Finite Group Databasebeta

← #275767

Subgroups — #275767

Order 512.

519 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29938C211yesmaximal, characteristic
2#27878C211yesmaximal, characteristic
3#22626C211yesmaximal, characteristic
4#21988C211yesmaximal, characteristic
5#10656C211yesmaximal, characteristic, complemented
6#10651C211yesmaximal, characteristic, complemented
7#10601C211yesmaximal, characteristic, complemented
8#10589C211yesmaximal, characteristic, complemented
9#9621C211yesmaximal, characteristic, complemented
10#9616C211yesmaximal, characteristic, complemented
11#9576C211yesmaximal, characteristic, complemented
12#9567C211yesmaximal, characteristic, complemented
13#9136C211yesmaximal, characteristic
14#8834C211yesmaximal, characteristic
15#8726C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17#2857C2211yescharacteristic
18#2285C2211yescharacteristic
19#2285C2211yescharacteristic
20#2099—21no—
21#2084—21no—
22#2077—21no—
23#2077—21no—
24#2076—21no—
25#2076—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1987C2211yescharacteristic
29#1987C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1921—21no—
39#1921—21no—
40#1917—21no—
41#1916—21no—
42#1854—21no—
43#1854—21no—
44#1847—21no—
45#1845C2211yescharacteristic
46#1836—21no—
47#1834C2211yescharacteristic
48#1798—21no—
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1750—21no—
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C22⋊Q8—21no—
69C2×#117—21no—
70C2×#117—21no—
71C22×C42C2311yescharacteristic
72#432—22no—
73#432—22no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#106—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8D411yescharacteristic
94C2×C22⋊C8D411yescharacteristic
95C2×#97—21no—
96C2×#97—21no—
97C2×C4×C8—21no—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#391—21no—
101#389—21no—
102#389—21no—
103#384D411yescharacteristic
104#384D411yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#382—21no—
109#382—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380D411yescharacteristic
116#380D411yescharacteristic
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
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—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146#335—22no—
147#334—22no—
148#333—22no—
149#333—22no—
150C22×Q8—21no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4#4111yescharacteristic
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C22⋊Q8—82no—
169#117—81no—
170#117—81no—
171#117—81no—
172#117—81no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
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—41no—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—41no—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42C2411yescharacteristic
208#107—42no—
209#107—42no—
210#107—81no—
211#107—81no—
212#106—81no—
213#106—81no—
214#106—81no—
215#106—81no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—81no—
227C4⋊C8—81no—
228C4⋊C8#4112yes—
229C4⋊C8#4112yes—
230C22⋊C8—24no—
231C22⋊C8—22no—
232C22⋊C8—41no—
233C22⋊C8—42no—
234C22⋊C8—81no—
235#97—24no—
236#97—81no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—81no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250C24#12111yescharacteristic
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—81no—
254C2×Q8—81no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
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×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8#13712yes—
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—42no—
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—41no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C4⋊C4—82no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—82no—
362C42C2×#4111yescharacteristic
363C42C2×#4111yescharacteristic
364C422−1+411yescharacteristic
365C422−1+411yescharacteristic
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23C2×#12111yescharacteristic
374Q8—161no—
375Q8—161no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
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—41no—
406C2×C4—41no—
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—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4#52511yescharacteristic
422C2×C4#53611yescharacteristic
423C2×C4#54711yescharacteristic
424C2×C4#54711yescharacteristic
425C2×C4#55211yescharacteristic
426C2×C4#55211yescharacteristic
427C2×C4C2×#13711yescharacteristic
428C2×C4#57511yescharacteristic
429C2×C4#57611yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57711yescharacteristic
432C2×C4#57711yescharacteristic
433C2×C4#57811yescharacteristic
434C2×C4#57811yescharacteristic
435C8—22no—
436C8—41no—
437C8—41no—
438C8—41no—
439C8—42no—
440C8—42no—
441C8—81no—
442C8—81no—
443C8—81no—
444C8—81no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22#241311yescharacteristic
453C22#303711yescharacteristic
454C22#304211yescharacteristic
455C22#305911yescharacteristic
456C22#306211yescharacteristic
457C22#306711yescharacteristic
458C22#306811yescharacteristic
459C4—21no—
460C4—21no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—81no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4#312011yescharacteristic
473C4#312411yescharacteristic
474C4#324211yescharacteristic
475C4#324311yescharacteristic
476C4#334411yescharacteristic
477C4#335011yescharacteristic
478C4#335111yescharacteristic
479C4#335611yescharacteristic
480C2—81no—
481C2#1934811yescharacteristic
482C2#1937411yescharacteristic
483C2#1951511yescharacteristic
484C2#1956311yescharacteristic
485C2#2459011yescharacteristic
486C2#2461411yescharacteristic
487C2#2462511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #10656C2semidirect
6. #10651C2semidirect
7. #10601C2semidirect
8. #10589C2semidirect
9. #9621C2semidirect
10. #9616C2semidirect
11. #9576C2semidirect
12. #9567C2semidirect

Maximal subgroups

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

subgroupindex
#299382
#278782
#226262
#219882
#106562
#106512
#106012
#105892
#96212
#96162
#95762
#95672
#91362
#88342
#87262