Finite Group Databasebeta

← #224932

Subgroups — #224932

Order 512.

883 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#22823C212yesmaximal, complemented
3#9707C212yesmaximal
4#9702C212yesmaximal, complemented
5#9699C212yesmaximal
6#8594C211yesmaximal, characteristic, complemented
7#8592C211yesmaximal, characteristic
8#8587C211yesmaximal, characteristic, complemented
9#8232C211yesmaximal, characteristic, complemented
10#8231C211yesmaximal, characteristic
11C4×#333C211yesmaximal, characteristic, complemented
12C2×#556C412yes—
13#2868C2211yescharacteristic
14#2858C2211yescharacteristic
15#2858C2211yescharacteristic
16#2547C412yes—
17#2545C412yes—
18#2545C412yes—
19#2538—22no—
20#2512—22no—
21#2512—22no—
22#2354—22no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#127C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27#1985—22no—
28#1917—22no—
29#1916—22no—
30#1842C2212yes—
31#1831C2212yes—
32#1831C2212yes—
33#1771—22no—
34#1767C2212yes—
35#1767C2212yes—
36C4×C4⋊C8C2212yes—
37C2×#333C2212yes—
38C2×#333C2212yes—
39C2×#333C2212yes—
40C2×#333C2212yes—
41C2×#331C2212yes—
42C2×#331C2212yes—
43#1296C2211yescharacteristic
44#1296C2211yescharacteristic
45#1296C2211yescharacteristic
46#1296C2211yescharacteristic
47#556—48no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×#127C2311yescharacteristic
53C2×#126—22no—
54C2×#126—22no—
55C2×#124C2×C412yes—
56C2×#124C2×C412yes—
57C2×#124C2×C412yes—
58C2×#124C2×C412yes—
59C2×C22⋊Q8—22no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8D412yes—
64C2×C4×D4—22no—
65C2×C4×D4C2×C412yes—
66C2×#117—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×#97—22no—
74C2×#97—22no—
75C2×C4×C8—22no—
76#397—24no—
77#396—24no—
78#396—24no—
79#391C2×C412yes—
80#391C2×C412yes—
81#390—22no—
82#390—22no—
83#390—22no—
84#390—22no—
85#390—22no—
86#390C2×C412yes—
87#389—22no—
88#389—22no—
89#388—24no—
90#388—24no—
91#384—22no—
92#384—22no—
93#384C2311yescharacteristic
94#380—24no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2×C411yescharacteristic
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4D412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C43C2311yescharacteristic
104C43C2311yescharacteristic
105C43C2311yescharacteristic
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#333—24no—
114#333—24no—
115#333—42no—
116#333—42no—
117#331—24no—
118#331—42no—
119C22×Q8C22⋊C411yescharacteristic
120C22×Q8C22×C412yes—
121C22×Q8C22×C412yes—
122C22×D4—22no—
123C22×D4C22×C412yes—
124C23×C4—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—44no—
136C4⋊Q8—82no—
137#127—22no—
138#127—21no—
139#127—21no—
140#126—48no—
141#126—48no—
142#124—44no—
143#124—44no—
144#124—82no—
145#124—82no—
146C22⋊Q8—48no—
147C4×Q8—24no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×Q8—44no—
151C4×Q8—44no—
152C4×D4—44no—
153C4×D4—44no—
154C4×D4—44no—
155#117—48no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2411yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—44no—
202C4⋊C8—82no—
203C4⋊C8#4114yes—
204#97—44no—
205#97—82no—
206C4×C8—82no—
207#95—24no—
208#95—24no—
209#95C22⋊C412yes—
210#95C22⋊C412yes—
211#95C22⋊C412yes—
212C24—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—48no—
237C22×C4—44no—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D412yes—
249C2×C8—24no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—84no—
278C4⋊C4#10014yes—
279C22⋊C4—48no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—84no—
286C22⋊C4—84no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—44no—
306C42—44no—
307C42C2×#4112yes—
308C42H(5,𝔽2)11yescharacteristic
309C422−1+411yescharacteristic
310C23—42no—
311C23—42no—
312C23—42no—
313C23—44no—
314C23—44no—
315C23C22×C22⋊C411yescharacteristic
316Q8—44no—
317Q8—82no—
318Q8—82no—
319Q8—82no—
320Q8—82no—
321D4—82no—
322D4—82no—
323D4—82no—
324D4—82no—
325D4—82no—
326D4—82no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—88no—
356C2×C4C2×#10012yes—
357C2×C4C2×#10012yes—
358C2×C4C2×#10012yes—
359C2×C4C4×#4111yescharacteristic
360C2×C4#52011yescharacteristic
361C2×C4#54812yes—
362C2×C4#55012yes—
363C2×C4#57512yes—
364C2×C4#57512yes—
365C2×C4#57712yes—
366C2×C4#57812yes—
367C8—44no—
368C8—82no—
369C8—82no—
370C8—82no—
371C22—82no—
372C22—84no—
373C22—82no—
374C22—84no—
375C22—82no—
376C22#234111yescharacteristic
377C22C22×#10011yescharacteristic
378C22#288811yescharacteristic
379C22#289612yes—
380C22#289812yes—
381C4—21no—
382C4—21no—
383C4—22no—
384C4—22no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—41no—
392C4—82no—
393C4—82no—
394C4—82no—
395C4#337612yes—
396C4#337712yes—
397C4#340112yes—
398C4#340212yes—
399C2—162no—
400C2#1773811yescharacteristic
401C2#1776812yes—
402C2#1777911yescharacteristic
403C2#2253611yescharacteristic
404C2#2254812yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22823C2semidirect
4. #9702C2semidirect
6. #8594C2semidirect
8. #8587C2semidirect
9. #8232C2semidirect
11. C4×#333C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#228232
#97072
#97022
#96992
#85942
#85922
#85872
#82322
#82312
C4×#3332