Finite Group Databasebeta

← #308017

Subgroups — #308017

Order 512.

648 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26465C211yesmaximal, characteristic, complemented
2#26106C211yesmaximal, characteristic
3#22962C211yesmaximal, characteristic
4D4×#106C211yesmaximal, characteristic
5#10675C211yesmaximal, characteristic, complemented
6#10653C211yesmaximal, characteristic, complemented
7#10602C211yesmaximal, characteristic, complemented
8#10579C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9535C211yesmaximal, characteristic, complemented
11#9219C211yesmaximal, characteristic
12#8981C211yesmaximal, characteristic
13#8923C211yesmaximal, characteristic, complemented
14#8882C211yesmaximal, characteristic
15#8843C211yesmaximal, characteristic
16C2×#479—21no—
17C2×#474C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C22×#106—21no—
20#2552C2211yescharacteristic
21#2365C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24#2099—21no—
25#2086—21no—
26#2066—21no—
27#2026—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1981C2211yescharacteristic
31#1956C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1942—21no—
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1918—21no—
41#1863—21no—
42#1862—21no—
43#1862—21no—
44#1853—21no—
45#1853—21no—
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1834C2211yescharacteristic
49#1807—21no—
50#1802—21no—
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#102—21no—
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×#125—21no—
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76#479—44no—
77#474—22no—
78#474Q814yes—
79C8×D4—22no—
80C8×D4—22no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—41no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—44no—
91C2×#106D411yescharacteristic
92C2×#106D411yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102D411yescharacteristic
106C2×#102D411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#398—41no—
113#391—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#387—41no—
120#385—41no—
121#385—41no—
122#384—21no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#380—21no—
127#380—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132#340—21no—
133#340—21no—
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—41no—
143#340D411yescharacteristic
144#340D411yescharacteristic
145#340D411yescharacteristic
146#340D411yescharacteristic
147#336—21no—
148#336—21no—
149#336—21no—
150#335—22no—
151#332—22no—
152D4⋊C8—22no—
153C22×Q8—21no—
154C22×D4C2×D411yescharacteristic
155C23×C4—21no—
156C23×C4—41no—
157C23×C4—41no—
158C22×C8—24no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—42no—
165C22×C8—42no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C2×Q812yes—
174#125—42no—
175#125—81no—
176C4×Q8—42no—
177C4×Q8—81no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—81no—
183C4×D4—81no—
184C4×D4C2×Q812yes—
185C4×D4#4112yes—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—81no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—41no—
221C2×C42C2411yescharacteristic
222#107—24no—
223#107—42no—
224#107—81no—
225#107—81no—
226#107—82no—
227#106—41no—
228#106—42no—
229#106—42no—
230#106—42no—
231#106—81no—
232#106D814yes—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—42no—
240C4⋊C8—81no—
241C4⋊C8C2×Q812yes—
242C4⋊C8#4112yes—
243#103—42no—
244#103—42no—
245#103—42no—
246#103—81no—
247#102—24no—
248#102—22no—
249#102—22no—
250#102—42no—
251C22⋊C8—24no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—81no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267C24—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C22⋊Q812yes—
278C2×D4#13512yes—
279C2×D4#13712yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—161no—
302C22×C4—161no—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4H(5,𝔽2)11yescharacteristic
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—22no—
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—48no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
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×C8C2×D812yes—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
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—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—161no—
370C4⋊C4C22⋊Q812yes—
371C4⋊C4C2×D812yes—
372C4⋊C4C2×D812yes—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—82no—
384C42C22×Q811yescharacteristic
385C42C2×#4111yescharacteristic
386C42C2×#4111yescharacteristic
387C422−1+411yescharacteristic
388C23—22no—
389C23—22no—
390C23—22no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23D4211yescharacteristic
396Q8—42no—
397Q8—42no—
398Q8—81no—
399Q8—81no—
400D4—22no—
401D4—22no—
402D4—22no—
403D4#47914yescomplemented
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—164no—
444C2×C4—164no—
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4#52511yescharacteristic
447C2×C4#53911yescharacteristic
448C2×C4#54711yescharacteristic
449C2×C4D4×Q811yescharacteristic
450C2×C4#55411yescharacteristic
451C2×C4C22×D811yescharacteristic
452C2×C4C2×#13511yescharacteristic
453C2×C4C2×#13711yescharacteristic
454C2×C4#57511yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57711yescharacteristic
458C2×C4#57811yescharacteristic
459C8—22no—
460C8—42no—
461C8—41no—
462C8—42no—
463C8—41no—
464C8—42no—
465C8—41no—
466C8—81no—
467C8—81no—
468C22—24no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22#244911yescharacteristic
476C22#326611yescharacteristic
477C22#326911yescharacteristic
478C22D4×D811yescharacteristic
479C22#328211yescharacteristic
480C22#328911yescharacteristic
481C22#329111yescharacteristic
482C4—21no—
483C4—21no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—82no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—161no—
493C4—161no—
494C4—161no—
495C4C2×#47911yescharacteristic
496C4#308411yescharacteristic
497C4#310011yescharacteristic
498C4#312411yescharacteristic
499C4#333711yescharacteristic
500C4#334411yescharacteristic
501C4Q8×D811yescharacteristic
502C4#338911yescharacteristic
503C2—44no—
504C2#1975711yescharacteristic
505C2#1979411yescharacteristic
506C2#1997011yescharacteristic
507C2#2010611yescharacteristic
508C2#2634611yescharacteristic
509C2#2636811yescharacteristic
510C2#2640111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×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. #26465C2semidirect
5. #10675C2semidirect
6. #10653C2semidirect
7. #10602C2semidirect
8. #10579C2semidirect
9. #9616C2semidirect
10. #9535C2semidirect
13. #8923C2semidirect
403. D4#479semidirect

Maximal subgroups

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

subgroupindex
#264652
#261062
#229622
D4×#1062
#106752
#106532
#106022
#105792
#96162
#95352
#92192
#89812
#89232
#88822
#88432