Finite Group Databasebeta

← #229535

Subgroups — #229535

Order 512.

684 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23220C211yesmaximal, characteristic
2#23101C211yesmaximal, characteristic
3#23004C211yesmaximal, characteristic, complemented
4#22742C211yesmaximal, characteristic, complemented
5C2×#1702C211yesmaximal, characteristic
6#9822C211yesmaximal, characteristic
7#9622C211yesmaximal, characteristic, complemented
8#9571C211yesmaximal, characteristic
9#9570C211yesmaximal, characteristic, complemented
10#8986C211yesmaximal, characteristic, complemented
11#8979C211yesmaximal, characteristic
12#8924C211yesmaximal, characteristic
13#8922C211yesmaximal, characteristic, complemented
14#8283C211yesmaximal, characteristic, complemented
15#8207C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#462C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×#441—21no—
21#2366C2211yescharacteristic
22#2365C2211yescharacteristic
23#2355C2211yescharacteristic
24Q8×C4⋊C4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1988C2211yescharacteristic
27#1982C2211yescharacteristic
28#1973—21no—
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1956C2211yescharacteristic
32#1949—21no—
33#1948—21no—
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1936—21no—
39#1935—21no—
40#1926—21no—
41#1925—21no—
42#1878—21no—
43#1877—21no—
44#1865—21no—
45#1864—21no—
46#1836—21no—
47#1834C2211yescharacteristic
48#1776—21no—
49C4×#106C2211yescharacteristic
50#1770C2211yescharacteristic
51#1770C2211yescharacteristic
52#1770C2211yescharacteristic
53#1762—21no—
54#1762—21no—
55C4×#103C2211yescharacteristic
56#1702C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×Q16—21no—
69C22×SD16—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77#486—22no—
78#486C2×C414yes—
79#462—22no—
80#462C2×C414yescomplemented
81#460—22no—
82#460C2×C414yes—
83#446—22no—
84#446C2×C414yescomplemented
85#441—44no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106D411yescharacteristic
91C2×#106D411yescharacteristic
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103D411yescharacteristic
101C2×#103D411yescharacteristic
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—41no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391—21no—
118#391—21no—
119#390—21no—
120#390—21no—
121#389—21no—
122#389—21no—
123#389—41no—
124#387—41no—
125#387—41no—
126#386—41no—
127#384—21no—
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#382—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
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—41no—
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#334C2×C414yescomplemented
155#332C2×C414yescomplemented
156#331C2×C414yes—
157Q8⋊C8—22no—
158C82—22no—
159C22×Q8—21no—
160C22×Q8C2×D411yescharacteristic
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C23×C4—41no—
165C2×Q16—24no—
166C2×Q16—42no—
167C2×Q16—42no—
168C2×SD16—24no—
169C2×SD16—81no—
170C2×SD16—81no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8C22×C412yes—
184#127—21no—
185#127—21no—
186#127—21no—
187#127C22×C412yes—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—81no—
193C4×Q8—81no—
194C4×Q8C22×C412yes—
195C4×Q8#4112yes—
196C4×D4—42no—
197C4×D4—42no—
198C4×D4—81no—
199C4×D4—81no—
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—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4C2×D411yescharacteristic
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×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42C2411yescharacteristic
230#107—42no—
231#107—81no—
232#106—41no—
233#106—41no—
234#106—42no—
235#106—42no—
236C4⋊C8—24no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8C22×C412yes—
243C4⋊C8C22×C412yes—
244C4⋊C8C22×C412yes—
245#103—22no—
246#103—24no—
247#103—41no—
248#103—42no—
249#103—81no—
250#102—24no—
251#102—24no—
252#102—24no—
253#102—42no—
254#102—42no—
255#102—81no—
256#102—81no—
257#102—82no—
258#97—24no—
259#97—42no—
260#97—81no—
261#97—81no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8C22×C412yes—
267C4×C8#4112yes—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277C24—41no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8C4×D412yes—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—41no—
294C2×D4—41no—
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—81no—
305C2×D4—81no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—161no—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4H(5,𝔽2)11yescharacteristic
323Q16—48no—
324SD16—84no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8C4×D412yes—
350C2×C8C4×D412yes—
351C2×C8AGL(1,ℤ/8)12yes—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—81no—
380C4⋊C4—81no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—161no—
384C4⋊C4C4×D412yes—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—161no—
388C42—22no—
389C42—22no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—82no—
397C42—82no—
398C42—161no—
399C42C23×C411yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C422−1+411yescharacteristic
403C23—42no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23D4211yescharacteristic
411Q8—22no—
412Q8—22no—
413Q8—41no—
414Q8—41no—
415Q8—42no—
416Q8—42no—
417Q8—81no—
418Q8—81no—
419D4—42no—
420D4—42no—
421D4—42no—
422D4—42no—
423D4—82no—
424D4—81no—
425D4—81no—
426D4—81no—
427D4—81no—
428D4—161no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
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—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—164no—
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×C4×D411yescharacteristic
467C2×C4#52011yescharacteristic
468C2×C4#54711yescharacteristic
469C2×C4#54811yescharacteristic
470C2×C4#55311yescharacteristic
471C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
472C2×C4#57511yescharacteristic
473C2×C4#57511yescharacteristic
474C2×C4#57511yescharacteristic
475C2×C4#57611yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57711yescharacteristic
478C2×C4#57811yescharacteristic
479C8—22no—
480C8—22no—
481C8—22no—
482C8—42no—
483C8—41no—
484C8—42no—
485C8—42no—
486C8—42no—
487C8—81no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—161no—
492C22—161no—
493C22—161no—
494C22—161no—
495C22#236111yescharacteristic
496C22#327411yescharacteristic
497C22#327511yescharacteristic
498C22#327511yescharacteristic
499C22#327711yescharacteristic
500C22#329011yescharacteristic
501C22#329111yescharacteristic
502C4—21no—
503C4—21no—
504C4—42no—
505C4—42no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4—82no—
513C4—161no—
514C4—161no—
515C4#294311yescharacteristic
516C4C4×AGL(1,ℤ/8)11yescharacteristic
517C4#294611yescharacteristic
518C4#294711yescharacteristic
519C4#334511yescharacteristic
520C4#334711yescharacteristic
521C4#337111yescharacteristic
522C4#337711yescharacteristic
523C2—162no—
524C2#2028911yescharacteristic
525C2#2029811yescharacteristic
526C2#2030011yescharacteristic
527C2#2031211yescharacteristic
528C2#2300611yescharacteristic
529C2#2303911yescharacteristic
530C2#2304411yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
3. #23004C2semidirect
4. #22742C2semidirect
7. #9622C2semidirect
9. #9570C2semidirect
10. #8986C2semidirect
13. #8922C2semidirect
14. #8283C2semidirect
15. #8207C2semidirect
56. #1702C4semidirect
80. #462C2×C4semidirect
84. #446C2×C4semidirect
154. #334C2×C4semidirect
155. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232202
#231012
#230042
#227422
C2×#17022
#98222
#96222
#95712
#95702
#89862
#89792
#89242
#89222
#82832
#82072