Finite Group Databasebeta

← #318976

Subgroups — #318976

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29288C211yesmaximal, characteristic, complemented
2#29256C211yesmaximal, characteristic
3#22301C211yesmaximal, characteristic, complemented
4#22268C211yesmaximal, characteristic
5#10600C211yesmaximal, characteristic, complemented
6#10599C211yesmaximal, characteristic
7#10588C211yesmaximal, characteristic, complemented
8#10587C211yesmaximal, characteristic
9#9744C211yesmaximal, characteristic
10#9732C211yesmaximal, characteristic
11#9622C211yesmaximal, characteristic, complemented
12#9616C211yesmaximal, characteristic, complemented
13#8812C211yesmaximal, characteristic, complemented
14#8589C211yesmaximal, characteristic, complemented
15#8585C211yesmaximal, characteristic
16C2×#492C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C4×#127C2211yescharacteristic
19C4×#125C2211yescharacteristic
20#2092—21no—
21#2088—21no—
22#2087—21no—
23#2025—21no—
24#1989—21no—
25#1988C2211yescharacteristic
26#1988C2211yescharacteristic
27#1982C2211yescharacteristic
28#1982C2211yescharacteristic
29#1957C2211yescharacteristic
30#1956C2211yescharacteristic
31#1949—21no—
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1940—21no—
37#1939—21no—
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1917—21no—
41#1916—21no—
42#1880—21no—
43#1880—21no—
44#1877—21no—
45#1877—21no—
46#1864—21no—
47#1862—21no—
48#1842C2211yescharacteristic
49#1842C2211yescharacteristic
50#1834C2211yescharacteristic
51#1831C2211yescharacteristic
52#1770C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54#1744C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67C22×SD16—21no—
68C22×SD16—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×#125D411yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76#492—22no—
77#492Q814yes—
78C2×#107—21no—
79C2×#107—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#103—21no—
95C2×#103—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#391—21no—
107#390—21no—
108#390—21no—
109#390—21no—
110#389—21no—
111#389—21no—
112#389D411yescharacteristic
113#389D411yescharacteristic
114#387—21no—
115#387—21no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#382D411yescharacteristic
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C43C2311yescharacteristic
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#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336D411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#336D411yescharacteristic
146#335—22no—
147#335—41no—
148#334—22no—
149#334—41no—
150#332—41no—
151#332—41no—
152#331—41no—
153#331—41no—
154C22×Q8—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C23×C4—41no—
159C2×SD16—24no—
160C2×SD16—24no—
161C2×SD16—81no—
162C2×SD16—81no—
163C2×SD16—81no—
164C2×SD16—81no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8C2×Q812yes—
179#127—21no—
180#127—21no—
181#127—21no—
182#127C2×Q812yes—
183#125—22no—
184#125—41no—
185C4×Q8—81no—
186C4×Q8—81no—
187C4×D4—81no—
188C4×D4—81no—
189C4×D4—81no—
190C4×D4—81no—
191C4×D4—82no—
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—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—41no—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#107—81no—
223#107—81no—
224#106—81no—
225#106—81no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—42no—
239C4⋊C8—81no—
240C4⋊C8—81no—
241C4⋊C8—81no—
242C4⋊C8—81no—
243C4⋊C8—81no—
244#103—81no—
245#103—81no—
246#102—42no—
247#102—42no—
248#102—81no—
249#102—81no—
250#102—81no—
251#102—81no—
252#97—81no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—81no—
256C4×C8C2×Q812yes—
257C4×C8#4112yes—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—41no—
265#95—41no—
266C24—41no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—81no—
288C2×D4—81no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4111yescharacteristic
306SD16—84no—
307SD16—84no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
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—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8C22⋊Q812yes—
337C2×C8C22⋊Q812yes—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C4⋊C4—82no—
364C4⋊C4#13712yes—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—22no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—82no—
383C42C22×Q811yescharacteristic
384C42C2×#4111yescharacteristic
385C42H(5,𝔽2)11yescharacteristic
386C422−1+411yescharacteristic
387C23—42no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23C2×#12111yescharacteristic
395Q8—42no—
396Q8—42no—
397Q8—81no—
398Q8—81no—
399D4—42no—
400D4—42no—
401D4—42no—
402D4—42no—
403D4—82no—
404D4—82no—
405D4—81no—
406D4—81no—
407D4—81no—
408D4—81no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—164no—
444C2×C4#47112yes—
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4#53711yescharacteristic
447C2×C4D4×Q811yescharacteristic
448C2×C4#55011yescharacteristic
449C2×C4#55411yescharacteristic
450C2×C4#55611yescharacteristic
451C2×C4C2×#13711yescharacteristic
452C2×C4#57511yescharacteristic
453C2×C4#57611yescharacteristic
454C2×C4#57611yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57711yescharacteristic
458C2×C4#57711yescharacteristic
459C8—22no—
460C8—22no—
461C8—41no—
462C8—41no—
463C8—81no—
464C8—81no—
465C8—81no—
466C8—81no—
467C8—81no—
468C8—81no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22#253811yescharacteristic
477C22#304511yescharacteristic
478C22#304511yescharacteristic
479C22#305611yescharacteristic
480C22#306211yescharacteristic
481C22C2×#47111yescharacteristic
482C22#306711yescharacteristic
483C4—22no—
484C4—21no—
485C4—21no—
486C4—41no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—82no—
494C4—161no—
495C4—161no—
496C4#308111yescharacteristic
497C4#308111yescharacteristic
498C4#325911yescharacteristic
499C4#326011yescharacteristic
500C4#338511yescharacteristic
501C4#338811yescharacteristic
502C4#340011yescharacteristic
503C4#340311yescharacteristic
504C2—162no—
505C2#1948211yescharacteristic
506C2#1949311yescharacteristic
507C2#1957611yescharacteristic
508C2#1958111yescharacteristic
509C2#2827711yescharacteristic
510C2#2829211yescharacteristic
511C2#2829611yescharacteristic
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
1. #29288C2semidirect
3. #22301C2semidirect
5. #10600C2semidirect
7. #10588C2semidirect
11. #9622C2semidirect
12. #9616C2semidirect
13. #8812C2semidirect
14. #8589C2semidirect

Maximal subgroups

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

subgroupindex
#292882
#292562
#223012
#222682
#106002
#105992
#105882
#105872
#97442
#97322
#96222
#96162
#88122
#85892
#85852