Finite Group Databasebeta

← #511729

Subgroups — #511729

Order 512.

453 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33267C211yesmaximal, characteristic, complemented
2#33266C211yesmaximal, characteristic
3#13516C211yesmaximal, characteristic, complemented
4#13515C211yesmaximal, characteristic, complemented
5#13514C211yesmaximal, characteristic, complemented
6#13477C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic
8#13475C211yesmaximal, characteristic, complemented
9#13475C211yesmaximal, characteristic, complemented
10#13474C211yesmaximal, characteristic
11#13474C211yesmaximal, characteristic
12#13449C211yesmaximal, characteristic
13#13449C211yesmaximal, characteristic
14#13449C211yesmaximal, characteristic
15#12898C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17#2863—21no—
18#2860—21no—
19C2×#367C2211yescharacteristic
20C2×#366C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1969C2211yescharacteristic
38#1969C2211yescharacteristic
39#1969C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1934—21no—
47#1934—21no—
48#1934—21no—
49#1933—21no—
50#1933—21no—
51#1933—21no—
52C42×C8C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×#126—41no—
62C2×SD32—41no—
63C2×SD32—41no—
64C2×SD32—41no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#103—21no—
76C2×#103—21no—
77C2×#103—21no—
78C2×#102—21no—
79C2×#102—21no—
80C2×#102—21no—
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8D411yescharacteristic
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#391—21no—
88#391—21no—
89#391—21no—
90#390—21no—
91#390—21no—
92#390—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#388—41no—
100#388—41no—
101#388—41no—
102#387—41no—
103#387—41no—
104#387—41no—
105#383—41no—
106C43D411yescharacteristic
107#367—41no—
108#366—41no—
109#366—41no—
110C4⋊C16—41no—
111C4⋊C16—41no—
112C4⋊C16—41no—
113C4⋊C16—41no—
114C4⋊C16—41no—
115C4⋊C16—41no—
116#358—41no—
117#358—41no—
118#358—41no—
119#357—41no—
120#357—41no—
121#357—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—
130C2×Q16—21no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2411yescharacteristic
146#126—84no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C42D811yescharacteristic
166C2×C42D811yescharacteristic
167C2×C42D811yescharacteristic
168C2×C42D811yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172SD32—44no—
173C2×C16—21no—
174C2×C16—21no—
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—
185#107—21no—
186#107—21no—
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#103—42no—
198#103—42no—
199#103—42no—
200#102—42no—
201#102—42no—
202#102—42no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—41no—
210C4×C8—41no—
211C4×C8—41no—
212#95—41no—
213#95—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—
224C24—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4C2×D811yescharacteristic
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C22×D411yescharacteristic
251Q16—41no—
252Q16—41no—
253Q16—41no—
254Q16—41no—
255D8—41no—
256D8—41no—
257D8—41no—
258D8—41no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8H(5,𝔽2)11yescharacteristic
269C2×C8H(5,𝔽2)11yescharacteristic
270C2×C8H(5,𝔽2)11yescharacteristic
271C2×C82−1+411yescharacteristic
272C2×C82−1+411yescharacteristic
273C2×C82−1+411yescharacteristic
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—82no—
297C22⋊C4—82no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—41no—
308C16—41no—
309C16—41no—
310C16—41no—
311C16—41no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23C22×D811yescharacteristic
320Q8—81no—
321Q8—81no—
322Q8—81no—
323Q8—81no—
324D4—81no—
325D4—81no—
326D4—81no—
327D4—81no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
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—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
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#50812yes—
365C2×C4#50912yes—
366C2×C4#51012yes—
367C2×C4#53711yescharacteristic
368C2×C4#53711yescharacteristic
369C2×C4#53711yescharacteristic
370C2×C4#57511yescharacteristic
371C2×C4#57511yescharacteristic
372C2×C4#57511yescharacteristic
373C8—41no—
374C8—41no—
375C8—41no—
376C8#56011yescharacteristic
377C8#56311yescharacteristic
378C8#56311yescharacteristic
379C8#56311yescharacteristic
380C22—161no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22—161no—
385C22—161no—
386C22—161no—
387C22#324511yescharacteristic
388C22#324511yescharacteristic
389C22#324511yescharacteristic
390C22C2×#50811yescharacteristic
391C22C2×#50911yescharacteristic
392C22C2×#51011yescharacteristic
393C22#341411yescharacteristic
394C4—22no—
395C4—22no—
396C4—22no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—161no—
401C4—161no—
402C4—161no—
403C4—161no—
404C4—161no—
405C4—161no—
406C4—161no—
407C4#282311yescharacteristic
408C4#325611yescharacteristic
409C4#325611yescharacteristic
410C4#325611yescharacteristic
411C2—321no—
412C2#3285811yescharacteristic
413C2#3381311yescharacteristic
414C2#3381411yescharacteristic
415C2#3381611yescharacteristic
416C2#3381711yescharacteristic
417C2#3381811yescharacteristic
418C2#3382011yescharacteristic
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. #13515C2semidirect
5. #13514C2semidirect
6. #13477C2semidirect
8. #13475C2semidirect
9. #13475C2semidirect
15. #12898C2semidirect

Maximal subgroups

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

subgroupindex
#332672
#332662
#135162
#135152
#135142
#134772
#134762
#134752
#134742
#134492
#128982