Finite Group Databasebeta

← #189134

Subgroups — #189134

Order 512.

1128 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic, complemented
3#20738C211yesmaximal, characteristic
4#20732C211yesmaximal, characteristic
5C2×#1572C211yesmaximal, characteristic, complemented
6#8648C212yesmaximal, complemented
7#8646C212yesmaximal, complemented
8#8644C212yesmaximal, complemented
9#8642C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic, complemented
11#8038C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13#2972—24no—
14#2972C418yescomplemented
15C2×C8×D4—22no—
16C2×C8×D4C2212yes—
17C2×C8×D4C2212yes—
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C22×C22⋊C8—22no—
23#2285C2211yescharacteristic
24#2285C2211yescharacteristic
25#1949—22no—
26#1854—22no—
27#1845C2211yescharacteristic, complemented
28#1844C2211yescharacteristic, complemented
29#1834C2211yescharacteristic, complemented
30#1834C2211yescharacteristic, complemented
31#1824—21no—
32#1823—21no—
33#1802C2212yes—
34#1802C2212yes—
35#1802C2212yes—
36#1802C2212yes—
37#1786—21no—
38#1786—21no—
39#1762—22no—
40C4×#102—22no—
41#1572—24no—
42C2×#335C2212yescomplemented
43C2×#334C2212yescomplemented
44#1296C2212yescomplemented
45#1296C2212yescomplemented
46#1296C2212yescomplemented
47#1296C2212yescomplemented
48C23×D4D411yescharacteristic
49C24×C4—21no—
50C23×C8—22no—
51C23×C8—22no—
52C2×C4×D4—21no—
53C2×C4×D4—22no—
54C2×C4×D4—21no—
55C2×C4×D4—22no—
56C2×C4×D4C2×C414yes—
57C2×C4×D4D412yes—
58C2×C4×D4Q814yes—
59C2×C4×D4C2312yes—
60C2×#117—21no—
61C2×#117—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C22⋊C4—21no—
64C22×C42C2311yescharacteristic
65C8×D4—216no—
66C8×D4—24no—
67C8×D4—24no—
68C8×D4C2×C418yes—
69C8×D4C2×C418yes—
70#432—22no—
71#432—22no—
72#431—22no—
73#431—22no—
74#431—22no—
75#431C2×C414yes—
76#431C2×C414yes—
77#431D414yes—
78C2×#107—22no—
79C2×#106—22no—
80C2×C4⋊C8D412yes—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—216no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8D412yes—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#382—21no—
105#382—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#335—24no—
121#334—24no—
122C25—21no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4C4⋊C414yes—
128C22×D4C2×D412yes—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—82no—
135C23×C4—82no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4#4111yescharacteristic
139C22×C8—22no—
140C22×C8—28no—
141C22×C8—28no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—82no—
145C22×C8—82no—
146C22×C8—82no—
147C22×C8—82no—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4C4⋊C418yes—
158C4×D4C22×C414yes—
159C4×D4C2×Q814yes—
160#117—82no—
161#117—82no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
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—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C4⋊C414yes—
177C2×C4⋊C4C2×D412yes—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42C22×C412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D412yes—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q812yes—
191C2×C42#4111yescharacteristic
192C2×C42#4112yes—
193C2×C42C2411yescharacteristic
194#107—82no—
195#106—82no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8C22⋊C418yes—
202C4⋊C8C22×C414yes—
203#102—44no—
204#102—44no—
205#102—44no—
206#102—44no—
207C22⋊C8—24no—
208C22⋊C8—28no—
209C22⋊C8—28no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212C22⋊C8C22⋊C418yes—
213#97—42no—
214C4×C8—24no—
215C4×C8—42no—
216C4×C8C22×C412yes—
217C4×C8C22×C412yes—
218C4×C8C2×D412yes—
219C4×C8C2×D412yes—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224C24—24no—
225C24—24no—
