Finite Group Databasebeta

← #308185

Subgroups — #308185

Order 512.

595 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26442C211yesmaximal, characteristic, complemented
2#26020C211yesmaximal, characteristic
3#22963C211yesmaximal, characteristic
4#22715C211yesmaximal, characteristic
5#10686C211yesmaximal, characteristic, complemented
6#10656C211yesmaximal, characteristic, complemented
7#10601C211yesmaximal, characteristic, complemented
8#10574C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9535C211yesmaximal, characteristic, complemented
11#9223C211yesmaximal, characteristic
12#8976C211yesmaximal, characteristic
13#8929C211yesmaximal, characteristic, complemented
14#8886C211yesmaximal, characteristic
15#8847C211yesmaximal, characteristic
16C2×#476C2211yescharacteristic
17C2×#436C2211yescharacteristic
18#2551C2211yescharacteristic
19#2369C2211yescharacteristic
20#2354C2211yescharacteristic
21D4×C4⋊C4C2211yescharacteristic
22#2088—21no—
23#2067—21no—
24#2058—21no—
25#2023—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1988—21no—
29#1988—21no—
30#1987C2211yescharacteristic
31#1956C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1941—21no—
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1919—21no—
41#1919—21no—
42#1918—21no—
43#1864—21no—
44#1854—21no—
45#1853—21no—
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1809—21no—
50#1802C2211yescharacteristic
51C4×#106C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55#1736—21no—
56C2×#335C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C22×C4⋊C4—21no—
75#476—22no—
76#476Q814yescomplemented
77#436—22no—
78#436—22no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—41no—
86C2×#106D411yescharacteristic
87C2×#106D411yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#103—41no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102D411yescharacteristic
98C2×#102D411yescharacteristic
99C2×C22⋊C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#395—41no—
105#391—41no—
106#390—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#388—41no—
113#385—41no—
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#380—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
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#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#336—21no—
146#336—21no—
147#336—21no—
148#335—22no—
149#333—22no—
150D4⋊C8—22no—
151C22×Q8—21no—
152C22×Q8—41no—
153C22×D4C2×D411yescharacteristic
154C23×C4—21no—
155C23×C4—41no—
156C2×M16—24no—
157C2×M16—81no—
158C2×M16—81no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—81no—
174C4⋊Q8—81no—
175C4⋊Q8C2×Q812yes—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—42no—
179C4×D4—81no—
180C4×D4C2×Q812yes—
181C4×D4#4112yes—
182#117—42no—
183#117—81no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
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—42no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—81no—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—41no—
216C2×C42—41no—
217C2×C42C2411yescharacteristic
218#107—42no—
219#107—81no—
220#106—24no—
221#106—22no—
222#106—22no—
223#106—81no—
224#106—81no—
225#106—81no—
226#106—82no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8C2×Q812yes—
236C4⋊C8#4112yes—
237#103—82no—
238#102—24no—
239#102—22no—
240#102—41no—
241#102—81no—
242C22⋊C8—24no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—81no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258C24—21no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×Q8—81no—
264C2×Q8—81no—
265C2×Q8—161no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4C22⋊Q812yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—161no—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4H(5,𝔽2)11yescharacteristic
295M16—84no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—161no—
356C4⋊C4C22⋊Q812yes—
357C4⋊C4#13512yes—
358C4⋊C4#13712yes—
359C4⋊C4#13712yes—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C22⋊C4—161no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—82no—
372C42—161no—
373C42C22×Q811yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C422−1+411yescharacteristic
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23D4211yescharacteristic
385Q8—82no—
386Q8—161no—
387Q8—164no—
388D4—22no—
389D4—22no—
390D4—41no—
391D4—41no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—164no—
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4#52511yescharacteristic
434C2×C4#53911yescharacteristic
435C2×C4#54711yescharacteristic
436C2×C4D4×Q811yescharacteristic
437C2×C4#55411yescharacteristic
438C2×C4C2×#13511yescharacteristic
439C2×C4C2×#13711yescharacteristic
440C2×C4C2×#13711yescharacteristic
441C2×C4#57511yescharacteristic
442C2×C4#57611yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57711yescharacteristic
446C8—42no—
447C8—42no—
448C8—41no—
449C8—41no—
450C8—41no—
451C8—41no—
452C8—42no—
453C8—81no—
454C8—81no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22#244911yescharacteristic
463C22#326611yescharacteristic
464C22#326611yescharacteristic
465C22#327511yescharacteristic
466C22#328011yescharacteristic
467C22#328811yescharacteristic
468C22#329111yescharacteristic
469C4—21no—
470C4—21no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4#307811yescharacteristic
483C4#308411yescharacteristic
484C4#310711yescharacteristic
485C4#311811yescharacteristic
486C4#334811yescharacteristic
487C4#335611yescharacteristic
488C4#338111yescharacteristic
489C4#338511yescharacteristic
490C2—82no—
491C2#1977411yescharacteristic
492C2#1990411yescharacteristic
493C2#2010711yescharacteristic
494C2#2017711yescharacteristic
495C2#2635611yescharacteristic
496C2#2637211yescharacteristic
497C2#2640711yescharacteristic
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. #26442C2semidirect
5. #10686C2semidirect
6. #10656C2semidirect
7. #10601C2semidirect
8. #10574C2semidirect
9. #9616C2semidirect
10. #9535C2semidirect
13. #8929C2semidirect
76. #476Q8semidirect

Maximal subgroups

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

subgroupindex
#264422
#260202
#229632
#227152
#106862
#106562
#106012
#105742
#96162
#95352
#92232
#89762
#89292
#88862
#88472