Finite Group Databasebeta

← #208465

Subgroups — #208465

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35086C211yesmaximal, characteristic
2#21904C211yesmaximal, characteristic
3#13780C211yesmaximal, characteristic
4#9910C212yesmaximal, complemented
5#9711C212yesmaximal, complemented
6#9154C212yesmaximal
7#9151C212yesmaximal
8#8221C212yesmaximal, complemented
9#8217C212yesmaximal, complemented
10#2857C2211yescharacteristic
11#2851C2211yescharacteristic
12#2848C2212yes—
13#2652—22no—
14#2505—22no—
15#2496C412yescomplemented
16#2489C412yescomplemented
17#2480—22no—
18#2429C412yescomplemented
19#2421C412yescomplemented
20#2334—22no—
21C4×C22⋊Q8C2212yes—
22C2×#381C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#1924—22no—
26#1923—22no—
27#1797C2212yes—
28#1779C2212yes—
29#1749C2211yescharacteristic
30C4×C22⋊C8C2211yescharacteristic
31C2×#328C2212yes—
32C2×#328C2212yes—
33C2×#328C2212yes—
34C2×#328C2212yes—
35#1437—22no—
36#1436—22no—
37#1434C2212yes—
38C4×#95C2212yes—
39#1294C2212yes—
40#1294C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43C24×C4C2311yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#126—22no—
46C2×#125C2×C411yescharacteristic
47C2×C22⋊Q8C2×C412yes—
48C2×C22⋊Q8C2×C412yes—
49C2×C22⋊Q8C2312yes—
50C2×C4×Q8C2×C412yes—
51C22×C4⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4D412yes—
55C22×C42—22no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8C2312yes—
62C2×C22⋊C8C2312yes—
63C2×#97—21no—
64C2×C4×C8—21no—
65#400—22no—
66#400—22no—
67#400—22no—
68#398—22no—
69#397—22no—
70#397—22no—
71#397—22no—
72#396—22no—
73#391C2×C412yes—
74#391C2×C412yes—
75#389—22no—
76#389—22no—
77#389C2×C412yes—
78#389C2×C412yes—
79#388—22no—
80#388—22no—
81#388—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#385—22no—
87#385—22no—
88#385—22no—
89#385—22no—
90#385—22no—
91#384C2×C411yescharacteristic
92#382—22no—
93#382C2×C411yescharacteristic
94#381—24no—
95#380—22no—
96#380—24no—
97#380C2×C412yes—
98#380C2×C412yes—
99#380C2312yes—
100#379—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C412yes—
103C4×C22⋊C4—22no—
104C4×C22⋊C4C2×C412yes—
105C4×C22⋊C4C2×C412yes—
106C4×C22⋊C4D412yes—
107C4×C22⋊C4D412yes—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D412yes—
113C2×#95C2312yes—
114C2×#95C2312yes—
115#336—22no—
116#336—21no—
117#336—21no—
118#328—24no—
119#328—24no—
120#328—24no—
121#328—24no—
122C25C2×D411yescharacteristic
123C22×Q8C22×C412yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—81no—
129C23×C4—81no—
130C23×C4C22×C412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C4⋊Q8—81no—
139#126—48no—
140#125—81no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—82no—
146C22⋊Q8—82no—
147C4×Q8—82no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—81no—
159C2×C4⋊C4—81no—
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⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4—81no—
174C2×C22⋊C4—81no—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C412yes—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—44no—
182C2×C42—81no—
183C2×C42—81no—
184C2×C42—82no—
185C2×C42—82no—
186C2×C42C22×C412yes—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D412yes—
192C4⋊C8—82no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—82no—
198C22⋊C8—82no—
199#97—81no—
200C4×C8—81no—
201#95—24no—
202#95—22no—
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#95—22no—
213#95—22no—
214#95—22no—
215#95—44no—
216#95—82no—
217#95—82no—
218#95C22⋊C412yes—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222#95#4114yes—
223#95#4114yes—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—22no—
228C24—42no—
229C24—41no—
230C24—41no—
231C24C2×C22⋊C412yes—
232C24C22×D411yescharacteristic
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—42no—
236C2×Q8—42no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C2×#4112yes—
285C22×C4C2×#4112yes—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—84no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—84no—
320C22⋊C4—84no—
321C42—41no—
322C42—41no—
323C42—42no—
324C42—42no—
325C42—41no—
326C42—41no—
327C42—44no—
328C42—82no—
329C42—88no—
330C23—21no—
331C23—21no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—82no—
348C23—82no—
349C23—82no—
350C23—82no—
351C23C22×C22⋊C411yescharacteristic
352C23#52511yescharacteristic
353C23#52511yescharacteristic
354C23#57712yes—
355C23#57812yes—
356Q8—82no—
357Q8—82no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4#42114yes—
413C2×C4C4×#4112yes—
414C2×C4#53212yes—
415C8—82no—
416C8—82no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—82no—
433C22—84no—
434C22—82no—
435C22#229711yescharacteristic
436C22#288311yescharacteristic
437C22#288311yescharacteristic
438C22#289812yes—
439C22C2×#42112yes—
440C22#312012yes—
441C22#312311yescharacteristic
442C22#312711yescharacteristic
443C4—22no—
444C4—22no—
445C4—24no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—84no—
451C4—84no—
452C4—82no—
453C4—82no—
454C4—82no—
455C4—82no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—82no—
460C2#1759812yes—
461C2#1760012yes—
462C2#2139811yescharacteristic
463C2#2141811yescharacteristic
464C2#2142211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9910C2semidirect
5. #9711C2semidirect
8. #8221C2semidirect
9. #8217C2semidirect
15. #2496C4semidirect
16. #2489C4semidirect
18. #2429C4semidirect
19. #2421C4semidirect

Maximal subgroups

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

subgroupindex
#350862
#219042
#137802
#99102
#97112
#91542
#91512
#82212
#82172