Finite Group Databasebeta

← #165538

Subgroups — #165538

Order 512.

735 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36509C211yesmaximal, characteristic
2#21998C212yesmaximal
3#7717C212yesmaximal, complemented
4#7695C212yesmaximal, complemented
5#7661C212yesmaximal, complemented
6#7653C211yesmaximal, characteristic, complemented
7#7617C212yesmaximal, complemented
8#7605C211yesmaximal, characteristic, complemented
9#7583C212yesmaximal, complemented
10#2864C2211yescharacteristic
11#2855C2212yes—
12#2590—22no—
13#2536C411yescharacteristic
14#2536C411yescharacteristic
15#2515—22no—
16#2496C412yes—
17#2466—22no—
18#2429C412yes—
19#2373—22no—
20#2356C411yescharacteristic
21#2351C411yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×C22⋊Q8C2212yes—
24C42×D4C2211yescharacteristic
25#1923—22no—
26#1916—22no—
27#1870—22no—
28#1851—22no—
29#1839C2212yes—
30#1831C2212yes—
31#1817C2212yes—
32#1797C2212yes—
33C4×C22⋊C8C2212yes—
34#1728C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1272C2212yes—
39#1272C2211yescharacteristic
40#1272C2211yescharacteristic
41#1271C2212yes—
42#1271C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45C22×M16—22no—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#126—22no—
48C2×C22⋊Q8C2×C412yes—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C412yes—
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4D412yes—
55C22×C42C2312yes—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C4×C8—22no—
62#400—22no—
63#398—21no—
64#398—21no—
65#398—22no—
66#397—22no—
67#396—22no—
68#394—21no—
69#394—21no—
70#391C2×C412yes—
71#391C2311yescharacteristic
72#391C2311yescharacteristic
73#389—22no—
74#389—22no—
75#389—22no—
76#389C2×C412yes—
77#389C2×C412yes—
78#389C2312yes—
79#388—22no—
80#388—22no—
81#388—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#385—22no—
90#385—22no—
91#384C2×C411yescharacteristic
92#383—21no—
93#382—21no—
94#382C2×C411yescharacteristic
95#382C2×C411yescharacteristic
96#381—22no—
97#380C2×C412yes—
98#380C2×C412yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4C2×C411yescharacteristic
104C4×C4⋊C4D412yes—
105C4×C22⋊C4—22no—
106C4×C22⋊C4C2×C412yes—
107C4×C22⋊C4D412yes—
108C4×C22⋊C4D412yes—
109#376—21no—
110C43C2311yescharacteristic
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336C2312yes—
117#336C2312yes—
118#336C2312yes—
119#336C2312yes—
120C22×Q8C22×C411yescharacteristic
121C22×Q8C22×C411yescharacteristic
122C22×D4C22⋊C411yescharacteristic
123C23×C4C22×C412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C2×M16—44no—
127C2×M16—82no—
128C2×M16—82no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—41no—
135#126—48no—
136C22⋊Q8—24no—
137C22⋊Q8—42no—
138C22⋊Q8—82no—
139C4×Q8—42no—
140C4×Q8—42no—
141C4×Q8—82no—
142C4×D4—42no—
143C4×D4—44no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
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⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C4⋊C4C22×C411yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C22⋊C412yes—
166C2×C22⋊C4C22⋊C412yes—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—82no—
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2411yescharacteristic
180C4⋊C8—82no—
181C4⋊C8—82no—
182C22⋊C8—44no—
183C22⋊C8—82no—
184C22⋊C8—82no—
185C4×C8—82no—
186#95—21no—
187#95—22no—
188#95—22no—
189#95—21no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—21no—
195#95—21no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95C22⋊C411yescharacteristic
201#95C22⋊C412yes—
202#95C22⋊C412yes—
203#95C22⋊C411yescharacteristic
204C24C2×C22⋊C412yes—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#13112yes—
213C2×D4—41no—
214C2×D4—42no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C412yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D412yes—
241M16—88no—
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—42no—
252C2×C8—42no—
253C2×C8—42no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—41no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—42no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—82no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—42no—
285C42—42no—
286C42—41no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—44no—
292C42—82no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23C22×C22⋊C411yescharacteristic
301Q8—42no—
302Q8—42no—
303Q8—81no—
304Q8—81no—
305D4—44no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4#41312yes—
347C2×C4#42114yes—
348C2×C4C2×#13111yescharacteristic
349C2×C4#56811yescharacteristic
350C2×C4#57712yes—
351C2×C4#57812yes—
352C8—82no—
353C8—82no—
354C8—82no—
355C8—82no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22#288011yescharacteristic
364C22#288311yescharacteristic
365C22C2×#41311yescharacteristic
366C22#289812yes—
367C22C2×#42112yes—
368C4—21no—
369C4—21no—
370C4—22no—
371C4—22no—
372C4—22no—
373C4—24no—
374C4—41no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—42no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—82no—
383C4—82no—
384C2—82no—
385C2#1742411yescharacteristic
386C2#1752512yes—
387C2#1759812yes—
388C2#1767311yescharacteristic
389C2#1778011yescharacteristic
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 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7717C2semidirect
4. #7695C2semidirect
5. #7661C2semidirect
6. #7653C2semidirect
7. #7617C2semidirect
8. #7605C2semidirect
9. #7583C2semidirect

Maximal subgroups

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

subgroupindex
#365092
#219982
#77172
#76952
#76612
#76532
#76172
#76052
#75832