Finite Group Databasebeta

← #306442

Subgroups — #306442

Order 512.

607 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26231C211yesmaximal, characteristic
2#24813C211yesmaximal, characteristic
3#22767C211yesmaximal, characteristic
4#22740C211yesmaximal, characteristic, complemented
5#10682C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10621C211yesmaximal, characteristic
8#10421C211yesmaximal, characteristic
9#9566C211yesmaximal, characteristic, complemented
10#9533C211yesmaximal, characteristic, complemented
11#9220C211yesmaximal, characteristic, complemented
12#8983C211yesmaximal, characteristic, complemented
13#8948C211yesmaximal, characteristic
14#8897C211yesmaximal, characteristic
15#8896C211yesmaximal, characteristic, complemented
16C2×#485—21no—
17C2×#475C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20#2542C2211yescharacteristic
21#2354C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23Q8×C4⋊C4C2211yescharacteristic
24#2084—21no—
25#2065—21no—
26#2036—21no—
27#2035—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1981C2211yescharacteristic
31#1956C2211yescharacteristic
32#1954C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947—21no—
35#1947—21no—
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1939C2211yescharacteristic
39#1934—21no—
40#1920—21no—
41#1880—21no—
42#1880—21no—
43#1873—21no—
44#1834—21no—
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1803—21no—
49#1803C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#107—21no—
53C4×#107C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#103—21no—
56C2×#332C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16D411yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76#485—44no—
77#475—22no—
78#475—22no—
79C8×Q8—22no—
80C8×Q8—22no—
81C4×SD16—22no—
82C4×SD16—22no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107D411yescharacteristic
86C2×#107D411yescharacteristic
87C2×#106—21no—
88C2×#106—21no—
89C2×#106D411yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103D411yescharacteristic
103C2×#102—21no—
104C2×#102—21no—
105C2×#102D411yescharacteristic
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#399—41no—
114#390—21no—
115#390—21no—
116#389—21no—
117#389—21no—
118#389—41no—
119#389—41no—
120#388—41no—
121#384—21no—
122#384—21no—
123#384—21no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#380—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
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—21no—
143#340—21no—
144#340—21no—
145#340—41no—
146#340—41no—
147#340D411yescharacteristic
148#340D411yescharacteristic
149#336—21no—
150#336—21no—
151#332—22no—
152Q8⋊C8—22no—
153D4⋊C8—22no—
154C22×Q8—21no—
155C22×Q8C2×D411yescharacteristic
156C22×D4C2×D411yescharacteristic
157C23×C4—21no—
158C2×SD16—22no—
159C2×SD16—22no—
160C2×SD16—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—81no—
177C22⋊Q8—81no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—81no—
183C4×Q8—81no—
184C4×Q8#4112yes—
185C4×Q8#4112yes—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190#117—42no—
191#117—81no—
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—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42C2411yescharacteristic
228#107—22no—
229#107—22no—
230#107—42no—
231#107—42no—
232#106—22no—
233#106—42no—
234#106—42no—
235C4⋊C8—24no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—81no—
244C4⋊C8#4112yes—
245#103—22no—
246#103—42no—
247#103—42no—
248#103—42no—
249#103—42no—
250#103—81no—
251#103—81no—
252#102—22no—
253#102—81no—
254#102—81no—
255C22⋊C8—42no—
256C22⋊C8—81no—
257C4×C8—24no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—81no—
265#95—21no—
266#95—21no—
267#95—41no—
268#95—41no—
269#95—41no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8#12312yes—
281C2×Q8#13512yes—
282C2×Q8#13712yes—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4H(5,𝔽2)11yescharacteristic
309SD16—44no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
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×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C4⋊C4—22no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
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—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—82no—
371C4⋊C4—161no—
372C4⋊C4—161no—
373C4⋊C4#12312yes—
374C4⋊C4#13512yes—
375C4⋊C4AGL(1,ℤ/8)12yes—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—161no—
382C42—22no—
383C42—22no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—82no—
389C42—82no—
390C42—82no—
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C422−1+411yescharacteristic
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23D4211yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—22no—
406Q8—81no—
407Q8—81no—
408Q8—81no—
409Q8—81no—
410Q8#48514yescomplemented
411D4—41no—
412D4—41no—
413D4—41no—
414D4—41no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
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—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
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—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4C2×#12311yescharacteristic
455C2×C4#52511yescharacteristic
456C2×C4#53911yescharacteristic
457C2×C4#54711yescharacteristic
458C2×C4#54811yescharacteristic
459C2×C4#55311yescharacteristic
460C2×C4C2×#13511yescharacteristic
461C2×C4C2×#13511yescharacteristic
462C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
463C2×C4C2×#13711yescharacteristic
464C2×C4#57611yescharacteristic
465C2×C4#57711yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57811yescharacteristic
468C8—41no—
469C8—41no—
470C8—41no—
471C8—42no—
472C8—41no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22#244311yescharacteristic
485C22#326511yescharacteristic
486C22#327011yescharacteristic
487C22#328011yescharacteristic
488C22#328211yescharacteristic
489C22#328911yescharacteristic
490C22#329011yescharacteristic
491C4—24no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—161no—
501C4—161no—
502C4—161no—
503C4—161no—
504C4C2×#48511yescharacteristic
505C4#309811yescharacteristic
506C4#310611yescharacteristic
507C4#312611yescharacteristic
508C4#333611yescharacteristic
509C4#334611yescharacteristic
510C4#336511yescharacteristic
511C4#336811yescharacteristic
512C2—161no—
513C2#1977911yescharacteristic
514C2#1983011yescharacteristic
515C2#1999111yescharacteristic
516C2#2008611yescharacteristic
517C2#2613411yescharacteristic
518C2#2615911yescharacteristic
519C2#2619211yescharacteristic
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
4. #22740C2semidirect
5. #10682C2semidirect
6. #10657C2semidirect
9. #9566C2semidirect
10. #9533C2semidirect
11. #9220C2semidirect
12. #8983C2semidirect
15. #8896C2semidirect
410. Q8#485semidirect

Maximal subgroups

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

subgroupindex
#262312
#248132
#227672
#227402
#106822
#106572
#106212
#104212
#95662
#95332
#92202
#89832
#89482
#88972
#88962