Finite Group Databasebeta

← #241394

Subgroups — #241394

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4#21364C211yesmaximal, characteristic
5#10581C212yesmaximal, complemented
6#10580C212yesmaximal, complemented
7#10575C212yesmaximal, complemented
8#10574C212yesmaximal, complemented
9#8040C211yesmaximal, characteristic
10#7902C211yesmaximal, characteristic
11#7900C211yesmaximal, characteristic
12C2×#544—21no—
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2868—21no—
17#2859—21no—
18#2660—22no—
19#2552—21no—
20#2552C2212yes—
21#2551—21no—
22#2551C2212yes—
23#2440—22no—
24#2440—22no—
25#2317—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1948—22no—
31#1948C2211yescharacteristic
32#1948C2212yes—
33#1948C2211yescharacteristic
34#1947C2212yes—
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1854—22no—
38#1854—21no—
39#1854—21no—
40#1853—22no—
41#1847—21no—
42#1844C2212yes—
43#1836—21no—
44#1834C2212yes—
45#1776—21no—
46#1776—21no—
47#1296C2212yes—
48#1296C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#544—44no—
54#544—44no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×C22⋊Q8—24no—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117D411yescharacteristic
65C2×#117D411yescharacteristic
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D412yes—
69C22×C42C2311yescharacteristic
70#432—22no—
71#432—22no—
72#432Q814yescomplemented
73#432Q814yescomplemented
74#431—22no—
75#431Q814yescomplemented
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—22no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—24no—
82C2×#106—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8C2312yes—
93#398—24no—
94#398—24no—
95#395—24no—
96#395—24no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#385—24no—
104#384—22no—
105#384—22no—
106#384—22no—
107#384—21no—
108#384—21no—
109#384D411yescharacteristic
110#384D411yescharacteristic
111#384D412yes—
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2312yes—
115#382—22no—
116#382—21no—
117#382—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380D412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2312yes—
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132#376—21no—
133C43—21no—
134#340—24no—
135#340—22no—
136#340—22no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141C22×Q8—24no—
142C23×C4—22no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C4⋊Q8—48no—
151C4⋊Q8—44no—
152C4⋊Q8—44no—
153C4⋊Q8—84no—
154C4⋊Q8—84no—
155#125—48no—
156#125—44no—
157#125—44no—
158C22⋊Q8—416no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—22no—
163#117—42no—
164#117—42no—
165#117—82no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D414yes—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—41no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×Q812yes—
213C2×C42C2×Q812yes—
214C2×C42C2×Q812yes—
215C2×C42#4112yes—
216C2×C42C2411yescharacteristic
217#107—42no—
218#107—42no—
219#107—82no—
220#106—42no—
221#106—42no—
222#106—82no—
223#106—84no—
224C4⋊C8—24no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—22no—
228C4⋊C8C2×Q812yes—
229C4⋊C8C2×Q812yes—
230C4⋊C8C2×Q814yes—
231C22⋊C8—24no—
232C22⋊C8—24no—
233C22⋊C8—24no—
234#97—24no—
235#97—24no—
236C4×C8—22no—
237C4×C8C2×Q814yes—
238#95—24no—
239#95—24no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—41no—
247#95—41no—
248C24#12011yescharacteristic
249C2×Q8—48no—
250C2×Q8—84no—
251C2×Q8—84no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—44no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4C22⋊Q812yes—
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q812yes—
278C22×C4C4⋊Q812yes—
279C22×C4C4⋊Q812yes—
280C22×C4C4⋊Q812yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—48no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—82no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C4⋊C4AGL(1,ℤ/8)12yes—
328C4⋊C4AGL(1,ℤ/8)12yes—
329C4⋊C4#13712yes—
330C4⋊C4#13712yes—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—44no—
334C22⋊C4—82no—
335C42—24no—
336C42—24no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—82no—
344C42C22⋊Q812yes—
345C42C22⋊Q812yes—
346C42C22⋊Q812yes—
347C42C4⋊Q812yes—
348C42C4⋊Q812yes—
349C42C4⋊Q812yes—
350C42C22×Q811yescharacteristic
351C42C22×Q811yescharacteristic
352C42C22×Q811yescharacteristic
353C42C2×#4111yescharacteristic
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23#39512yes—
361C23C2×#12011yescharacteristic
362Q8—168no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—164no—
407C2×C4#39512yes—
408C2×C4#39512yes—
409C2×C4#39512yes—
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4C2×C22⋊Q811yescharacteristic
413C2×C4C2×C4⋊Q811yescharacteristic
414C2×C4C2×C4⋊Q811yescharacteristic
415C2×C4C2×C4⋊Q811yescharacteristic
416C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
417C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
418C2×C4C2×#13711yescharacteristic
419C2×C4C2×#13711yescharacteristic
420C2×C4#57712yes—
421C2×C4#57712yes—
422C8—44no—
423C8—42no—
424C8—42no—
425C8—44no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22C2×#39511yescharacteristic
434C22#300311yescharacteristic
435C22#300411yescharacteristic
436C22#300911yescharacteristic
437C22#301011yescharacteristic
438C22#301911yescharacteristic
439C22#302011yescharacteristic
440C4—22no—
441C4—22no—
442C4—21no—
443C4—21no—
444C4—22no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—84no—
450C4—82no—
451C4—82no—
452C4—161no—
453C4—161no—
454C4#307911yescharacteristic
455C4#307911yescharacteristic
456C4#308412yes—
457C4#316312yes—
458C4#316511yescharacteristic
459C4#316511yescharacteristic
460C2—42no—
461C2#1881511yescharacteristic
462C2#1881611yescharacteristic
463C2#1882611yescharacteristic
464C2#1882911yescharacteristic
465C2#2355211yescharacteristic
466C2#2355611yescharacteristic
467C2#2356111yescharacteristic
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. #10581C2semidirect
6. #10580C2semidirect
7. #10575C2semidirect
8. #10574C2semidirect
72. #432Q8semidirect
73. #432Q8semidirect
75. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#219882
#219812
#213642
#105812
#105802
#105752
#105742
#80402
#79022
#79002