Finite Group Databasebeta

← #241298

Subgroups — #241298

Order 512.

1178 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21358C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic
5#10575C214yesmaximal, complemented
6#10573C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7901C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C22×C4⋊Q8—21no—
11C22×C4⋊Q8—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#428—21no—
15C2×#423C2211yescharacteristic
16#2551—22no—
17#2551—22no—
18#2551—22no—
19#2551C2214yes—
20C2×#395—22no—
21C2×#389—21no—
22C2×#389—21no—
23#2285C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1949—22no—
28#1948C2212yes—
29#1947—22no—
30#1947C2212yes—
31#1947C2212yes—
32#1947C2212yes—
33#1852—21no—
34#1852—21no—
35#1852—22no—
36#1845C2212yes—
37#1834C2212yes—
38C4×#107—22no—
39#1736—21no—
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×Q8—22no—
44C22×M16—21no—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—48no—
50C2×C4⋊Q8—44no—
51C2×C4⋊Q8—44no—
52C2×#117—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D412yes—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65#431—22no—
66#431—22no—
67#431Q814yescomplemented
68#431Q814yescomplemented
69#428—44no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—24no—
74C2×#106—22no—
75#423—22no—
76#423Q814yescomplemented
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2312yes—
85#395—28no—
86#395—28no—
87#389—24no—
88#389—24no—
89#389—24no—
90#389—24no—
91#384—21no—
92#384—21no—
93#384D412yes—
94#384C2311yescharacteristic
95#384C2311yescharacteristic
96#384C2312yes—
97#382—21no—
98#380—21no—
99#380—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4C2312yes—
109C4×C4⋊C4C2312yes—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113#340—24no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118C22×Q8—28no—
119C22×Q8—48no—
120C22×Q8—48no—
121C22×Q8—84no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×M16—24no—
131C2×M16—42no—
132C2×M16—42no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—416no—
137C4⋊Q8—48no—
138C4⋊Q8—48no—
139C4⋊Q8—416no—
140C4⋊Q8—48no—
141C4⋊Q8—48no—
142#117—42no—
143#117—42no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4D814yes—
172C2×C4⋊C4Q1614yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D414yes—
178C2×C22⋊C4—41no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—44no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42—41no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q812yes—
191C2×C42C2×Q812yes—
192C2×C42C2×Q812yes—
193C2×C42#4112yes—
194C2×C42C2411yescharacteristic
195#107—44no—
196#107—44no—
197#107—44no—
198#107—84no—
199#106—44no—
200C4⋊C8—24no—
201C4⋊C8—24no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8C2×Q814yescomplemented
205C4⋊C8C2×Q814yescomplemented
206C22⋊C8—24no—
207C22⋊C8—24no—
208C4×C8—24no—
209C4×C8—22no—
210C4×C8C2×Q814yescomplemented
211#95—28no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—41no—
217#95—41no—
218C24#12011yescharacteristic
219C2×Q8—416no—
220C2×Q8—416no—
221C2×Q8—88no—
222C2×Q8—832no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4C22⋊Q812yes—
253C22×C4C22⋊Q812yes—
254C22×C4C22⋊Q812yes—
255C22×C4C4⋊Q812yes—
256C22×C4C4⋊Q812yes—
257C22×C4C4⋊Q812yes—
258C22×C4C2×D812yes—
259C22×C4C2×Q1612yes—
260C22×C4AGL(1,ℤ/8)12yes—
261C22×C4#13712yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265M16—48no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—416no—
286C4⋊C4—416no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—82no—
300C4⋊C4C2×D814yes—
301C4⋊C4C2×Q1614yes—
302C4⋊C4AGL(1,ℤ/8)14yes—
303C4⋊C4#13714yes—
304C22⋊C4—82no—
305C42—44no—
306C42—42no—
307C42—42no—
308C42—44no—
309C42—44no—
310C42—82no—
311C42C22⋊Q812yes—
312C42C22⋊Q812yes—
313C42C22⋊Q812yes—
314C42C4⋊Q812yes—
315C42C4⋊Q812yes—
316C42C4⋊Q812yes—
317C42C22×Q811yescharacteristic
318C42C22×Q811yescharacteristic
319C42C22×Q811yescharacteristic
320C42C2×#4111yescharacteristic
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23#39512yes—
326C23#44712yes—
327C23#45112yes—
328C23C2×#12011yescharacteristic
329Q8—832no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—88no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4#39512yes—
367C2×C4#39512yes—
368C2×C4#39512yes—
369C2×C4#44712yes—
370C2×C4#45112yes—
371C2×C4#47414yes—
372C2×C4#47714yes—
373C2×C4#50014yes—
374C2×C4#50114yes—
375C2×C4C2×C22⋊Q811yescharacteristic
376C2×C4C2×C22⋊Q811yescharacteristic
377C2×C4C2×C22⋊Q811yescharacteristic
378C2×C4C2×C4⋊Q811yescharacteristic
379C2×C4C2×C4⋊Q811yescharacteristic
380C2×C4C2×C4⋊Q811yescharacteristic
381C2×C4C22×D811yescharacteristic
382C2×C4C22×Q1611yescharacteristic
383C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
384C2×C4C2×AGL(1,ℤ/8)12yes—
385C2×C4C2×#13712yes—
386C2×C4C2×#13711yescharacteristic
387C8—44no—
388C8—44no—
389C8—44no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22#205714yes—
397C22C2×#39511yescharacteristic
398C22C2×#44711yescharacteristic
399C22C2×#45111yescharacteristic
400C22#300311yescharacteristic
401C22#300411yescharacteristic
402C22#301911yescharacteristic
403C22#302011yescharacteristic
404C4—22no—
405C4—21no—
406C4—21no—
407C4—22no—
408C4—44no—
409C4—44no—
410C4—44no—
411C4—44no—
412C4—84no—
413C4—161no—
414C4—161no—
415C4#205714yes—
416C4C2×#47411yescharacteristic
417C4C2×#47711yescharacteristic
418C4#307912yes—
419C4C2×#50011yescharacteristic
420C4C2×#50111yescharacteristic
421C4#316512yes—
422C2—24no—
423C2#1874611yescharacteristic
424C2#1875911yescharacteristic
425C2#1876911yescharacteristic
426C2#1877611yescharacteristic
427C2C2×#205711yescharacteristic
428C2#2355211yescharacteristic
429C2#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. #10575C2semidirect
6. #10573C2semidirect
67. #431Q8semidirect
68. #431Q8semidirect
76. #423Q8semidirect
204. C4⋊C8C2×Q8semidirect
205. C4⋊C8C2×Q8semidirect
210. C4×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#219822
#213582
#211702
#105752
#105732
#80432
#79012
#78922