Finite Group Databasebeta

← #238185

Subgroups — #238185

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4#21259C211yesmaximal, characteristic
5#10357C214yesmaximal, complemented
6#10353C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7897C211yesmaximal, characteristic
9#7890C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#419—21no—
14C2×#419—21no—
15C22×C4×C8C2211yescharacteristic
16#2859—21no—
17#2859—21no—
18#2647—22no—
19#2571—21no—
20#2571C2212yes—
21#2552—21no—
22#2552C2212yes—
23#2442—22no—
24#2442—22no—
25#2317—21no—
26#2316—21no—
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1990—22no—
31#1990—21no—
32#1990—21no—
33#1990C2214yes—
34#1988C2214yes—
35#1981—21no—
36#1981—21no—
37#1981C2212yes—
38#1981C2212yes—
39#1823—22no—
40C2×#340—22no—
41#1296C2214yes—
42#1279C2212yes—
43#1279C2214yes—
44#1279C2212yes—
45C23×Q8—21no—
46C22×M16—21no—
47C23×C8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8Q814yes—
55C2×C4⋊Q8C2312yes—
56C2×#125—22no—
57C2×#125—22no—
58C2×#117—21no—
59C2×#117D411yescharacteristic
60C2×#117D411yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D412yes—
66C22×C42C2311yescharacteristic
67#431—22no—
68#431—22no—
69#423—22no—
70#423—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74#419—44no—
75#419—44no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×C22⋊C8—21no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2312yes—
86#400—24no—
87#400—24no—
88#398—24no—
89#398—24no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#385—24no—
101#384—22no—
102#384—21no—
103#384—21no—
104#384D411yescharacteristic
105#384D411yescharacteristic
106#384D412yes—
107#384C2312yes—
108#382—22no—
109#382—22no—
110#382—22no—
111#382—21no—
112#380—24no—
113#380—21no—
114#380D412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2312yes—
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123#376—21no—
124C43—21no—
125#340—24no—
126#340—24no—
127#340—28no—
128C22×Q8—24no—
129C22×Q8—22no—
130C22×Q8—41no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C23×C4—22no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—42no—
142C2×M16—42no—
143C2×M16—42no—
144C22×C8—24no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—42no—
149C22×C8—42no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8C2×Q814yes—
160#125—84no—
161#125—84no—
162#117—22no—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—42no—
167#117—42no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D414yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—21no—
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—
223#103—44no—
224#103—44no—
225#103—44no—
226#103—44no—
227C22⋊C8—42no—
228C4×C8—22no—
229C4×C8—22no—
230C4×C8—42no—
231C4×C8—42no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95—22no—
242C24#12011yescharacteristic
243C2×Q8—24no—
244C2×Q8—42no—
245C2×Q8—44no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—48no—
251C2×Q8—48no—
252C2×Q8—82no—
253C2×Q8—82no—
254C2×Q8—84no—
255C2×Q8—84no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4C22⋊Q812yes—
278C22×C4C22⋊Q812yes—
279C22×C4C22⋊Q812yes—
280C22×C4#12412yes—
281C22×C4#12412yes—
282C22×C4#12412yes—
283C22×C4#13512yes—
284C22×C4#13512yes—
285C22×C4AGL(1,ℤ/8)12yes—
286C22×C4AGL(1,ℤ/8)12yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290M16—84no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—48no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—82no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C4⋊C4—84no—
331C4⋊C4—84no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—44no—
335C42—42no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42C22⋊Q812yes—
344C42C22⋊Q812yes—
345C42C22⋊Q812yes—
346C42#12412yes—
347C42#12412yes—
348C42#12412yes—
349C42C22×Q811yescharacteristic
350C42C2×#4111yescharacteristic
351C42C2×#4111yescharacteristic
352C42C2×#4111yescharacteristic
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23#39312yes—
358C23#44912yes—
359C23#44912yes—
360C23C2×#12011yescharacteristic
361Q8—44no—
362Q8—44no—
363Q8—82no—
364Q8—82no—
365Q8—88no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—44no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—88no—
399C2×C4—164no—
400C2×C4#39312yes—
401C2×C4#39312yes—
402C2×C4#39312yes—
403C2×C4#44912yes—
404C2×C4#44912yes—
405C2×C4#45512yes—
406C2×C4#45612yes—
407C2×C4#47814yes—
408C2×C4#47814yes—
409C2×C4#48814yes—
410C2×C4#49014yes—
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4C2×C22⋊Q811yescharacteristic
413C2×C4C2×C22⋊Q811yescharacteristic
414C2×C4C2×#12411yescharacteristic
415C2×C4C2×#12411yescharacteristic
416C2×C4C2×#12411yescharacteristic
417C2×C4C2×#13511yescharacteristic
418C2×C4C2×#13511yescharacteristic
419C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
420C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
421C2×C4#57712yes—
422C2×C4#57812yes—
423C8—44no—
424C8—82no—
425C8—82no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22#202514yes—
433C22C2×#39311yescharacteristic
434C22C2×#44911yescharacteristic
435C22C2×#44911yescharacteristic
436C22#300911yescharacteristic
437C22#301111yescharacteristic
438C22C2×#45511yescharacteristic
439C22C2×#45611yescharacteristic
440C4—22no—
441C4—21no—
442C4—21no—
443C4—84no—
444C4—81no—
445C4—84no—
446C4—81no—
447C4—82no—
448C4—81no—
449C4—81no—
450C4—82no—
451C4#202514yes—
452C4#202814yes—
453C4C2×#47811yescharacteristic
454C4C2×#47811yescharacteristic
455C4#308212yes—
456C4C2×#48811yescharacteristic
457C4C2×#49011yescharacteristic
458C4#313712yes—
459C2—24no—
460C2#1863411yescharacteristic
461C2#1863611yescharacteristic
462C2#1865411yescharacteristic
463C2#1865611yescharacteristic
464C2C2×#202511yescharacteristic
465C2#2345711yescharacteristic
466C2C2×#202811yescharacteristic
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. #10357C2semidirect
6. #10353C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#218332
C2×#19902
#212592
#103572
#103532
#80432
#78972
#78902