Finite Group Databasebeta

← #279385

Subgroups — #279385

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33198C211yesmaximal, characteristic
2#29219C211yesmaximal, characteristic
3#22734C211yesmaximal, characteristic
4#21506C211yesmaximal, characteristic
5#10655C212yesmaximal
6#10594C212yesmaximal
7#9905C211yesmaximal, characteristic
8#9604C211yesmaximal, characteristic
9#9218C212yesmaximal
10#8649C212yesmaximal
11#8589C211yesmaximal, characteristic
12C2×#477—21no—
13C2×#462C2211yescharacteristic
14C2×C8×Q8C2212yes—
15C2×#441—21no—
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C42×Q8C2211yescharacteristic
19#2099—21no—
20#2099—21no—
21#1990C2212yes—
22#1988—22no—
23#1987—21no—
24#1981C2211yescharacteristic
25#1948C2211yescharacteristic
26#1947—21no—
27#1947C2212yes—
28#1939—22no—
29#1939C2212yes—
30#1927—21no—
31#1878—21no—
32#1842C2211yescharacteristic
33#1834C2212yes—
34#1831—21no—
35#1831C2211yescharacteristic
36#1803—21no—
37#1803C2212yes—
38#1803C2211yescharacteristic
39#1767C2211yescharacteristic
40C4×C4⋊C8—21no—
41C4×C4⋊C8C2211yescharacteristic
42#1763—21no—
43C2×#335C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C22×Q16—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4×Q8—21no—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58#477—44no—
59#462—22no—
60#462Q814yescomplemented
61C8×Q8—24no—
62C8×Q8—42no—
63#441—44no—
64C2×#107—22no—
65C2×#106—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8D412yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2311yescharacteristic
75C2×#103—21no—
76C2×#103—22no—
77C2×#103—21no—
78C2×#103—21no—
79C2×#103—21no—
80C2×#97—21no—
81C2×C4×C8—22no—
82C2×C4×C8C2312yes—
83#391—21no—
84#389—21no—
85#389—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384C2311yescharacteristic
90#384C2312yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4D411yescharacteristic
97C4×C4⋊C4D411yescharacteristic
98C4×C4⋊C4C2311yescharacteristic
99C43—21no—
100C43C2311yescharacteristic
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—21no—
107#340—22no—
108#340—21no—
109#336—22no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336D411yescharacteristic
114#336D411yescharacteristic
115#335—42no—
116#332—42no—
117C22×Q8—21no—
118C22×Q8—41no—
119C22×Q8C2×D411yescharacteristic
120C2×Q16—24no—
121C2×Q16—42no—
122C2×Q16—42no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8C2×D412yes—
128C22×C8C2×D412yes—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—44no—
133C4⋊Q8—42no—
134C4⋊Q8—42no—
135C4⋊Q8—82no—
136C4⋊Q8C2×Q812yes—
137C4×Q8—22no—
138C4×Q8—21no—
139C4×Q8—21no—
140C4×Q8—22no—
141C4×Q8—41no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×Q8D814yes—
145C4×Q8C2×Q812yes—
146C4×Q8#4112yes—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—41no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42C2411yescharacteristic
179#107—44no—
180#106—42no—
181C4⋊C8—21no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—21no—
186C4⋊C8—22no—
187C4⋊C8—21no—
188C4⋊C8—42no—
189C4⋊C8—44no—
190C4⋊C8—42no—
191C4⋊C8—44no—
192C4⋊C8—42no—
193C4⋊C8C2×Q812yes—
194#103—24no—
195#103—42no—
196#103—42no—
197#103—44no—
198#103—42no—
199#97—42no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—44no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—41no—
207#95—41no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—82no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×Q8C22⋊Q812yes—
221C2×Q8C2×D812yes—
222C2×Q8#13712yes—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C2×#4111yescharacteristic
237Q16—48no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—42no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—41no—
251C2×C8—42no—
252C2×C8—41no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C4⋊C4—21no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—81no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4C22⋊Q812yes—
281C4⋊C4C2×D812yes—
282C42—22no—
283C42—22no—
284C42—21no—
285C42—21no—
286C42—22no—
287C42—22no—
288C42—24no—
289C42—22no—
290C42—41no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—81no—
295C42—82no—
296C42C2×D812yes—
297C42#13512yes—
298C42C22×Q811yescharacteristic
299C42C2×#4111yescharacteristic
300C42H(5,𝔽2)11yescharacteristic
301C422−1+411yescharacteristic
302C23C2×#12111yescharacteristic
303Q8—22no—
304Q8—22no—
305Q8—22no—
306Q8—42no—
307Q8—42no—
308Q8—81no—
309Q8—81no—
310Q8—164no—
311Q8#47414yes—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4#46612yes—
345C2×C4#46612yes—
346C2×C4#47112yes—
347C2×C4C2×C22⋊Q811yescharacteristic
348C2×C4#53711yescharacteristic
349C2×C4#54611yescharacteristic
350C2×C4D4×Q811yescharacteristic
351C2×C4#55211yescharacteristic
352C2×C4#55411yescharacteristic
353C2×C4C22×D811yescharacteristic
354C2×C4C2×#13511yescharacteristic
355C2×C4C2×#13711yescharacteristic
356C2×C4#57511yescharacteristic
357C2×C4#57512yes—
358C2×C4#57812yes—
359C8—42no—
360C8—42no—
361C8—42no—
362C8—81no—
363C8—82no—
364C8—82no—
365C22#241711yescharacteristic
366C22#304011yescharacteristic
367C22#304411yescharacteristic
368C22C2×#46611yescharacteristic
369C22#305611yescharacteristic
370C22C2×#47111yescharacteristic
371C22#306911yescharacteristic
372C4—24no—
373C4—22no—
374C4—21no—
375C4—21no—
376C4—22no—
377C4—22no—
378C4—42no—
379C4—42no—
380C4—41no—
381C4—82no—
382C4—81no—
383C4—162no—
384C4—161no—
385C4C2×#47411yescharacteristic
386C4#307611yescharacteristic
387C4#324511yescharacteristic
388C4#324811yescharacteristic
389C4#331312yes—
390C4#339012yes—
391C2#1927611yescharacteristic
392C2#1932511yescharacteristic
393C2#1940411yescharacteristic
394C2#1947511yescharacteristic
395C2#2483211yescharacteristic
396C2#2483511yescharacteristic
397C2#2485811yescharacteristic
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
60. #462Q8semidirect

Maximal subgroups

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

subgroupindex
#331982
#292192
#227342
#215062
#106552
#105942
#99052
#96042
#92182
#86492
#85892