Finite Group Databasebeta

← #223843

Subgroups — #223843

Order 512.

803 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36192C211yesmaximal, characteristic
2#21108C212yesmaximal
3#9894C212yesmaximal, complemented
4#9125C212yesmaximal, complemented
5#9122C212yesmaximal, complemented
6#8722C211yesmaximal, characteristic, complemented
7#8722C211yesmaximal, characteristic, complemented
8#8584C211yesmaximal, characteristic, complemented
9#8225C211yesmaximal, characteristic, complemented
10#8031C211yesmaximal, characteristic, complemented
11#8028C211yesmaximal, characteristic, complemented
12C2×#432C2212yes—
13C2×C4×M16C2212yes—
14#2859C2211yescharacteristic
15#2851C2211yescharacteristic
16#2851C2211yescharacteristic
17#2552C412yes—
18#2522—22no—
19#2522—22no—
20#2454—22no—
21#2441C412yes—
22#2441C412yes—
23#2370—22no—
24#2354C412yes—
25C4×#125C2211yescharacteristic
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1923—22no—
30#1916—22no—
31#1870—22no—
32#1851—22no—
33#1767C2212yes—
34#1749C2212yes—
35#1749C2212yes—
36#1745C2212yes—
37C2×#330C2212yes—
38C2×#330C2212yes—
39#1295C2211yescharacteristic
40#1295C2211yescharacteristic
41#1295C2211yescharacteristic
42#1295C2212yes—
43#1295C2212yes—
44#1295C2211yescharacteristic
45#1295C2212yes—
46#1295C2212yes—
47C22×M16—22no—
48C2×#125—22no—
49C2×#125C2×C412yes—
50C2×#125C2311yescharacteristic
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4D412yes—
53C2×#117C2×C412yes—
54C2×#117C2×C412yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#432—24no—
58#432—42no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C4×M16—24no—
65C4×M16—42no—
66C2×#97C2312yes—
67C2×#97C2312yes—
68C2×#97C2312yes—
69C2×#97C2312yes—
70C2×C4×C8—22no—
71#399—22no—
72#399—22no—
73#398—22no—
74#398—22no—
75#396—22no—
76#396—22no—
77#390—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2×C412yes—
86#388—22no—
87#388—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#385—22no—
94#385—22no—
95#384C2×C412yes—
96#384C2×C412yes—
97#384C2×C412yes—
98#382C2311yescharacteristic
99#382C2311yescharacteristic
100#382C2311yescharacteristic
101#380—22no—
102#380—22no—
103#380C2×C412yes—
104#380C2×C412yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C4⋊C4D412yes—
112C4×C22⋊C4C2×C411yescharacteristic
113C4×C22⋊C4C2×C411yescharacteristic
114C4×C22⋊C4D412yes—
115C4×C22⋊C4D412yes—
116C43C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#330—24no—
123#330—42no—
124C22×D4C22⋊C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C2×M16—44no—
130C2×M16—44no—
131C2×M16—44no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136#125—22no—
137#125—21no—
138#125—21no—
139#125—48no—
140#125—82no—
141C4×D4—24no—
142C4×D4—44no—
143#117—82no—
144#117—82no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4C22⋊C411yescharacteristic
168C2×C22⋊C4C22⋊C411yescharacteristic
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22⋊C412yes—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2411yescharacteristic
189C4⋊C8—44no—
190C4⋊C8—44no—
191C22⋊C8—44no—
192C22⋊C8—44no—
193C22⋊C8—44no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—22no—
201#97#4114yes—
202C4×C8—44no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95C22⋊C412yes—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215C24C2×C22⋊C411yescharacteristic
216C24C2×C22⋊C411yescharacteristic
217C2×D4—22no—
218C2×D4—42no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C412yes—
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D412yes—
248M16—416no—
249C2×C8—24no—
250C2×C8—24no—
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—24no—
262C4⋊C4—22no—
263C4⋊C4—22no—
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—44no—
273C4⋊C4—44no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4#10114yes—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—24no—
284C22⋊C4—42no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C22⋊C4—82no—
288C22⋊C4—82no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—42no—
299C42—41no—
300C42—42no—
301C42—44no—
302C42—82no—
303C42—82no—
304C42C2×#4112yes—
305C42H(5,𝔽2)11yescharacteristic
306C422−1+411yescharacteristic
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23C22×C22⋊C411yescharacteristic
318D4—44no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4C2×#10112yes—
359C2×C4C2×#10112yes—
360C2×C4C2×#10112yes—
361C2×C4#42114yes—
362C2×C4C4×#4111yescharacteristic
363C2×C4#52011yescharacteristic
364C2×C4#54612yes—
365C2×C4#54712yes—
366C8—44no—
367C8—44no—
368C8—82no—
369C8—82no—
370C22—41no—
371C22—41no—
372C22—42no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—42no—
380C22#233911yescharacteristic
381C22C22×#10111yescharacteristic
382C22#288811yescharacteristic
383C22C2×#42112yes—
384C22#290112yes—
385C4—22no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—24no—
393C4—42no—
394C4—41no—
395C4—42no—
396C4—82no—
397C4—82no—
398C4—82no—
399C4—82no—
400C2—81no—
401C2—81no—
402C2#1766411yescharacteristic
403C2#1768412yes—
404C2#1769211yescharacteristic
405C2#2248111yescharacteristic
406C2#2250612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9894C2semidirect
4. #9125C2semidirect
5. #9122C2semidirect
6. #8722C2semidirect
7. #8722C2semidirect
8. #8584C2semidirect
9. #8225C2semidirect
10. #8031C2semidirect
11. #8028C2semidirect

Maximal subgroups

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

subgroupindex
#361922
#211082
#98942
#91252
#91222
#87222
#85842
#82252
#80312
#80282