Finite Group Databasebeta

← #238130

Subgroups — #238130

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
1C2×#2551C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3#21385C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5#10350C214yesmaximal, complemented
6#10349C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7901C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8—21no—
12C22×C4⋊Q8—21no—
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2551—22no—
17#2551—22no—
18#2551—22no—
19#2551C2214yes—
20C2×#395—22no—
21C2×#389—21no—
22C2×#389—21no—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1990C2212yes—
27#1989—22no—
28#1988C2212yes—
29#1987C2212yes—
30#1987C2212yes—
31#1981C2212yes—
32#1956C2212yes—
33#1872—22no—
34#1872—21no—
35#1872—21no—
36#1803—22no—
37C4×#103—22no—
38#1736—21no—
39#1736—21no—
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×Q8—22no—
44C22×M16—21no—
45#533—22no—
46#533—22no—
47#533—22no—
48#533Q814yescomplemented
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—48no—
54C2×C4⋊Q8—44no—
55C2×C4⋊Q8—44no—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×#125—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×#117—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D412yes—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#423—22no—
81#423—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—24no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8C2312yes—
93#395—28no—
94#395—28no—
95#389—24no—
96#389—24no—
97#389—24no—
98#389—24no—
99#384—21no—
100#384—21no—
101#384D412yes—
102#384C2312yes—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4C2312yes—
110C4×C4⋊C4C2312yes—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121C22×Q8—28no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—48no—
125C22×Q8—48no—
126C22×Q8—84no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C2×M16—42no—
136C2×M16—42no—
137C2×M16—42no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—416no—
148C4⋊Q8—48no—
149C4⋊Q8—48no—
150C4⋊Q8—416no—
151C4⋊Q8—48no—
152C4⋊Q8—48no—
153C4⋊Q8C2×Q812yescomplemented
154C4⋊Q8C2×Q812yescomplemented
155#125—24no—
156#125—42no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—42no—
160C22⋊Q8—42no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—42no—
164C4×Q8—42no—
165#117—24no—
166#117—42no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4SD1614yes—
195C2×C4⋊C4Q1614yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D414yes—
201C2×C22⋊C4—41no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—44no—
207C2×C42—41no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×Q812yes—
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224#103—44no—
225#103—44no—
226#103—84no—
227C22⋊C8—42no—
228C22⋊C8—42no—
229C4×C8—22no—
230C4×C8—22no—
231C4×C8—42no—
232#95—28no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237C24#12011yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—416no—
241C2×Q8—416no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—88no—
249C2×Q8—832no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—41no—
262C22×C4—41no—
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—44no—
272C22×C4—44no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—44no—
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12011yescharacteristic
279C22×C4C22⋊Q812yes—
280C22×C4C22⋊Q812yes—
281C22×C4C22⋊Q812yes—
282C22×C4#12412yes—
283C22×C4#12412yes—
284C22×C4#12412yes—
285C22×C4C2×SD1612yes—
286C22×C4C2×Q1612yes—
287C22×C4AGL(1,ℤ/8)12yes—
288C22×C4#13712yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292M16—84no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—416no—
317C4⋊C4—416no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—82no—
329C4⋊C4C2×SD1614yes—
330C4⋊C4C2×Q1614yes—
331C4⋊C4AGL(1,ℤ/8)14yes—
332C4⋊C4#13714yes—
333C22⋊C4—44no—
334C42—44no—
335C42—42no—
336C42—42no—
337C42—44no—
338C42—44no—
339C42—44no—
340C42C22⋊Q812yes—
341C42C22⋊Q812yes—
342C42C22⋊Q812yes—
343C42#12412yes—
344C42#12412yes—
345C42#12412yes—
346C42C22×Q811yescharacteristic
347C42C2×#4111yescharacteristic
348C42C2×#4111yescharacteristic
349C42C2×#4111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23#39312yes—
355C23#45012yes—
356C23#45112yes—
357C23C2×#12011yescharacteristic
358Q8—44no—
359Q8—44no—
360Q8—82no—
361Q8—832no—
362Q8—82no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—88no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4#39312yes—
400C2×C4#39312yes—
401C2×C4#39312yes—
402C2×C4#45012yes—
403C2×C4#45112yes—
404C2×C4#47514yes—
405C2×C4#47714yes—
406C2×C4#48714yes—
407C2×C4#48914yes—
408C2×C4C2×C22⋊Q811yescharacteristic
409C2×C4C2×C22⋊Q811yescharacteristic
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×#12411yescharacteristic
412C2×C4C2×#12411yescharacteristic
413C2×C4C2×#12411yescharacteristic
414C2×C4C22×SD1611yescharacteristic
415C2×C4C22×Q1611yescharacteristic
416C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
417C2×C4C2×AGL(1,ℤ/8)12yes—
418C2×C4C2×#13711yescharacteristic
419C2×C4C2×#13712yes—
420C8—82no—
421C8—82no—
422C8—82no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22#202414yes—
430C22C2×#39311yescharacteristic
431C22C2×#45011yescharacteristic
432C22C2×#45111yescharacteristic
433C22#300311yescharacteristic
434C22#300411yescharacteristic
435C22#301911yescharacteristic
436C22#302011yescharacteristic
437C4—22no—
438C4—21no—
439C4—21no—
440C4—22no—
441C4—44no—
442C4—44no—
443C4—44no—
444C4—44no—
445C4—84no—
446C4—82no—
447C4—82no—
448C4#202414yes—
449C4C2×#47511yescharacteristic
450C4C2×#47711yescharacteristic
451C4#307912yes—
452C4C2×#48711yescharacteristic
453C4C2×#48911yescharacteristic
454C4#314112yes—
455C2—24no—
456C2#1874711yescharacteristic
457C2#1875911yescharacteristic
458C2#1876911yescharacteristic
459C2#1877711yescharacteristic
460C2C2×#202411yescharacteristic
461C2#2345211yescharacteristic
462C2#2346211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10350C2semidirect
6. #10349C2semidirect
48. #533Q8semidirect
153. C4⋊Q8C2×Q8semidirect
154. C4⋊Q8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#218602
#213852
#213122
#103502
#103492
#80432
#79012
#78922