Finite Group Databasebeta

← #413850

Subgroups — #413850

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#544C211yesmaximal, characteristic
2C2×#2061C212yesmaximal
3#21171C212yesmaximal
4#13911C211yesmaximal, characteristic
5#11217C214yesmaximal
6#11211C212yesmaximal
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×#384C2211yescharacteristic
21C2×#382C2212yes—
22C4×#117C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#2061—28no—
26#1948—22no—
27#1948—22no—
28#1947—22no—
29#1947—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32#1736C2214yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35#1442—22no—
36C4⋊#95—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—21no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C42D412yes—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57#428—28no—
58#428—28no—
59C2×#107—24no—
60C2×#106—24no—
61#423—24no—
62#423Q818yescomplemented
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65#391—24no—
66#391—22no—
67#389—24no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#387—24no—
74#387—24no—
75#387—24no—
76#385—24no—
77#384—22no—
78#384—22no—
79#384—22no—
80#382—22no—
81#382—24no—
82#382—24no—
83#380—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D412yes—
91C4×C4⋊C4D412yes—
92C4×C4⋊C4D412yes—
93C4×C4⋊C4Q814yescomplemented
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96#376—22no—
97C2×#95—21no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95D412yes—
101C43—22no—
102#343—24no—
103#343—42no—
104#340—24no—
105#340—24no—
106#339—28no—
107#339—44no—
108C22×Q8—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16C2×Q818yes—
122C22×C8—24no—
123C4⋊Q8—28no—
124C4⋊Q8—28no—
125#125—28no—
126#125—28no—
127C22⋊Q8—28no—
128C22⋊Q8—28no—
129C4×Q8—84no—
130#117—22no—
131#117—22no—
132#117—22no—
133#117—22no—
134#117—22no—
135#117—22no—
136#117C2×D414yes—
137#117#4114yes—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4#4112yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—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—44no—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×Q814yes—
187C2×C42C2×Q814yes—
188C2×C42C2×Q812yes—
189C2×C42#4112yes—
190#107—48no—
191#106—48no—
192C4⋊C8—28no—
193C4⋊C8—28no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—24no—
203#95—48no—
204#95—84no—
205C24C22×D411yescharacteristic
206C2×Q8—28no—
207C2×Q8—82no—
208C2×Q8—82no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4#12011yescharacteristic
239C22×C4#12012yes—
240C22×C4#12011yescharacteristic
241C22×C4#12114yes—
242C22×C4C22⋊Q812yes—
243C22×C4C22⋊Q812yes—
244C22×C4C4⋊Q812yes—
245C22×C4C4⋊Q812yes—
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D412yes—
248C22×C4C22×Q812yes—
249C22×C4C22×Q811yescharacteristic
250C22×C4C2×#4112yes—
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4H(5,𝔽2)11yescharacteristic
254C22×C42−1+411yescharacteristic
255C22×C42−1+412yes—
256M16—28no—
257M16—28no—
258M16—44no—
259M16—44no—
260C2×C8—28no—
261C2×C8—44no—
262C2×C8—44no—
263C4⋊C4—28no—
264C4⋊C4—216no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—48no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—82no—
279C4⋊C4—84no—
280C4⋊C4—84no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—84no—
284C4⋊C4—82no—
285C4⋊C4#12114yes—
286C4⋊C4#12114yes—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C42—24no—
292C42—24no—
293C42—24no—
294C42—24no—
295C42—42no—
296C42—42no—
297C42—42no—
298C42—42no—
299C42—44no—
300C42—44no—
301C42—44no—
302C42—82no—
303C42—82no—
304C42—82no—
305C42C22⋊Q814yes—
306C42C22⋊Q814yes—
307C42C4⋊Q814yes—
308C42C4⋊Q814yes—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23C2×#12011yescharacteristic
314C23C2×#12111yescharacteristic
315C23C2×C22⋊Q811yescharacteristic
316C23#52511yescharacteristic
317C23C2×C4⋊Q811yescharacteristic
318C23#53611yescharacteristic
319C23#53711yescharacteristic
320Q8—88no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
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—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4#39512yes—
364C2×C4#39514yes—
365C2×C4#39512yes—
366C2×C4#45412yes—
367C2×C4#45512yes—
368C2×C4C2×#12112yes—
369C2×C4C2×C22⋊Q812yes—
370C2×C4#52512yes—
371C2×C4C2×C4⋊Q812yes—
372C2×C4#53612yes—
373C2×C4#53712yes—
374C2×C4D4×Q812yes—
375C2×C4D4×Q811yescharacteristic
376C2×C4#55011yescharacteristic
377C2×C4#55411yescharacteristic
378C2×C4#55412yes—
379C2×C4#55611yescharacteristic
380C2×C4#55612yes—
381C2×C4#55712yes—
382C2×C4#55712yes—
383C2×C4#55712yes—
384C8—48no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22C2×#39511yescharacteristic
398C22#253811yescharacteristic
399C22#254611yescharacteristic
400C22#258111yescharacteristic
401C22#258411yescharacteristic
402C22#261611yescharacteristic
403C22#284711yescharacteristic
404C22C2×#45511yescharacteristic
405C22C2×#45411yescharacteristic
406C22#302111yescharacteristic
407C22#302211yescharacteristic
408C4—22no—
409C4—24no—
410C4—44no—
411C4—44no—
412C4—44no—
413C4—84no—
414C4—82no—
415C4—82no—
416C4—82no—
417C4—162no—
418C4#206114yescomplemented
419C4#253812yes—
420C4#254612yes—
421C4#258112yes—
422C4#258412yes—
423C2—21no—
424C2—21no—
425C2—21no—
426C2—21no—
427C2#1515811yescharacteristic
428C2C2×#206111yescharacteristic
429C2#2356211yescharacteristic
430C2#2970411yescharacteristic
431C2#2970511yescharacteristic
432C2#3292111yescharacteristic
433C2#3292211yescharacteristic
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
62. #423Q8semidirect
93. C4×C4⋊C4Q8semidirect
418. C4#2061semidirect

Maximal subgroups

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

subgroupindex
C4×#5442
C2×#20612
#211712
#139112
#112172
#112112
#108522
#108472