Finite Group Databasebeta

← #288729

Subgroups — #288729

Order 512.

650 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28532C211yesmaximal, characteristic
2#28265C211yesmaximal, characteristic
3#25294C212yesmaximal
4C2×#1674C211yesmaximal, characteristic
5C2×C8×Q16C211yesmaximal, characteristic
6#10422C212yesmaximal
7#10234C212yesmaximal
8#9823C211yesmaximal, characteristic
9#9614C211yesmaximal, characteristic
10#9220C212yesmaximal
11C8×#103C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#475C2212yes—
14C2×#462C2212yes—
15C2×C8×Q8C2211yescharacteristic
16C2×C8×Q8C2212yes—
17C2×C4×Q16C2211yescharacteristic
18#2582C2212yes—
19#2568C2212yes—
20#2097—22no—
21#2068—22no—
22#2068—22no—
23#2056—22no—
24#1990C2212yes—
25#1969—22no—
26#1969—21no—
27#1969—21no—
28#1956C2212yes—
29#1947C2212yes—
30#1942—21no—
31#1933—22no—
32#1846—21no—
33#1845C2211yescharacteristic
34#1803C2212yes—
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37#1674—24no—
38C8×Q16—24no—
39C2×#335C2211yescharacteristic
40C2×#332C2212yes—
41C2×Q8⋊C8C2211yescharacteristic
42C2×Q8⋊C8C2212yes—
43C2×C82C2211yescharacteristic
44#1296C2212yes—
45#1279C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×Q16—22no—
48C22×Q16D411yescharacteristic
49C2×C4⋊Q8C2312yes—
50C2×C4×Q8C2312yes—
51C2×C4×Q8C2311yescharacteristic
52#500—22no—
53#500—22no—
54#475—24no—
55#475—42no—
56#462—24no—
57#462—42no—
58C8×Q8—24no—
59C8×Q8—22no—
60C8×Q8—22no—
61C8×Q8—42no—
62C4×Q16—22no—
63C4×Q16—22no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#107—22no—
69C2×#107D411yescharacteristic
70C2×#106—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103D411yescharacteristic
82C2×#103D412yes—
83C2×#103D411yescharacteristic
84C2×C4×C8—21no—
85C2×C4×C8—22no—
86C2×C4×C8—21no—
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#400—42no—
91#398—42no—
92#391—22no—
93#391—22no—
94#391—22no—
95#391—22no—
96#391—22no—
97#389—22no—
98#389—22no—
99#384C2312yes—
100#382—22no—
101C4×C4⋊C4C2311yescharacteristic
102#340—22no—
103#340—22no—
104#340—42no—
105#340D411yescharacteristic
106#340D411yescharacteristic
107#336—21no—
108#335D414yes—
109#332—24no—
110Q8⋊C8—24no—
111Q8⋊C8—22no—
112C82—22no—
113C22×Q8—22no—
114C22×Q8C2×D412yes—
115C22×Q8C2×D411yescharacteristic
116C2×Q16—22no—
117C2×Q16—22no—
118C2×Q16—44no—
119C2×Q16—44no—
120C2×Q16—82no—
121C2×Q16Q1614yescomplemented
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4×Q8—22no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—22no—
138C4×Q8—21no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C42—21no—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42C2411yescharacteristic
158#107—22no—
159#107—42no—
160#107—42no—
161#107—42no—
162#107—82no—
163#107—81no—
164#106—44no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—22no—
168C4⋊C8—21no—
169C4⋊C8—44no—
170C4⋊C8—42no—
171C4⋊C8C2×D414yes—
172C4⋊C8#4112yes—
173#103—22no—
174#103—24no—
175#103—22no—
176#103—44no—
177#103—44no—
178#103—44no—
179#103—82no—
180#103—82no—
181C4×C8—21no—
182C4×C8—21no—
183C4×C8—21no—
184C4×C8—21no—
185C4×C8—44no—
186C4×C8—42no—
187C4×C8—42no—
188C4×C8C2×D412yes—
189C4×C8#4112yes—
190#95—21no—
191#95—42no—
192#95—42no—
193#95—42no—
194#95—42no—
195#95—42no—
196#95—42no—
197#95—42no—
198C2×Q8—21no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—22no—
204C2×Q8—21no—
205C2×Q8—22no—
206C2×Q8—21no—
207C2×Q8—22no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8C2×Q1614yes—
215C2×Q8#13512yes—
216C22×C4—21no—
217C22×C4—22no—
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—24no—
236C2×C8—22no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8#12112yes—
249C2×C8#12112yes—
250C2×C8C2×Q1612yes—
251C2×C8#13512yes—
252C2×C8#13512yes—
253C4⋊C4—22no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—82no—
274C4⋊C4—162no—
275C42—41no—
276C42—42no—
277C42—42no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42C22×D411yescharacteristic
282C42C2×#4111yescharacteristic
283C42C2×#4111yescharacteristic
284C42H(5,𝔽2)11yescharacteristic
285C23D4211yescharacteristic
286Q8—22no—
287Q8—22no—
288Q8—24no—
289Q8—24no—
290Q8—41no—
291Q8—41no—
292Q8—42no—
293Q8—42no—
294Q8—82no—
295Q8—82no—
296Q8—82no—
297Q8—82no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4C2×#12111yescharacteristic
325C2×C4#53211yescharacteristic
326C2×C4#54011yescharacteristic
327C2×C4D4211yescharacteristic
328C2×C4#54611yescharacteristic
329C2×C4#55011yescharacteristic
330C2×C4C22×Q1611yescharacteristic
331C2×C4C2×#13511yescharacteristic
332C2×C4C2×#13511yescharacteristic
333C2×C4C2×#13511yescharacteristic
334C2×C4#57611yescharacteristic
335C2×C4#57711yescharacteristic
336C2×C4#57811yescharacteristic
337C2×C4#57811yescharacteristic
338C8—22no—
339C8—22no—
340C8—42no—
341C8—41no—
342C8—44no—
343C8—42no—
344C8—81no—
345C8#46714yes—
346C22#242811yescharacteristic
347C22#328411yescharacteristic
348C22#328511yescharacteristic
349C22D4×Q1611yescharacteristic
350C22#328711yescharacteristic
351C22#329011yescharacteristic
352C22#329311yescharacteristic
353C4—21no—
354C4—21no—
355C4—44no—
356C4—42no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—82no—
361C4—162no—
362C4—162no—
363C4C2×#46711yescharacteristic
364C4#306311yescharacteristic
365C4#316811yescharacteristic
366C4#317511yescharacteristic
367C4#328011yescharacteristic
368C4#328511yescharacteristic
369C4#329911yescharacteristic
370C4#330211yescharacteristic
371C2#2000811yescharacteristic
372C2#2001011yescharacteristic
373C2#2001411yescharacteristic
374C2#2001811yescharacteristic
375C2#2538611yescharacteristic
376C2#2541211yescharacteristic
377C2#2544211yescharacteristic
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
121. C2×Q16Q16semidirect

Maximal subgroups

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

subgroupindex
#285322
#282652
#252942
C2×#16742
C2×C8×Q162
#104222
#102342
#98232
#96142
#92202
C8×#1032