Finite Group Databasebeta

← #238268

Subgroups — #238268

Order 512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35330C211yesmaximal, characteristic, complemented
2#21212C211yesmaximal, characteristic
3#17574C211yesmaximal, characteristic
4C2×#1517C211yesmaximal, characteristic
5#10374C218yesmaximal, complemented
6#8026C211yesmaximal, characteristic, complemented
7#7981C211yesmaximal, characteristic, complemented
8#7924C211yesmaximal, characteristic, complemented
9C2×#544—21no—
10C2×#532C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#410—21no—
14C2×#410—21no—
15#2862—21no—
16#2859—21no—
17#2660—22no—
18#2550—22no—
19#2550—22no—
20#2549—22no—
21#2549C2214yes—
22#2317—21no—
23#2285C2211yescharacteristic
24#2285C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1964—22no—
28#1964C2214yes—
29#1519C2214yes—
30#1517—22no—
31#1517—22no—
32#1517—21no—
33#1517—21no—
34#1502—22no—
35C2×#327C2214yes—
36#1294C2214yes—
37#1289C2214yes—
38#1282C2214yes—
39C22×#41—21no—
40#544—44no—
41#544—44no—
42#532—21no—
43#532—21no—
44#532—21no—
45#532—21no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8Q811yescharacteristic, complemented
53C2×#127Q811yescharacteristic, complemented
54C2×#125—22no—
55C2×#125—22no—
56C2×#124—21no—
57C2×C22⋊Q8—24no—
58C2×#121—21no—
59C2×#121C2312yes—
60C2×C4×D4—21no—
61C2×C4×D4Q812yescomplemented
62C2×#117—21no—
63C2×#117—21no—
64C2×#117D411yescharacteristic
65C2×#117D411yescharacteristic
66C22×C4⋊C4—21no—
67C22×C42C2311yescharacteristic
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78#410—44no—
79#410—44no—
80C2×#99—24no—
81C2×#99—24no—
82C2×C22⋊C8C2312yes—
83C2×C22⋊C8C2312yes—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86#398—24no—
87#398—24no—
88#395—24no—
89#395—24no—
90#391—24no—
91#391—24no—
92#389—22no—
93#389—22no—
94#389—24no—
95#389—24no—
96#385—24no—
97#384—22no—
98#384—21no—
99#384—21no—
100#382—22no—
101#382—21no—
102#382—21no—
103#382—21no—
104#382D411yescharacteristic
105#382D411yescharacteristic
106#382D412yes—
107#382D412yes—
108#380—21no—
109#380—21no—
110#380C2312yes—
111#380C2312yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4D412yes—
118C4×C22⋊C4D412yes—
119C4×C22⋊C4C2312yes—
120#376—21no—
121C2×#95—21no—
122C2×#95C2312yes—
123C43—21no—
124#327—28no—
125#327—24no—
126#327—24no—
127C2×#41—41no—
128C2×#41—41no—
129C2×#41—41no—
130C2×#41—41no—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41—42no—
134C2×#41—42no—
135C22×Q8—24no—
136C22×Q8—21no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C2×Q812yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—48no—
153C4⋊Q8—48no—
154#127—21no—
155#127—21no—
156#127—21no—
157#127—21no—
158#125—48no—
159#125—48no—
160#124—41no—
161#124—41no—
162#124—41no—
163#124—41no—
164C22⋊Q8—48no—
165C22⋊Q8—48no—
166#121—22no—
167#121—22no—
168#121—22no—
169#121—22no—
170#121—41no—
171#121—41no—
172#121—41no—
173#121—41no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182#117—22no—
183#117—22no—
184#117—42no—
185#117—42no—
186#117—42no—
187#117—42no—
188#117D814yes—
189#117SD1614yes—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C2×Q812yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4C2×D412yes—
215C2×C22⋊C4C2×D412yes—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—41no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×Q811yescharacteristic
227C2×C42C2×Q811yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C4⋊C8—41no—
235C4⋊C8—41no—
236C4⋊C8—41no—
237C4⋊C8—41no—
238#99—48no—
239#99—48no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C4×C8—41no—
245C4×C8—41no—
246C4×C8—41no—
247C4×C8—41no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—21no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—41no—
259#95C2×D414yes—
260#95C2×D412yes—
261#95C2×D412yes—
