Finite Group Databasebeta

← #242487

Subgroups — #242487

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21961C211yesmaximal, characteristic, complemented
2#21952C211yesmaximal, characteristic, complemented
3#21863C211yesmaximal, characteristic
4#21809C211yesmaximal, characteristic
5C2×#1573C211yesmaximal, characteristic, complemented
6C2×#1540C211yesmaximal, characteristic
7#10594C212yesmaximal, complemented
8#10591C212yesmaximal, complemented
9#10590C212yesmaximal, complemented
10#10587C212yesmaximal, complemented
11#8046C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17#2285C2211yescharacteristic
18#2285C2211yescharacteristic
19#1990C2212yes—
20#1982C2211yescharacteristic
21#1982C2211yescharacteristic
22#1981C2211yescharacteristic
23#1981C2211yescharacteristic
24#1959—22no—
25#1957C2212yes—
26#1947—22no—
27#1947C2212yes—
28#1947C2212yes—
29#1939—22no—
30#1939C2212yes—
31#1939C2212yes—
32#1875—21no—
33#1875—21no—
34#1808—22no—
35#1804—22no—
36#1798—21no—
37#1798—21no—
38#1788—22no—
39#1573—24no—
40C2×#335C2212yes—
41C2×#334C2212yes—
42#1540—24no—
43C2×#332C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47C22×#41—21no—
48C23×C8—21no—
49#532—22no—
50#532—22no—
51#532—22no—
52#532Q814yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×#127C2311yescharacteristic
55C2×#124—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×#117—21no—
61C2×#117—21no—
62C22×C42C2311yescharacteristic
63#432—22no—
64#432—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69C2×#107—22no—
70C2×#107—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8D412yes—
76C2×C4⋊C8D414yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2312yes—
81C2×#103—22no—
82C2×#103—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8D411yescharacteristic
90C2×C22⋊C8D411yescharacteristic
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8C2312yes—
94#384D412yes—
95#384C2312yes—
96#382—21no—
97#382—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380D412yes—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2311yescharacteristic
106C4×C4⋊C4C2311yescharacteristic
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#336—22no—
112#336—22no—
113#335—24no—
114#334—24no—
115#332—24no—
116#331—24no—
117C2×#41—24no—
118C2×#41—42no—
119C2×#41—82no—
120C2×#41—82no—
121C22×Q8—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—41no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4111yescharacteristic
130C22×C8—22no—
131C22×C8—24no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—82no—
136C22×C8C2×D412yes—
137C22×C8C2×D412yes—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8C2×Q812yes—
142#127—21no—
143#127—21no—
144#127—21no—
145#127C2×Q812yes—
146#124—24no—
147#124—42no—
148#121—24no—
149#121—24no—
150#121—42no—
151#121—42no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—42no—
155C4×D4—42no—
156#117—82no—
157#117—82no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D412yes—
177C2×C42C2×Q812yes—
178C2×C42#4112yes—
179C2×C42#4112yes—
180C2×C42#4111yescharacteristic
181C2×C42C2411yescharacteristic
182#107—82no—
183#107—82no—
184C4⋊C8—24no—
185C4⋊C8—24no—
186C4⋊C8—22no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—82no—
194C4⋊C8—82no—
195C4⋊C8C2×D414yes—
196#103—82no—
197#103—82no—
198#102—82no—
199#102—82no—
200#102—82no—
201#102—82no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206#97—42no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—42no—
210#95—22no—
211#95—21no—
212#95—21no—
213#95—41no—
214#95—41no—
215C24—41no—
216C24#12111yescharacteristic
217#41—88no—
218#41—84no—
219#41—84no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—44no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—82no—
236C2×D4—84no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—84no—
248C22×C4#12112yes—
249C22×C4#12112yes—
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4C22⋊Q812yes—
254C22×C4C22⋊Q812yes—
255C22×C4C22⋊Q812yes—
256C22×C4#12612yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—24no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—84no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—44no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C42—42no—
297C42—42no—
298C42—82no—
299C42#12112yes—
300C42#12112yes—
301C42C22⋊Q812yes—
302C42C22⋊Q812yes—
303C42C22⋊Q812yes—
304C42#12612yes—
305C42C22×D411yescharacteristic
306C42C22×Q811yescharacteristic
307C42C2×#4111yescharacteristic
308C42C2×#4111yescharacteristic
309C23—21no—
310C23—21no—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—82no—
317C23—82no—
318C23#39612yes—
319C23C2×#12111yescharacteristic
320Q8—44no—
321Q8—81no—
322Q8—81no—
323D4—44no—
324D4—44no—
325D4—82no—
326D4—82no—
327D4—88no—
328D4—81no—
329D4—81no—
330D4—82no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—88no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4#39612yes—
356C2×C4#39612yes—
357C2×C4#39612yes—
358C2×C4C2×#12111yescharacteristic
359C2×C4C2×#12111yescharacteristic
360C2×C4C2×C22⋊Q811yescharacteristic
361C2×C4C2×C22⋊Q811yescharacteristic
362C2×C4C2×C22⋊Q811yescharacteristic
363C2×C4C2×#12611yescharacteristic
364C2×C4#57612yes—
365C2×C4#57612yes—
366C2×C4#57712yes—
367C2×C4#57812yes—
368C8—44no—
369C8—42no—
370C8—42no—
371C8—81no—
372C8—81no—
373C8—82no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—42no—
378C22—42no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—162no—
383C22—162no—
384C22C2×#39611yescharacteristic
385C22#304111yescharacteristic
386C22#304411yescharacteristic
387C22#306212yes—
388C22#306712yes—
389C4—21no—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—82no—
395C4—82no—
396C4—82no—
397C4—162no—
398C4#304111yescharacteristic
399C4#304311yescharacteristic
400C4#306112yes—
401C4#308112yes—
402C4#308312yes—
403C2—42no—
404C2—162no—
405C2#1955712yes—
406C2#1957512yes—
407C2#2360111yescharacteristic
408C2#2361111yescharacteristic
409C2#2361411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
1. #21961C2semidirect
2. #21952C2semidirect
5. C2×#1573C2semidirect
7. #10594C2semidirect
8. #10591C2semidirect
9. #10590C2semidirect
10. #10587C2semidirect
11. #8046C2semidirect

Maximal subgroups

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

subgroupindex
#219612
#219522
#218632
#218092
C2×#15732
C2×#15402
#105942
#105912
#105902
#105872
#80462