Finite Group Databasebeta

← #241371

Subgroups — #241371

Order 512.

912 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic
3#21358C211yesmaximal, characteristic
4#20824C211yesmaximal, characteristic
5#10581C212yesmaximal, complemented
6#10580C212yesmaximal, complemented
7#10575C212yesmaximal, complemented
8#10573C212yesmaximal, complemented
9#8046C211yesmaximal, characteristic
10#7904C211yesmaximal, characteristic
11#7896C211yesmaximal, characteristic
12C2×#544—21no—
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2857—22no—
17#2659—22no—
18#2552—21no—
19#2552C2212yes—
20#2551—21no—
21#2551C2212yes—
22#2550—22no—
23#2550—22no—
24#2316—21no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1948C2212yes—
30#1947C2212yes—
31#1947C2212yes—
32#1947C2212yes—
33#1854—22no—
34#1852—21no—
35#1852—21no—
36#1847—21no—
37#1845C2211yescharacteristic
38#1844C2211yescharacteristic
39#1836—21no—
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1809—22no—
43#1776—22no—
44C4×#107—22no—
45#1296C2212yes—
46#1296C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×M16—21no—
52#544—48no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×C22⋊Q8—24no—
59C2×#117—21no—
60C2×#117—21no—
61C2×#117—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66#432—22no—
67#432Q814yescomplemented
68#431—22no—
69#431Q814yescomplemented
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#106—22no—
75C2×#106—22no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C4×M16—22no—
81C4×M16Q814yescomplemented
82C2×#97—21no—
83C2×#97—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
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#384C2311yescharacteristic
105#384C2311yescharacteristic
106#382—21no—
107#382—21no—
108#382—21no—
109#380—24no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380D412yes—
115#380D412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4C2312yes—
125C4×C4⋊C4C2312yes—
126C2×#95—21no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133C22×Q8—24no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—42no—
141C2×M16—42no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—48no—
147C4⋊Q8—44no—
148C4⋊Q8—44no—
149C4⋊Q8—82no—
150C4⋊Q8—82no—
151#125—48no—
152#125—44no—
153#125—44no—
154C22⋊Q8—416no—
155#117—44no—
156#117—44no—
157#117—82no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
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⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—41no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—41no—
195C2×C42—82no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×Q812yes—
200C2×C42C2×Q812yes—
201C2×C42C2×Q812yes—
202C2×C42#4112yes—
203C2×C42C2411yescharacteristic
204#107—44no—
205#107—44no—
206#107—44no—
207#107—82no—
208#106—44no—
209#106—82no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8C2×Q814yes—
213C4⋊C8C2×Q814yes—
214C22⋊C8—24no—
215C22⋊C8—24no—
216#97—24no—
217#97—24no—
218C4×C8—24no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8C2×Q812yes—
222C4×C8C2×Q812yes—
223#95—24no—
224#95—24no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—41no—
232#95—41no—
233C24#12011yescharacteristic
234C2×Q8—48no—
235C2×Q8—84no—
236C2×Q8—84no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4C22⋊Q812yes—
261C22×C4C22⋊Q812yes—
262C22×C4C22⋊Q812yes—
263C22×C4C4⋊Q812yes—
264C22×C4C4⋊Q812yes—
265C22×C4C4⋊Q812yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269M16—48no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—24no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—48no—
294C4⋊C4—48no—
295C4⋊C4—48no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C4⋊C4AGL(1,ℤ/8)14yes—
311C4⋊C4#13714yes—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—82no—
315C42—44no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—82no—
321C42—82no—
322C42C22⋊Q812yes—
323C42C22⋊Q812yes—
324C42C22⋊Q812yes—
325C42C4⋊Q812yes—
326C42C4⋊Q812yes—
327C42C4⋊Q812yes—
328C42C22×Q811yescharacteristic
329C42C22×Q811yescharacteristic
330C42C22×Q811yescharacteristic
331C42C2×#4111yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23#39512yes—
339C23C2×#12011yescharacteristic
340Q8—168no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
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—84no—
374C2×C4—84no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4#39512yes—
382C2×C4#39512yes—
383C2×C4#39512yes—
384C2×C4C2×C22⋊Q811yescharacteristic
385C2×C4C2×C22⋊Q811yescharacteristic
386C2×C4C2×C22⋊Q811yescharacteristic
387C2×C4C2×C4⋊Q811yescharacteristic
388C2×C4C2×C4⋊Q811yescharacteristic
389C2×C4C2×C4⋊Q811yescharacteristic
390C2×C4C2×AGL(1,ℤ/8)12yes—
391C2×C4C2×#13712yes—
392C2×C4#57612yes—
393C2×C4#57812yes—
394C8—24no—
395C8—42no—
396C8—44no—
397C8—44no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22C2×#39511yescharacteristic
406C22#300311yescharacteristic
407C22#300411yescharacteristic
408C22#300811yescharacteristic
409C22#301111yescharacteristic
410C22#301811yescharacteristic
411C22#301811yescharacteristic
412C4—22no—
413C4—22no—
414C4—21no—
415C4—21no—
416C4—22no—
417C4—44no—
418C4—44no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4—82no—
423C4—161no—
424C4—161no—
425C4#307912yes—
426C4#308312yes—
427C4#316412yes—
428C4#316512yes—
429C2—42no—
430C2#1859811yescharacteristic
431C2#1860011yescharacteristic
432C2#1861111yescharacteristic
433C2#1861311yescharacteristic
434C2#2355211yescharacteristic
435C2#2355511yescharacteristic
436C2#2356011yescharacteristic
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. #10573C2semidirect
67. #432Q8semidirect
69. #431Q8semidirect
81. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#219882
#213582
#208242
#105812
#105802
#105752
#105732
#80462
#79042
#78962