Finite Group Databasebeta

← #306166

Subgroups — #306166

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28797C211yesmaximal, characteristic, complemented
2#26107C211yesmaximal, characteristic
3#26020C211yesmaximal, characteristic
4#22713C211yesmaximal, characteristic
5#10656C211yesmaximal, characteristic, complemented
6#10655C211yesmaximal, characteristic, complemented
7#10578C211yesmaximal, characteristic, complemented
8#10575C211yesmaximal, characteristic, complemented
9#9619C211yesmaximal, characteristic, complemented
10#9594C211yesmaximal, characteristic, complemented
11#8893C211yesmaximal, characteristic
12#8890C211yesmaximal, characteristic
13#8817C211yesmaximal, characteristic, complemented
14#8654C211yesmaximal, characteristic
15#8206C211yesmaximal, characteristic
16C2×#436C2211yescharacteristic
17#2571C2211yescharacteristic
18#2551C2211yescharacteristic
19#2354C2211yescharacteristic
20D4×C4⋊C4C2211yescharacteristic
21#2101—21no—
22#2088—21no—
23#2085—21no—
24#2067—21no—
25#2057—21no—
26#2023—21no—
27#1993—21no—
28#1988—21no—
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1944—21no—
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1918—21no—
39#1864—21no—
40#1854—21no—
41#1852—21no—
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1809—21no—
49#1804—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#107C2211yescharacteristic
53#1770C2211yescharacteristic
54C4×#102C2211yescharacteristic
55#1736—21no—
56C2×#335C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4×D4—21no—
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C22×C4⋊C4—21no—
74#436—22no—
75#436Q814yescomplemented
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107D411yescharacteristic
80C2×#107D411yescharacteristic
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102—21no—
91C2×#102—21no—
92C2×#102D411yescharacteristic
93C2×#102D411yescharacteristic
94C2×C22⋊C8—21no—
95C2×#97—21no—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#400—41no—
100#395—41no—
101#390—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#388—41no—
108#385—41no—
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#382—21no—
114#382—21no—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
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—41no—
138#340—41no—
139#340D411yescharacteristic
140#340D411yescharacteristic
141#340D411yescharacteristic
142#340D411yescharacteristic
143#336—21no—
144#336—21no—
145#336—21no—
146#335—22no—
147D4⋊C8Q814yescomplemented
148C82—22no—
149C22×Q8—41no—
150C22×D4C2×D411yescharacteristic
151C23×C4—21no—
152C23×C4—41no—
153C2×M16—24no—
154C2×M16—81no—
155C2×M16—81no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—81no—
168C4⋊Q8—81no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—42no—
172C4×D4—81no—
173C4×D4C2×Q812yes—
174C4×D4C2×Q812yes—
175#117—42no—
176#117—81no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—41no—
210C2×C42C2411yescharacteristic
211#107—22no—
212#107—41no—
213#107—42no—
214#107—81no—
215#107—81no—
216#106—42no—
217#106—81no—
218#106—81no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—81no—
226C4⋊C8—81no—
227C4⋊C8C2×Q812yes—
228C4⋊C8C2×Q812yes—
229#102—41no—
230#102—41no—
231#102—42no—
232#102—81no—
233C22⋊C8—24no—
234#97—81no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8C2×Q812yes—
239C4×C8C2×Q812yes—
240C4×C8#4112yes—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250C24—21no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×Q8—161no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4C4⋊Q812yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—161no—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4H(5,𝔽2)11yescharacteristic
283M16—84no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8C22⋊Q812yes—
310C2×C8C22⋊Q812yes—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—82no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—161no—
347C4⋊C4C4⋊Q812yes—
348C4⋊C4#13512yes—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—161no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—82no—
361C42C22×Q811yescharacteristic
362C42C22×Q811yescharacteristic
363C42C2×#4111yescharacteristic
364C422−1+411yescharacteristic
365C23—22no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23D4211yescharacteristic
373Q8—164no—
374D4—22no—
375D4—22no—
376D4—41no—
377D4—41no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—164no—
418C2×C4C2×C22⋊Q811yescharacteristic
419C2×C4C2×C4⋊Q811yescharacteristic
420C2×C4#54411yescharacteristic
421C2×C4#54711yescharacteristic
422C2×C4D4×Q811yescharacteristic
423C2×C4#55411yescharacteristic
424C2×C4C2×#13511yescharacteristic
425C2×C4#57511yescharacteristic
426C2×C4#57511yescharacteristic
427C2×C4#57511yescharacteristic
428C2×C4#57611yescharacteristic
429C2×C4#57611yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57811yescharacteristic
432C8—22no—
433C8—22no—
434C8—42no—
435C8—42no—
436C8—41no—
437C8—41no—
438C8—42no—
439C8—81no—
440C8—81no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22#244011yescharacteristic
449C22#327311yescharacteristic
450C22#327511yescharacteristic
451C22#327711yescharacteristic
452C22#328011yescharacteristic
453C22#329011yescharacteristic
454C22#329011yescharacteristic
455C4—21no—
456C4—21no—
457C4—42no—
458C4—42no—
459C4—81no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4—161no—
468C4#307611yescharacteristic
469C4#308111yescharacteristic
470C4#316411yescharacteristic
471C4#316611yescharacteristic
472C4#334711yescharacteristic
473C4#334911yescharacteristic
474C4#338111yescharacteristic
475C4#338911yescharacteristic
476C2—82no—
477C2#1989911yescharacteristic
478C2#1990611yescharacteristic
479C2#2028711yescharacteristic
480C2#2029611yescharacteristic
481C2#2604911yescharacteristic
482C2#2607911yescharacteristic
483C2#2608211yescharacteristic
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. #28797C2semidirect
5. #10656C2semidirect
6. #10655C2semidirect
7. #10578C2semidirect
8. #10575C2semidirect
9. #9619C2semidirect
10. #9594C2semidirect
13. #8817C2semidirect
75. #436Q8semidirect
147. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#287972
#261072
#260202
#227132
#106562
#106552
#105782
#105752
#96192
#95942
#88932
#88902
#88172
#86542
#82062