Finite Group Databasebeta

← #290078

Subgroups — #290078

Order 512.

603 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29286C211yesmaximal, characteristic
2#28689C211yesmaximal, characteristic
3#28512C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5#10653C212yesmaximal
6#10588C212yesmaximal
7#9898C211yesmaximal, characteristic
8#9611C212yesmaximal
9#9602C211yesmaximal, characteristic
10#9561C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12C2×#500—21no—
13C2×#475—21no—
14C2×#475—21no—
15C2×C8×Q8—21no—
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2096—22no—
20#2086—22no—
21#2083—22no—
22#1990—21no—
23#1988—21no—
24#1988C2212yes—
25#1984—21no—
26#1956C2212yes—
27#1947—21no—
28#1947C2212yes—
29#1947C2212yes—
30#1947C2212yes—
31#1939C2212yes—
32#1916—22no—
33#1844C2212yes—
34#1842C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#334C2212yes—
40C2×#332C2211yescharacteristic
41C2×#332C2211yescharacteristic
42#1296C2211yescharacteristic
43#1296C2211yescharacteristic
44#1296C2212yes—
45#1296C2212yes—
46#1277C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54#500—44no—
55#475—44no—
56#475—44no—
57C8×Q8—44no—
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#106—22no—
62C2×#106—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8D411yescharacteristic
67C2×C4⋊C8D411yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×#103—22no—
74C2×#103—22no—
75C2×C4×C8—22no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8C2312yes—
79#391—21no—
80#391—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389D412yes—
85#384—21no—
86#384—21no—
87#384—21no—
88#384D411yescharacteristic
89#384C2311yescharacteristic
90#384C2312yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4C2312yes—
95C43C2311yescharacteristic
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#336—22no—
102#336—22no—
103#336—21no—
104#336—21no—
105#336D412yes—
106#334—24no—
107#332—22no—
108#332Q814yescomplemented
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—41no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8C2×D412yes—
118C22×C8C2×D412yes—
119C4⋊Q8—22no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—42no—
124C4⋊Q8—42no—
125C4⋊Q8—42no—
126C4⋊Q8—42no—
127C4⋊Q8—82no—
128C4⋊Q8SD1614yes—
129C4⋊Q8C2×Q812yes—
130C4×Q8—42no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D412yes—
152C2×C42—22no—
153C2×C42—41no—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42C2411yescharacteristic
161#107—44no—
162#107—44no—
163#107—82no—
164#106—44no—
165#106—44no—
166C4⋊C8—22no—
167C4⋊C8—21no—
168C4⋊C8—22no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—22no—
174C4⋊C8—44no—
175C4⋊C8—42no—
176C4⋊C8—81no—
177C4⋊C8D814yes—
178C4⋊C8C2×Q812yes—
179C4⋊C8C2×Q812yes—
180C4⋊C8#4112yes—
181#103—44no—
182#103—44no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—44no—
186C4×C8—42no—
187C4×C8—42no—
188#95—22no—
189#95—22no—
190#95—41no—
191#95—41no—
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—41no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—82no—
202C2×Q8—82no—
203C2×Q8—82no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D411yescharacteristic
215C22×C4C2×#4111yescharacteristic
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8C2×D814yes—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4C2×SD1614yes—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—22no—
268C42—41no—
269C42—42no—
270C42—42no—
271C42—82no—
272C42C2×D812yes—
273C42C2×SD1612yes—
274C42#13512yes—
275C42AGL(1,ℤ/8)12yes—
276C42C22×Q811yescharacteristic
277C42C2×#4111yescharacteristic
278C422−1+411yescharacteristic
279C422−1+411yescharacteristic
280C23C2×#12111yescharacteristic
281Q8—48no—
282Q8—42no—
283Q8—42no—
284Q8—81no—
285Q8—81no—
286Q8—164no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—24no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4#45912yes—
312C2×C4#45912yes—
313C2×C4#46512yes—
314C2×C4#46512yes—
315C2×C4#47112yes—
316C2×C4#53311yescharacteristic
317C2×C4#54111yescharacteristic
318C2×C4#54711yescharacteristic
319C2×C4#54811yescharacteristic
320C2×C4D4×Q811yescharacteristic
321C2×C4D4×Q811yescharacteristic
322C2×C4C22×D811yescharacteristic
323C2×C4C22×SD1611yescharacteristic
324C2×C4C2×#13511yescharacteristic
325C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
326C2×C4#57711yescharacteristic
327C2×C4#57811yescharacteristic
328C2×C4#57811yescharacteristic
329C2×C4#57811yescharacteristic
330C8—24no—
331C8—42no—
332C8—42no—
333C8—42no—
334C8—42no—
335C8—81no—
336C8—82no—
337C22#242911yescharacteristic
338C22C2×#45911yescharacteristic
339C22#304411yescharacteristic
340C22C2×#46511yescharacteristic
341C22#306311yescharacteristic
342C22C2×#47111yescharacteristic
343C22#306911yescharacteristic
344C4—22no—
345C4—21no—
346C4—21no—
347C4—22no—
348C4—41no—
349C4—44no—
350C4—82no—
351C4—82no—
352C4—161no—
353C4—162no—
354C4#166514yes—
355C4#333411yescharacteristic
356C4#335311yescharacteristic
357C4#335911yescharacteristic
358C4#336511yescharacteristic
359C4Q8×D811yescharacteristic
360C4Q8×SD1611yescharacteristic
361C4#338311yescharacteristic
362C4#338611yescharacteristic
363C2C2×#166511yescharacteristic
364C2#1923111yescharacteristic
365C2#1940211yescharacteristic
366C2#1957111yescharacteristic
367C2#2544911yescharacteristic
368C2#2547211yescharacteristic
369C2#2549211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
108. #332Q8semidirect

Maximal subgroups

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

subgroupindex
#292862
#286892
#285122
#231492
#106532
#105882
#98982
#96112
#96022
#95612
C4×#3322