Finite Group Databasebeta

← #226074

Subgroups — #226074

Order 512.

707 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22770C211yesmaximal, characteristic
2#22742C211yesmaximal, characteristic, complemented
3#22671C211yesmaximal, characteristic
4D4×#106C211yesmaximal, characteristic
5C2×#1585C211yesmaximal, characteristic
6#9619C211yesmaximal, characteristic, complemented
7#9615C211yesmaximal, characteristic, complemented
8#8896C211yesmaximal, characteristic, complemented
9#8894C211yesmaximal, characteristic
10#8891C211yesmaximal, characteristic, complemented
11#8890C211yesmaximal, characteristic
12#8821C211yesmaximal, characteristic, complemented
13#8651C211yesmaximal, characteristic, complemented
14#8648C211yesmaximal, characteristic
15#8289C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×#440—21no—
18C2×C4×SD16C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C22×#106—21no—
21#2355C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1949—21no—
26#1948—21no—
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1947C2211yescharacteristic
31#1944—21no—
32#1933—21no—
33#1919—21no—
34#1877—21no—
35#1863—21no—
36#1862—21no—
37#1853—21no—
38#1853—21no—
39#1845C2211yescharacteristic
40#1844C2211yescharacteristic
41#1836—21no—
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1807—21no—
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1776—21no—
52C4×#106C2211yescharacteristic
53#1770C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102—21no—
56#1585C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×SD16D411yescharacteristic
69C23×C8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77#446—22no—
78#446C2×C414yescomplemented
79#440—44no—
80C4×SD16C2×C414yes—
81C4×SD16Q814yes—
82C8×D4—22no—
83C8×D4C2×C414yes—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—44no—
93C2×#106D411yescharacteristic
94C2×#106D411yescharacteristic
95C2×#106D411yescharacteristic
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#103D411yescharacteristic
104C2×#102—21no—
105C2×#102—21no—
106C2×#102D411yescharacteristic
107C2×C22⋊C8—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—41no—
119#387—41no—
120#385—41no—
121#385—41no—
122#384—21no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#382—21no—
128#380—21no—
129#380—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
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—21no—
145#340—21no—
146#340—21no—
147#340—41no—
148#340—41no—
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#335—22no—
154#335C2×C414yescomplemented
155Q8⋊C8C2×C414yescomplemented
156D4⋊C8C2×C414yes—
157#322C2×C414yescomplemented
158C22×Q8C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—41no—
162C23×C4—41no—
163C2×SD16—22no—
164C2×SD16—22no—
165C2×SD16C4⋊C414yes—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—24no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—42no—
181C4×Q8—81no—
182C4×Q8C22×C412yes—
183C4×Q8C2×Q812yes—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—81no—
189C4×D4—81no—
190C4×D4C22×C412yes—
191C4×D4C2×Q812yes—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
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—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—81no—
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×C22⋊C4—21no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42C2411yescharacteristic
228#107—42no—
229#107—42no—
230#107—81no—
231#106—22no—
232#106—41no—
233#106—42no—
234#106—42no—
235#106—42no—
236#106—42no—
237#106—81no—
238#106C4⋊C414yes—
239C4⋊C8—24no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8C22×C412yes—
247C4⋊C8C22×C412yes—
248#103—42no—
249#103—81no—
250#103C4⋊C414yes—
251#102—42no—
252#102—42no—
253#102C4⋊C414yes—
254C22⋊C8—24no—
255#97—24no—
256#97—42no—
257C4×C8—21no—
258C4×C8—81no—
259C4×C8C22×C412yes—
260C4×C8C22×C412yes—
261C4×C8C22×C412yes—
262C4×C8C2×Q812yes—
263C4×C8#4112yes—
264#95—21no—
265#95—21no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272C24—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8C2×C4⋊C412yes—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4C2×C4⋊C412yes—
283C2×D4AGL(1,ℤ/8)12yes—
284C2×D4#13712yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—161no—
307C22×C4—161no—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4H(5,𝔽2)11yescharacteristic
311SD16—28no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—48no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8C2×C4⋊C412yes—
337C2×C8C2×C4⋊C412yes—
338C2×C8C4×D412yes—
339C2×C8C4×D412yes—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—82no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—161no—
375C4⋊C4C2×C4⋊C412yes—
376C4⋊C4C2×C4⋊C412yes—
377C4⋊C4AGL(1,ℤ/8)12yes—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C42—22no—
383C42—22no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—82no—
389C42—161no—
390C42C23×C411yescharacteristic
391C42C22×Q811yescharacteristic
392C42C2×#4111yescharacteristic
393C422−1+411yescharacteristic
394C23—22no—
395C23—22no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23D4211yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—41no—
405Q8—41no—
406D4—22no—
407D4—22no—
408D4—22no—
409D4#42814yes—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—164no—
450C2×C4—164no—
451C2×C4C22×C4⋊C411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4#52011yescharacteristic
454C2×C4#54711yescharacteristic
455C2×C4D4×Q811yescharacteristic
456C2×C4#55411yescharacteristic
457C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
458C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57511yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57711yescharacteristic
463C2×C4#57711yescharacteristic
464C2×C4#57811yescharacteristic
465C8—22no—
466C8—22no—
467C8—22no—
468C8—22no—
469C8—42no—
470C8—42no—
471C8—42no—
472C8—81no—
473C8—81no—
474C22—24no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22D4×C4⋊C411yescharacteristic
482C22#326511yescharacteristic
483C22#326511yescharacteristic
484C22#326611yescharacteristic
485C22#326911yescharacteristic
486C22#329011yescharacteristic
487C22#329011yescharacteristic
488C4—21no—
489C4—21no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—161no—
498C4—161no—
499C4—161no—
500C4—161no—
501C4C2×#42811yescharacteristic
502C4#291211yescharacteristic
503C4C4×AGL(1,ℤ/8)11yescharacteristic
504C4#294711yescharacteristic
505C4#334411yescharacteristic
506C4#334511yescharacteristic
507C4#338611yescharacteristic
508C4#338811yescharacteristic
509C2—44no—
510C2#2003911yescharacteristic
511C2#2004411yescharacteristic
512C2#2007211yescharacteristic
513C2#2008311yescharacteristic
514C2#2265911yescharacteristic
515C2#2267711yescharacteristic
516C2#2269011yescharacteristic
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. #22742C2semidirect
6. #9619C2semidirect
7. #9615C2semidirect
8. #8896C2semidirect
10. #8891C2semidirect
12. #8821C2semidirect
13. #8651C2semidirect
15. #8289C2semidirect
56. #1585C4semidirect
78. #446C2×C4semidirect
154. #335C2×C4semidirect
155. Q8⋊C8C2×C4semidirect
157. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#227702
#227422
#226712
D4×#1062
C2×#15852
#96192
#96152
#88962
#88942
#88912
#88902
#88212
#86512
#86482
#82892