Finite Group Databasebeta

← #238146

Subgroups — #238146

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2#21835C211yesmaximal, characteristic
3#21394C211yesmaximal, characteristic
4#21389C211yesmaximal, characteristic
5#10357C212yesmaximal, complemented
6#10356C212yesmaximal, complemented
7#10350C212yesmaximal, complemented
8#10350C212yesmaximal, complemented
9#8046C211yesmaximal, characteristic
10#7902C211yesmaximal, characteristic
11#7892C211yesmaximal, characteristic
12C2×#533—21no—
13C2×#533C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2552C2212yes—
18#2551C2212yes—
19#2550—22no—
20#2549—22no—
21#2440—22no—
22#2399—22no—
23C4×C4⋊Q8—22no—
24#2296—21no—
25#2285C2211yescharacteristic
26C4×#117C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990C2212yes—
29#1990C2212yes—
30#1989—22no—
31#1988C2211yescharacteristic
32#1988C2212yes—
33#1988C2211yescharacteristic
34#1981C2211yescharacteristic
35#1981C2212yes—
36#1981C2211yescharacteristic
37#1873—21no—
38#1873—21no—
39#1872—22no—
40#1825—22no—
41#1763—22no—
42C4×#103—22no—
43#1296C2212yes—
44#1296C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#533—22no—
50#533—22no—
51#533—22no—
52#533—48no—
53#533Q814yescomplemented
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×#125—22no—
59C2×#125—21no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C4×Q8—24no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117D411yescharacteristic
69C2×#117D411yescharacteristic
70C22×C4⋊C4D412yes—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#432—22no—
74#432—22no—
75#431—22no—
76#431—22no—
77#431—22no—
78#431—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×#97—21no—
91C2×C4×C8—21no—
92C2×C4×C8C2312yes—
93#398—24no—
94#398—24no—
95#395—24no—
96#395—24no—
97#391—24no—
98#389—24no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#385—24no—
104#384—22no—
105#384C2312yes—
106#382—21no—
107#380—21no—
108#380D412yes—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4C2311yescharacteristic
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2312yes—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121#376—21no—
122C2×#95—21no—
123C43—21no—
124#340—24no—
125#340—24no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—21no—
133C23×C4—22no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—44no—
148C4⋊Q8—44no—
149C4⋊Q8—44no—
150C4⋊Q8—44no—
151C4⋊Q8C2×Q812yes—
152C4⋊Q8C2×Q812yes—
153#125—24no—
154#125—44no—
155#125—44no—
156#125—42no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—416no—
160C22⋊Q8—42no—
161C22⋊Q8—42no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—416no—
165C4×Q8—42no—
166C4×Q8—42no—
167#117—24no—
168#117—24no—
169#117—24no—
170#117—44no—
171#117—42no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D414yes—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—44no—
206C2×C42—41no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×Q812yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42C2411yescharacteristic
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—42no—
222#103—44no—
223#103—44no—
224#103—44no—
225#103—44no—
226C22⋊C8—42no—
227C22⋊C8—42no—
228C22⋊C8—42no—
229#97—42no—
230C4×C8—22no—
231C4×C8—22no—
232C4×C8—42no—
233#95—24no—
234#95—24no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95—22no—
242C24#12011yescharacteristic
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—48no—
246C2×Q8—48no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—84no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—82no—
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4C22⋊Q812yes—
273C22×C4C22⋊Q812yes—
274C22×C4C22⋊Q812yes—
275C22×C4#12412yes—
276C22×C4#12412yes—
277C22×C4#12412yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C2×C8—44no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—48no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—84no—
316C4⋊C4#13514yes—
317C4⋊C4AGL(1,ℤ/8)14yes—
318C4⋊C4#13714yes—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C42—24no—
324C42—24no—
325C42—44no—
326C42—44no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—44no—
332C42—82no—
333C42C22⋊Q812yes—
334C42C22⋊Q812yes—
335C42C22⋊Q812yes—
336C42#12412yes—
337C42#12412yes—
338C42#12412yes—
339C42C22×Q811yescharacteristic
340C42C2×#4111yescharacteristic
341C42C2×#4111yescharacteristic
342C42C2×#4111yescharacteristic
343C23—22no—
344C23—21no—
345C23—21no—
346C23—22no—
347C23#39312yes—
348C23C2×#12011yescharacteristic
349Q8—44no—
350Q8—44no—
351Q8—816no—
352Q8—81no—
353Q8—81no—
354Q8—82no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—88no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4#39312yes—
389C2×C4#39312yes—
390C2×C4#39312yes—
391C2×C4C2×C22⋊Q811yescharacteristic
392C2×C4C2×C22⋊Q811yescharacteristic
393C2×C4C2×C22⋊Q811yescharacteristic
394C2×C4C2×#12411yescharacteristic
395C2×C4C2×#12411yescharacteristic
396C2×C4C2×#12411yescharacteristic
397C2×C4C2×#13512yes—
398C2×C4C2×AGL(1,ℤ/8)12yes—
399C2×C4C2×#13712yes—
400C2×C4#57512yes—
401C8—82no—
402C8—81no—
403C8—81no—
404C8—82no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—42no—
409C22—42no—
410C22C2×#39311yescharacteristic
411C22#300212yes—
412C22#300311yescharacteristic
413C22#300411yescharacteristic
414C22#302212yes—
415C4—22no—
416C4—22no—
417C4—21no—
418C4—21no—
419C4—22no—
420C4—44no—
421C4—44no—
422C4—44no—
423C4—82no—
424C4—84no—
425C4—82no—
426C4—82no—
427C4#307612yes—
428C4#307912yes—
429C4#313111yescharacteristic
430C4#313511yescharacteristic
431C4#314112yes—
432C2—42no—
433C2#1880612yes—
434C2#1880912yes—
435C2#2345111yescharacteristic
436C2#2345211yescharacteristic
437C2#2346411yescharacteristic
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. #10356C2semidirect
7. #10350C2semidirect
8. #10350C2semidirect
53. #533Q8semidirect

Maximal subgroups

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

subgroupindex
#349382
#218352
#213942
#213892
#103572
#103562
#103502
#80462
#79022
#78922