Finite Group Databasebeta

← #413849

Subgroups — #413849

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#544C211yesmaximal, characteristic
2C2×#2060C212yesmaximal
3#21171C212yesmaximal
4#13910C211yesmaximal, characteristic
5#11217C214yesmaximal
6#11210C212yesmaximal
7#10852C212yesmaximal
8#10847C211yesmaximal, characteristic
9C2×#544C2211yescharacteristic
10C2×#428C2214yes—
11C2×#423C2212yes—
12#2331—22no—
13#2330—22no—
14#2329—22no—
15#2317C2212yes—
16#2316C2211yescharacteristic
17C4×C4⋊Q8—22no—
18C4×#125—22no—
19C4×C22⋊Q8—22no—
20C2×#389—22no—
21C2×#384C2212yes—
22C2×#384C2211yescharacteristic
23C4×#117C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#2060—28no—
27#1948—22no—
28#1948—22no—
29#1947—22no—
30#1947—22no—
31C2×#343C2212yes—
32C2×#343C2212yes—
33#1736C2214yes—
34C2×#339C2214yes—
35C2×#339C2214yes—
36#1444—22no—
37C4⋊#95C2212yes—
38C22×M16C2314yes—
39#544—24no—
40#544—22no—
41#544D414yes—
42C2×C4⋊Q8—22no—
43C2×#125—22no—
44C2×C22⋊Q8—22no—
45C2×C4×Q8—22no—
46C2×#117C2312yes—
47C2×#117C2312yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C42D412yes—
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58#428—28no—
59#428—28no—
60C2×#107—24no—
61C2×#106—24no—
62#423—24no—
63#423Q818yescomplemented
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66#391—24no—
67#391—22no—
68#389—24no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—48no—
75#387—24no—
76#387—24no—
77#387—24no—
78#385—24no—
79#384—22no—
80#384—24no—
81#384—24no—
82#384—22no—
83#384—22no—
84#382—22no—
85#380—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4D412yes—
92C4×C4⋊C4D412yes—
93C4×C4⋊C4D412yes—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4Q814yescomplemented
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98#376—22no—
99C2×#95—21no—
100C2×#95—24no—
101C2×#95D412yes—
102C43—22no—
103#343—24no—
104#343—42no—
105#340—24no—
106#340—24no—
107#339—28no—
108#339—44no—
109C22×Q8—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2411yescharacteristic
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16C2×Q818yes—
123C22×C8—24no—
124C4⋊Q8—28no—
125C4⋊Q8—28no—
126#125—28no—
127#125—28no—
128C22⋊Q8—28no—
129C22⋊Q8—28no—
130C4×Q8—84no—
131#117—22no—
132#117—22no—
133#117—22no—
134#117—22no—
135#117—22no—
136#117—22no—
137#117C2×D414yes—
138#117#4114yes—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4#4112yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—44no—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×Q814yes—
191C2×C42C2×Q814yes—
192C2×C42C2×Q812yes—
193C2×C42#4112yes—
194#107—48no—
195#106—48no—
196C4⋊C8—28no—
197C4⋊C8—28no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—24no—
207#95—48no—
208C24C22×D411yescharacteristic
209C2×Q8—28no—
210C2×Q8—82no—
211C2×Q8—82no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4#12011yescharacteristic
242C22×C4#12012yes—
243C22×C4#12011yescharacteristic
244C22×C4#12114yes—
245C22×C4C22⋊Q812yes—
246C22×C4C22⋊Q812yes—
247C22×C4C4⋊Q812yes—
248C22×C4C4⋊Q812yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D412yes—
251C22×C4C22×Q812yes—
252C22×C4C22×Q811yescharacteristic
253C22×C4C2×#4112yes—
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4H(5,𝔽2)11yescharacteristic
257C22×C42−1+411yescharacteristic
258C22×C42−1+412yes—
259M16—28no—
260M16—28no—
261M16—44no—
262M16—44no—
263C2×C8—28no—
264C2×C8—44no—
265C2×C8—44no—
266C4⋊C4—216no—
267C4⋊C4—28no—
268C4⋊C4—216no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—82no—
282C4⋊C4—84no—
283C4⋊C4—82no—
284C4⋊C4—84no—
285C4⋊C4—82no—
286C4⋊C4—84no—
287C4⋊C4—816no—
288C4⋊C4—82no—
289C4⋊C4#12114yes—
290C4⋊C4#12114yes—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C42—24no—
296C42—24no—
297C42—24no—
298C42—24no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—44no—
304C42—44no—
305C42—44no—
306C42—82no—
307C42—82no—
308C42—82no—
309C42C22⋊Q814yes—
310C42C22⋊Q814yes—
311C42C4⋊Q814yes—
312C42C4⋊Q814yes—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23C2×#12011yescharacteristic
318C23C2×#12111yescharacteristic
319C23C2×C22⋊Q811yescharacteristic
320C23#52511yescharacteristic
321C23C2×C4⋊Q811yescharacteristic
322C23#53611yescharacteristic
323C23#53711yescharacteristic
324Q8—88no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4#39512yes—
368C2×C4#39514yes—
369C2×C4#39512yes—
370C2×C4#45312yes—
371C2×C4#45612yes—
372C2×C4C2×#12112yes—
373C2×C4C2×C22⋊Q812yes—
374C2×C4#52512yes—
375C2×C4C2×C4⋊Q812yes—
376C2×C4#53612yes—
377C2×C4#53712yes—
378C2×C4D4×Q812yes—
379C2×C4D4×Q811yescharacteristic
380C2×C4#55011yescharacteristic
381C2×C4#55411yescharacteristic
382C2×C4#55412yes—
383C2×C4#55611yescharacteristic
384C2×C4#55612yes—
385C2×C4#55712yes—
386C2×C4#55712yes—
387C2×C4#55712yes—
388C8—48no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22C2×#39511yescharacteristic
402C22#253811yescharacteristic
403C22#254611yescharacteristic
404C22#258111yescharacteristic
405C22#258411yescharacteristic
406C22#261611yescharacteristic
407C22#284711yescharacteristic
408C22C2×#45311yescharacteristic
409C22C2×#45611yescharacteristic
410C22#302111yescharacteristic
411C22#302211yescharacteristic
412C4—22no—
413C4—24no—
414C4—48no—
415C4—44no—
416C4—44no—
417C4—44no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—162no—
422C4#206014yescomplemented
423C4#253812yes—
424C4#254612yes—
425C4#258112yes—
426C4#258412yes—
427C2—21no—
428C2—21no—
429C2—21no—
430C2—21no—
431C2#1515811yescharacteristic
432C2C2×#206011yescharacteristic
433C2#2356211yescharacteristic
434C2#2970311yescharacteristic
435C2#2970611yescharacteristic
436C2#3292011yescharacteristic
437C2#3292311yescharacteristic
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
63. #423Q8semidirect
95. C4×C4⋊C4Q8semidirect
422. C4#2060semidirect

Maximal subgroups

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

subgroupindex
C4×#5442
C2×#20602
#211712
#139102
#112172
#112102
#108522
#108472