Finite Group Databasebeta

← #507802

Subgroups — #507802

Order 512.

514 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33267C211yesmaximal, characteristic, complemented
2#21275C211yesmaximal, characteristic
3#13516C211yesmaximal, characteristic, complemented
4#13477C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic
6#13449C211yesmaximal, characteristic
7#13237C211yesmaximal, characteristic
8#13166C211yesmaximal, characteristic, complemented
9#13148C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic
11#13132C211yesmaximal, characteristic, complemented
12#13111C211yesmaximal, characteristic, complemented
13#13014C211yesmaximal, characteristic
14#12937C211yesmaximal, characteristic
15C4×C4⋊C16C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2863—21no—
19#2293—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1970C2211yescharacteristic
34#1970C2211yescharacteristic
35#1969C2211yescharacteristic
36#1969C2211yescharacteristic
37#1969C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1934—21no—
44#1934—21no—
45#1933—21no—
46#1933—21no—
47#1933—21no—
48C4×#107C2211yescharacteristic
49C4×#102—21no—
50C42×C8C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×#117—41no—
63C2×SD32—22no—
64C2×SD32—22no—
65C2×SD32—41no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C4×D8—22no—
71C4×D8—22no—
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#103—21no—
79C2×#103—21no—
80C2×#103—21no—
81C2×#102—21no—
82C2×#102—21no—
83C2×#102—21no—
84C2×C4×C8D411yescharacteristic
85C2×C4×C8D411yescharacteristic
86C2×C4×C8D411yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#391—21no—
91#391—21no—
92#391—21no—
93#390—21no—
94#390—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#388—41no—
101#388—41no—
102#387—41no—
103#387—41no—
104#387—41no—
105#383—41no—
106C4×C4⋊C4—21no—
107C4×C22⋊C4—41no—
108C43D411yescharacteristic
109#367—22no—
110#367—22no—
111#367—41no—
112C4⋊C16—22no—
113C4⋊C16—22no—
114#358—22no—
115#358—22no—
116#358—41no—
117#357—22no—
118#357—22no—
119#357—41no—
120C4×C16—41no—
121C4×C16—41no—
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—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8#4112yes—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2411yescharacteristic
147C4×D4—24no—
148C4×D4—42no—
149#117—84no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C42—41no—
167C2×C42D811yescharacteristic
168C2×C42D811yescharacteristic
169C2×C42D811yescharacteristic
170C2×C42D811yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174SD32—44no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16#4112yes—
186C2×C16#4112yes—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107#4112yes—
199#103—42no—
200#103—42no—
201#103—42no—
202#102—24no—
203#102—42no—
204#102—42no—
205C4×C8—22no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—41no—
213C4×C8—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—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—
226C24—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×D4—22no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—44no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C22×D411yescharacteristic
256Q16—41no—
257Q16—41no—
258Q16—41no—
259Q16—41no—
260D8—22no—
261D8—22no—
262D8—41no—
263D8—41no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8#13512yes—
273C2×C8C2×#4111yescharacteristic
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8C2×#4111yescharacteristic
277C2×C8H(5,𝔽2)11yescharacteristic
278C2×C82−1+411yescharacteristic
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C22⋊C4—44no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—22no—
308C42—41no—
309C42—41no—
310C42—82no—
311C42SD3214yescomplemented
312C16—22no—
313C16—22no—
314C16—41no—
315C16—41no—
316C23—42no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23C22×D811yescharacteristic
324Q8—81no—
325Q8—81no—
326Q8—81no—
327Q8—81no—
328D4—42no—
329D4—42no—
330D4—81no—
331D4—81no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—88no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4C2×SD3212yes—
369C2×C4C2×SD3212yes—
370C2×C4C2×SD3212yes—
371C2×C4#50812yes—
372C2×C4#52511yescharacteristic
373C2×C4#53211yescharacteristic
374C2×C4#53711yescharacteristic
375C2×C4C2×#13511yescharacteristic
376C2×C4#57511yescharacteristic
377C2×C4#57511yescharacteristic
378C8—22no—
379C8—41no—
380C8—41no—
381C8#55111yescharacteristic
382C8#55211yescharacteristic
383C8#55311yescharacteristic
384C8#55611yescharacteristic
385C22—82no—
386C22—82no—
387C22—82no—
388C22—161no—
389C22—161no—
390C22—161no—
391C22—161no—
392C22#310011yescharacteristic
393C22#316911yescharacteristic
394C22#324511yescharacteristic
395C22C22×SD3211yescharacteristic
396C22C2×#50811yescharacteristic
397C22#341411yescharacteristic
398C22#341411yescharacteristic
399C4—22no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—41no—
404C4—41no—
405C4—82no—
406C4—161no—
407C4—161no—
408C4—161no—
409C4—161no—
410C4—161no—
411C4—161no—
412C4#260111yescharacteristic
413C4#310211yescharacteristic
414C4#318311yescharacteristic
415C4#325011yescharacteristic
416C2—162no—
417C2#2942811yescharacteristic
418C2#3366111yescharacteristic
419C2#3366211yescharacteristic
420C2#3373611yescharacteristic
421C2#3375011yescharacteristic
422C2#3381211yescharacteristic
423C2#3381611yescharacteristic
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. #13516C2semidirect
4. #13477C2semidirect
8. #13166C2semidirect
9. #13148C2semidirect
11. #13132C2semidirect
12. #13111C2semidirect
15. C4×C4⋊C16C2semidirect
311. C42SD32semidirect

Maximal subgroups

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

subgroupindex
#332672
#212752
#135162
#134772
#134762
#134492
#132372
#131662
#131482
#131472
#131322
#131112
#130142
#129372
C4×C4⋊C162