Finite Group Databasebeta

← #238225

Subgroups — #238225

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5#10356C214yesmaximal, complemented
6#10350C214yesmaximal, complemented
7#8039C211yesmaximal, characteristic
8#7893C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#544—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103—21no—
16C22×#103—21no—
17#2857—22no—
18#2659—22no—
19#2552—21no—
20#2552C2212yes—
21#2551—21no—
22#2551C2212yes—
23#2550—22no—
24#2550—22no—
25#2316—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—
45#544—44no—
46#544—44no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8Q814yescomplemented
57C2×C4⋊Q8C2312yes—
58C2×#125—22no—
59C2×#125—22no—
60C2×C22⋊Q8—24no—
61C2×#117—21no—
62C2×#117—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#431—22no—
70#431—22no—
71#431—22no—
72#431—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—44no—
82C2×#103—44no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8C2312yes—
87#398—24no—
88#398—24no—
89#395—24no—
90#395—24no—
91#391—24no—
92#389—22no—
93#389—22no—
94#389—24no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#384—22no—
100#384D411yescharacteristic
101#384D411yescharacteristic
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#382—21no—
105#382—21no—
106#380—24no—
107#380—21no—
108#380—21no—
109#380D412yes—
110#380D412yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4C2312yes—
118C4×C4⋊C4C2312yes—
119C2×#95—21no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124C22×Q8—24no—
125C22×Q8—24no—
126C22×Q8—22no—
127C22×Q8—41no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8—42no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—44no—
152C4⋊Q8—48no—
153C4⋊Q8—44no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—84no—
158C4⋊Q8C2×Q814yes—
159#125—44no—
160#125—48no—
161#125—44no—
162C22⋊Q8—416no—
163#117—42no—
164#117—42no—
165#117—42no—
166#117—42no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4D814yes—
193C2×C4⋊C4SD1614yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—82no—
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—22no—
218C4⋊C8—22no—
219C4⋊C8—42no—
220C4⋊C8—42no—
221#103—48no—
222#103—48no—
223C22⋊C8—42no—
224C22⋊C8—42no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—42no—
228#95—24no—
229#95—24no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236C24#12011yescharacteristic
237C2×Q8—24no—
238C2×Q8—48no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—48no—
246C2×Q8—48no—
247C2×Q8—84no—
248C2×Q8—84no—
249C2×Q8—82no—
250C2×Q8—82no—
251C2×Q8—84no—
252C2×Q8—84no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q812yes—
278C22×C4C22⋊Q812yes—
279C22×C4#12412yes—
280C22×C4#12412yes—
281C22×C4#12412yes—
282C22×C4C2×D812yes—
283C22×C4C2×SD1612yes—
284C22×C4AGL(1,ℤ/8)12yes—
285C22×C4#13712yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—84no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—48no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—81no—
325C4⋊C4—81no—
326C4⋊C4—84no—
327C4⋊C4—84no—
328C4⋊C4C2×D814yes—
329C4⋊C4C2×SD1614yes—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C42—42no—
333C42—42no—
334C42—44no—
335C42—44no—
336C42—82no—
337C42C22⋊Q812yes—
338C42C22⋊Q812yes—
339C42C22⋊Q812yes—
340C42#12412yes—
341C42#12412yes—
342C42#12412yes—
343C42C22×Q811yescharacteristic
344C42C2×#4111yescharacteristic
345C42C2×#4111yescharacteristic
346C42C2×#4111yescharacteristic
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23#39312yes—
352C23#44712yes—
353C23#44812yes—
354C23C2×#12011yescharacteristic
355Q8—44no—
356Q8—44no—
357Q8—82no—
358Q8—82no—
359Q8—88no—
360Q8—168no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
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—84no—
391C2×C4—84no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—88no—
395C2×C4#39312yes—
396C2×C4#39312yes—
397C2×C4#39312yes—
398C2×C4#44712yes—
399C2×C4#44812yes—
400C2×C4#47414yes—
401C2×C4#47614yes—
402C2×C4#48614yes—
403C2×C4#48914yes—
404C2×C4C2×C22⋊Q811yescharacteristic
405C2×C4C2×C22⋊Q811yescharacteristic
406C2×C4C2×C22⋊Q811yescharacteristic
407C2×C4C2×#12411yescharacteristic
408C2×C4C2×#12411yescharacteristic
409C2×C4C2×#12411yescharacteristic
410C2×C4C22×D811yescharacteristic
411C2×C4C22×SD1611yescharacteristic
412C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
413C2×C4C2×#13711yescharacteristic
414C2×C4#57712yes—
415C2×C4#57812yes—
416C8—82no—
417C8—82no—
418C8—82no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22#202314yes—
426C22C2×#39311yescharacteristic
427C22C2×#44711yescharacteristic
428C22C2×#44811yescharacteristic
429C22#300911yescharacteristic
430C22#301111yescharacteristic
431C22#302211yescharacteristic
432C22#302211yescharacteristic
433C4—22no—
434C4—21no—
435C4—21no—
436C4—22no—
437C4—44no—
438C4—44no—
439C4—84no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—82no—
444C4#202314yes—
445C4C2×#47411yescharacteristic
446C4C2×#47611yescharacteristic
447C4#308212yes—
448C4C2×#48611yescharacteristic
449C4C2×#48911yescharacteristic
450C4#313712yes—
451C2—24no—
452C2#1890211yescharacteristic
453C2#1890411yescharacteristic
454C2#1892211yescharacteristic
455C2#1892411yescharacteristic
456C2C2×#202311yescharacteristic
457C2#2345711yescharacteristic
458C2#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. #10356C2semidirect
6. #10350C2semidirect
56. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#218332
C2×#19812
#103562
#103502
#80392
#78932
#78922