Finite Group Databasebeta

← #277316

Subgroups — #277316

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28676C211yesmaximal, characteristic
2#24817C212yesmaximal
3Q8×#107C211yesmaximal, characteristic
4C2×#1676C211yesmaximal, characteristic
5C2×C8×Q16C211yesmaximal, characteristic
6#10593C212yesmaximal
7#10414C212yesmaximal
8#9823C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#8975C212yesmaximal
11#8645C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#477C2212yes—
14C2×C8×Q8C2211yescharacteristic
15C2×C8×Q8C2211yescharacteristic
16C2×C4×Q16C2211yescharacteristic
17#2567C2212yes—
18Q8×C4⋊C4C2212yes—
19#2087—22no—
20#2083—22no—
21#2068—22no—
22#1990C2212yes—
23#1987C2212yes—
24#1956C2212yes—
25#1947—22no—
26#1947C2212yes—
27#1942—21no—
28#1942—21no—
29#1939C2212yes—
30#1878—22no—
31#1845—21no—
32#1845C2211yescharacteristic
33#1803—22no—
34#1803C2212yes—
35C4×#107—21no—
36C4×#107C2211yescharacteristic
37#1676—24no—
38C8×Q16—24no—
39C2×#335C2211yescharacteristic
40C2×#335C2211yescharacteristic
41C2×#333C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×C82C2211yescharacteristic
45#1296C2212yes—
46#1279C2212yes—
47C22×Q16D411yescharacteristic
48C2×C4⋊Q8C2312yes—
49C2×C4×Q8—22no—
50C2×C4×Q8C2311yescharacteristic
51C2×C4×Q8C2311yescharacteristic
52#500—22no—
53#500—22no—
54#477—24no—
55#477—24no—
56C8×Q8—22no—
57C8×Q8—22no—
58C8×Q8—22no—
59C8×Q8D414yes—
60C4×Q16—22no—
61C4×Q16—22no—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—22no—
66C2×#107—21no—
67C2×#107—22no—
68C2×#107D411yescharacteristic
69C2×#107D411yescharacteristic
70C2×#107D411yescharacteristic
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—42no—
80C2×#103D411yescharacteristic
81C2×#103D411yescharacteristic
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#395—42no—
88#391—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—42no—
93#384—22no—
94#384—22no—
95#384C2312yes—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2312yes—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340D412yes—
105#336—22no—
106#335—22no—
107#335—22no—
108#333—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×Q16Q1614yes—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4×Q8—21no—
131C4×Q8—21no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×Q8—21no—
136C4×Q8—44no—
137C4×Q8—82no—
138C4×Q8C2×D412yes—
139C4×Q8#4112yes—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D412yes—
155C2×C42—21no—
156C2×C42—22no—
157C2×C42—21no—
158C2×C42—42no—
159C2×C42C2411yescharacteristic
160#107—22no—
161#107—22no—
162#107—42no—
163#107—42no—
164#107—42no—
165#107—44no—
166#107—82no—
167#107—81no—
168#107Q1614yescomplemented
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—82no—
182#103—84no—
183C4×C8—24no—
184C4×C8—21no—
185C4×C8—21no—
186C4×C8—21no—
187C4×C8—21no—
188C4×C8—42no—
189C4×C8C2×D412yes—
190C4×C8#4112yes—
191#95—22no—
192#95—42no—
193#95—42no—
194#95—42no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8#12112yes—
208C2×Q8C2×Q1612yes—
209C2×Q8C2×Q1612yes—
210C2×Q8#13512yes—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4H(5,𝔽2)11yescharacteristic
222Q16—28no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—22no—
228C2×C8—21no—
229C2×C8—22no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—42no—
242C2×C8#12112yes—
243C2×C8#12112yes—
244C2×C8C2×Q1612yes—
245C2×C8C2×Q1612yes—
246C2×C8#13712yes—
247C4⋊C4—22no—
248C4⋊C4—21no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—84no—
266C4⋊C4—82no—
267C4⋊C4—162no—
268C4⋊C4—162no—
269C4⋊C4—162no—
270C4⋊C4#12112yes—
271C4⋊C4C2×Q1614yes—
272C42—22no—
273C42—22no—
274C42—41no—
275C42—41no—
276C42—42no—
277C42—42no—
278C42—84no—
279C42C22×D411yescharacteristic
280C42C2×#4111yescharacteristic
281C42C2×#4111yescharacteristic
282C422−1+411yescharacteristic
283C23D4211yescharacteristic
284Q8—22no—
285Q8—22no—
286Q8—22no—
287Q8—22no—
288Q8—22no—
289Q8—41no—
290Q8—41no—
291Q8—84no—
292Q8—162no—
293Q8#46714yes—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4C2×#12111yescharacteristic
321C2×C4C2×#12111yescharacteristic
322C2×C4#53711yescharacteristic
323C2×C4#54711yescharacteristic
324C2×C4#54811yescharacteristic
325C2×C4#55311yescharacteristic
326C2×C4C22×Q1611yescharacteristic
327C2×C4C22×Q1611yescharacteristic
328C2×C4C2×#13511yescharacteristic
329C2×C4C2×#13711yescharacteristic
330C2×C4#57611yescharacteristic
331C2×C4#57711yescharacteristic
332C2×C4#57811yescharacteristic
333C2×C4#57811yescharacteristic
334C8—22no—
335C8—22no—
336C8—42no—
337C8—42no—
338C8—41no—
339C8—81no—
340C8—82no—
341C8#46214yes—
342C22#241611yescharacteristic
343C22#327811yescharacteristic
344C22#328011yescharacteristic
345C22D4×Q1611yescharacteristic
346C22D4×Q1611yescharacteristic
347C22#329011yescharacteristic
348C22#329211yescharacteristic
349C4—24no—
350C4—21no—
351C4—21no—
352C4—44no—
353C4—42no—
354C4—42no—
355C4—82no—
356C4—81no—
357C4—162no—
358C4—162no—
359C4C2×#46211yescharacteristic
360C4#304611yescharacteristic
361C4C2×#46711yescharacteristic
362C4#306111yescharacteristic
363C4#333611yescharacteristic
364C4#333811yescharacteristic
365C4#336311yescharacteristic
366C4#337411yescharacteristic
367C2#1990911yescharacteristic
368C2#1993211yescharacteristic
369C2#1999311yescharacteristic
370C2Q16211yescharacteristic
371C2#2473211yescharacteristic
372C2#2478111yescharacteristic
373C2#2479711yescharacteristic
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
168. #107Q16semidirect

Maximal subgroups

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

subgroupindex
#286762
#248172
Q8×#1072
C2×#16762
C2×C8×Q162
#105932
#104142
#98232
#95942
#89752
#86452