Finite Group Databasebeta

← #413236

Subgroups — #413236

Order 512.

1226 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2061C212yesmaximal
3C2×#2060C212yesmaximal
4#20801C212yesmaximal
5#13909C211yesmaximal, characteristic
6#11342C212yesmaximal
7#11342C212yesmaximal
8#10851C212yesmaximal
9C4×#343C211yesmaximal, characteristic
10C2×#544C2212yes—
11C2×#428C2214yes—
12C2×#428C2214yes—
13C2×C4×M16C2212yes—
14#2857—22no—
15#2659—22no—
16#2552—22no—
17#2551—22no—
18#2550—24no—
19#2316C2212yes—
20#2285C2212yes—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#2061—28no—
27#2060—28no—
28#1847—22no—
29#1847—22no—
30#1846—24no—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35#1441—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C22×M16C2314yes—
40#544—24no—
41#544—24no—
42#544—44no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—24no—
45C2×#125—24no—
46C2×C22⋊Q8—24no—
47C2×#117C2314yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#428—28no—
59#428—28no—
60#428—44no—
61#428—44no—
62C2×#107—24no—
63C2×#107—24no—
64C2×#106—24no—
65C2×#106—24no—
66C4×M16—24no—
67C4×M16Q818yescomplemented
68C2×#97—22no—
69C2×C4×C8—22no—
70#398—28no—
71#395—28no—
72#391—24no—
73#389—24no—
74#389—24no—
75#389—24no—
76#389—24no—
77#384—22no—
78#384—24no—
79#384D412yes—
80#384D412yes—
81#382—24no—
82#382—22no—
83#382—22no—
84#382D414yes—
85#380—24no—
86#380—22no—
87#380—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4Q814yescomplemented
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95D412yes—
101#343D418yes—
102#343Q818yescomplemented
103#339—28no—
104#339—28no—
105C22×Q8—24no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16C2×D418yes—
119C2×M16C2×Q818yes—
120C2×M16C2×Q818yes—
121C22×C8—24no—
122C4⋊Q8—48no—
123C4⋊Q8—48no—
124C4⋊Q8—84no—
125#125—48no—
126#125—48no—
127C22⋊Q8—48no—
128C22⋊Q8—48no—
129#117—24no—
130#117—24no—
131#117—24no—
132#117—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4C2×D414yes—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×Q814yes—
161C2×C22⋊C4—24no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—24no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—82no—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×Q812yes—
175C2×C42C2×Q812yes—
176C2×C42C2×Q814yes—
177C2×C42C2×Q812yes—
178C2×C42#4112yes—
179#107—48no—
180#107—48no—
181#106—48no—
182#106—48no—
183#97—28no—
184C4×C8—28no—
185#95—28no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—44no—
191#95—41no—
192#95—42no—
193#95—41no—
194#95—44no—
195#95—82no—
196C24C22×D411yescharacteristic
197C2×Q8—48no—
198C2×Q8—84no—
199C2×Q8—84no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4#12011yescharacteristic
239C22×C4#12012yes—
240C22×C4#12011yescharacteristic
241C22×C4C22⋊Q812yes—
242C22×C4C22⋊Q812yes—
243C22×C4#12714yes—
244C22×C4C4⋊Q814yes—
245C22×C4C4⋊Q812yes—
246C22×C4C4⋊Q812yes—
247C22×C4C22×D412yes—
248C22×C4C22×D412yes—
249C22×C4C22×Q812yes—
250C22×C4C22×Q812yes—
251C22×C4C22×Q811yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4H(5,𝔽2)12yes—
254C22×C42−1+412yes—
255M16—28no—
256M16—28no—
257M16—28no—
258M16C4⋊Q8116yescomplemented
259C2×C8—28no—
260C2×C8—28no—
261C2×C8—28no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—44no—
269C4⋊C4—48no—
270C4⋊C4—48no—
271C4⋊C4—416no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—84no—
278C4⋊C4—84no—
279C4⋊C4—88no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C42—24no—
283C42—24no—
284C42—44no—
285C42—44no—
286C42—44no—
287C42—42no—
288C42—42no—
289C42—82no—
290C42C22⋊Q814yes—
291C42C22⋊Q814yes—
292C42C4⋊Q814yes—
293C42C4⋊Q814yes—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23C2×#12011yescharacteristic
299C23C2×C22⋊Q811yescharacteristic
300C23C2×#12711yescharacteristic
301C23C2×C4⋊Q811yescharacteristic
302C23C2×C4⋊Q811yescharacteristic
303C23#53511yescharacteristic
304C23#53611yescharacteristic
305Q8—168no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4#39514yes—
357C2×C4#39512yes—
358C2×C4#39512yes—
359C2×C4C2×C22⋊Q812yes—
360C2×C4C2×C4⋊Q814yes—
361C2×C4C2×C4⋊Q812yes—
362C2×C4#54414yes—
363C2×C4D4214yes—
364C2×C4#54712yes—
365C2×C4D4×Q814yes—
366C2×C4D4×Q812yes—
367C2×C4#55412yes—
368C2×C4Q8212yes—
369C8—216no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22C2×#39511yescharacteristic
383C22#243311yescharacteristic
384C22#243411yescharacteristic
385C22#254211yescharacteristic
386C22#256711yescharacteristic
387C22#266011yescharacteristic
388C22#286811yescharacteristic
389C22#301811yescharacteristic
390C22#301811yescharacteristic
391C22#301911yescharacteristic
392C22#302011yescharacteristic
393C4—24no—
394C4—22no—
395C4—22no—
396C4—44no—
397C4—44no—
398C4—44no—
399C4—82no—
400C4—84no—
401C4—84no—
402C4—161no—
403C4—161no—
404C4#244014yes—
405C4#255114yes—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2#1514311yescharacteristic
411C2#2356011yescharacteristic
412C2#2356111yescharacteristic
413C2#2574211yescharacteristic
414C2#2574311yescharacteristic
415C2#2582711yescharacteristic
416C2#2582811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
67. C4×M16Q8semidirect
93. C4×C4⋊C4Q8semidirect
102. #343Q8semidirect
258. M16C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#20612
C2×#20602
#208012
#139092
#113422
#108512
C4×#3432