Finite Group Databasebeta

← #338829

Subgroups — #338829

Order 512.

499 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28830C211yesmaximal, characteristic, complemented
2#27728C211yesmaximal, characteristic
3#26299C211yesmaximal, characteristic
4#26087C211yesmaximal, characteristic
5#10680C211yesmaximal, characteristic, complemented
6#10619C211yesmaximal, characteristic
7#10593C211yesmaximal, characteristic, complemented
8#10578C211yesmaximal, characteristic, complemented
9#10413C211yesmaximal, characteristic
10#10352C211yesmaximal, characteristic, complemented
11#9614C211yesmaximal, characteristic, complemented
12#9567C211yesmaximal, characteristic, complemented
13#9541C211yesmaximal, characteristic
14#9535C211yesmaximal, characteristic, complemented
15#8897C211yesmaximal, characteristic
16#2571C2211yescharacteristic
17#2571C2211yescharacteristic
18#2563C2211yescharacteristic
19#2354C2211yescharacteristic
20#2101—21no—
21#2092—21no—
22#2085—21no—
23#2085—21no—
24#2077—21no—
25#2075—21no—
26#2073—21no—
27#2067—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1956C2211yescharacteristic
33#1954C2211yescharacteristic
34#1949—21no—
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1941—21no—
40#1939C2211yescharacteristic
41#1934—21no—
42#1921—21no—
43#1854—21no—
44#1846—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1835—21no—
48#1834C2211yescharacteristic
49#1825—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#106—21no—
76C2×#106—21no—
77C2×#106—21no—
78C2×#106D411yescharacteristic
79C2×#106D411yescharacteristic
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×#103—41no—
86C2×#102—21no—
87C2×#102—21no—
88C2×#102D411yescharacteristic
89C2×#102D411yescharacteristic
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#400—41no—
98#400—41no—
99#393—41no—
100#390—21no—
101#390—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#388—41no—
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#382—21no—
111#382—21no—
112#382—21no—
113#382—21no—
114#382—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2311yescharacteristic
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—41no—
137#340D411yescharacteristic
138#340D411yescharacteristic
139#340D411yescharacteristic
140#340D411yescharacteristic
141#336—21no—
142#336—21no—
143#336—21no—
144#335—22no—
145#333—22no—
146D4⋊C8—22no—
147C22×Q8—21no—
148C22×D4C2×D411yescharacteristic
149C23×C4—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4×D4—21no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—21no—
167#117—42no—
168#117—81no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
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—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—41no—
198C2×C42C2411yescharacteristic
199#107—42no—
200#107—81no—
201#107—81no—
202#107—81no—
203#107—81no—
204#106—22no—
205#106—41no—
206#106—42no—
207#106—81no—
208#106—81no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—42no—
217C4⋊C8#4112yes—
218#103—82no—
219#102—22no—
220#102—41no—
221#102—42no—
222#102—81no—
223C22⋊C8—42no—
224C22⋊C8—81no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—42no—
232C4×C8—42no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245C24—21no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—81no—
249C2×Q8—81no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4H(5,𝔽2)11yescharacteristic
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8#13512yes—
302C2×C8#13512yes—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
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—81no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C4⋊C4—161no—
337C4⋊C4—161no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—82no—
341C22⋊C4—161no—
342C22⋊C4—161no—
343C22⋊C4—161no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—82no—
351C42C2×#4111yescharacteristic
352C422−1+411yescharacteristic
353C422−1+411yescharacteristic
354C422−1+411yescharacteristic
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23D4211yescharacteristic
363Q8—82no—
364Q8—161no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4#53611yescharacteristic
409C2×C4#53611yescharacteristic
410C2×C4#54211yescharacteristic
411C2×C4#54711yescharacteristic
412C2×C4#54811yescharacteristic
413C2×C4#56311yescharacteristic
414C2×C4C2×#13511yescharacteristic
415C2×C4C2×#13511yescharacteristic
416C2×C4#57511yescharacteristic
417C2×C4#57511yescharacteristic
418C2×C4#57611yescharacteristic
419C2×C4#57711yescharacteristic
420C2×C4#57811yescharacteristic
421C2×C4#57811yescharacteristic
422C8—22no—
423C8—22no—
424C8—41no—
425C8—42no—
426C8—41no—
427C8—81no—
428C8—81no—
429C8—81no—
430C8—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22#267411yescharacteristic
439C22#326811yescharacteristic
440C22#326911yescharacteristic
441C22#328311yescharacteristic
442C22#328411yescharacteristic
443C22#328811yescharacteristic
444C22#329011yescharacteristic
445C4—21no—
446C4—21no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4#323211yescharacteristic
459C4#323311yescharacteristic
460C4#324211yescharacteristic
461C4#324311yescharacteristic
462C4#333611yescharacteristic
463C4#335111yescharacteristic
464C4#336111yescharacteristic
465C4#337711yescharacteristic
466C2—161no—
467C2#1985511yescharacteristic
468C2#1986411yescharacteristic
469C2#2001611yescharacteristic
470C2#2015311yescharacteristic
471C2#3078211yescharacteristic
472C2#3081111yescharacteristic
473C2#3083911yescharacteristic
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
1. #28830C2semidirect
5. #10680C2semidirect
7. #10593C2semidirect
8. #10578C2semidirect
10. #10352C2semidirect
11. #9614C2semidirect
12. #9567C2semidirect
14. #9535C2semidirect

Maximal subgroups

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

subgroupindex
#288302
#277282
#262992
#260872
#106802
#106192
#105932
#105782
#104132
#103522
#96142
#95672
#95412
#95352
#88972