Finite Group Databasebeta

← #238241

Subgroups — #238241

Order 512.

884 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#35192C211yesmaximal, characteristic
2#21840C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5#10352C214yesmaximal, complemented
6#10349C214yesmaximal, complemented
7#8042C211yesmaximal, characteristic
8#7902C211yesmaximal, characteristic
9#7899C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2864—21no—
15#2862—21no—
16#2665—22no—
17#2605—22no—
18#2571C2212yes—
19#2551C2212yes—
20#2439—22no—
21#2439—22no—
22#2331—22no—
23#2285C2211yescharacteristic
24#2283C2211yescharacteristic
25C4×#117C2211yescharacteristic
26#1993—21no—
27#1993—21no—
28#1988—22no—
29#1987C2212yes—
30#1987C2212yes—
31#1987C2214yes—
32#1981—22no—
33#1956C2214yes—
34#1823—22no—
35#1787—22no—
36#1736—22no—
37#1296C2214yes—
38#1279C2212yes—
39#1279C2214yes—
40#1279C2212yes—
41C23×Q8—21no—
42C22×M16—21no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8Q814yescomplemented
52C2×C4⋊Q8C2312yes—
53C2×#117—21no—
54C2×#117D412yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C42C2311yescharacteristic
58#432—22no—
59#432—22no—
60#431—22no—
61#431—22no—
62#423—22no—
63#423—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×#103—28no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×#97—21no—
72C2×C4×C8C2312yes—
73#400—24no—
74#396—24no—
75#395—24no—
76#393—24no—
77#391—24no—
78#391—24no—
79#391—24no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#387—24no—
85#387—24no—
86#387—24no—
87#384C2312yes—
88#383—21no—
89#382—22no—
90#382—22no—
91#382—21no—
92#382D411yescharacteristic
93#382D411yescharacteristic
94#381—21no—
95#380—21no—
96#380D412yes—
97#380D412yes—
98C4×C4⋊C4—21no—
99C4×C4⋊C4D412yes—
100C4×C4⋊C4D412yes—
101C4×C4⋊C4C2311yescharacteristic
102C4×C4⋊C4C2312yes—
103C4×C4⋊C4C2311yescharacteristic
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107#376—21no—
108#376—21no—
109C43—21no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—22no—
117C22×Q8—41no—
118C22×Q8—42no—
119C22×Q8—84no—
120C23×C4—41no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×M16—44no—
125C2×M16—42no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C4⋊Q8—24no—
130C4⋊Q8—24no—
131C4⋊Q8—24no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—84no—
141C4⋊Q8C2×Q814yes—
142#117—24no—
143#117—42no—
144#117—82no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D414yes—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×Q812yes—
176C2×C42#4112yes—
177C2×C42#4112yes—
178C2×C42#4112yes—
179C2×C42C2411yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—42no—
185C4⋊C8—42no—
186#103—88no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189#97—42no—
190C4×C8—22no—
191C4×C8—22no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—22no—
197#95—22no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—22no—
203#95—22no—
204#95—21no—
205#95—21no—
206#95—22no—
207C24#12011yescharacteristic
208C2×Q8—24no—
209C2×Q8—48no—
210C2×Q8—48no—
211C2×Q8—42no—
212C2×Q8—44no—
213C2×Q8—84no—
214C2×Q8—84no—
215C2×Q8—84no—
216C2×Q8—88no—
217C2×Q8—88no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—84no—
231C22×C4#12011yescharacteristic
232C22×C4#12011yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4C22⋊Q812yes—
235C22×C4C22⋊Q812yes—
236C22×C4C22⋊Q812yes—
237C22×C4#12412yes—
238C22×C4#12412yes—
239C22×C4#12412yes—
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243M16—84no—
244C2×C8—44no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—44no—
249C2×C8—42no—
250C2×C8—42no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—44no—
259C4⋊C4—48no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C4⋊C4#13714yes—
270C22⋊C4—44no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C42—44no—
274C42—42no—
275C42—42no—
276C42—42no—
277C42—42no—
278C42—82no—
279C42—82no—
280C42C22⋊Q812yes—
281C42C22⋊Q812yes—
282C42C22⋊Q812yes—
283C42#12412yes—
284C42#12412yes—
285C42#12412yes—
286C42C22×Q811yescharacteristic
287C42C2×#4111yescharacteristic
288C42C2×#4111yescharacteristic
289C42C2×#4111yescharacteristic
290C23—22no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23#39312yes—
295C23C2×#12011yescharacteristic
296Q8—48no—
297Q8—84no—
298Q8—88no—
299Q8—168no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—88no—
329C2×C4—84no—
330C2×C4#39312yes—
331C2×C4#39312yes—
332C2×C4#39312yes—
333C2×C4C2×C22⋊Q811yescharacteristic
334C2×C4C2×C22⋊Q811yescharacteristic
335C2×C4C2×C22⋊Q811yescharacteristic
336C2×C4C2×#12411yescharacteristic
337C2×C4C2×#12411yescharacteristic
338C2×C4C2×#12411yescharacteristic
339C2×C4C2×#13712yes—
340C2×C4#57512yes—
341C2×C4#57712yes—
342C2×C4#57812yes—
343C8—82no—
344C8—82no—
345C8—82no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22C2×#39311yescharacteristic
352C22#300712yes—
353C22#300911yescharacteristic
354C22#301111yescharacteristic
355C22#302112yes—
356C4—22no—
357C4—22no—
358C4—21no—
359C4—21no—
360C4—22no—
361C4—44no—
362C4—84no—
363C4—82no—
364C4—82no—
365C4—82no—
366C4—84no—
367C4#307812yes—
368C4#308212yes—
369C4#313712yes—
370C4#313911yescharacteristic
371C4#314011yescharacteristic
372C2—42no—
373C2#1883512yes—
374C2#1885312yes—
375C2#2345511yescharacteristic
376C2#2345711yescharacteristic
377C2#2346311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #10352C2semidirect
6. #10349C2semidirect
51. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351922
#218402
#218042
#213132
#103522
#103492
#80422
#79022
#78992