Finite Group Databasebeta

← #227047

Subgroups — #227047

Order 512.

724 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#23061C211yesmaximal, characteristic
2#22965C211yesmaximal, characteristic, complemented
3#22522C211yesmaximal, characteristic
4#22502C211yesmaximal, characteristic
5C2×#1632C211yesmaximal, characteristic
6#9839C211yesmaximal, characteristic, complemented
7#9568C211yesmaximal, characteristic, complemented
8#9567C211yesmaximal, characteristic, complemented
9#9224C211yesmaximal, characteristic, complemented
10#9223C211yesmaximal, characteristic
11#9066C211yesmaximal, characteristic, complemented
12#8889C211yesmaximal, characteristic
13#8888C211yesmaximal, characteristic, complemented
14#8213C211yesmaximal, characteristic, complemented
15#8212C211yesmaximal, characteristic
16C2×#554C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×#477C2211yescharacteristic
19C2×#476C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×#436C2211yescharacteristic
22#2369C2211yescharacteristic
23#2354C2211yescharacteristic
24Q8×C4⋊C4C2211yescharacteristic
25#1992—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1989—21no—
31#1987C2211yescharacteristic
32#1987C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1942—21no—
36#1934—21no—
37#1926—21no—
38#1921—21no—
39#1920—21no—
40#1894—21no—
41#1873—21no—
42#1863—21no—
43#1844C2211yescharacteristic
44#1823—21no—
45#1803C2211yescharacteristic
46#1802—21no—
47#1802C2211yescharacteristic
48C4×#106—21no—
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1762—21no—
52C4×#103—21no—
53C4×#103C2211yescharacteristic
54C4×#102C2211yescharacteristic
55#1736—21no—
56#1632C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×Q16—41no—
69C22×M16—21no—
70#554—22no—
71#554—22no—
72#554—41no—
73#554C2×C414yes—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×#125—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—41no—
79C2×C4×Q8—21no—
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4C2311yescharacteristic
83C2×#117—21no—
84C22×C4⋊C4—21no—
85#494—41no—
86#494C2×C414yes—
87#477—22no—
88#477C2×C414yes—
89#476—22no—
90#476C2×C414yes—
91#446—22no—
92#446—41no—
93#436—22no—
94#436—41no—
95C2×#107—21no—
96C2×#107—21no—
97C2×#106—21no—
98C2×#106—21no—
99C2×#106—41no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103D411yescharacteristic
108C2×#103D411yescharacteristic
109C2×#102—21no—
110C2×#102—21no—
111C2×#102D411yescharacteristic
112C2×#102D411yescharacteristic
113C2×C22⋊C8—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#391—41no—
121#390—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—41no—
126#388—41no—
127#384—21no—
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340D411yescharacteristic
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#336—21no—
158#336—21no—
159#333—41no—
160#333C2×C414yes—
161Q8⋊C8C2×C414yes—
162D4⋊C8C2×C414yes—
163#322—41no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8C2×D411yescharacteristic
167C22×D4C2×D411yescharacteristic
168C23×C4—21no—
169C23×C4—41no—
170C2×Q16—44no—
171C2×Q16—82no—
172C2×Q16—82no—
173C2×M16—42no—
174C2×M16—42no—
175C2×M16—81no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8C22×C412yes—
192C4⋊Q8C22×C412yes—
193#125—24no—
194#125—42no—
195C22⋊Q8—24no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C22⋊Q8—42no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—42no—
202C4×Q8—81no—
203C4×Q8C22×C412yes—
204C4×Q8#4112yes—
205C4×D4—24no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—42no—
209C4×D4C22×C412yes—
210C4×D4#4112yes—
211#117—42no—
212#117—81no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42C2411yescharacteristic
248#107—24no—
249#107—81no—
250#106—24no—
251#106—42no—
252#106—82no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—42no—
260C4⋊C8C22×C412yes—
261C4⋊C8C22×C412yes—
262#103—24no—
263#103—22no—
264#103—41no—
265#103—81no—
266#102—24no—
267#102—22no—
268#102—41no—
269#102—42no—
270C22⋊C8—42no—
271#97—42no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—42no—
278C4×C8—81no—
279C4×C8C22×C412yes—
280#95—21no—
281#95—21no—
282#95—41no—
283#95—41no—
284#95—41no—
285#95—41no—
286C24—21no—
287C2×Q8—21no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—81no—
297C2×Q8—81no—
298C2×Q8—81no—
299C2×Q8—81no—
300C2×Q8C4×D412yes—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4C4×D412yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—44no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—161no—
326C22×C4—161no—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330Q16—88no—
331M16—84no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C4⋊C4—82no—
390C4⋊C4—161no—
391C4⋊C4—161no—
392C4⋊C4C4×D412yes—
393C4⋊C4C4×D412yes—
394C4⋊C4#13712yes—
395C4⋊C4#13712yes—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—44no—
399C22⋊C4—82no—
400C22⋊C4—161no—
401C42—22no—
402C42—22no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—82no—
408C42—82no—
409C42—161no—
410C42C23×C411yescharacteristic
411C42C2×#4111yescharacteristic
412C42C2×#4111yescharacteristic
413C42H(5,𝔽2)11yescharacteristic
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23D4211yescharacteristic
422Q8—22no—
423Q8—22no—
424Q8—41no—
425Q8—41no—
426Q8—82no—
427Q8—82no—
428Q8—161no—
429Q8—161no—
430D4—22no—
431D4—22no—
432D4—41no—
433D4—41no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—164no—
474C2×C4C2×C4×D411yescharacteristic
475C2×C4C2×C4×D411yescharacteristic
476C2×C4#51811yescharacteristic
477C2×C4#54611yescharacteristic
478C2×C4#54611yescharacteristic
479C2×C4#55111yescharacteristic
480C2×C4C2×#13711yescharacteristic
481C2×C4C2×#13711yescharacteristic
482C2×C4#57511yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57711yescharacteristic
485C2×C4#57711yescharacteristic
486C2×C4#57811yescharacteristic
487C2×C4#57811yescharacteristic
488C8—24no—
489C8—41no—
490C8—41no—
491C8—42no—
492C8—42no—
493C8—41no—
494C8—41no—
495C8—81no—
496C8—81no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22#235811yescharacteristic
505C22#326611yescharacteristic
506C22#326711yescharacteristic
507C22#326911yescharacteristic
508C22#327011yescharacteristic
509C22#329111yescharacteristic
510C22#329211yescharacteristic
511C4—21no—
512C4—21no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—82no—
519C4—82no—
520C4—81no—
521C4—81no—
522C4—161no—
523C4—161no—
524C4C4×#13711yescharacteristic
525C4C4×#13711yescharacteristic
526C4#295011yescharacteristic
527C4#295211yescharacteristic
528C4#332311yescharacteristic
529C4#332711yescharacteristic
530C4#332811yescharacteristic
531C4#333111yescharacteristic
532C2—82no—
533C2#2009211yescharacteristic
534C2#2009611yescharacteristic
535C2#2010611yescharacteristic
536C2#2012211yescharacteristic
537C2#2288111yescharacteristic
538C2#2288211yescharacteristic
539C2#2289711yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22965C2semidirect
6. #9839C2semidirect
7. #9568C2semidirect
8. #9567C2semidirect
9. #9224C2semidirect
11. #9066C2semidirect
13. #8888C2semidirect
14. #8213C2semidirect
56. #1632C4semidirect

Maximal subgroups

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

subgroupindex
#230612
#229652
#225222
#225022
C2×#16322
#98392
#95682
#95672
#92242
#92232
#90662
#88892
#88882
#82132
#82122