Finite Group Databasebeta

← #240312

Subgroups — #240312

Order 512.

1016 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2#21324C211yesmaximal, characteristic
3#21303C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1565C211yesmaximal, characteristic
6C2×#1548C211yesmaximal, characteristic
7#10483C214yesmaximal, complemented
8#10482C214yesmaximal, complemented
9#8045C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#477—24no—
13C2×#477—24no—
14C2×#477—22no—
15C2×#477C2214yes—
16C2×#431C2211yescharacteristic
17C22×#107—21no—
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#103—21no—
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23#1993—22no—
24#1990C2214yes—
25#1985—22no—
26#1984C2214yes—
27#1824—21no—
28#1803C2212yes—
29#1803C2212yes—
30#1787—21no—
31C4×#103—22no—
32#1736—21no—
33#1736—21no—
34#1728—21no—
35#1728—21no—
36#1565—24no—
37C2×#335C2212yes—
38C2×#334C2212yes—
39#1548—24no—
40C2×#333C2212yes—
41C2×#333C2212yes—
42#1296C2214yes—
43C23×Q8—21no—
44C23×Q8D411yescharacteristic
45C22×M16—21no—
46C22×M16D411yescharacteristic
47C22×M16D411yescharacteristic
48C23×C8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8D414yes—
56C2×C4⋊Q8C2312yes—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8D412yes—
64C2×C4×Q8C2312yes—
65C2×#117—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#477—216no—
73#477—28no—
74#477—28no—
75#431—22no—
76#431—22no—
77C2×#107—28no—
78C2×#107—24no—
79#423—22no—
80#423—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8D414yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#103—28no—
90C2×#103—24no—
91C2×#103—22no—
92C2×#103—22no—
93C2×C22⋊C8—21no—
94C4×M16—22no—
95C4×M16—22no—
96C2×#97D412yes—
97C2×C4×C8—21no—
98C2×C4×C8C2312yes—
99#384C2312yes—
100#382—21no—
101#380—21no—
102#380—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105#340—24no—
106#340—24no—
107#340—22no—
108#340—22no—
109#336—24no—
110#336—24no—
111#335—24no—
112#334—24no—
113#333—24no—
114#333—24no—
115C22×Q8—21no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—24no—
120C22×Q8—22no—
121C22×Q8—42no—
122C22×Q8—41no—
123C22×Q8—42no—
124C22×Q8—84no—
125C22×Q8C2×D412yes—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4#4111yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—42no—
138C2×M16—42no—
139C2×M16—42no—
140C2×M16Q1614yes—
141C22×C8—24no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—44no—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8C2×D414yes—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—42no—
169#117—42no—
170#117—42no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—41no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—41no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42#4111yescharacteristic
203C2×C42C2411yescharacteristic
204#107—216no—
205C4⋊C8—24no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—22no—
213C4⋊C8—42no—
214C4⋊C8C2×D414yes—
215#103—216no—
216#103—44no—
217#103—44no—
218C22⋊C8—42no—
219#97—24no—
220C4×C8—22no—
221C4×C8—22no—
222C4×C8—42no—
223#95—41no—
224#95—41no—
225C24#12111yescharacteristic
226C2×Q8—22no—
227C2×Q8—28no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—24no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—44no—
234C2×Q8—48no—
235C2×Q8—44no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—44no—
239C2×Q8—84no—
240C2×Q8—84no—
241C2×Q8—88no—
242C2×Q8—88no—
243C2×Q8#13714yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—82no—
256C22×C4—84no—
257C22×C4#12012yes—
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12112yes—
261C22×C4#12112yes—
262C22×C4#12111yescharacteristic
263C22×C4#12312yes—
264C22×C4#12312yes—
265C22×C4#12412yes—
266C22×C4C2×Q1612yes—
267C22×C4#13512yes—
268C22×C4#13712yes—
269C22×C4#13712yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C2×#4111yescharacteristic
273M16—28no—
274M16—44no—
275M16—84no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—48no—
283C2×C8—44no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8C2×Q1614yes—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—82no—
309C22⋊C4—82no—
310C42—42no—
311C42—42no—
312C42—44no—
313C42—82no—
314C42#12012yes—
315C42#12112yes—
316C42#12112yes—
317C42#12312yes—
318C42#12312yes—
319C42#12412yes—
320C42C22×D411yescharacteristic
321C42C22×D411yescharacteristic
322C42C2×#4111yescharacteristic
323C42C2×#4111yescharacteristic
324C23—22no—
325C23—22no—
326C23#39412yes—
327C23#46212yes—
328C23#46412yes—
329C23C2×#12111yescharacteristic
330Q8—24no—
331Q8—24no—
332Q8—48no—
333Q8—42no—
334Q8—42no—
335Q8—84no—
336Q8—88no—
337Q8—84no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—84no—
354C2×C4—88no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4#39412yes—
360C2×C4#39412yes—
361C2×C4#39412yes—
362C2×C4#46212yes—
363C2×C4#46214yes—
364C2×C4#46412yes—
365C2×C4#46414yes—
366C2×C4#46714yes—
367C2×C4#47014yes—
368C2×C4C2×#12011yescharacteristic
369C2×C4C2×#12111yescharacteristic
370C2×C4C2×#12111yescharacteristic
371C2×C4C2×#12311yescharacteristic
372C2×C4C2×#12311yescharacteristic
373C2×C4C2×#12411yescharacteristic
374C2×C4C22×Q1611yescharacteristic
375C2×C4C2×#13511yescharacteristic
376C2×C4C2×#13712yes—
377C2×C4C2×#13711yescharacteristic
378C2×C4C2×#13711yescharacteristic
379C2×C4#57512yes—
380C8—24no—
381C8—44no—
382C8—42no—
383C8—81no—
384C8—81no—
385C8—82no—
386C22—22no—
387C22—24no—
388C22—24no—
389C22—22no—
390C22#205614yes—
391C22C2×#39411yescharacteristic
392C22C2×#46211yescharacteristic
393C22C2×#46411yescharacteristic
394C22#305612yes—
395C22#306812yes—
396C4—22no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—44no—
401C4—82no—
402C4—84no—
403C4—162no—
404C4#205614yes—
405C4#300411yescharacteristic
406C4#300511yescharacteristic
407C4C2×#46211yescharacteristic
408C4C2×#46411yescharacteristic
409C4C2×#46711yescharacteristic
410C4C2×#47011yescharacteristic
411C4#309512yes—
412C2—24no—
413C2#1917312yes—
414C2#1917612yes—
415C2#2348111yescharacteristic
416C2#2353111yescharacteristic
417C2C2×#205611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
7. #10483C2semidirect
8. #10482C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
#213242
#213032
#207342
C2×#15652
C2×#15482
#104832
#104822
#80452