Finite Group Databasebeta

← #318917

Subgroups — #318917

Order 512.

660 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29287C211yesmaximal, characteristic, complemented
2#29258C211yesmaximal, characteristic
3#21713C211yesmaximal, characteristic, complemented
4#21692C211yesmaximal, characteristic
5#10600C211yesmaximal, characteristic, complemented
6#10599C211yesmaximal, characteristic
7#10596C211yesmaximal, characteristic, complemented
8#10595C211yesmaximal, characteristic
9#9744C211yesmaximal, characteristic
10#9732C211yesmaximal, characteristic
11#9621C211yesmaximal, characteristic, complemented
12#9615C211yesmaximal, characteristic, complemented
13#8808C211yesmaximal, characteristic, complemented
14#8589C211yesmaximal, characteristic, complemented
15#8585C211yesmaximal, characteristic
16C2×#492C2211yescharacteristic
17C2×#446—21no—
18C2×#440—21no—
19C2×#440—21no—
20C4×C4⋊Q8C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C4×#127C2211yescharacteristic
23#2094—21no—
24#2088—21no—
25#2088—21no—
26#2023—21no—
27#1988C2211yescharacteristic
28#1988C2211yescharacteristic
29#1987—21no—
30#1985—21no—
31#1982C2211yescharacteristic
32#1982C2211yescharacteristic
33#1973—21no—
34#1973—21no—
35#1957C2211yescharacteristic
36#1956C2211yescharacteristic
37#1948—21no—
38#1948C2211yescharacteristic
39#1948C2211yescharacteristic
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1940—21no—
43#1939C2211yescharacteristic
44#1939C2211yescharacteristic
45#1864—21no—
46#1864—21no—
47#1845C2211yescharacteristic
48#1842C2211yescharacteristic
49#1842C2211yescharacteristic
50#1831C2211yescharacteristic
51#1771—21no—
52#1770C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#97C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67C22×SD16—21no—
68C22×SD16—21no—
69C2×C4⋊Q8D411yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77#492—22no—
78#492Q814yescomplemented
79#446—44no—
80#440—44no—
81#440—44no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103—21no—
101C2×#103—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#97—21no—
109C2×#97D411yescharacteristic
110C2×#97D411yescharacteristic
111C2×#97D411yescharacteristic
112C2×#97D411yescharacteristic
113C2×C4×C8—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391—21no—
117#391—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389D411yescharacteristic
125#389D411yescharacteristic
126#384D411yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C43C2311yescharacteristic
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#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#335—22no—
150#335—41no—
151#334—22no—
152#334—41no—
153#332—41no—
154#332—41no—
155#331—41no—
156#331—41no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C23×C4—41no—
163C2×SD16—24no—
164C2×SD16—24no—
165C2×SD16—42no—
166C2×SD16—42no—
167C2×SD16—42no—
168C2×SD16—42no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C4⋊Q8—22no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—41no—
184C4⋊Q8C2×Q812yes—
185#127—21no—
186#127—21no—
187#127—21no—
188#127C2×Q812yes—
189C4×Q8—42no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—81no—
196C4×D4—81no—
197C4×D4—82no—
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—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—41no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42C2411yescharacteristic
229#107—42no—
230#107—42no—
231#106—81no—
232#106—81no—
233#106—81no—
234#106—81no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—81no—
253#103—42no—
254#103—42no—
255#102—42no—
256#102—42no—
257#102—42no—
258#102—42no—
259#102—81no—
260#102—81no—
261#97—22no—
262#97—22no—
263#97—41no—
264#97—42no—
265#97SD1614yes—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—42no—
269C4×C8C2×Q812yes—
270C4×C8#4112yes—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—41no—
276#95—41no—
277C24—41no—
278C2×Q8—22no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—81no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
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—81no—
302C2×D4—81no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320SD16—48no—
321SD16—48no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
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×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8C22⋊Q812yes—
348C2×C8C22⋊Q812yes—
349C2×C8C2×SD1612yes—
350C2×C8C2×SD1612yes—
351C2×C8AGL(1,ℤ/8)12yes—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
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—81no—
378C4⋊C4—82no—
379C4⋊C4#13712yes—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—82no—
396C42C2×SD1612yes—
397C42AGL(1,ℤ/8)12yes—
398C42C22×Q811yescharacteristic
399C42C2×#4111yescharacteristic
400C42H(5,𝔽2)11yescharacteristic
401C422−1+411yescharacteristic
402C23—42no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23C2×#12111yescharacteristic
410Q8—42no—
411Q8—42no—
412Q8—84no—
413Q8—81no—
414Q8—81no—
415D4—42no—
416D4—42no—
417D4—42no—
418D4—42no—
419D4—82no—
420D4—82no—
421D4—81no—
422D4—81no—
423D4—81no—
424D4—81no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
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×C4#46012yes—
460C2×C4#46012yes—
461C2×C4C2×C22⋊Q811yescharacteristic
462C2×C4#53711yescharacteristic
463C2×C4D4×Q811yescharacteristic
464C2×C4#55011yescharacteristic
465C2×C4#55411yescharacteristic
466C2×C4#55611yescharacteristic
467C2×C4C22×SD1611yescharacteristic
468C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
469C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
470C2×C4C2×#13711yescharacteristic
471C2×C4#57511yescharacteristic
472C2×C4#57511yescharacteristic
473C2×C4#57611yescharacteristic
474C2×C4#57711yescharacteristic
475C8—22no—
476C8—22no—
477C8—41no—
478C8—81no—
479C8—81no—
480C8—81no—
481C8—81no—
482C8—81no—
483C8—81no—
484C8#47614yes—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—161no—
489C22—161no—
490C22—161no—
491C22—161no—
492C22#253811yescharacteristic
493C22C2×#46011yescharacteristic
494C22#303611yescharacteristic
495C22#305611yescharacteristic
496C22#306211yescharacteristic
497C22#306811yescharacteristic
498C22#306911yescharacteristic
499C4—22no—
500C4—21no—
501C4—21no—
502C4—22no—
503C4—41no—
504C4—42no—
505C4—41no—
506C4—81no—
507C4—82no—
508C4—81no—
509C4—81no—
510C4—161no—
511C4—161no—
512C4C2×#47611yescharacteristic
513C4#307711yescharacteristic
514C4#324611yescharacteristic
515C4#325311yescharacteristic
516C4#338511yescharacteristic
517C4#338811yescharacteristic
518C4#340011yescharacteristic
519C4#340311yescharacteristic
520C2—162no—
521C2#1917111yescharacteristic
522C2#1923811yescharacteristic
523C2#1930511yescharacteristic
524C2#1936811yescharacteristic
525C2#2827711yescharacteristic
526C2#2828011yescharacteristic
527C2#2829811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29287C2semidirect
3. #21713C2semidirect
5. #10600C2semidirect
7. #10596C2semidirect
11. #9621C2semidirect
12. #9615C2semidirect
13. #8808C2semidirect
14. #8589C2semidirect
78. #492Q8semidirect

Maximal subgroups

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

subgroupindex
#292872
#292582
#217132
#216922
#106002
#105992
#105962
#105952
#97442
#97322
#96212
#96152
#88082
#85892
#85852