Finite Group Databasebeta

← #219795

Subgroups — #219795

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2512C211yesmaximal, characteristic
2#22601C212yesmaximal, complemented
3C2×#1473C212yesmaximal, complemented
4#9653C212yesmaximal
5#9084C212yesmaximal, complemented
6#9080C212yesmaximal, complemented
7#8560C211yesmaximal, characteristic, complemented
8#8309C212yesmaximal, complemented
9#8153C211yesmaximal, characteristic, complemented
10C22×#126C2212yes—
11C2×#432C2212yes—
12C2×#431C2212yes—
13#2512—22no—
14#2512—22no—
15#2512—22no—
16#2512—22no—
17C2×#397C412yes—
18C2×#396C412yes—
19C2×#390C411yescharacteristic
20C2×#390C411yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C411yescharacteristic
23C2×#388C2212yes—
24C2×#384C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1940—22no—
28#1917—22no—
29#1916—22no—
30#1817C2212yes—
31#1817C2212yes—
32#1797C2212yes—
33#1771—22no—
34#1749C2212yes—
35#1473—28no—
36C2×#324C2212yes—
37C2×#324C2211yescharacteristic
38C2×#324C2211yescharacteristic
39C2×#323C2212yes—
40C2×#323C2212yes—
41C2×#323C2212yes—
42C2×#323C2212yes—
43#1280C2211yescharacteristic
44#1280C2211yescharacteristic
45#1280C2212yes—
46C23×D4C2×C411yescharacteristic
47C2×#126—24no—
48C2×#126—24no—
49C2×#126—22no—
50C2×#126—22no—
51C2×#126—22no—
52C2×#126—22no—
53C2×#126—42no—
54C2×#126—42no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2312yes—
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73#432—24no—
74#432—42no—
75#431—24no—
76#431—42no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82C2×C22⋊C8C2312yes—
83C2×C22⋊C8C2312yes—
84C2×#97—22no—
85C2×#97—22no—
86C2×C4×C8—22no—
87#397—22no—
88#397—22no—
89#397—22no—
90#397—22no—
91#396—22no—
92#396—22no—
93#396—22no—
94#396—22no—
95#390—21no—
96#390—21no—
97#390—22no—
98#390—22no—
99#390—21no—
100#390—21no—
101#390—22no—
102#390—22no—
103#389—22no—
104#389—21no—
105#389—21no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#384—22no—
111#384—21no—
112#384—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4D412yes—
119C2×#95C2×C411yescharacteristic
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#324—24no—
130#324—22no—
131#324D414yescomplemented
132#323—24no—
133#323—24no—
134#323—42no—
135#323—42no—
136C25C22×C412yes—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—41no—
140C22×D4—41no—
141C22×D4—42no—
142C22×D4—42no—
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157#126—44no—
158#126—44no—
159#126—44no—
160#126—44no—
161#126—44no—
162#126—44no—
163#126—44no—
164#126—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4#4112yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—42no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42—42no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C4⋊C8—44no—
238C4⋊C8—44no—
239C4⋊C8—44no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8C2×D414yes—
248#97—44no—
249#97—44no—
250C4×C8—44no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24C23×C411yescharacteristic
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—84no—
278C2×D4—84no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C4×D412yes—
331C22×C4C4×D412yes—
332C22×C4#13112yes—
333C22×C4C22×D412yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4H(5,𝔽2)12yes—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—24no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23C22×C22⋊C411yescharacteristic
422C23C2×C4×D411yescharacteristic
423C23#51811yescharacteristic
424C23C2×#13111yescharacteristic
425C23#56811yescharacteristic
426C23#56811yescharacteristic
427C23#56811yescharacteristic
428D4—84no—
429D4—84no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4#41012yes—
492C2×C4#41312yes—
493C2×C4D4212yes—
494C2×C4#54612yes—
495C8—44no—
496C8—82no—
497C8—82no—
498C8—82no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—162no—
519C22—162no—
520C22—162no—
521C22—162no—
522C22D4×C22⋊C411yescharacteristic
523C22#287911yescharacteristic
524C22#288011yescharacteristic
525C22C2×#41011yescharacteristic
526C22#288411yescharacteristic
527C22C2×#41311yescharacteristic
528C22#288811yescharacteristic
529C22#293211yescharacteristic
530C22#293511yescharacteristic
531C22#298011yescharacteristic
532C22#298111yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—81no—
546C4—82no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—162no—
552C2#1739811yescharacteristic
553C2#1740911yescharacteristic
554C2#1741511yescharacteristic
555C2#1742811yescharacteristic
556C2#2240311yescharacteristic
557C2#2240611yescharacteristic
558C2#2241211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
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. #22601C2semidirect
3. C2×#1473C2semidirect
5. #9084C2semidirect
6. #9080C2semidirect
7. #8560C2semidirect
8. #8309C2semidirect
9. #8153C2semidirect
131. #324D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25122
#226012
C2×#14732
#96532
#90842
#90802
#85602
#83092
#81532