Finite Group Databasebeta

← #240475

Subgroups — #240475

Order 512.

946 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4#21302C211yesmaximal, characteristic
5C2×#1565C211yesmaximal, characteristic
6C22×#333C211yesmaximal, characteristic
7#10483C214yesmaximal, complemented
8#10482C214yesmaximal, complemented
9#8039C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3230—24no—
13#3230—24no—
14C2×#500—21no—
15C2×#487—21no—
16C2×#484—22no—
17C2×#477C2212yes—
18C2×#477C2212yes—
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2285C2211yescharacteristic
24#1993—22no—
25#1990C2214yes—
26#1987—21no—
27#1985—22no—
28#1984C2214yes—
29#1981—21no—
30#1892—22no—
31#1803C2214yes—
32#1736—21no—
33#1736—21no—
34#1729—21no—
35#1729—21no—
36#1565—24no—
37C2×#335C2212yes—
38C2×#334C2212yes—
39C2×#333—24no—
40C2×#333C2214yes—
41#1296C2212yes—
42#1296C2212yes—
43C23×Q8—21no—
44C22×M16D411yescharacteristic
45C22×M16D411yescharacteristic
46C23×C8—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8D414yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8C2312yes—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8D412yes—
65C2×C4×Q8—21no—
66C2×C4×Q8C2312yes—
67C2×#117—21no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C42C2311yescharacteristic
72#500—28no—
73#487—28no—
74#484—216no—
75#477—24no—
76#477—24no—
77#477—24no—
78#477—24no—
79#431—22no—
80#431D414yes—
81C2×#107—22no—
82C2×#107—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—24no—
96C2×C22⋊C8—21no—
97C4×M16—22no—
98C4×M16—22no—
99C2×#97D412yes—
100C2×C4×C8—21no—
101C2×C4×C8C2312yes—
102#384C2312yes—
103#382—21no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108#340—24no—
109#340—24no—
110#340—24no—
111#336—24no—
112#336—24no—
113#335—24no—
114#334—24no—
115#333—28no—
116C22×Q8—21no—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—41no—
120C22×Q8—42no—
121C22×Q8—84no—
122C22×Q8C2×D412yes—
123C23×C4—41no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4#4111yescharacteristic
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16Q1614yes—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—82no—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C4⋊Q8—24no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—24no—
148C4⋊Q8—22no—
149C4⋊Q8—24no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8C2×D414yes—
158#125—42no—
159#125—42no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—42no—
169C4×Q8—42no—
170#117—42no—
171#117—42no—
172#117—42no—
173#117—42no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—41no—
193C2×C42—22no—
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42#4112yes—
200C2×C42#4112yes—
201C2×C42#4111yescharacteristic
202C2×C42C2411yescharacteristic
203#107—28no—
204#107—28no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—42no—
212C4⋊C8—42no—
213C4⋊C8C2×D412yes—
214C4⋊C8C2×D412yes—
215#103—28no—
216#103—28no—
217#103—84no—
218C22⋊C8—24no—
219#97—24no—
220C4×C8—24no—
221C4×C8—22no—
222C4×C8—22no—
223#95—41no—
224#95—41no—
225C24#12111yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—44no—
234C2×Q8—82no—
235C2×Q8—84no—
236C2×Q8—84no—
237C2×Q8—88no—
238C2×Q8—88no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—84no—
248C22×C4#12012yes—
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12112yes—
252C22×C4#12112yes—
253C22×C4#12111yescharacteristic
254C22×C4#12312yes—
255C22×C4#12312yes—
256C22×C4#12412yes—
257C22×C4C2×Q1612yes—
258C22×C4#13512yes—
259C22×C4#13712yes—
260C22×C4#13712yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C2×#4111yescharacteristic
264M16—28no—
265M16—44no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—44no—
280C2×C8—84no—
281C2×C8—84no—
282C2×C8C2×Q1614yes—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—82no—
302C22⋊C4—44no—
303C22⋊C4—82no—
304C42—42no—
305C42—42no—
306C42—82no—
307C42#12012yes—
308C42#12112yes—
309C42#12112yes—
310C42#12312yes—
311C42#12312yes—
312C42#12412yes—
313C42C22×D411yescharacteristic
314C42C22×D411yescharacteristic
315C42C2×#4111yescharacteristic
316C42C2×#4111yescharacteristic
317C23—22no—
318C23—22no—
319C23#39412yes—
320C23#46212yes—
321C23#46412yes—
322C23C2×#12111yescharacteristic
323Q8—44no—
324Q8—42no—
325Q8—42no—
326Q8—84no—
327Q8—84no—
328Q8—84no—
329Q8—88no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—88no—
344C2×C4—84no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4#39412yes—
349C2×C4#39412yes—
350C2×C4#39412yes—
351C2×C4#46212yes—
352C2×C4#46214yes—
353C2×C4#46412yes—
354C2×C4#46414yes—
355C2×C4#46714yes—
356C2×C4#47014yes—
357C2×C4C2×#12011yescharacteristic
358C2×C4C2×#12111yescharacteristic
359C2×C4C2×#12111yescharacteristic
360C2×C4C2×#12311yescharacteristic
361C2×C4C2×#12311yescharacteristic
362C2×C4C2×#12411yescharacteristic
363C2×C4C22×Q1611yescharacteristic
364C2×C4C2×#13511yescharacteristic
365C2×C4C2×#13711yescharacteristic
366C2×C4C2×#13711yescharacteristic
367C2×C4#57712yes—
368C2×C4#57812yes—
369C8—24no—
370C8—42no—
371C8—42no—
372C8—42no—
373C8—82no—
374C8—82no—
375C22—24no—
376C22—24no—
377C22—22no—
378C22—22no—
379C22#205614yes—
380C22C2×#39411yescharacteristic
381C22C2×#46211yescharacteristic
382C22C2×#46411yescharacteristic
383C22#306312yes—
384C22#306812yes—
385C4—22no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—82no—
390C4—82no—
391C4—84no—
392C4—162no—
393C4#205614yes—
394C4#301011yescharacteristic
395C4#301111yescharacteristic
396C4C2×#46211yescharacteristic
397C4C2×#46411yescharacteristic
398C4C2×#46711yescharacteristic
399C4C2×#47011yescharacteristic
400C4#309712yes—
401C2—24no—
402C2#1924112yes—
403C2#1924412yes—
404C2#2348411yescharacteristic
405C2#2353111yescharacteristic
406C2C2×#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
C2×#32302
#218852
#213242
#213022
C2×#15652
C22×#3332
#104832
#104822
#80392