Finite Group Databasebeta

← #341536

Subgroups — #341536

Order 512.

529 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28828C211yesmaximal, characteristic, complemented
2#28790C211yesmaximal, characteristic, complemented
3#26533C211yesmaximal, characteristic
4#26526C211yesmaximal, characteristic
5#10671C211yesmaximal, characteristic
6#10655C211yesmaximal, characteristic, complemented
7#10652C211yesmaximal, characteristic, complemented
8#10582C211yesmaximal, characteristic, complemented
9#10578C211yesmaximal, characteristic, complemented
10#10351C211yesmaximal, characteristic
11#9564C211yesmaximal, characteristic
12#9564C211yesmaximal, characteristic
13#9536C211yesmaximal, characteristic, complemented
14#9533C211yesmaximal, characteristic, complemented
15#8924C211yesmaximal, characteristic
16#2571C2211yescharacteristic
17#2571C2211yescharacteristic
18#2552C2211yescharacteristic
19#2366C2211yescharacteristic
20#2093—21no—
21#2092—21no—
22#2085—21no—
23#2085—21no—
24#2085—21no—
25#2067—21no—
26#2065—21no—
27#2059—21no—
28#1991—21no—
29#1982C2211yescharacteristic
30#1982C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1948C2211yescharacteristic
39#1947C2211yescharacteristic
40#1942—21no—
41#1941—21no—
42#1939C2211yescharacteristic
43#1939C2211yescharacteristic
44#1862—21no—
45#1862—21no—
46#1846—21no—
47#1845C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1804—21no—
51#1802—21no—
52C4×#107C2211yescharacteristic
53C4×#107C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C8×C4⋊C4C2211yescharacteristic
56C2×#331C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#125—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107D411yescharacteristic
75C2×#107D411yescharacteristic
76C2×#107D411yescharacteristic
77C2×#107D411yescharacteristic
78C2×#106—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—41no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#400—41no—
104#400—41no—
105#398—41no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#386—41no—
115#384—21no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120#382—21no—
121#382—21no—
122#382—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
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—41no—
136#340D411yescharacteristic
137#340D411yescharacteristic
138#340D411yescharacteristic
139#340D411yescharacteristic
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#331Q814yes—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C23×C4—41no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158#127—21no—
159#127—21no—
160#127—21no—
161#127C2×Q812yes—
162#125—42no—
163#125—81no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—81no—
167C4×D4—81no—
168C2×C4⋊C4—21no—
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—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
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×C4⋊C4C2×D411yescharacteristic
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C2411yescharacteristic
198#107—22no—
199#107—22no—
200#107—41no—
201#107—41no—
202#107—42no—
203#107—81no—
204#107—81no—
205#106—42no—
206#106—81no—
207#106—81no—
208#106—81no—
209#106—81no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—81no—
217C4⋊C8—81no—
218C4⋊C8C2×Q812yes—
219C4⋊C8C2×Q812yes—
220#102—42no—
221#102—42no—
222#102—81no—
223#102—81no—
224#102—81no—
225#102—81no—
226#102—81no—
227#102—81no—
228#102—82no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—42no—
236C4×C8—81no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248C24—41no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—81no—
264C2×D4—81no—
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—82no—
277C22×C4—82no—
278C22×C4—161no—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
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#13512yes—
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—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
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—82no—
337C4⋊C4—82no—
338C4⋊C4—81no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C4⋊C4—161no—
342C4⋊C4—161no—
343C4⋊C4—161no—
344C4⋊C4#13512yes—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—161no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42C22×Q811yescharacteristic
357C422−1+411yescharacteristic
358C422−1+411yescharacteristic
359C422−1+411yescharacteristic
360C23—42no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23D4211yescharacteristic
368D4—42no—
369D4—42no—
370D4—42no—
371D4—42no—
372D4—82no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—161no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
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—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—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—164no—
414C2×C4#53611yescharacteristic
415C2×C4#53611yescharacteristic
416C2×C4#54411yescharacteristic
417C2×C4D4×Q811yescharacteristic
418C2×C4D4×Q811yescharacteristic
419C2×C4#56311yescharacteristic
420C2×C4C2×#13511yescharacteristic
421C2×C4C2×#13511yescharacteristic
422C2×C4#57511yescharacteristic
423C2×C4#57511yescharacteristic
424C2×C4#57611yescharacteristic
425C2×C4#57611yescharacteristic
426C2×C4#57711yescharacteristic
427C2×C4#57711yescharacteristic
428C8—22no—
429C8—42no—
430C8—42no—
431C8—41no—
432C8—41no—
433C8—41no—
434C8—81no—
435C8—81no—
436C8—81no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22#267711yescharacteristic
445C22#326711yescharacteristic
446C22#327111yescharacteristic
447C22#328011yescharacteristic
448C22#328711yescharacteristic
449C22#328811yescharacteristic
450C22#329011yescharacteristic
451C4—21no—
452C4—21no—
453C4—42no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4#323211yescharacteristic
465C4#323311yescharacteristic
466C4#324111yescharacteristic
467C4#324111yescharacteristic
468C4#338111yescharacteristic
469C4#338111yescharacteristic
470C4#338911yescharacteristic
471C4#338911yescharacteristic
472C2—162no—
473C2#1977511yescharacteristic
474C2#1987411yescharacteristic
475C2#1999511yescharacteristic
476C2#2013611yescharacteristic
477C2#3095011yescharacteristic
478C2#3095911yescharacteristic
479C2#3098111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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. #28828C2semidirect
2. #28790C2semidirect
6. #10655C2semidirect
7. #10652C2semidirect
8. #10582C2semidirect
9. #10578C2semidirect
13. #9536C2semidirect
14. #9533C2semidirect

Maximal subgroups

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

subgroupindex
#288282
#287902
#265332
#265262
#106712
#106552
#106522
#105822
#105782
#103512
#95642
#95362
#95332
#89242