Finite Group Databasebeta

← #220096

Subgroups — #220096

Order 512.

1223 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic, complemented
3#22650C211yesmaximal, characteristic, complemented
4C2×#1557C212yesmaximal, complemented
5#9694C211yesmaximal, characteristic, complemented
6#9692C211yesmaximal, characteristic, complemented
7#9140C212yesmaximal, complemented
8#9137C212yesmaximal, complemented
9#8585C211yesmaximal, characteristic, complemented
10#8341C212yesmaximal, complemented
11C4×#331C211yesmaximal, characteristic, complemented
12C2×#547—21no—
13C2×#532C2212yes—
14C2×#436—21no—
15C2×#432C2212yes—
16C2×#423C2212yes—
17#2868C2211yescharacteristic
18#2857C2212yes—
19#2547—21no—
20#2537—21no—
21#2537C412yes—
22#2432—22no—
23#2422—22no—
24#2422C414yes—
25D4×C4⋊C4—21no—
26D4×C4⋊C4C412yes—
27C4×#127C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1939—21no—
31#1924—22no—
32#1851—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1797C2212yes—
37#1770—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1557—28no—
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#331C2211yescharacteristic
46C2×#331C2212yes—
47C2×#331C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2211yescharacteristic
50#1296C2212yes—
51C22×#41—22no—
52C22×M16—22no—
53#547—44no—
54#547—44no—
55#532—24no—
56#532—24no—
57#532—42no—
58#532—42no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8C2312yes—
62C2×#127C2311yescharacteristic
63C2×#124—22no—
64C2×#123—24no—
65C2×C22⋊Q8—22no—
66C2×#121—22no—
67C2×#121—22no—
68C2×#121C2×C414yes—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—22no—
72C2×C4×D4C2×C414yes—
73C2×C4×D4C2×C412yes—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C22×C4⋊C4—22no—
78C22×C42C2312yes—
79#436—44no—
80#432—24no—
81#432—42no—
82#423—24no—
83#423—42no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×#97—22no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101#397—24no—
102#394—24no—
103#394—24no—
104#391—22no—
105#390—21no—
106#390—21no—
107#390C2×C412yes—
108#389—22no—
109#389C2×C414yes—
110#385—24no—
111#385—24no—
112#384—21no—
113#384—21no—
114#384C2×C412yes—
115#384C2×C412yes—
116#384C2311yescharacteristic
117#380—22no—
118#380—22no—
119#380—22no—
120#380C2×C414yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4C2×C412yes—
125C4×C22⋊C4D412yes—
126C4×C22⋊C4D412yes—
127C43C2311yescharacteristic
128#336—22no—
129#336—22no—
130#336—21no—
131#336—21no—
132#332—24no—
133#332—24no—
134#332—42no—
135#332—42no—
136#331—24no—
137#331—22no—
138#331D414yescomplemented
139C2×#41—44no—
140C2×#41—44no—
141C2×#41—44no—
142C2×#41—44no—
143C2×#41—82no—
144C2×#41—82no—
145C22×Q8—22no—
146C22×D4—24no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22⋊C411yescharacteristic
150C22×D4C22×C412yes—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4C22×C412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C2×M16—44no—
157C2×M16—44no—
158C2×M16—82no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—81no—
172C4⋊Q8—81no—
173#127—21no—
174#127—21no—
175#127C2×D412yes—
176#127#4112yes—
177#124—44no—
178#124—44no—
179#123—416no—
180C22⋊Q8—44no—
181C22⋊Q8—44no—
182#121—28no—
183#121—44no—
184#121—44no—
185#121—44no—
186#121—48no—
187C4×D4—28no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—42no—
193C4×D4—44no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—42no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22×C414yes—
214C2×C4⋊C4C22×C412yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22⋊C412yes—
223C2×C22⋊C4C22⋊C412yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C412yes—
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2411yescharacteristic
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—44no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—81no—
