Finite Group Databasebeta

← #230215

Subgroups — #230215

Order 512.

760 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23223C211yesmaximal, characteristic, complemented
2#22995C211yesmaximal, characteristic
3#22969C211yesmaximal, characteristic, complemented
4D4×#107C211yesmaximal, characteristic, complemented
5C2×#1677C211yesmaximal, characteristic
6#9830C211yesmaximal, characteristic, complemented
7#9620C211yesmaximal, characteristic, complemented
8#9571C211yesmaximal, characteristic, complemented
9#9568C211yesmaximal, characteristic, complemented
10#8977C211yesmaximal, characteristic
11#8973C211yesmaximal, characteristic, complemented
12#8929C211yesmaximal, characteristic, complemented
13#8920C211yesmaximal, characteristic, complemented
14#8288C211yesmaximal, characteristic, complemented
15#8204C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×C4×D8—21no—
20C2×C8×D4C2211yescharacteristic
21C22×#107—21no—
22#2369C2211yescharacteristic
23#2366C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25D4×C4⋊C4C2211yescharacteristic
26#1990C2211yescharacteristic
27#1987C2211yescharacteristic
28#1982C2211yescharacteristic
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1956C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1944—21no—
37#1936—21no—
38#1934—21no—
39#1925—21no—
40#1920—21no—
41#1879—21no—
42#1862—21no—
43#1862—21no—
44#1862—21no—
45#1852—21no—
46#1852—21no—
47#1845C2211yescharacteristic
48#1808—21no—
49C4×#107C2211yescharacteristic
50#1770C2211yescharacteristic
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1762—21no—
54#1762—21no—
55C4×#102C2211yescharacteristic
56#1677C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×D8—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2311yescharacteristic
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79#486—22no—
80#486C2×C414yescomplemented
81#476—22no—
82#476C2×C414yescomplemented
83#459—22no—
84#459C2×C414yes—
85C4×D8—44no—
86C8×D4—22no—
87C8×D4C2×C414yescomplemented
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#107—44no—
95C2×#107D411yescharacteristic
96C2×#107D411yescharacteristic
97C2×#106—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#103—41no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—41no—
112C2×#102D411yescharacteristic
113C2×#102D411yescharacteristic
114C2×C22⋊C8—21no—
115C2×#97—21no—
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—41no—
122#390—21no—
123#390—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#386—41no—
129#385—41no—
130#385—41no—
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#380—21no—
134#380—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
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#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#340D411yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#334C2×C414yescomplemented
156#333C2×C414yescomplemented
157#331C2×C414yescomplemented
158D4⋊C8—22no—
159#322—22no—
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C2×D411yescharacteristic
165C23×C4—21no—
166C23×C4—41no—
167C23×C4—41no—
168C23×C4—41no—
169C2×D8—24no—
170C2×D8—42no—
171C2×D8—42no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—24no—
177C22×C8—21no—
178C22×C8—42no—
179C22×C8—42no—
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8C22×C412yes—
188#127—21no—
189#127—21no—
190#127—21no—
191#127C22×C412yes—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—42no—
198C4×D4—81no—
199C4×D4—81no—
200C4×D4—81no—
201C4×D4—81no—
202C4×D4C22×C412yes—
203C4×D4#4112yes—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—41no—
236C2×C42C2411yescharacteristic
237#107—22no—
238#107—41no—
239#107—42no—
240#107—42no—
241#107—42no—
242#107—42no—
243#107—81no—
244#107—81no—
245#106—24no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8C22×C412yes—
252C4⋊C8C22×C412yes—
253C4⋊C8C22×C412yes—
254#103—82no—
255#102—24no—
256#102—22no—
257#102—22no—
258#102—24no—
