Finite Group Databasebeta

← #208452

Subgroups — #208452

Order 512.

848 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35087C211yesmaximal, characteristic
2#21914C211yesmaximal, characteristic
3#13781C211yesmaximal, characteristic
4#9913C212yesmaximal
5#9714C212yesmaximal
6#9153C212yesmaximal
7#9152C212yesmaximal
8#8219C212yesmaximal
9C4×#328C212yesmaximal
10#2857C2211yescharacteristic
11#2855C2212yes—
12#2851C2211yescharacteristic
13#2655—22no—
14#2514C412yescomplemented
15#2506C412yescomplemented
16#2493—22no—
17#2473—22no—
18#2453C412yescomplemented
19#2446C412yescomplemented
20#2336—22no—
21C4×C22⋊Q8C2212yes—
22C2×#383C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1917—22no—
26#1916—22no—
27#1817C2212yes—
28#1797C2212yes—
29#1749C2211yescharacteristic
30C4×C22⋊C8C2211yescharacteristic
31C2×#328C2212yes—
32C2×#328C2212yes—
33C2×#328C2212yes—
34C2×#328C2212yes—
35#1436—22no—
36#1436—22no—
37#1434C2212yes—
38C4×#95C2212yes—
39#1294C2212yes—
40#1294C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—22no—
45C2×#125C2×C411yescharacteristic
46C2×C22⋊Q8C2312yes—
47C2×C4×Q8C2×C412yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D412yes—
50C22×C42—22no—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×C4⋊C8—22no—
55C2×C22⋊C8C2312yes—
56C2×C22⋊C8C2312yes—
57C2×#97—21no—
58C2×C4×C8—21no—
59#401—22no—
60#400—22no—
61#400—22no—
62#400—22no—
63#400—22no—
64#398—22no—
65#398—22no—
66#398—22no—
67#391C2×C412yes—
68#391C2×C412yes—
69#391C2×C412yes—
70#391C2×C412yes—
71#389—22no—
72#389—22no—
73#389C2×C412yes—
74#389C2×C412yes—
75#389C2×C412yes—
76#389C2×C412yes—
77#387—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#384C2×C411yescharacteristic
90#383—22no—
91#383—24no—
92#382—24no—
93#382C2×C411yescharacteristic
94#381—22no—
95#380—22no—
96#380C2312yes—
97C4×C4⋊C4—22no—
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C22⋊C4—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D412yes—
111C2×#95C2312yes—
112C2×#95C2312yes—
113#336—22no—
114#336—24no—
115#336—21no—
116#336—21no—
117#328—24no—
118#328—24no—
119#328—24no—
120#328—24no—
121C22×Q8C22×C412yes—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2411yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
131C4⋊Q8—81no—
132#125—48no—
133#125—81no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C4×Q8—82no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—44no—
165C2×C42—81no—
166C2×C42—81no—
167C2×C42—82no—
168C2×C42—82no—
169C2×C42C22×C412yes—
170C2×C42C22×C412yes—
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C2×D412yes—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D412yes—
177C4⋊C8—82no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182#97—81no—
183C4×C8—81no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—44no—
205#95—82no—
206#95—82no—
207#95C22⋊C412yes—
208#95C22⋊C412yes—
209#95C22⋊C412yes—
210#95C22⋊C412yes—
211#95#4114yes—
212#95#4114yes—
213C24C22×D411yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—42no—
217C2×Q8—42no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C2×#4112yes—
262C22×C4C2×#4112yes—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—84no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C42—24no—
293C42—24no—
294C42—41no—
295C42—41no—
296C42—42no—
297C42—42no—
298C42—41no—
299C42—41no—
300C42—44no—
301C42—82no—
302C42—88no—
303C23—21no—
304C23—21no—
305C23—22no—
306C23C22×C22⋊C411yescharacteristic
307C23#52511yescharacteristic
308C23#52511yescharacteristic
309C23#57712yes—
310C23#57812yes—
311Q8—82no—
312Q8—82no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
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—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4#42114yes—
371C2×C4C4×#4112yes—
372C2×C4#53212yes—
373C8—82no—
374C8—82no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22#229711yescharacteristic
384C22#288311yescharacteristic
385C22#288411yescharacteristic
386C22#289812yes—
387C22C2×#42112yes—
388C22#312012yes—
389C22#312311yescharacteristic
390C22#312711yescharacteristic
391C4—22no—
392C4—22no—
393C4—24no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—84no—
400C4—84no—
401C4—82no—
402C4—82no—
403C4—82no—
404C4—82no—
405C2—21no—
406C2—22no—
407C2—21no—
408C2#1759812yes—
409C2#1760112yes—
410C2#2139811yescharacteristic
411C2#2141611yescharacteristic
412C2#2142311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2514C4semidirect
15. #2506C4semidirect
18. #2453C4semidirect
19. #2446C4semidirect

Maximal subgroups

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

subgroupindex
#350872
#219142
#137812
#99132
#97142
#91532
#91522
#82192
C4×#3282