Finite Group Databasebeta

← #222894

Subgroups — #222894

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2363C211yesmaximal, characteristic
2D4×C22⋊C8C212yesmaximal, complemented
3#9879C212yesmaximal, complemented
4#9079C214yesmaximal, complemented
5#8700C212yesmaximal, complemented
6#8557C211yesmaximal, characteristic, complemented
7C4×#323C211yesmaximal, characteristic, complemented
8#7907C212yesmaximal, complemented
9C22×#121C411yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C22×#117C411yescharacteristic
12C2×C8×D4—22no—
13C22×C22⋊C8C2214yes—
14#2363—22no—
15#2363—24no—
16#2363—22no—
17C2×#390C412yes—
18C2×#390C411yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#388C412yes—
21C2×#386C2212yes—
22C2×C4×C4⋊C4C411yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1954—22no—
26#1869—24no—
27#1797C2212yes—
28#1779C2214yes—
29C4×C22⋊C8C2212yes—
30C2×#323C2212yes—
31C2×#323C2212yes—
32#1280C2212yes—
33#1280C2214yes—
34#1280C2214yes—
35#1280C2211yescharacteristic
36#1280C2211yescharacteristic
37C23×D4C2×C412yes—
38C23×D4C2×C411yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2312yes—
41C23×C8—24no—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—24no—
45C2×#121C818yescomplemented
46C2×C4×D4—44no—
47C2×C4×D4D412yes—
48C2×C4×D4D414yes—
49C2×C4×D4D412yes—
50C2×#117—21no—
51C2×#117—22no—
52C2×#117—21no—
53C2×#117—44no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2311yescharacteristic
65C22×C42C2×C412yes—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C412yes—
68C22×C42C2311yescharacteristic
69C8×D4—48no—
70C2×C4⋊C8—22no—
71C2×C22⋊C8—28no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—44no—
76C2×C22⋊C8C2314yes—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79C2×C4×C8—22no—
80#390—22no—
81#390—22no—
82#390—24no—
83#390—24no—
84#389—22no—
85#389—22no—
86#388—24no—
87#388—24no—
88#386—24no—
89#386—24no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—21no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C412yes—
99C2×#95C2311yescharacteristic
100C2×#95C2312yes—
101#336—22no—
102#336—22no—
103#323—24no—
104#323—42no—
105C25C22×C411yescharacteristic
106C25C22×C412yes—
107C22×D4—24no—
108C22×D4—24no—
109C22×D4—42no—
110C22×D4—42no—
111C22×D4—82no—
112C22×D4—82no—
113C22×D4C22⋊C412yes—
114C22×D4C22⋊C412yes—
115C22×D4C2×C814yes—
116C22×D4C2×C818yescomplemented
117C22×D4M1614yes—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—44no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—82no—
128C23×C4C2×C814yescomplemented
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D414yes—
138C23×C4C2×D414yes—
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—24no—
142C22×C8—22no—
143C22×C8—48no—
144C22×C8—48no—
145C22×C8—84no—
146#121—216no—
147#121—216no—
148C4×D4—28no—
149C4×D4—44no—
150C4×D4—48no—
151#117—44no—
152#117—48no—
153#117—44no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C2×C814yescomplemented
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4C22⋊C414yes—
186C2×C22⋊C4C22⋊C414yes—
187C2×C22⋊C4C2×C818yescomplemented
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—42no—
195C2×C42—44no—
196C2×C42—42no—
197C2×C42—82no—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C4⋊C8—44no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—24no—
204C22⋊C8—24no—
205C22⋊C8—22no—
206C22⋊C8—48no—
207C22⋊C8—48no—
208C22⋊C8—84no—
209C22⋊C8#4114yes—
210C4×C8—44no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—24no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—41no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24C2×C22⋊C414yes—
235C24C22×C812yes—
236C24C22×C814yes—
237C24C2×M1614yes—
238C24C23×C411yescharacteristic
239C2×D4—216no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—48no—
243C2×D4—48no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—84no—
247C2×D4—84no—
248C2×D4—88no—
249C2×D4—88no—
250C2×D4C22⋊C818yes—
251C2×D4C22⋊C818yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—41no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—48no—
289C22×C4—48no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—88no—
303C22×C4—84no—
304C22×C4—84no—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C22×C812yes—
311C22×C4C22×C812yes—
312C22×C4C22×C814yes—
313C22×C4C2×M1612yes—
314C22×C4#13112yes—
315C22×C4C22×D412yes—
316C22×C4C2×#4112yes—
317C22×C4H(5,𝔽2)11yescharacteristic
318C22×C42−1+411yescharacteristic
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—44no—
324C2×C8—42no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—42no—
328C2×C8—816no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—84no—
336C4⋊C4#9914yes—
337C22⋊C4—216no—
338C22⋊C4—28no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—48no—
342C22⋊C4—44no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—44no—
346C22⋊C4—84no—
347C42—24no—
348C42—42no—
349C42—42no—
350C42—44no—
351C42—44no—
352C42—42no—
353C42—82no—
354C42—82no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23C2×C22⋊C818yes—
405C23C22×C22⋊C411yescharacteristic
406C23C4×#4111yescharacteristic
407C23#52011yescharacteristic
408C23C23×C811yescharacteristic
409C23C22×M1611yescharacteristic
410C23C2×#13111yescharacteristic
411C23#56811yescharacteristic
412D4—216no—
413D4—48no—
414D4—816no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—48no—
436C2×C4—48no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—48no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4C2×C22⋊C812yes—
468C2×C4C2×C22⋊C812yes—
469C2×C4C2×#9914yes—
470C2×C4C2×#9912yes—
471C2×C4#41312yes—
472C2×C4#54612yes—
473C2×C4#54712yes—
474C8—44no—
475C8—82no—
476C8—84no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—28no—
490C22—28no—
491C22—28no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—84no—
500C22—84no—
501C22—84no—
502C22—84no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22#233911yescharacteristic
512C22C22×C22⋊C811yescharacteristic
513C22#288011yescharacteristic
514C22C22×#9911yescharacteristic
515C22#288311yescharacteristic
516C22C2×#41311yescharacteristic
517C22#288811yescharacteristic
518C22C8×#4111yescharacteristic
519C22#297011yescharacteristic
520C22#298811yescharacteristic
521C22#299011yescharacteristic
522C4—22no—
523C4—22no—
524C4—22no—
525C4—24no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—44no—
530C4—42no—
531C4—82no—
532C4—81no—
533C4#145914yes—
534C2—21no—
535C2—21no—
536C2—21no—
537C2—21no—
538C2—48no—
539C2—82no—
540C2C2×#145911yescharacteristic
541C2#1735011yescharacteristic
542C2#1741911yescharacteristic
543C2#1742411yescharacteristic
544C2#2246911yescharacteristic
545C2#2247611yescharacteristic
546C2#2248311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×C22⋊C8C2semidirect
3. #9879C2semidirect
4. #9079C2semidirect
5. #8700C2semidirect
6. #8557C2semidirect
7. C4×#323C2semidirect
8. #7907C2semidirect
45. C2×#121C8semidirect
116. C22×D4C2×C8semidirect
128. C23×C4C2×C8semidirect
166. C2×C4⋊C4C2×C8semidirect
187. C2×C22⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23632
D4×C22⋊C82
#98792
#90792
#87002
#85572
C4×#3232
#79072