Finite Group Databasebeta

← #229532

Subgroups — #229532

Order 512.

667 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23222C211yesmaximal, characteristic, complemented
2#23100C211yesmaximal, characteristic
3#22968C211yesmaximal, characteristic
4#22713C211yesmaximal, characteristic
5C2×#1699C211yesmaximal, characteristic
6#9821C211yesmaximal, characteristic, complemented
7#9619C211yesmaximal, characteristic, complemented
8#9574C211yesmaximal, characteristic, complemented
9#9570C211yesmaximal, characteristic, complemented
10#8985C211yesmaximal, characteristic
11#8984C211yesmaximal, characteristic
12#8923C211yesmaximal, characteristic, complemented
13#8923C211yesmaximal, characteristic, complemented
14#8282C211yesmaximal, characteristic, complemented
15#8206C211yesmaximal, characteristic
16C2×#487C2211yescharacteristic
17C2×#474C2211yescharacteristic
18C2×#461C2211yescharacteristic
19C2×#436C2211yescharacteristic
20#2365C2211yescharacteristic
21#2365C2211yescharacteristic
22#2354C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1988C2211yescharacteristic
28#1981C2211yescharacteristic
29#1973—21no—
30#1956C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1943—21no—
36#1935—21no—
37#1928—21no—
38#1928—21no—
39#1921—21no—
40#1880—21no—
41#1864—21no—
42#1863—21no—
43#1854—21no—
44#1852—21no—
45#1834C2211yescharacteristic
46#1809—21no—
47C4×#107C2211yescharacteristic
48#1770C2211yescharacteristic
49#1770C2211yescharacteristic
50#1770C2211yescharacteristic
51#1763—21no—
52#1763—21no—
53#1762—21no—
54C4×#102C2211yescharacteristic
55#1736—21no—
56#1699C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×M16—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C2×#117—21no—
77C22×C4⋊C4—21no—
78#487—22no—
79#487C2×C414yescomplemented
80#474—22no—
81#474C2×C414yes—
82#461—22no—
83#461C2×C414yes—
84#436—22no—
85#436C2×C414yes—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107D411yescharacteristic
90C2×#107D411yescharacteristic
91C2×#106—21no—
92C2×#106—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#102—21no—
105C2×#102D411yescharacteristic
106C2×#102D411yescharacteristic
107C2×C22⋊C8—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#391—21no—
116#390—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#388—41no—
121#387—41no—
122#387—41no—
123#385—41no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#380—21no—
127#380—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—41no—
145#340—41no—
146#340D411yescharacteristic
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#335C2×C414yescomplemented
154#332C2×C414yescomplemented
155#332C2×C414yescomplemented
156D4⋊C8—22no—
157C82—22no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×D4C2×D411yescharacteristic
161C23×C4—21no—
162C23×C4—41no—
163C2×SD16—24no—
164C2×SD16—81no—
165C2×SD16—81no—
166C2×M16—24no—
167C2×M16—81no—
168C2×M16—81no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8C22×C412yes—
185C4⋊Q8C22×C412yes—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—81no—
189C4×Q8—81no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—42no—
193C4×D4—81no—
194C4×D4C22×C412yes—
195C4×D4#4112yes—
196#117—42no—
197#117—81no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42C2411yescharacteristic
231#107—24no—
232#107—41no—
233#107—41no—
234#107—81no—
235#107—81no—
236#106—81no—
237#106—81no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8C22×C412yes—
244C4⋊C8C22×C412yes—
245C4⋊C8C22×C412yes—
246#103—24no—
247#103—24no—
248#103—24no—
249#103—42no—
250#103—42no—
251#103—81no—
252#103—81no—
253#102—24no—
254#102—22no—
255#102—41no—
256C22⋊C8—24no—
257#97—81no—
258#97—81no—
259#97—81no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8C22×C412yes—
265C4×C8#4112yes—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275C24—21no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4C4×D412yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—161no—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
317SD16—84no—
318M16—84no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8C4×D412yes—
344C2×C8C4×D412yes—
345C2×C8#13712yes—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4C4×D412yes—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—161no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—82no—
393C42—82no—
394C42—82no—
395C42C23×C411yescharacteristic
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C422−1+411yescharacteristic
399C23—22no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23D4211yescharacteristic
407Q8—42no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—81no—
412Q8—81no—
413Q8—81no—
414Q8—81no—
415D4—22no—
416D4—22no—
417D4—41no—
418D4—41no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—164no—
459C2×C4C2×C4×D411yescharacteristic
460C2×C4C2×C4×D411yescharacteristic
461C2×C4#52011yescharacteristic
462C2×C4#54711yescharacteristic
463C2×C4#54811yescharacteristic
464C2×C4#55311yescharacteristic
465C2×C4C2×#13711yescharacteristic
466C2×C4#57511yescharacteristic
467C2×C4#57511yescharacteristic
468C2×C4#57511yescharacteristic
469C2×C4#57611yescharacteristic
470C2×C4#57611yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57711yescharacteristic
473C8—22no—
474C8—22no—
475C8—22no—
476C8—42no—
477C8—41no—
478C8—42no—
479C8—42no—
480C8—42no—
481C8—81no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22#236111yescharacteristic
490C22#327311yescharacteristic
491C22#327311yescharacteristic
492C22#327511yescharacteristic
493C22#327811yescharacteristic
494C22#329011yescharacteristic
495C22#329311yescharacteristic
496C4—21no—
497C4—21no—
498C4—42no—
499C4—82no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—82no—
507C4—161no—
508C4—161no—
509C4#294311yescharacteristic
510C4C4×#13711yescharacteristic
511C4#294611yescharacteristic
512C4#294611yescharacteristic
513C4#334611yescharacteristic
514C4#334811yescharacteristic
515C4#337011yescharacteristic
516C4#337611yescharacteristic
517C2—82no—
518C2#2028711yescharacteristic
519C2#2029711yescharacteristic
520C2#2030411yescharacteristic
521C2#2030711yescharacteristic
522C2#2300511yescharacteristic
523C2#2304111yescharacteristic
524C2#2304511yescharacteristic
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
1. #23222C2semidirect
6. #9821C2semidirect
7. #9619C2semidirect
8. #9574C2semidirect
9. #9570C2semidirect
12. #8923C2semidirect
13. #8923C2semidirect
14. #8282C2semidirect
56. #1699C4semidirect
79. #487C2×C4semidirect
153. #335C2×C4semidirect
154. #332C2×C4semidirect
155. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232222
#231002
#229682
#227132
C2×#16992
#98212
#96192
#95742
#95702
#89852
#89842
#89232
#82822
#82062