259#102—24no—
260#102—24no—
261#102—42no—
262#102—42no—
263#102—81no—
264#102—81no—
265#102—82no—
266C22⋊C8—24no—
267#97—81no—
268#97—81no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—42no—
274C4×C8C22×C412yes—
275C4×C8#4112yes—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—41no—
280#95—41no—
281#95—41no—
282#95—41no—
283C24—21no—
284C24—41no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—81no—
288C2×Q8—81no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—81no—
307C2×D4—81no—
308C2×D4C4×D412yes—
309C2×D4AGL(1,ℤ/8)12yes—
310C2×D4#13712yes—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—161no—
334C22×C4—161no—
335C22×C4—161no—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339D8—48no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—48no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8C4×D412yes—
367C2×C8C4×D412yes—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—82no—
393C4⋊C4—81no—
394C4⋊C4—81no—
395C4⋊C4—82no—
396C4⋊C4—161no—
397C4⋊C4C4×D412yes—
398C4⋊C4AGL(1,ℤ/8)12yes—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C22⋊C4—161no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—161no—
413C42C23×C411yescharacteristic
414C42C2×#4111yescharacteristic
415C42C2×#4111yescharacteristic
416C422−1+411yescharacteristic
417C23—22no—
418C23—22no—
419C23—22no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23D4211yescharacteristic
432Q8—82no—
433Q8—161no—
434D4—22no—
435D4—22no—
436D4—22no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—42no—
441D4—82no—
442D4—81no—
443D4—81no—
444D4—81no—
445D4—81no—
446D4—161no—
447D4#44014yes—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—164no—
485C2×C4—164no—
486C2×C4—164no—
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4#52011yescharacteristic
490C2×C4#54711yescharacteristic
491C2×C4#54811yescharacteristic
492C2×C4#55311yescharacteristic
493C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
494C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
495C2×C4C2×#13711yescharacteristic
496C2×C4#57511yescharacteristic
497C2×C4#57611yescharacteristic
498C2×C4#57611yescharacteristic
499C2×C4#57811yescharacteristic
500C2×C4#57811yescharacteristic
501C8—22no—
502C8—22no—
503C8—42no—
504C8—41no—
505C8—41no—
506C8—42no—
507C8—42no—
508C8—42no—
509C8—81no—
510C22—24no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—161no—
521C22—161no—
522C22—161no—
523C22—161no—
524C22#236111yescharacteristic
525C22#326411yescharacteristic
526C22#326411yescharacteristic
527C22#326911yescharacteristic
528C22#327011yescharacteristic
529C22#329011yescharacteristic
530C22#329211yescharacteristic
531C4—21no—
532C4—21no—
533C4—42no—
534C4—42no—
535C4—82no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—161no—
542C4—161no—
543C4C2×#44011yescharacteristic
544C4C4×AGL(1,ℤ/8)11yescharacteristic
545C4#295111yescharacteristic
546C4#295111yescharacteristic
547C4#334311yescharacteristic
548C4#334611yescharacteristic
549C4#336711yescharacteristic
550C4#337011yescharacteristic
551C2—44no—
552C2—162no—
553C2#2005311yescharacteristic
554C2#2005811yescharacteristic
555C2#2006111yescharacteristic
556C2#2009511yescharacteristic
557C2#2301111yescharacteristic
558C2#2303211yescharacteristic
559C2#2304611yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
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
1. #23223C2semidirect
3. #22969C2semidirect
4. D4×#107C2semidirect
6. #9830C2semidirect
7. #9620C2semidirect
8. #9571C2semidirect
9. #9568C2semidirect
11. #8973C2semidirect
12. #8929C2semidirect
13. #8920C2semidirect
14. #8288C2semidirect
15. #8204C2semidirect
56. #1677C4semidirect
80. #486C2×C4semidirect
82. #476C2×C4semidirect
87. C8×D4C2×C4semidirect
155. #334C2×C4semidirect
156. #333C2×C4semidirect
157. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232232
#229952
#229692
D4×#1072
C2×#16772
#98302
#96202
#95712
#95682
#89772
#89732
#89292
#89202
#82882
#82042