252C4⋊C8—81no—
253C4⋊C8C2×D412yes—
254C4⋊C8C2×D412yes—
255C22⋊C8—44no—
256C22⋊C8—44no—
257C22⋊C8—82no—
258#97—44no—
259C4×C8—42no—
260C4×C8—42no—
261#95—24no—
262#95—24no—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266C24—22no—
267C24C2×C22⋊C412yes—
268#41—88no—
269#41—88no—
270#41—88no—
271#41—88no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—48no—
289C2×D4—48no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—84no—
299C2×D4C4×D414yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—84no—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334M16—88no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—41no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—42no—
359C4⋊C4—48no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—42no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—44no—
376C22⋊C4—48no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—22no—
390C42—41no—
391C42AGL(1,ℤ/8)12yes—
392C42AGL(1,ℤ/8)12yes—
393C42AGL(1,ℤ/8)12yes—
394C42#13712yes—
395C42C22×D411yescharacteristic
396C42C2×#4111yescharacteristic
397C42H(5,𝔽2)11yescharacteristic
398C42H(5,𝔽2)11yescharacteristic
399C23—24no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23C22×C22⋊C411yescharacteristic
414Q8—82no—
415Q8—82no—
416Q8—82no—
417Q8—82no—
418D4—44no—
419D4—44no—
420D4—42no—
421D4—42no—
422D4—44no—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—42no—
427D4—42no—
428D4—42no—
429D4—816no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—816no—
465C2×C4—88no—
466C2×C4#41312yes—
467C2×C4#41712yes—
468C2×C4#41712yes—
469C2×C4#41812yes—
470C2×C4#41812yes—
471C2×C4C2×C4×D411yescharacteristic
472C2×C4#51811yescharacteristic
473C2×C4D4211yescharacteristic
474C2×C4D4211yescharacteristic
475C2×C4#54611yescharacteristic
476C2×C4#54611yescharacteristic
477C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
478C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
479C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
480C2×C4C2×#13711yescharacteristic
481C2×C4#57611yescharacteristic
482C2×C4#57711yescharacteristic
483C2×C4#57711yescharacteristic
484C2×C4#57811yescharacteristic
485C8—42no—
486C8—42no—
487C8—81no—
488C8—82no—
489C8—82no—
490C8—81no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—42no—
498C22—44no—
499C22—42no—
500C22—44no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22D4×C22⋊C411yescharacteristic
506C22C2×#41311yescharacteristic
507C22#288911yescharacteristic
508C22C2×#41811yescharacteristic
509C22C2×#41711yescharacteristic
510C22#289711yescharacteristic
511C22#289811yescharacteristic
512C4—22no—
513C4—22no—
514C4—21no—
515C4—21no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—82no—
521C4—84no—
522C4#155714yescomplemented
523C4#327211yescharacteristic
524C4#327211yescharacteristic
525C4#327411yescharacteristic
526C4#327611yescharacteristic
527C4#332411yescharacteristic
528C4#332511yescharacteristic
529C4#333011yescharacteristic
530C4#333111yescharacteristic
531C2—82no—
532C2—84no—
533C2C2×#155711yescharacteristic
534C2#1775911yescharacteristic
535C2#1776511yescharacteristic
536C2#1777211yescharacteristic
537C2#2241111yescharacteristic
538C2#2242811yescharacteristic
539C2#2243111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22799C2semidirect
3. #22650C2semidirect
4. C2×#1557C2semidirect
5. #9694C2semidirect
6. #9692C2semidirect
7. #9140C2semidirect
8. #9137C2semidirect
9. #8585C2semidirect
10. #8341C2semidirect
11. C4×#331C2semidirect
138. #331D4semidirect
522. C4#1557semidirect

Maximal subgroups

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

subgroupindex
#361272
#227992
#226502
C2×#15572
#96942
#96922
#91402
#91372
#85852
#83412
C4×#3312