Finite Group Databasebeta

← #308955

Subgroups — #308955

Order 512.

571 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28792C211yesmaximal, characteristic, complemented
2#26533C211yesmaximal, characteristic
3#26512C211yesmaximal, characteristic, complemented
4#23215C211yesmaximal, characteristic
5#10593C211yesmaximal, characteristic, complemented
6#10591C211yesmaximal, characteristic
7#10354C211yesmaximal, characteristic
8#10352C211yesmaximal, characteristic, complemented
9#9822C211yesmaximal, characteristic
10#9735C211yesmaximal, characteristic
11#9613C211yesmaximal, characteristic, complemented
12#9593C211yesmaximal, characteristic, complemented
13#8927C211yesmaximal, characteristic, complemented
14#8920C211yesmaximal, characteristic
15C8×#107C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17#2571C2211yescharacteristic
18#2552C2211yescharacteristic
19#2369C2211yescharacteristic
20#2366C2211yescharacteristic
21#2100—21no—
22#2092—21no—
23#2087—21no—
24#2085—21no—
25#2065—21no—
26#2059—21no—
27#1991—21no—
28#1990C2211yescharacteristic
29#1987C2211yescharacteristic
30#1987C2211yescharacteristic
31#1982C2211yescharacteristic
32#1982C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1956C2211yescharacteristic
36#1948C2211yescharacteristic
37#1947—21no—
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1947C2211yescharacteristic
41#1942—21no—
42#1939C2211yescharacteristic
43#1939C2211yescharacteristic
44#1934—21no—
45#1926—21no—
46#1921—21no—
47#1864—21no—
48#1862—21no—
49#1834C2211yescharacteristic
50#1804—21no—
51C4×#107C2211yescharacteristic
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102—21no—
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72#486—22no—
73#486Q814yes—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107D411yescharacteristic
78C2×#107D411yescharacteristic
79C2×#106—21no—
80C2×#106D411yescharacteristic
81C2×#106D411yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#103—41no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—41no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#400—41no—
104#398—41no—
105#391—41no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#386—41no—
115#384—21no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#382—21no—
119#382—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
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#340D411yescharacteristic
139#340D411yescharacteristic
140#340D411yescharacteristic
141#340D411yescharacteristic
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#335—22no—
147#334—22no—
148#333—22no—
149#331—22no—
150C82—22no—
151C22×Q8—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C23×C4—41no—
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—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8C2×Q812yes—
169#127—21no—
170#127—21no—
171#127—21no—
172#127C2×Q812yes—
173#125—42no—
174#125—81no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—81no—
178C4×D4—81no—
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—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—41no—
208C2×C42C2411yescharacteristic
209#107—22no—
210#107—22no—
211#107—42no—
212#107—42no—
213#107—81no—
214#106—22no—
215#106—41no—
216#106—81no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—42no—
225C4⋊C8—81no—
226C4⋊C8#4112yes—
227#103—82no—
228#102—24no—
229#102—24no—
230#102—42no—
231#102—42no—
232#102—81no—
233#102—81no—
234#102—81no—
235#102—81no—
236#102—82no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—81no—
242C4×C8—81no—
243C4×C8C2×Q812yes—
244C4×C8#4112yes—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255C24—41no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—81no—
259C2×Q8—81no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—81no—
275C2×D4—81no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—161no—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4H(5,𝔽2)11yescharacteristic
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8C22⋊Q812yes—
318C2×C8C22⋊Q812yes—
319C2×C8#13512yes—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
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—81no—
347C4⋊C4—81no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—161no—
351C4⋊C4—161no—
352C4⋊C4—161no—
353C4⋊C4#13512yes—
354C4⋊C4#13512yes—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—161no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—161no—
367C42C22×Q811yescharacteristic
368C42C2×#4111yescharacteristic
369C42C2×#4111yescharacteristic
370C422−1+411yescharacteristic
371C23—42no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23D4211yescharacteristic
379Q8—82no—
380Q8—161no—
381D4—42no—
382D4—42no—
383D4—42no—
384D4—42no—
385D4—82no—
386D4—81no—
387D4—81no—
388D4—81no—
389D4—81no—
390D4—161no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—164no—
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4#53211yescharacteristic
429C2×C4#54211yescharacteristic
430C2×C4#54811yescharacteristic
431C2×C4D4×Q811yescharacteristic
432C2×C4#55411yescharacteristic
433C2×C4C2×#13511yescharacteristic
434C2×C4C2×#13511yescharacteristic
435C2×C4C2×#13511yescharacteristic
436C2×C4#57511yescharacteristic
437C2×C4#57611yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57811yescharacteristic
441C8—22no—
442C8—22no—
443C8—22no—
444C8—42no—
445C8—42no—
446C8—41no—
447C8—81no—
448C8—81no—
449C8—81no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22#245211yescharacteristic
458C22#327311yescharacteristic
459C22#328211yescharacteristic
460C22#328411yescharacteristic
461C22#328511yescharacteristic
462C22#329011yescharacteristic
463C22#329211yescharacteristic
464C4—21no—
465C4—21no—
466C4—42no—
467C4—42no—
468C4—81no—
469C4—82no—
470C4—81no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—161no—
476C4—161no—
477C4#307611yescharacteristic
478C4#308311yescharacteristic
479C4#316811yescharacteristic
480C4#317411yescharacteristic
481C4#336111yescharacteristic
482C4#337711yescharacteristic
483C4#338111yescharacteristic
484C4#338111yescharacteristic
485C2—162no—
486C2#1991111yescharacteristic
487C2#1994611yescharacteristic
488C2#2000511yescharacteristic
489C2#2001211yescharacteristic
490C2#2646811yescharacteristic
491C2#2648111yescharacteristic
492C2#2650111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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. #28792C2semidirect
3. #26512C2semidirect
5. #10593C2semidirect
8. #10352C2semidirect
11. #9613C2semidirect
12. #9593C2semidirect
13. #8927C2semidirect
15. C8×#107C2semidirect

Maximal subgroups

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

subgroupindex
#287922
#265332
#265122
#232152
#105932
#105912
#103542
#103522
#98222
#97352
#96132
#95932
#89272
#89202
C8×#1072