Finite Group Databasebeta

← #242865

Subgroups — #242865

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21982C211yesmaximal, characteristic
2#21818C211yesmaximal, characteristic
3#20647C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#10623C214yesmaximal, complemented
6#10617C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#8039C211yesmaximal, characteristic
9#7877C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#419—21no—
16C22×#103—21no—
17#2285C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#1984—22no—
20#1984C2214yes—
21#1947—22no—
22#1852—22no—
23#1845C2212yes—
24#1834C2212yes—
25#1817—22no—
26#1805—22no—
27#1803—22no—
28#1803C2212yes—
29#1803C2214yes—
30#1803C2212yes—
31#1736—21no—
32C2×#340—21no—
33#1296C2212yes—
34#1296C2212yes—
35#1296C2214yes—
36#1277C2214yes—
37C23×Q8D411yescharacteristic
38C22×M16—21no—
39C23×C8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×C4×Q8D412yes—
47C2×C4×Q8C2312yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#431—22no—
57#431—22no—
58#431—22no—
59#431—22no—
60C2×#107—24no—
61#423—22no—
62#423—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8D412yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71#419—44no—
72C2×#103—24no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—44no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8D412yes—
78C2×C4×C8—21no—
79C2×C4×C8C2312yes—
80#384—21no—
81#384—21no—
82#384—21no—
83#384C2311yescharacteristic
84#384C2312yes—
85#384C2311yescharacteristic
86#382—21no—
87#380—21no—
88#380—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91#340—24no—
92#340—24no—
93#340—22no—
94#340—22no—
95#336—28no—
96C22×Q8—21no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8D814yes—
103C22×Q8C2×D412yes—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—41no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—42no—
119C22×C8C2×D414yes—
120C4×Q8—24no—
121C4×Q8—24no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—42no—
127C4×Q8—42no—
128C4×Q8—42no—
129C4×Q8—42no—
130#117—42no—
131#117—42no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
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—21no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4C2×D412yes—
156C2×C22⋊C4—41no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42—41no—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42#4112yes—
165C2×C42#4112yes—
166C2×C42#4112yes—
167C2×C42#4112yes—
168C2×C42#4111yescharacteristic
169C2×C42C2411yescharacteristic
170#107—84no—
171C4⋊C8—24no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—42no—
179C4⋊C8—42no—
180#103—48no—
181#103—44no—
182#103—44no—
183C22⋊C8—24no—
184C22⋊C8—42no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—42no—
188#95—41no—
189#95—41no—
190C24#12111yescharacteristic
191C2×Q8—22no—
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—28no—
198C2×Q8—28no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8C2×D814yes—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12312yes—
226C22×C4#12312yes—
227C22×C4#12312yes—
228C22×C4#12412yes—
229C22×C4#12612yes—
230C22×C4#12612yes—
231C22×C4C2×D812yes—
232C22×C4#13512yes—
233C22×C4#13712yes—
234C22×C4#13712yes—
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D411yescharacteristic
237C22×C4C2×#4111yescharacteristic
238M16—84no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—84no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—42no—
267C4⋊C4—81no—
268C4⋊C4—81no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C22⋊C4—82no—
274C42—42no—
275C42—42no—
276C42—44no—
277C42—82no—
278C42#12312yes—
279C42#12312yes—
280C42#12312yes—
281C42#12412yes—
282C42#12612yes—
283C42#12612yes—
284C42C2×#4111yescharacteristic
285C42C2×#4111yescharacteristic
286C42C2×#4111yescharacteristic
287C42C2×#4111yescharacteristic
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23#39712yes—
293C23#46612yes—
294C23#47012yes—
295C23C2×#12111yescharacteristic
296Q8—24no—
297Q8—24no—
298Q8—42no—
299Q8—42no—
300Q8—48no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—48no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—84no—
319C2×C4—82no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—164no—
329C2×C4#39712yes—
330C2×C4#39712yes—
331C2×C4#39712yes—
332C2×C4#46612yes—
333C2×C4#47012yes—
334C2×C4#48014yes—
335C2×C4#48014yes—
336C2×C4#48514yes—
337C2×C4#48514yes—
338C2×C4C2×#12311yescharacteristic
339C2×C4C2×#12311yescharacteristic
340C2×C4C2×#12311yescharacteristic
341C2×C4C2×#12411yescharacteristic
342C2×C4C2×#12611yescharacteristic
343C2×C4C2×#12611yescharacteristic
344C2×C4C22×D811yescharacteristic
345C2×C4C2×#13511yescharacteristic
346C2×C4C2×#13711yescharacteristic
347C2×C4C2×#13711yescharacteristic
348C2×C4#57712yes—
349C2×C4#57812yes—
350C8—44no—
351C8—82no—
352C8—82no—
353C8—82no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22#207514yes—
361C22C2×#39711yescharacteristic
362C22#304311yescharacteristic
363C22#304411yescharacteristic
364C22C2×#46611yescharacteristic
365C22C2×#47011yescharacteristic
366C22#306811yescharacteristic
367C22#306911yescharacteristic
368C4—22no—
369C4—21no—
370C4—21no—
371C4—22no—
372C4—44no—
373C4—82no—
374C4—161no—
375C4—162no—
376C4—161no—
377C4#207514yes—
378C4C2×#48011yescharacteristic
379C4C2×#48011yescharacteristic
380C4C2×#48511yescharacteristic
381C4C2×#48511yescharacteristic
382C4#309712yes—
383C4#313712yes—
384C2—24no—
385C2#1939811yescharacteristic
386C2#1940411yescharacteristic
387C2#1943511yescharacteristic
388C2#1944111yescharacteristic
389C2#2363311yescharacteristic
390C2C2×#207511yescharacteristic
391C2#2364711yescharacteristic
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. #10623C2semidirect
6. #10617C2semidirect

Maximal subgroups

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

subgroupindex
#219822
#218182
#206472
C2×#18032
#106232
#106172
#80432
#80392
#78772