Finite Group Databasebeta

← #238226

Subgroups — #238226

Order 512.

934 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5#10357C214yesmaximal, complemented
6#10353C214yesmaximal, complemented
7#8039C211yesmaximal, characteristic
8#7893C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103—21no—
15C22×#103—21no—
16#2851—22no—
17#2646—22no—
18#2571—21no—
19#2571C2212yes—
20#2570—22no—
21#2570—22no—
22#2552—21no—
23#2552C2212yes—
24#2317—21no—
25#2317—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1990—21no—
30#1990—21no—
31#1990C2212yes—
32#1990C2212yes—
33#1988—22no—
34#1988C2214yes—
35#1981—22no—
36#1981C2214yes—
37#1823—22no—
38#1823—22no—
39#1296C2212yes—
40#1296C2212yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×Q8—21no—
44C23×C8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8Q814yes—
52C2×C4⋊Q8C2312yes—
53C2×#125—22no—
54C2×#117—21no—
55C2×#117—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#431—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—44no—
75C2×#103—44no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8C2312yes—
80#400—24no—
81#400—24no—
82#398—24no—
83#398—24no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#387—24no—
91#387—24no—
92#385—24no—
93#385—24no—
94#384—22no—
95#384—22no—
96#384D411yescharacteristic
97#384D411yescharacteristic
98#384C2311yescharacteristic
99#384C2311yescharacteristic
100#382—22no—
101#382—22no—
102#382—22no—
103#382—21no—
104#382—21no—
105#380—21no—
106#380—21no—
107#380D412yes—
108#380D412yes—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4C2312yes—
116C4×C4⋊C4C2312yes—
117C2×#95—21no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122C22×Q8—24no—
123C22×Q8—22no—
124C22×Q8—41no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—42no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—42no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8C2×Q814yes—
152#125—84no—
153#117—42no—
154#117—42no—
155#117—42no—
156#117—42no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—82no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×Q812yes—
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203C2×C42C2411yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—42no—
209C4⋊C8—42no—
210#103—48no—
211#103—48no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—42no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227C24#12011yescharacteristic
228C2×Q8—24no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—84no—
240C2×Q8—84no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4C22⋊Q812yes—
265C22×C4C22⋊Q812yes—
266C22×C4C22⋊Q812yes—
267C22×C4#12412yes—
268C22×C4#12412yes—
269C22×C4#12412yes—
270C22×C4#13512yes—
271C22×C4#13512yes—
272C22×C4AGL(1,ℤ/8)12yes—
273C22×C4AGL(1,ℤ/8)12yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—84no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—81no—
312C4⋊C4—81no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C42—42no—
320C42—42no—
321C42—44no—
322C42—44no—
323C42—82no—
324C42C22⋊Q812yes—
325C42C22⋊Q812yes—
326C42C22⋊Q812yes—
327C42#12412yes—
328C42#12412yes—
329C42#12412yes—
330C42C22×Q811yescharacteristic
331C42C2×#4111yescharacteristic
332C42C2×#4111yescharacteristic
333C42C2×#4111yescharacteristic
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23#39312yes—
339C23#44912yes—
340C23#44912yes—
341C23C2×#12011yescharacteristic
342Q8—44no—
343Q8—44no—
344Q8—82no—
345Q8—82no—
346Q8—88no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
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—84no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—88no—
381C2×C4#39312yes—
382C2×C4#39312yes—
383C2×C4#39312yes—
384C2×C4#44912yes—
385C2×C4#44912yes—
386C2×C4#47814yes—
387C2×C4#47814yes—
388C2×C4#48814yes—
389C2×C4#49014yes—
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4C2×C22⋊Q811yescharacteristic
392C2×C4C2×C22⋊Q811yescharacteristic
393C2×C4C2×#12411yescharacteristic
394C2×C4C2×#12411yescharacteristic
395C2×C4C2×#12411yescharacteristic
396C2×C4C2×#13511yescharacteristic
397C2×C4C2×#13511yescharacteristic
398C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
399C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
400C2×C4#57712yes—
401C2×C4#57812yes—
402C8—82no—
403C8—82no—
404C8—82no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22#202514yes—
412C22C2×#39311yescharacteristic
413C22C2×#44911yescharacteristic
414C22C2×#44911yescharacteristic
415C22#300911yescharacteristic
416C22#301111yescharacteristic
417C22#302211yescharacteristic
418C22#302211yescharacteristic
419C4—22no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—84no—
424C4—82no—
425C4—82no—
426C4—82no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4#202514yes—
431C4C2×#47811yescharacteristic
432C4C2×#47811yescharacteristic
433C4#308212yes—
434C4C2×#48811yescharacteristic
435C4C2×#49011yescharacteristic
436C4#313712yes—
437C2—24no—
438C2#1890611yescharacteristic
439C2#1890811yescharacteristic
440C2#1892611yescharacteristic
441C2#1892811yescharacteristic
442C2C2×#202511yescharacteristic
443C2#2345711yescharacteristic
444C2#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. #10353C2semidirect

Maximal subgroups

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

subgroupindex
#351812
#218332
C2×#19812
#103572
#103532
#80392
#78932
#78922