Finite Group Databasebeta

← #512243

Subgroups — #512243

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33267C211yesmaximal, characteristic, complemented
2C42×D8C211yesmaximal, characteristic
3#13133C211yesmaximal, characteristic, complemented
4#13133C211yesmaximal, characteristic, complemented
5#13132C211yesmaximal, characteristic, complemented
6#13112C211yesmaximal, characteristic, complemented
7#13110C211yesmaximal, characteristic, complemented
8#13109C211yesmaximal, characteristic, complemented
9#13014C211yesmaximal, characteristic
10#13014C211yesmaximal, characteristic
11#13014C211yesmaximal, characteristic
12#12937C211yesmaximal, characteristic
13#12935C211yesmaximal, characteristic
14#12935C211yesmaximal, characteristic
15#12894C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2863—21no—
21C42×D4—21no—
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31#1969C2211yescharacteristic
32#1969C2211yescharacteristic
33#1969C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1933—21no—
38#1933—21no—
39#1933—21no—
40C4×#107C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#102—21no—
44C4×#102—21no—
45C4×#102—21no—
46C42×C8C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1374C2211yescharacteristic
54#1374C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C22×C42—41no—
65C2×SD32—22no—
66C2×SD32—22no—
67C2×SD32—22no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C4×D8—22no—
73C4×D8—22no—
74C4×D8—22no—
75C4×D8—22no—
76C4×D8—22no—
77C4×D8—22no—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#103—21no—
85C2×#103—21no—
86C2×#103—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#391—21no—
97#391—21no—
98#391—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#387—41no—
103#387—41no—
104#387—41no—
105#383—41no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C22⋊C4—41no—
110C4×C22⋊C4—41no—
111C4×C22⋊C4—41no—
112C43D411yescharacteristic
113#367—22no—
114#366—22no—
115#366—22no—
116#358—22no—
117#358—22no—
118#358—22no—
119#357—22no—
120#357—22no—
121#357—22no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×Q8—21no—
129C22×D4—21no—
130C23×C4—41no—
131C23×C4—41no—
132C23×C4—41no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8#4112yes—
143C2×D8#4112yes—
144C2×D8#4112yes—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2411yescharacteristic
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42—84no—
171C2×C42D811yescharacteristic
172C2×C42D811yescharacteristic
173C2×C42D811yescharacteristic
174C2×C42D811yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178SD32—44no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107#4112yes—
201#107#4112yes—
202#107#4112yes—
203#103—42no—
204#103—42no—
205#103—42no—
206#102—24no—
207#102—24no—
208#102—24no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—41no—
216C4×C8—41no—
217C4×C8—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230C24—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C22×D411yescharacteristic
266Q16—41no—
267Q16—41no—
268Q16—41no—
269Q16—41no—
270D8—22no—
271D8—22no—
272D8—22no—
273D8#12614yes—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—41no—
317C42—82no—
318C42—82no—
319C42—82no—
320C16—41no—
321C16—41no—
322C16—41no—
323C16—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23C22×D811yescharacteristic
332Q8—81no—
333Q8—81no—
334Q8—81no—
335Q8—81no—
336D4—24no—
337D4—42no—
338D4—42no—
339D4—42no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—88no—
359C2×C4—88no—
360C2×C4—88no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4#50812yes—
380C2×C4#50912yes—
381C2×C4#51012yes—
382C2×C4#52511yescharacteristic
383C2×C4#52511yescharacteristic
384C2×C4#52511yescharacteristic
385C2×C4#57511yescharacteristic
386C2×C4#57511yescharacteristic
387C2×C4#57511yescharacteristic
388C8—41no—
389C8—41no—
390C8—41no—
391C8C2×#12611yescharacteristic
392C8#52911yescharacteristic
393C8#52911yescharacteristic
394C8#52911yescharacteristic
395C22—44no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22#310011yescharacteristic
403C22#310011yescharacteristic
404C22#310011yescharacteristic
405C22C2×#50811yescharacteristic
406C22C2×#50911yescharacteristic
407C22C2×#51011yescharacteristic
408C22#341411yescharacteristic
409C4—22no—
410C4—22no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—82no—
416C4—82no—
417C4—82no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4#285211yescharacteristic
423C4#311111yescharacteristic
424C4#311111yescharacteristic
425C4#311111yescharacteristic
426C2—84no—
427C2#3307811yescharacteristic
428C2#3366211yescharacteristic
429C2#3366511yescharacteristic
430C2#3366711yescharacteristic
431C2#3367011yescharacteristic
432C2#3367411yescharacteristic
433C2#3367611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33267C2semidirect
3. #13133C2semidirect
4. #13133C2semidirect
5. #13132C2semidirect
6. #13112C2semidirect
7. #13110C2semidirect
8. #13109C2semidirect
15. #12894C2semidirect

Maximal subgroups

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

subgroupindex
#332672
C42×D82
#131332
#131322
#131122
#131102
#131092
#130142
#129372
#129352
#128942