Finite Group Databasebeta

← #202867

Subgroups — #202867

Order 512.

828 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21834C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20533C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#9568C214yesmaximal, complemented
7#9567C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic
9#7877C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#428—21no—
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2285C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#1990C2212yes—
19#1990C2212yes—
20#1987—22no—
21#1987—22no—
22#1987C2214yes—
23#1949—22no—
24#1948C2212yes—
25#1947C2212yes—
26#1824—22no—
27C8×C4⋊C4—22no—
28C8×C4⋊C4C2214yes—
29C4×C22⋊C8—22no—
30#1736—21no—
31C2×#340—21no—
32C2×#340—21no—
33#1548C418yescomplemented
34C2×#333C2214yes—
35#1296C2212yes—
36#1296C2212yes—
37#1277C2214yes—
38C23×Q8—21no—
39C22×M16—21no—
40C23×C8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2×C414yes—
48C2×C4⋊Q8C2312yes—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#431—22no—
57#431—22no—
58#431—22no—
59#431C2×C414yes—
60#428—44no—
61C2×#107—22no—
62C2×#106—22no—
63#423—22no—
64#423C2×C414yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8D412yes—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8D412yes—
75C2×C4×C8—21no—
76C2×C4×C8—24no—
77C2×C4×C8C2312yes—
78#384C2311yescharacteristic
79#384C2311yescharacteristic
80#382—21no—
81#380—21no—
82#380—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4D412yes—
88C4×C4⋊C4C2312yes—
89C2×#95—21no—
90#340—24no—
91#340—24no—
92#340—22no—
93#340—22no—
94#340—24no—
95#340—24no—
96#336—24no—
97#333C2×C4116yescomplemented
98C22×Q8—24no—
99C22×Q8—22no—
100C22×Q8—41no—
101C22×Q8—42no—
102C22×Q8—82no—
103C22×Q8—82no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—41no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—24no—
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—24no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—42no—
127C4⋊Q8—42no—
128C4⋊Q8—42no—
129C4⋊Q8C22×C414yes—
130#117—42no—
131#117—42no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4SD1614yes—
151C2×C4⋊C4Q1614yes—
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×D412yes—
154C2×C22⋊C4—41no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—41no—
158C2×C42—82no—
159C2×C42C22×C412yes—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4112yes—
164C2×C42#4112yes—
165C2×C42#4112yes—
166C2×C42C2411yescharacteristic
167#107—44no—
168#106—44no—
169C4⋊C8—24no—
170C4⋊C8—24no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—42no—
176C4⋊C8C22×C414yes—
177C4⋊C8C22×C414yes—
178C22⋊C8—24no—
179C22⋊C8—24no—
180C4×C8—24no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—84no—
184#95—22no—
185#95—22no—
186#95—41no—
187#95—41no—
188C24#12111yescharacteristic
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8—44no—
192C2×Q8—82no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8—88no—
197C2×Q8—88no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4#11712yes—
217C22×C4C4×D412yes—
218C22×C4C4×D412yes—
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12312yes—
223C22×C4#12412yes—
224C22×C4#12612yes—
225C22×C4C2×SD1612yes—
226C22×C4C2×Q1612yes—
227C22×C4#13512yes—
228C22×C4#13512yes—
229C22×C4C22×D411yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4C2×#4111yescharacteristic
232M16—48no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—84no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4C2×SD1614yes—
265C4⋊C4C2×Q1614yes—
266C22⋊C4—82no—
267C42—44no—
268C42—82no—
269C42—82no—
270C42#11712yes—
271C42C4×D412yes—
272C42C4×D412yes—
273C42#12312yes—
274C42#12412yes—
275C42#12612yes—
276C42C23×C411yescharacteristic
277C42C2×#4111yescharacteristic
278C42C2×#4111yescharacteristic
279C42C2×#4111yescharacteristic
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23#38812yes—
285C23#46512yes—
286C23#46712yes—
287C23C2×#12111yescharacteristic
288Q8—84no—
289Q8—88no—
290Q8—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4—88no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4#38812yes—
320C2×C4#38812yes—
321C2×C4#38812yes—
322C2×C4C4×SD1614yes—
323C2×C4C4×Q1614yes—
324C2×C4#46512yes—
325C2×C4#46712yes—
326C2×C4#48714yes—
327C2×C4#48814yes—
328C2×C4C2×#11711yescharacteristic
329C2×C4C2×C4×D411yescharacteristic
330C2×C4C2×C4×D411yescharacteristic
331C2×C4C2×#12311yescharacteristic
332C2×C4C2×#12411yescharacteristic
333C2×C4C2×#12611yescharacteristic
334C2×C4C22×SD1611yescharacteristic
335C2×C4C22×Q1611yescharacteristic
336C2×C4C2×#13511yescharacteristic
337C2×C4C2×#13511yescharacteristic
338C2×C4#57712yes—
339C2×C4#57812yes—
340C8—44no—
341C8—44no—
342C8—44no—
343C8—82no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22#193314yes—
351C22C2×#38811yescharacteristic
352C22#304211yescharacteristic
353C22#304411yescharacteristic
354C22C2×#46511yescharacteristic
355C22C2×#46711yescharacteristic
356C22#306811yescharacteristic
357C22#306811yescharacteristic
358C4—22no—
359C4—21no—
360C4—21no—
361C4—22no—
362C4—44no—
363C4—44no—
364C4—84no—
365C4—161no—
366C4—161no—
367C4#193314yes—
368C4C2×C4×SD1611yescharacteristic
369C4C2×C4×Q1611yescharacteristic
370C4#294712yes—
371C4#309712yes—
372C4C2×#48711yescharacteristic
373C4C2×#48811yescharacteristic
374C2—24no—
375C2#1939111yescharacteristic
376C2#1939311yescharacteristic
377C2#1940311yescharacteristic
378C2#1940511yescharacteristic
379C2#2106411yescharacteristic
380C2#2107811yescharacteristic
381C2C2×#193311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ 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
6. #9568C2semidirect
7. #9567C2semidirect
33. #1548C4semidirect
97. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
#218342
C2×#19872
#211702
#205332
C2×#15482
#95682
#95672
#80392
#78772