Finite Group Databasebeta

← #275270

Subgroups — #275270

Order 512.

637 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33147C211yesmaximal, characteristic
2#27972C211yesmaximal, characteristic
3#22625C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic
5#10657C212yesmaximal, complemented
6#10588C212yesmaximal, complemented
7#9613C212yesmaximal, complemented
8#9561C212yesmaximal, complemented
9#9129C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic
11#8724C211yesmaximal, characteristic
12C22×#107—21no—
13C22×C4⋊C8C2211yescharacteristic
14#2857C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#2072—22no—
18#1988C2211yescharacteristic
19#1988C2211yescharacteristic
20#1956C2212yes—
21#1948C2211yescharacteristic
22#1948C2211yescharacteristic
23#1947—21no—
24#1947—21no—
25#1947C2212yes—
26#1947C2212yes—
27#1939C2212yes—
28#1919—21no—
29#1919—21no—
30#1916—22no—
31#1872—21no—
32#1872—21no—
33#1853—21no—
34#1834C2212yes—
35#1824—21no—
36#1797—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C8×C4⋊C4C2212yes—
40C4×C22⋊C8—21no—
41C4×C22⋊C8C2211yescharacteristic
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×C8—21no—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C22⋊Q8—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59C2×#107—22no—
60C2×#107—22no—
61C2×#107—44no—
62C2×#106—21no—
63C2×#106—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2312yes—
71C2×#103—22no—
72C2×#103—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8D411yescharacteristic
76C2×C22⋊C8D411yescharacteristic
77C2×C4×C8—22no—
78C2×C4×C8C2312yes—
79#391—21no—
80#389—21no—
81#389—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384C2311yescharacteristic
86#384C2312yes—
87#380—21no—
88#380—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4D411yescharacteristic
92C4×C4⋊C4D411yescharacteristic
93C4×C4⋊C4C2311yescharacteristic
94C4×C4⋊C4C2311yescharacteristic
95C2×#95—21no—
96#340—22no—
97#340—22no—
98#340—21no—
99#340—21no—
100#340—21no—
101#340—21no—
102#340—21no—
103#340—21no—
104#340—22no—
105#340—22no—
106#336—22no—
107#336—22no—
108#336—21no—
109#336—21no—
110#336D412yes—
111#334—24no—
112#332—24no—
113C22×Q8—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—41no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4#4111yescharacteristic
120C22×C8—24no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—42no—
126C22×C8—42no—
127C22×C8C2×D412yes—
128C22×C8C2×D412yes—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C22⋊Q8—42no—
132C22⋊Q8—42no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4SD1614yes—
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C22⋊C4—41no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—82no—
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42C2411yescharacteristic
171#107—44no—
172#107—44no—
173#106—42no—
174#106—42no—
175C4⋊C8—24no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—82no—
182C4⋊C8#4114yes—
183#103—82no—
184#103—82no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—82no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—41no—
197#95—41no—
198C24#12111yescharacteristic
199C2×Q8—42no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—42no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4C2×SD1612yes—
232C22×C4#13512yes—
233C22×C4#13512yes—
234C22×C4#13712yes—
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D411yescharacteristic
237C22×C4C2×#4111yescharacteristic
238C2×C8—24no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—48no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—42no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—24no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—81no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4C2×SD1612yes—
282C4⋊C4C2×SD1612yes—
283C4⋊C4#13712yes—
284C22⋊C4—82no—
285C42—21no—
286C42—21no—
287C42—22no—
288C42—22no—
289C42—42no—
290C42—42no—
291C42—81no—
292C42—81no—
293C42C2×#4112yes—
294C422−1+412yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#46412yes—
300C23#46512yes—
301C23#47312yes—
302C23C2×#12111yescharacteristic
303Q8—82no—
304Q8—82no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—44no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—164no—
345C2×C4#46412yes—
346C2×C4#46512yes—
347C2×C4#52511yescharacteristic
348C2×C4#53611yescharacteristic
349C2×C4#54712yes—
350C2×C4#55212yes—
351C2×C4C22×SD1611yescharacteristic
352C2×C4C2×#13511yescharacteristic
353C2×C4C2×#13511yescharacteristic
354C2×C4C2×#13711yescharacteristic
355C2×C4C2×#13711yescharacteristic
356C2×C4#57511yescharacteristic
357C2×C4#57612yes—
358C8—42no—
359C8—42no—
360C8—44no—
361C8—82no—
362C8—81no—
363C8—81no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22#168014yes—
371C22#241311yescharacteristic
372C22C2×#46411yescharacteristic
373C22#304111yescharacteristic
374C22C2×#46511yescharacteristic
375C22#305611yescharacteristic
376C22C2×#47311yescharacteristic
377C22#306811yescharacteristic
378C4—22no—
379C4—22no—
380C4—22no—
381C4—42no—
382C4—41no—
383C4—42no—
384C4—42no—
385C4—81no—
386C4—161no—
387C4—162no—
388C4—161no—
389C4#310111yescharacteristic
390C4#310811yescharacteristic
391C4#322911yescharacteristic
392C4#323211yescharacteristic
393C4#333912yes—
394C4#334612yes—
395C2—24no—
396C2C2×#168011yescharacteristic
397C2#1917611yescharacteristic
398C2#1938411yescharacteristic
399C2#1945911yescharacteristic
400C2#2458311yescharacteristic
401C2#2460111yescharacteristic
402C2#2462211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10657C2semidirect
6. #10588C2semidirect
7. #9613C2semidirect
8. #9561C2semidirect

Maximal subgroups

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

subgroupindex
#331472
#279722
#226252
C2×#19472
#106572
#105882
#96132
#95612
#91292
#88302
#87242