226C24—22no—
227C24—24no—
228C24—81no—
229C24—81no—
230C24—81no—
231C24—81no—
232C24#12111yescharacteristic
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—216no—
236C2×D4—216no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—24no—
241C2×D4C2×C4⋊C414yes—
242C2×D4#13514yes—
243C2×D4#13514yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—28no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—88no—
264C22×C4—88no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C4⋊C412yes—
271C22×C4#11712yes—
272C22×C4#12112yes—
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q812yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—24no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—88no—
295C2×C8—88no—
296C2×C8C2×C22⋊C414yes—
297C2×C8C2×C22⋊C414yes—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—81no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C4⋊C4C2×C4⋊C414yes—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—28no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C42—82no—
319C42—82no—
320C42C2×C22⋊C412yes—
321C42C2×C4⋊C412yes—
322C42#11712yes—
323C42#12112yes—
324C42C22⋊Q812yes—
325C42C22⋊Q812yes—
326C42C23×C411yescharacteristic
327C42C22×D411yescharacteristic
328C42C22×Q811yescharacteristic
329C42C2×#4111yescharacteristic
330C23—216no—
331C23—21no—
332C23—21no—
333C23—24no—
334C23—21no—
335C23—21no—
336C23—24no—
337C23—24no—
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—84no—
345C23—84no—
346C23—84no—
347C23—84no—
348C23—84no—
349C23—84no—
350C23#38012yes—
351C23C2×#12111yescharacteristic
352D4—216no—
353D4—24no—
354D4—24no—
355D4—24no—
356D4#42718yes—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—88no—
379C2×C4—84no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4#38012yes—
393C2×C4#38012yes—
394C2×C4#38012yes—
395C2×C4C22×C22⋊C411yescharacteristic
396C2×C4C22×C4⋊C411yescharacteristic
397C2×C4C2×#11711yescharacteristic
398C2×C4C2×#12111yescharacteristic
399C2×C4C2×C22⋊Q811yescharacteristic
400C2×C4C2×C22⋊Q811yescharacteristic
401C2×C4C2×#13512yes—
402C2×C4C2×#13512yes—
403C2×C4#57612yes—
404C2×C4#57712yes—
405C8—24no—
406C8—24no—
407C8—44no—
408C8—82no—
409C8—82no—
410C22—22no—
411C22—22no—
412C22—28no—
413C22—22no—
414C22—44no—
415C22—41no—
416C22—44no—
417C22—41no—
418C22—44no—
419C22—41no—
420C22—44no—
421C22—41no—
422C22—44no—
423C22—44no—
424C22—816no—
425C22C2×#38011yescharacteristic
426C22#304111yescharacteristic
427C22#304211yescharacteristic
428C22#305011yescharacteristic
429C22#305011yescharacteristic
430C22#306811yescharacteristic
431C22#306911yescharacteristic
432C4—22no—
433C4—22no—
434C4—21no—
435C4—21no—
436C4—22no—
437C4—44no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4#289712yes—
443C4C2×#42712yes—
444C4#305012yes—
445C4#308112yes—
446C2—42no—
447C2—48no—
448C2#1938811yescharacteristic
449C2#1938911yescharacteristic
450C2#1939411yescharacteristic
451C2#1939511yescharacteristic
452C2#2065011yescharacteristic
453C2#2065811yescharacteristic
454C2#2067911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ 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
2. #21984C2semidirect
5. C2×#1572C2semidirect
6. #8648C2semidirect
7. #8646C2semidirect
8. #8644C2semidirect
9. #8642C2semidirect
10. #8046C2semidirect
11. #8038C2semidirect
14. #2972C4semidirect
27. #1845C22semidirect
28. #1844C22semidirect
29. #1834C22semidirect
30. #1834C22semidirect
42. C2×#335C22semidirect
43. C2×#334C22semidirect
44. #1296C22semidirect
45. #1296C22semidirect
46. #1296C22semidirect
47. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
#219842
#207382
#207322
C2×#15722
#86482
#86462
#86442
#86422
#80462
#80382