Finite Group Databasebeta

← #325081

Subgroups — #325081

Order 512.

552 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30203C211yesmaximal, characteristic
2#29354C211yesmaximal, characteristic
3#29208C211yesmaximal, characteristic, complemented
4#22309C211yesmaximal, characteristic, complemented
5#10593C211yesmaximal, characteristic, complemented
6#10591C211yesmaximal, characteristic
7#10589C211yesmaximal, characteristic, complemented
8#10587C211yesmaximal, characteristic
9#9905C211yesmaximal, characteristic, complemented
10#9699C211yesmaximal, characteristic
11#9576C211yesmaximal, characteristic, complemented
12#9571C211yesmaximal, characteristic
13#9568C211yesmaximal, characteristic, complemented
14#9563C211yesmaximal, characteristic
15#8279C211yesmaximal, characteristic, complemented
16#2868C2211yescharacteristic
17#2858C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2071—21no—
20#2071—21no—
21#2071—21no—
22#2057—21no—
23#2057—21no—
24#2023—21no—
25#2023—21no—
26#2023—21no—
27#1990—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1985—21no—
33#1984—21no—
34#1982C2211yescharacteristic
35#1982C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1947C2211yescharacteristic
41#1947C2211yescharacteristic
42#1939C2211yescharacteristic
43#1939C2211yescharacteristic
44#1936—21no—
45#1936—21no—
46#1917—21no—
47#1916—21no—
48#1845—21no—
49#1831C2211yescharacteristic
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1744C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#331C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8D411yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×#124—21no—
74C2×#124—21no—
75C2×C4×Q8—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#391—21no—
103#390—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#389—21no—
108#389D411yescharacteristic
109#389D411yescharacteristic
110#384—21no—
111#384—21no—
112#384—21no—
113#384D411yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C43C2311yescharacteristic
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#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336D411yescharacteristic
141#336D411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144#335—22no—
145#334—22no—
146#333—22no—
147#333—22no—
148#331—22no—
149#331—22no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—41no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C4⋊Q8—22no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—41no—
172C4⋊Q8—81no—
173C4⋊Q8—81no—
174C4⋊Q8—81no—
175C4⋊Q8—81no—
176C4⋊Q8—82no—
177#127—21no—
178#127—21no—
179#127—21no—
180#127—21no—
181#124—82no—
182#124—82no—
183C4×Q8—81no—
184C4×Q8—81no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42C2411yescharacteristic
218#107—81no—
219#107—81no—
220#107—81no—
221#107—81no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—81no—
233C4⋊C8—81no—
234C4⋊C8#4112yes—
235C4⋊C8#4112yes—
236#102—42no—
237#102—42no—
238#102—81no—
239#102—81no—
240#102—81no—
241#102—81no—
242#102—81no—
243#102—81no—
244#97—81no—
245#97—81no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—42no—
251C4×C8—81no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256C24—41no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—81no—
262C2×Q8—82no—
263C2×Q8—82no—
264C2×Q8—82no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—81no—
282C2×D4—81no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—42no—
335C4⋊C4—42no—
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—41no—
347C4⋊C4—41no—
348C4⋊C4—81no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4#13712yes—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—22no—
370C42—22no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42C2×#4111yescharacteristic
376C42C2×#4111yescharacteristic
377C422−1+411yescharacteristic
378C422−1+411yescharacteristic
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23C2×#12111yescharacteristic
387Q8—84no—
388Q8—161no—
389Q8—161no—
390Q8—164no—
391D4—81no—
392D4—81no—
393D4—81no—
394D4—81no—
395D4—81no—
396D4—81no—
397D4—81no—
398D4—81no—
399D4—161no—
400D4—161no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4#47112yes—
435C2×C4#47212yes—
436C2×C4#53211yescharacteristic
437C2×C4#53611yescharacteristic
438C2×C4#54811yescharacteristic
439C2×C4#54811yescharacteristic
440C2×C4#55611yescharacteristic
441C2×C4#55611yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4#57511yescharacteristic
444C2×C4#57611yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57611yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57811yescharacteristic
449C2×C4#57811yescharacteristic
450C8—42no—
451C8—42no—
452C8—41no—
453C8—41no—
454C8—41no—
455C8—41no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8—81no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22—161no—
466C22—161no—
467C22#254511yescharacteristic
468C22#304111yescharacteristic
469C22#304411yescharacteristic
470C22#305611yescharacteristic
471C22#306211yescharacteristic
472C22C2×#47111yescharacteristic
473C22C2×#47211yescharacteristic
474C4—22no—
475C4—22no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—41no—
480C4—41no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4#317511yescharacteristic
488C4#317511yescharacteristic
489C4#324311yescharacteristic
490C4#324311yescharacteristic
491C4#336911yescharacteristic
492C4#337011yescharacteristic
493C4#337511yescharacteristic
494C4#337611yescharacteristic
495C2—321no—
496C2#1945711yescharacteristic
497C2#1947711yescharacteristic
498C2#1955411yescharacteristic
499C2#1957011yescharacteristic
500C2#2865111yescharacteristic
501C2#2866311yescharacteristic
502C2#2867011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29208C2semidirect
4. #22309C2semidirect
5. #10593C2semidirect
7. #10589C2semidirect
9. #9905C2semidirect
11. #9576C2semidirect
13. #9568C2semidirect
15. #8279C2semidirect

Maximal subgroups

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

subgroupindex
#302032
#293542
#292082
#223092
#105932
#105912
#105892
#105872
#99052
#96992
#95762
#95712
#95682
#95632
#82792