Finite Group Databasebeta

← #194852

Subgroups — #194852

Order 512.

1390 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2633C211yesmaximal, characteristic
2#34057C211yesmaximal, characteristic
3#21836C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic
5#9028C212yesmaximal, complemented
6#9025C212yesmaximal, complemented
7#9024C212yesmaximal, complemented
8#9021C212yesmaximal, complemented
9#8025C212yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#431C2212yes—
13#2633—24no—
14#2633C418yescomplemented
15C2×#399C2211yescharacteristic
16C2×#399C2211yescharacteristic
17C2×#398—21no—
18C2×#397C2211yescharacteristic
19C2×#397C2211yescharacteristic
20C2×#394—21no—
21#2318—24no—
22#2315C2212yes—
23#2313—21no—
24#2312C2212yes—
25#2308—21no—
26#2290—24no—
27C2×#385—21no—
28C2×#385—21no—
29#2283C2212yes—
30C2×#383C2211yescharacteristic
31#2280—21no—
32C2×C4×C22⋊C4—21no—
33#1991—24no—
34#1989—24no—
35#1824C2214yes—
36#1823C2214yes—
37#1460—24no—
38C2×#323C2212yes—
39C2×#323C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×D4D411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—24no—
49C2×#125C2×C414yescomplemented
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×C4×D4C2×C414yes—
54C2×C4×D4D414yes—
55C2×#117—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4C2311yescharacteristic
69C22×C22⋊C4C2311yescharacteristic
70C22×C42—22no—
71C22×C42C2311yescharacteristic
72#431—24no—
73#431—24no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8C2314yes—
76C2×C4×C8—22no—
77#399—22no—
78#399—22no—
79#399C2×C414yes—
80#399C2×C414yescomplemented
81#398—28no—
82#397—22no—
83#397—22no—
84#397C2×C414yes—
85#397C2×C414yescomplemented
86#394—28no—
87#388—28no—
88#387—28no—
89#386—28no—
90#385—28no—
91#385—28no—
92#385—28no—
93#383—22no—
94#383—22no—
95#383C2×C414yescomplemented
96#382D414yes—
97#381—22no—
98#381—22no—
99#380—22no—
100#380—22no—
101#380C2314yes—
102#379—22no—
103C4×C4⋊C4D414yes—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109#376—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95C2311yescharacteristic
116C2×#95C2311yescharacteristic
117#340—28no—
118#340—28no—
119#323—24no—
120#323—24no—
121C25C2×D411yescharacteristic
122C25C2×D411yescharacteristic
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—42no—
127C22×D4C22⋊C414yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—24no—
139C23×C4—82no—
140C23×C4—82no—
141C23×C4C22⋊C414yes—
142C23×C4C22⋊C414yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—24no—
149#125—28no—
150#125—28no—
151#125—44no—
152#125—44no—
153C4×D4—28no—
154C4×D4—28no—
155C4×D4—28no—
156C4×D4—28no—
157#117—84no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C22⋊C414yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C2×D418yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4C22×C412yes—
201C2×C22⋊C4C22×C412yes—
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4C2×D412yes—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—84no—
207C2×C42C22×C412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42#4114yes—
212C4⋊C8—44no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215C4×C8—44no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—21no—
226#95—21no—
227#95—24no—
228#95C22×C412yes—
229#95C22×C412yes—
230C24—21no—
231C24—21no—
232C24—22no—
233C24—21no—
234C24—22no—
235C24—21no—
236C24—22no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24C2×C22⋊C412yes—
242C24C2×C22⋊C412yes—
243C24#13514yes—
244C24C22×D411yescharacteristic
245C2×D4—28no—
246C2×D4—28no—
247C2×D4—44no—
248C2×D4—48no—
249C2×D4—48no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—28no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—48no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4#12012yes—
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12312yes—
302C22×C4#12312yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D412yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4112yes—
307C2×C8—48no—
308C2×C8—44no—
309C4⋊C4—28no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—48no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—48no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—48no—
326C22⋊C4—48no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—48no—
330C22⋊C4—84no—
331C22⋊C4—84no—
332C22⋊C4—84no—
333C22⋊C4—84no—
334C42—48no—
335C42—84no—
336C42C4×D414yes—
337C42C4×D414yes—
338C42#12114yes—
339C42#12114yes—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—22no—
344C23—24no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—48no—
354C23—48no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—81no—
362C23—81no—
363C23—84no—
364C23—81no—
365C23—81no—
366C23—84no—
367C23#41614yes—
368C23C22×C22⋊C411yescharacteristic
369C23C2×#12011yescharacteristic
370C23C2×#12311yescharacteristic
371C23C2×#13512yes—
372C23#57512yes—
373D4—416no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—48no—
389C2×C4—48no—
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—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—82no—
416C2×C4—816no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4#38614yes—
421C2×C4#38612yes—
422C2×C4#38612yes—
423C2×C4C2×C4×D412yes—
424C2×C4C2×#12112yes—
425C8—84no—
426C22—24no—
427C22—24no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—24no—
433C22—44no—
434C22—44no—
435C22—48no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—82no—
442C22—84no—
443C22—82no—
444C22—84no—
445C22—84no—
446C22C2×#38611yescharacteristic
447C22C2×#41611yescharacteristic
448C22#289611yescharacteristic
449C22#300212yes—
450C22#301912yes—
451C22#302812yes—
452C22#309112yes—
453C4—24no—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—22no—
458C4—42no—
459C4—42no—
460C4—82no—
461C4—82no—
462C4—88no—
463C4—82no—
464C4—82no—
465C2—21no—
466C2—21no—
467C2—22no—
468C2—44no—
469C2—82no—
470C2#1831414yes—
471C2#2090811yescharacteristic
472C2#2095311yescharacteristic
473C2#2096211yescharacteristic
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
3. #21836C2semidirect
5. #9028C2semidirect
6. #9025C2semidirect
7. #9024C2semidirect
8. #9021C2semidirect
9. #8025C2semidirect
14. #2633C4semidirect
49. C2×#125C2×C4semidirect
80. #399C2×C4semidirect
85. #397C2×C4semidirect
95. #383C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26332
#340572
#218362
C2×#14602
#90282
#90252
#90242
#90212
#80252