Finite Group Databasebeta

← #275361

Subgroups — #275361

Order 512.

544 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#33153C211yesmaximal, characteristic
2#27962C211yesmaximal, characteristic
3#22632C211yesmaximal, characteristic
4#21364C211yesmaximal, characteristic
5#10657C211yesmaximal, characteristic, complemented
6#10655C211yesmaximal, characteristic, complemented
7#10601C211yesmaximal, characteristic, complemented
8#10597C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9613C211yesmaximal, characteristic, complemented
11#9568C211yesmaximal, characteristic, complemented
12#9568C211yesmaximal, characteristic, complemented
13#9136C211yesmaximal, characteristic
14#8834C211yesmaximal, characteristic
15#8727C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17#2857C2211yescharacteristic
18#2285C2211yescharacteristic
19C4×#117C2211yescharacteristic
20#2101—21no—
21#2101—21no—
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1987C2211yescharacteristic
25#1987C2211yescharacteristic
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1923—21no—
35#1923—21no—
36#1919—21no—
37#1918—21no—
38#1854—21no—
39#1854—21no—
40#1854—21no—
41#1834C2211yescharacteristic
42#1834C2211yescharacteristic
43#1831—21no—
44#1825—21no—
45#1825—21no—
46#1825—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1776—21no—
50#1776—21no—
51C8×C4⋊C4C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54#1744—21no—
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×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C42C2311yescharacteristic
73#432—22no—
74#432—22no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#106—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8D411yescharacteristic
93C2×C22⋊C8D411yescharacteristic
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#391—21no—
100#389—21no—
101#389—21no—
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#382—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4C2311yescharacteristic
115C4×C4⋊C4C2311yescharacteristic
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118#376—21no—
119C43—21no—
120#340—21no—
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—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—22no—
170#117—22no—
171#117—22no—
172#117—41no—
173#117—42no—
174#117—42no—
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—41no—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42C2411yescharacteristic
210#107—42no—
211#107—42no—
212#107—42no—
213#106—42no—
214#106—42no—
215#106—42no—
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—41no—
232C22⋊C8—41no—
233C22⋊C8—42no—
234C22⋊C8—81no—
235#97—24no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—42no—
241C4×C8—42no—
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—22no—
286C2×C8—22no—
287C2×C8—21no—
288C2×C8—21no—
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—21no—
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—41no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
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—81no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—82no—
340C4⋊C4#13512yes—
341C4⋊C4#13512yes—
342C4⋊C4#13712yes—
343C22⋊C4—24no—
344C22⋊C4—42no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C42—24no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—42no—
355C42—42no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—81no—
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—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
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×#13511yescharacteristic
428C2×C4C2×#13511yescharacteristic
429C2×C4C2×#13711yescharacteristic
430C2×C4#57511yescharacteristic
431C2×C4#57511yescharacteristic
432C2×C4#57511yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57811yescharacteristic
435C8—41no—
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#304011yescharacteristic
454C22#304611yescharacteristic
455C22#305011yescharacteristic
456C22#305611yescharacteristic
457C22#306711yescharacteristic
458C22#306811yescharacteristic
459C4—21no—
460C4—21no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—81no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4#310211yescharacteristic
473C4#310711yescharacteristic
474C4#323111yescharacteristic
475C4#323311yescharacteristic
476C4#334811yescharacteristic
477C4#334811yescharacteristic
478C4#335211yescharacteristic
479C4#335211yescharacteristic
480C2—81no—
481C2#1927811yescharacteristic
482C2#1931811yescharacteristic
483C2#1941911yescharacteristic
484C2#1949911yescharacteristic
485C2#2459511yescharacteristic
486C2#2460411yescharacteristic
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. #10657C2semidirect
6. #10655C2semidirect
7. #10601C2semidirect
8. #10597C2semidirect
9. #9616C2semidirect
10. #9613C2semidirect
11. #9568C2semidirect
12. #9568C2semidirect

Maximal subgroups

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

subgroupindex
#331532
#279622
#226322
#213642
#106572
#106552
#106012
#105972
#96162
#96132
#95682
#91362
#88342
#87272