Finite Group Databasebeta

← #165078

Subgroups — #165078

Order 512.

687 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36394C211yesmaximal, characteristic
2#22830C212yesmaximal, complemented
3#7713C212yesmaximal, complemented
4#7709C212yesmaximal, complemented
5#7657C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic
7#7628C212yesmaximal
8#7594C212yesmaximal
9#7567C211yesmaximal, characteristic
10#2864C2211yescharacteristic
11#2863C2212yes—
12#2599—22no—
13#2581C411yescharacteristic
14#2580C411yescharacteristic
15#2577—22no—
16#2575—22no—
17#2508C412yes—
18#2447C412yes—
19#2356C411yescharacteristic
20#2349C411yescharacteristic
21#2343—22no—
22C4×#124C2211yescharacteristic
23#2293C2212yes—
24C42×Q8C2211yescharacteristic
25#1985—22no—
26#1917—22no—
27#1916—22no—
28#1839C2212yes—
29#1839C2212yes—
30#1831C2212yes—
31#1771—22no—
32#1767C2212yes—
33#1745C2212yes—
34#1744C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1271C2212yes—
39#1271C2212yes—
40#1271C2212yes—
41#1271C2212yes—
42#1270C2211yescharacteristic
43#1270C2211yescharacteristic
44#1270C2212yes—
45C2×#124C2×C411yescharacteristic
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8D412yes—
49C2×C4×D4C2×C412yes—
50C2×C4×D4C2×C411yescharacteristic
51C2×#117—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×#97—22no—
55C2×#97—22no—
56C2×C4×C8—22no—
57#400—21no—
58#400—22no—
59#399—21no—
60#398—22no—
61#397—22no—
62#397—22no—
63#395—21no—
64#393—21no—
65#391C2×C412yes—
66#391C2×C412yes—
67#391C2311yescharacteristic
68#390C2×C412yes—
69#390C2311yescharacteristic
70#389C2×C412yes—
71#389C2×C412yes—
72#389C2×C412yes—
73#389C2×C412yes—
74#389C2312yes—
75#388—22no—
76#388—22no—
77#388—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#385—22no—
87#385—22no—
88#385—22no—
89#383—22no—
90#383—21no—
91#382—22no—
92#382C2×C411yescharacteristic
93#382C2×C411yescharacteristic
94#382C2×C411yescharacteristic
95#381—22no—
96#380—21no—
97C4×C4⋊C4C2×C412yes—
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4D412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105#376—21no—
106C2×#95—21no—
107C43C2312yes—
108C43C2311yescharacteristic
109#336—22no—
110#336—22no—
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×C4—22no—
124C23×C4—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129#124—22no—
130#124—41no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×D4—24no—
136C4×D4—42no—
137C4×D4—42no—
138#117—44no—
139#117—44no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C411yescharacteristic
153C2×C4⋊C4C22×C411yescharacteristic
154C2×C4⋊C4C22×C412yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C412yes—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2411yescharacteristic
174C4⋊C8—44no—
175C4⋊C8—44no—
176#97—44no—
177#97—44no—
178C4×C8—44no—
179#95—21no—
180#95—22no—
181#95—22no—
182#95—21no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—21no—
189#95—21no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95C22⋊C411yescharacteristic
196#95C22⋊C411yescharacteristic
197#95C22⋊C412yes—
198#95C22⋊C412yes—
199C24—21no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—41no—
203C2×Q8—42no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×D4—21no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4#13112yes—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C23×C411yescharacteristic
228C22×C4C22×D412yes—
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—82no—
265C42—22no—
266C42—21no—
267C42—22no—
268C42—21no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42—41no—
281C42—44no—
282C42—42no—
283C23—22no—
284C23—41no—
285C23—41no—
286C23—42no—
287C23—42no—
288C23C22×C22⋊C411yescharacteristic
289Q8—44no—
290Q8—81no—
291Q8—81no—
292D4—42no—
293D4—42no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—88no—
330C2×C4—162no—
331C2×C4#41312yes—
332C2×C4C2×#13111yescharacteristic
333C2×C4#56811yescharacteristic
334C2×C4#57512yes—
335C2×C4#57512yes—
336C8—44no—
337C8—82no—
338C8—82no—
339C8—82no—
340C22—42no—
341C22—44no—
342C22—81no—
343C22—81no—
344C22—82no—
345C22#288011yescharacteristic
346C22#288411yescharacteristic
347C22C2×#41311yescharacteristic
348C22#289612yes—
349C22#290112yes—
350C4—21no—
351C4—21no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—42no—
356C4—41no—
357C4—42no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—82no—
365C4—82no—
366C2—82no—
367C2#1742311yescharacteristic
368C2#1751312yes—
369C2#1762412yes—
370C2#1768611yescharacteristic
371C2#1774011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22830C2semidirect
3. #7713C2semidirect
4. #7709C2semidirect
5. #7657C2semidirect

Maximal subgroups

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

subgroupindex
#363942
#228302
#77132
#77092
#76572
#76492
#76282
#75942
#75672