Finite Group Databasebeta

← #277313

Subgroups — #277313

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28675C211yesmaximal, characteristic
2#24632C212yesmaximal
3#22727C211yesmaximal, characteristic
4C2×#1674C211yesmaximal, characteristic
5C2×C8×Q16C211yesmaximal, characteristic
6#10600C212yesmaximal
7#10418C212yesmaximal
8#9823C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#8986C212yesmaximal
11#8645C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#462C2212yes—
14C2×C8×Q8C2211yescharacteristic
15C2×C8×Q8C2211yescharacteristic
16C2×C4×Q16C2211yescharacteristic
17#2546C2212yes—
18#2355C2212yes—
19#2086—22no—
20#2056—22no—
21#2016—22no—
22#1990C2212yes—
23#1988C2212yes—
24#1981—22no—
25#1956C2212yes—
26#1949—22no—
27#1948C2212yes—
28#1942—21no—
29#1942—21no—
30#1939C2212yes—
31#1928—22no—
32#1846—21no—
33#1845C2211yescharacteristic
34#1803C2212yes—
35C4×#107—21no—
36C4×#107C2211yescharacteristic
37#1674—24no—
38C8×Q16—24no—
39C2×#335C2211yescharacteristic
40C2×#335C2211yescharacteristic
41C2×#332C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×C82C2211yescharacteristic
45#1296C2212yes—
46#1279C2212yes—
47C22×Q16—22no—
48C22×Q16D411yescharacteristic
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2312yes—
51C2×C4×Q8C2311yescharacteristic
52C2×C4×Q8C2311yescharacteristic
53#500—22no—
54#500—22no—
55#462—24no—
56#462—24no—
57C8×Q8—22no—
58C8×Q8—22no—
59C8×Q8—22no—
60C8×Q8D414yes—
61C4×Q16—22no—
62C4×Q16—22no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107D411yescharacteristic
68C2×#107D411yescharacteristic
69C2×#107D411yescharacteristic
70C2×#106—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103D411yescharacteristic
82C2×#103D411yescharacteristic
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#398—42no—
89#391—22no—
90#391—22no—
91#389—22no—
92#389—22no—
93#387—42no—
94#384C2312yes—
95#382—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4C2312yes—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—42no—
104#340D412yes—
105#336—22no—
106#335—22no—
107#335—22no—
108#332—24no—
109Q8⋊C8—22no—
110Q8⋊C8—22no—
111C82—22no—
112C22×Q8—22no—
113C22×Q8C2×D411yescharacteristic
114C22×Q8C2×D411yescharacteristic
115C2×Q16—22no—
116C2×Q16—22no—
117C2×Q16—22no—
118C2×Q16—44no—
119C2×Q16—82no—
120C2×Q16—82no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—44no—
134C4⋊Q8—82no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—21no—
138C4×Q8—21no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8C2×D412yes—
142C4×Q8#4112yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C4⋊C4C2×D412yes—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—42no—
160C2×C42C2411yescharacteristic
161#107—22no—
162#107—22no—
163#107—42no—
164#107—42no—
165#107—42no—
166#107—81no—
167#107Q1614yescomplemented
168#106—82no—
169C4⋊C8—24no—
170C4⋊C8—21no—
171C4⋊C8—22no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—22no—
175C4⋊C8—42no—
176C4⋊C8—82no—
177C4⋊C8C2×D412yes—
178#103—22no—
179#103—22no—
180#103—44no—
181#103—44no—
182#103—82no—
183#103—82no—
184C4×C8—24no—
185C4×C8—21no—
186C4×C8—21no—
187C4×C8—21no—
188C4×C8—21no—
189C4×C8—42no—
190C4×C8C2×D412yes—
191C4×C8#4112yes—
192#95—22no—
193#95—42no—
194#95—42no—
195#95—42no—
196#95—42no—
197#95—42no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8#12112yes—
215C2×Q8#13512yes—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4H(5,𝔽2)11yescharacteristic
227Q16—28no—
228Q16—88no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—22no—
234C2×C8—21no—
235C2×C8—22no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8#12112yes—
249C2×C8#12112yes—
250C2×C8C2×Q1612yes—
251C2×C8#13512yes—
252C2×C8#13712yes—
253C4⋊C4—22no—
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C4⋊C4—82no—
274C4⋊C4#12112yes—
275C4⋊C4C2×Q1614yes—
276C42—22no—
277C42—22no—
278C42—41no—
279C42—41no—
280C42—42no—
281C42—42no—
282C42—162no—
283C42C22×D411yescharacteristic
284C42C2×#4111yescharacteristic
285C42C2×#4111yescharacteristic
286C422−1+411yescharacteristic
287C23D4211yescharacteristic
288Q8—22no—
289Q8—22no—
290Q8—22no—
291Q8—22no—
292Q8—41no—
293Q8—41no—
294Q8—41no—
295Q8—41no—
296Q8—82no—
297Q8—82no—
298Q8—82no—
299Q8—82no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4C2×#12111yescharacteristic
327C2×C4C2×#12111yescharacteristic
328C2×C4#53711yescharacteristic
329C2×C4#54711yescharacteristic
330C2×C4#54811yescharacteristic
331C2×C4#55311yescharacteristic
332C2×C4C22×Q1611yescharacteristic
333C2×C4C2×#13511yescharacteristic
334C2×C4C2×#13511yescharacteristic
335C2×C4C2×#13711yescharacteristic
336C2×C4#57611yescharacteristic
337C2×C4#57611yescharacteristic
338C2×C4#57711yescharacteristic
339C2×C4#57811yescharacteristic
340C8—22no—
341C8—22no—
342C8—42no—
343C8—42no—
344C8—41no—
345C8—81no—
346C8—82no—
347C8#46214yes—
348C22#241611yescharacteristic
349C22#327811yescharacteristic
350C22#328011yescharacteristic
351C22D4×Q1611yescharacteristic
352C22#328711yescharacteristic
353C22#329011yescharacteristic
354C22#329311yescharacteristic
355C4—21no—
356C4—21no—
357C4—44no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4—82no—
362C4—81no—
363C4—162no—
364C4—162no—
365C4C2×#46211yescharacteristic
366C4#304611yescharacteristic
367C4#305011yescharacteristic
368C4#305811yescharacteristic
369C4#333911yescharacteristic
370C4#334111yescharacteristic
371C4#336311yescharacteristic
372C4#337411yescharacteristic
373C2#1990711yescharacteristic
374C2#1996611yescharacteristic
375C2#1999311yescharacteristic
376C2#2001011yescharacteristic
377C2#2473211yescharacteristic
378C2#2477811yescharacteristic
379C2#2479811yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
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
167. #107Q16semidirect

Maximal subgroups

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

subgroupindex
#286752
#246322
#227272
C2×#16742
C2×C8×Q162
#106002
#104182
#98232
#95942
#89862
#86452