262C24—22no—
263C24C22×D411yescharacteristic
264#41—82no—
265#41—82no—
266#41—82no—
267#41—82no—
268#41—82no—
269#41—82no—
270#41—82no—
271#41—82no—
272C2×Q8—48no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—82no—
294C2×D4—82no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4C22⋊Q811yescharacteristic
317C22×C4C22⋊Q811yescharacteristic
318C22×C4C22⋊Q811yescharacteristic
319C22×C4C22⋊Q811yescharacteristic
320C22×C4#12411yescharacteristic
321C22×C4#12411yescharacteristic
322C22×C4#12411yescharacteristic
323C22×C4#12411yescharacteristic
324C22×C4C2×D812yes—
325C22×C4C2×SD1612yes—
326C22×C4AGL(1,ℤ/8)12yes—
327C22×C4#13712yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×Q811yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—82no—
337C2×C8—82no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—48no—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—84no—
354C4⋊C4—84no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4C2×D814yes—
369C22⋊C4C2×SD1614yes—
370C42—24no—
371C42—24no—
372C42—44no—
373C42—82no—
374C42—82no—
375C42—82no—
376C42C22⋊Q811yescharacteristic
377C42C22⋊Q811yescharacteristic
378C42C22⋊Q811yescharacteristic
379C42C22⋊Q811yescharacteristic
380C42C22⋊Q811yescharacteristic
381C42C22⋊Q811yescharacteristic
382C42C22⋊Q811yescharacteristic
383C42C22⋊Q811yescharacteristic
384C42#12411yescharacteristic
385C42#12411yescharacteristic
386C42#12411yescharacteristic
387C42#12411yescharacteristic
388C42#12411yescharacteristic
389C42#12411yescharacteristic
390C42#12411yescharacteristic
391C42#12411yescharacteristic
392C23—41no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—82no—
398C23—82no—
399C23C2×#12011yescharacteristic
400C23C2×C22⋊Q811yescharacteristic
401C23C2×#12411yescharacteristic
402C23C22×D811yescharacteristic
403C23C22×SD1611yescharacteristic
404C23C2×AGL(1,ℤ/8)11yescharacteristic
405C23C2×#13711yescharacteristic
406Q8—82no—
407Q8—82no—
408Q8—168no—
409D4—82no—
410D4—82no—
411D4—82no—
412D4—82no—
413D4—82no—
414D4—82no—
415D4—82no—
416D4—82no—
417D4—82no—
418D4—82no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—162no—
455C2×C4—162no—
456C2×C4#39311yescharacteristic
457C2×C4#39311yescharacteristic
458C2×C4#39311yescharacteristic
459C2×C4#39311yescharacteristic
460C2×C4#39311yescharacteristic
461C2×C4#39311yescharacteristic
462C2×C4#39311yescharacteristic
463C2×C4#39311yescharacteristic
464C2×C4#44712yes—
465C2×C4#44712yes—
466C2×C4#44812yes—
467C2×C4#44812yes—
468C2×C4#47412yes—
469C2×C4#47412yes—
470C2×C4#47612yes—
471C2×C4#47612yes—
472C2×C4#48612yes—
473C2×C4#48612yes—
474C2×C4#48912yes—
475C2×C4#48912yes—
476C2×C4C2×C22⋊Q811yescharacteristic
477C2×C4C2×C22⋊Q811yescharacteristic
478C2×C4C2×#12411yescharacteristic
479C2×C4C2×#12411yescharacteristic
480C8—82no—
481C8—82no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—162no—
496C22—162no—
497C22C2×#39311yescharacteristic
498C22C2×#44711yescharacteristic
499C22C2×#44811yescharacteristic
500C22C2×#47411yescharacteristic
501C22C2×#47611yescharacteristic
502C22C2×#48611yescharacteristic
503C22C2×#48911yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—44no—
513C4—44no—
514C4—82no—
515C4—84no—
516C4—82no—
517C4—82no—
518C4—162no—
519C4#202312yes—
520C4#202312yes—
521C4#202312yes—
522C4#202312yes—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—21no—
527C2—21no—
528C2—21no—
529C2—162no—
530C2C2×#202311yescharacteristic
531C2#2347211yescharacteristic
532C2#2347411yescharacteristic
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35330C2semidirect
5. #10374C2semidirect
6. #8026C2semidirect
7. #7981C2semidirect
8. #7924C2semidirect
52. C2×C4⋊Q8Q8semidirect
53. C2×#127Q8semidirect
61. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#353302
#212122
#175742
C2×#15172
#103742
#80262
#79812
#79242