Finite Group Databasebeta

← #192133

Subgroups — #192133

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34057C211yesmaximal, characteristic
2#21983C212yesmaximal, complemented
3C2×#1460C211yesmaximal, characteristic
4#14345C211yesmaximal, characteristic
5#8858C214yesmaximal, complemented
6#8854C212yesmaximal, complemented
7#8853C212yesmaximal, complemented
8#8025C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#431C2212yes—
11#2318—22no—
12#2318—22no—
13#2315C2212yes—
14#2313—21no—
15#2312C2212yes—
16#2308—21no—
17#2290—24no—
18#2283C2212yes—
19C2×#383C2211yescharacteristic
20C2×#382C2211yescharacteristic
21#2280—21no—
22C2×C4×C22⋊C4—21no—
23#1949—24no—
24#1853C2214yes—
25#1852C2214yes—
26#1846—22no—
27#1835—22no—
28#1460C418yescomplemented
29C2×#323C2211yescharacteristic
30C2×#323C2211yescharacteristic
31C2×#323C2212yes—
32#1445—21no—
33#1439C2212yes—
34#1439C2212yes—
35#1438—21no—
36C22⋊#95—21no—
37C22⋊#95—21no—
38#1294C2214yes—
39#1294C2214yes—
40C23×D4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4C2×C414yes—
47C2×C4×D4Q814yes—
48C2×#117—22no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4C2311yescharacteristic
55C22×C22⋊C4C2311yescharacteristic
56C22×C42—22no—
57C22×C42C2311yescharacteristic
58#431—24no—
59#431C2×C418yescomplemented
60C2×#107—24no—
61C2×#106—24no—
62C2×C4⋊C8—22no—
63C2×C22⋊C8C2312yes—
64C2×C22⋊C8C2312yes—
65C2×C4×C8—22no—
66#388—24no—
67#388—24no—
68#387—24no—
69#387—24no—
70#386—24no—
71#386—24no—
72#385—24no—
73#385—24no—
74#383—22no—
75#383—24no—
76#382—22no—
77#382—22no—
78#382D412yes—
79#382D412yes—
80#381—22no—
81#381—22no—
82#380—21no—
83#380—21no—
84#380—22no—
85#380C2314yes—
86#379—22no—
87C4×C4⋊C4D414yes—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93#376—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95C2312yes—
103#340—24no—
104#340—24no—
105#323C2×C414yes—
106#323C2×C414yes—
107#323C2×C418yescomplemented
108C25C2×D411yescharacteristic
109C25C2×D411yescharacteristic
110C22×D4—24no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—42no—
114C22×D4C4⋊C414yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C4⋊C414yes—
128C23×C4C4⋊C414yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C4×D4—28no—
137C4×D4—28no—
138C4×D4—28no—
139C4×D4—28no—
140#117—84no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4C4⋊C414yes—
148C2×C4⋊C4C2×D414yes—
149C2×C4⋊C4C2×D414yes—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—82no—
164C2×C22⋊C4—82no—
165C2×C22⋊C4—82no—
166C2×C22⋊C4—82no—
167C2×C22⋊C4C22×C412yes—
168C2×C22⋊C4C22×C412yes—
169C2×C22⋊C4C2×Q812yes—
170C2×C22⋊C4C2×Q812yes—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—84no—
174C2×C42C22×C412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×Q812yes—
178C2×C42#4114yes—
179#107—84no—
180#106—84no—
181C4⋊C8—28no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8C22×C414yes—
185C22⋊C8C22×C414yes—
186C4×C8—28no—
187#95—22no—
188#95—22no—
189#95—24no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—42no—
198#95—41no—
199#95—41no—
200#95—42no—
201#95—42no—
202#95—82no—
203#95—82no—
204C24—21no—
205C24—21no—
206C24—22no—
207C24—21no—
208C24—22no—
209C24—21no—
210C24—22no—
211C24—41no—
212C24—41no—
213C24—42no—
214C24—42no—
215C24—41no—
216C24—41no—
217C24C2×C4⋊C412yes—
218C24C2×C4⋊C412yes—
219C24#13514yes—
220C24C22×D411yescharacteristic
221C2×D4—28no—
222C2×D4—28no—
223C2×D4—44no—
224C2×D4—48no—
225C2×D4—48no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—84no—
264C22×C4—84no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4C2×C4⋊C412yes—
272C22×C4C2×C4⋊C412yes—
273C22×C4C2×C4⋊C412yes—
274C22×C4C2×C4⋊C412yes—
275C22×C4#12012yes—
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12312yes—
279C22×C4#12312yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D412yes—
282C22×C4C22×Q811yescharacteristic
283C22×C4C2×#4112yes—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—28no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—48no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—84no—
307C22⋊C4—88no—
308C22⋊C4—88no—
309C42—48no—
310C42—84no—
311C42C4×D414yes—
312C42C4×D414yes—
313C42C22⋊Q814yes—
314C42C22⋊Q814yes—
315C23—22no—
316C23—21no—
317C23—21no—
318C23—24no—
319C23—22no—
320C23—24no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—44no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23—42no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—81no—
342C23—81no—
343C23—82no—
344C23—82no—
345C23—81no—
346C23—81no—
347C23—82no—
348C23—82no—
349C23#42714yes—
350C23C22×C4⋊C411yescharacteristic
351C23C2×#12011yescharacteristic
352C23C2×#12311yescharacteristic
353C23C2×#13512yes—
354C23#57512yes—
355D4—416no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—88no—
388C2×C4—88no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4#38514yes—
404C2×C4#38512yes—
405C2×C4#38512yes—
406C2×C4C2×C4×D412yes—
407C2×C4C2×C22⋊Q812yes—
408C8—44no—
409C8—44no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—21no—
417C22—21no—
418C22—24no—
419C22—44no—
420C22—44no—
421C22—42no—
422C22—44no—
423C22—44no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—84no—
429C22—82no—
430C22—82no—
431C22—84no—
432C22—82no—
433C22—82no—
434C22C2×#38511yescharacteristic
435C22C2×#42711yescharacteristic
436C22#291111yescharacteristic
437C22#300212yes—
438C22#301912yes—
439C22#302711yescharacteristic
440C22#302811yescharacteristic
441C22#309112yes—
442C4—24no—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—42no—
448C4—42no—
449C4—84no—
450C4—84no—
451C4—162no—
452C4—162no—
453C2—22no—
454C2—21no—
455C2—21no—
456C2—44no—
457C2—82no—
458C2#1830612yes—
459C2#1831412yes—
460C2#2084311yescharacteristic
461C2#2085611yescharacteristic
462C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21983C2semidirect
5. #8858C2semidirect
6. #8854C2semidirect
7. #8853C2semidirect
8. #8025C2semidirect
28. #1460C4semidirect
59. #431C2×C4semidirect
107. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340572
#219832
C2×#14602
#143452
#88582
#88542
#88532
#80252