteristic
96#336—22no—
97#336—24no—
98#336—22no—
99#336—24no—
100#323—28no—
101#323—24no—
102#323—44no—
103#323—42no—
104C25C22×C412yes—
105C25C22×C411yescharacteristic
106C22×D4—24no—
107C22×D4—24no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—24no—
111C22×D4—22no—
112C22×D4—42no—
113C22×D4—42no—
114C22×D4—82no—
115C22×D4—82no—
116C22×D4M1614yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—44no—
127C23×C4—82no—
128C23×C4M1614yes—
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—24no—
141#121—216no—
142#121—216no—
143C4×D4—48no—
144C4×D4—44no—
145C4×D4—44no—
146#117—44no—
147#117—48no—
148#117—44no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—82no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—42no—
189C2×C42—44no—
190C2×C42—42no—
191C2×C42—82no—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D414yes—
195C2×C42C2×D414yes—
196C4⋊C8—44no—
197C4⋊C8—84no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—24no—
201C22⋊C8—22no—
202C22⋊C8—24no—
203C22⋊C8#4114yes—
204#97—84no—
205C4×C8—44no—
206#95—24no—
207#95—22no—
208#95—22no—
209#95C22⋊C414yes—
210#95C22⋊C414yes—
211C24—22no—
212C24—21no—
213C24—22no—
214C24—21no—
215C24—21no—
216C24—22no—
217C24—21no—
218C24—22no—
219C24—21no—
220C24—22no—
221C24—21no—
222C24—22no—
223C24—41no—
224C24—42no—
225C24—42no—
226C24—41no—
227C24—42no—
228C24—42no—
229C24—41no—
230C24—41no—
231C24C2×M1614yes—
232C24C2×M1612yes—
233C24C23×C411yescharacteristic
234C2×D4—216no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—48no—
240C2×D4—48no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—84no—
244C2×D4—84no—
245C2×D4—88no—
246C2×D4—88no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—88no—
296C22×C4—88no—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C414yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×M1612yes—
304C22×C4C2×M1612yes—
305C22×C4C2×M1612yes—
306C22×C4#13112yes—
307C22×C4C22×D412yes—
308C22×C4C2×#4112yes—
309C22×C4H(5,𝔽2)11yescharacteristic
310C22×C42−1+411yescharacteristic
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—44no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—84no—
327C4⋊C4#10014yes—
328C22⋊C4—216no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—84no—
339C42—24no—
340C42—44no—
341C42—44no—
342C42—42no—
343C42—44no—
344C42—42no—
345C42—44no—
346C42—42no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—22no—
356C23—24no—
357C23—42no—
358C23—44no—
359C23—42no—
360C23—44no—
361C23—42no—
362C23—44no—
363C23—42no—
364C23—44no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—42no—
378C23—41no—
379C23—42no—
380C23—41no—
381C23—82no—
382C23—82no—
383C23—81no—
384C23—82no—
385C23—81no—
386C23—82no—
387C23—81no—
388C23—82no—
389C23—81no—
390C23—82no—
391C23—81no—
392C23—82no—
393C23—81no—
394C23—82no—
395C23—81no—
396C23—81no—
397C23C22×C22⋊C411yescharacteristic
398C23C4×#4111yescharacteristic
399C23#52011yescharacteristic
400C23C22×M1611yescharacteristic
401C23C22×M1611yescharacteristic
402C23C2×#13111yescharacteristic
403C23#56811yescharacteristic
404D4—48no—
405D4—48no—
406D4—816no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—48no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—88no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—88no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4#40712yes—
458C2×C4#40712yes—
459C2×C4#41012yes—
460C2×C4C2×#10014yes—
461C2×C4C2×#10012yes—
462C2×C4#54812yes—
463C2×C4#55012yes—
464C8—44no—
465C8—82no—
466C8—84no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—44no—
480C22—42no—
481C22—42no—
482C22—44no—
483C22—42no—
484C22—44no—
485C22—82no—
486C22—84no—
487C22—82no—
488C22—84no—
489C22—84no—
490C22—82no—
491C22—84no—
492C22—82no—
493C22—81no—
494C22—82no—
495C22—81no—
496C22—82no—
497C22—81no—
498C22—82no—
499C22—82no—
500C22—81no—
501C22#234111yescharacteristic
502C22C2×#40711yescharacteristic
503C22#288011yescharacteristic
504C22C2×#41011yescharacteristic
505C22#288311yescharacteristic
506C22C22×#10011yescharacteristic
507C22#288811yescharacteristic
508C22#296511yescharacteristic
509C22#297011yescharacteristic
510C22#298911yescharacteristic
511C22#299011yescharacteristic
512C4—22no—
513C4—22no—
514C4—24no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—44no—
521C4—81no—
522C4—82no—
523C4#146614yes—
524C2—21no—
525C2—21no—
526C2—21no—
527C2—21no—
528C2—82no—
529C2—84no—
530C2C2×#146611yescharacteristic
531C2#1736911yescharacteristic
532C2#1741811yescharacteristic
533C2#1742111yescharacteristic
534C2#2253211yescharacteristic
535C2#2253511yescharacteristic
536C2#2253611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ 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. #22629C2semidirect
3. #9653C2semidirect
4. #9649C2semidirect
5. #8557C2semidirect
6. #8557C2semidirect
7. #8152C2semidirect
8. C4×#323C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23632
#226292
#96532
#96492
#85572
#81522
C4×#3232