Finite Group Databasebeta

← #343471

Subgroups — #343471

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26110C211yesmaximal, characteristic
2#26023C211yesmaximal, characteristic, complemented
3#24993C211yesmaximal, characteristic
4#24969C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10626C211yesmaximal, characteristic, complemented
8#10625C211yesmaximal, characteristic
9#10579C211yesmaximal, characteristic, complemented
10#10434C211yesmaximal, characteristic
11#9554C211yesmaximal, characteristic
12#9553C211yesmaximal, characteristic, complemented
13#8883C211yesmaximal, characteristic, complemented
14#8882C211yesmaximal, characteristic
15#8816C211yesmaximal, characteristic, complemented
16#2552C2211yescharacteristic
17#2538C2211yescharacteristic
18#2355C2211yescharacteristic
19D4×C4⋊C4C2211yescharacteristic
20#2101—21no—
21#2100—21no—
22#2088—21no—
23#2077—21no—
24#2073—21no—
25#2068—21no—
26#2035—21no—
27#2026—21no—
28#1989—21no—
29#1984C2211yescharacteristic
30#1954C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1925—21no—
38#1921—21no—
39#1918—21no—
40#1872—21no—
41#1864—21no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1823—21no—
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1802—21no—
52#1802C2211yescharacteristic
53#1802C2211yescharacteristic
54C4×#103C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×#125—21no—
68C2×C22⋊Q8—21no—
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2311yescharacteristic
72C22×C4⋊C4—21no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107—41no—
76C2×#107—41no—
77C2×#106—21no—
78C2×#106—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×#103—21no—
86C2×#103—21no—
87C2×#103D411yescharacteristic
88C2×#103D411yescharacteristic
89C2×#102—21no—
90C2×#102—21no—
91C2×#102D411yescharacteristic
92C2×#102D411yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#398—41no—
99#396—41no—
100#391—21no—
101#390—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#387—41no—
107#385—41no—
108#384—21no—
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#382—21no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
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#340D411yescharacteristic
137#340D411yescharacteristic
138#340D411yescharacteristic
139#340D411yescharacteristic
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144Q8⋊C8—22no—
145D4⋊C8—22no—
146C82—22no—
147C22×Q8C2×D411yescharacteristic
148C22×D4C2×D411yescharacteristic
149C23×C4—21no—
150C23×C4—41no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160#125—42no—
161#125—81no—
162C22⋊Q8—42no—
163C22⋊Q8—81no—
164C4×Q8—21no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—42no—
173C4×D4—81no—
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—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—81no—
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—21no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—41no—
208C2×C42C2411yescharacteristic
209#107—42no—
210#107—42no—
211#107—82no—
212#107—82no—
213#106—81no—
214#106—81no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—81no—
224C4⋊C8—81no—
225#103—22no—
226#103—41no—
227#103—81no—
228#103—81no—
229#102—22no—
230#102—41no—
231#102—42no—
232#102—42no—
233C22⋊C8—81no—
234C22⋊C8—81no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8#4112yes—
241#95—21no—
242#95—21no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249C24—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4#13512yes—
262C22×C4—21no—
263C22×C4—21no—
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—41no—
278C22×C4—41no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—161no—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4H(5,𝔽2)11yescharacteristic
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
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—21no—
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—
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—41no—
340C4⋊C4—41no—
341C4⋊C4—82no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—161no—
350C4⋊C4#13512yes—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—161no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—161no—
363C42C2×#4111yescharacteristic
364C422−1+411yescharacteristic
365C422−1+411yescharacteristic
366C422−1+411yescharacteristic
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23D4211yescharacteristic
375Q8—41no—
376Q8—41no—
377Q8—41no—
378Q8—41no—
379D4—22no—
380D4—22no—
381D4—41no—
382D4—41no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—41no—
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—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—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—164no—
423C2×C4#53611yescharacteristic
424C2×C4#53611yescharacteristic
425C2×C4#53911yescharacteristic
426C2×C4#54711yescharacteristic
427C2×C4#54711yescharacteristic
428C2×C4#56311yescharacteristic
429C2×C4C2×#13511yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4#57511yescharacteristic
432C2×C4#57511yescharacteristic
433C2×C4#57611yescharacteristic
434C2×C4#57711yescharacteristic
435C2×C4#57711yescharacteristic
436C2×C4#57811yescharacteristic
437C8—24no—
438C8—41no—
439C8—41no—
440C8—41no—
441C8—41no—
442C8—81no—
443C8—81no—
444C8—81no—
445C8—81no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22#268011yescharacteristic
454C22#327311yescharacteristic
455C22#327711yescharacteristic
456C22#328211yescharacteristic
457C22#328211yescharacteristic
458C22#329211yescharacteristic
459C22#329311yescharacteristic
460C4—21no—
461C4—21no—
462C4—42no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4—161no—
473C4#323211yescharacteristic
474C4#323311yescharacteristic
475C4#324111yescharacteristic
476C4#324211yescharacteristic
477C4#333611yescharacteristic
478C4#334111yescharacteristic
479C4#335011yescharacteristic
480C4#335011yescharacteristic
481C2—82no—
482C2#1991411yescharacteristic
483C2#1992011yescharacteristic
484C2#1999611yescharacteristic
485C2#2030811yescharacteristic
486C2#3108711yescharacteristic
487C2#3109011yescharacteristic
488C2#3111011yescharacteristic
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
2. #26023C2semidirect
5. #10658C2semidirect
6. #10657C2semidirect
7. #10626C2semidirect
9. #10579C2semidirect
12. #9553C2semidirect
13. #8883C2semidirect
15. #8816C2semidirect

Maximal subgroups

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

subgroupindex
#261102
#260232
#249932
#249692
#106582
#106572
#106262
#106252
#105792
#104342
#95542
#95532
#88832
#88822
#88162