Finite Group Databasebeta

← #240528

Subgroups — #240528

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4#21300C211yesmaximal, characteristic
5C2×#1564C211yesmaximal, characteristic
6C2×#1549C211yesmaximal, characteristic
7#10479C214yesmaximal, complemented
8#10478C214yesmaximal, complemented
9#8043C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209—24no—
14C2×#494—21no—
15C2×#487—21no—
16C2×#462C2214yes—
17C2×#451—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#1990—21no—
24#1990—21no—
25#1990—21no—
26#1990C2212yes—
27#1990C2212yes—
28#1984—21no—
29#1984—21no—
30#1984C2212yes—
31#1984C2212yes—
32#1956—21no—
33#1872—22no—
34#1803C2214yes—
35C2×#340—21no—
36C2×#340—21no—
37#1729—21no—
38#1729—21no—
39#1564—24no—
40C2×#335C2212yes—
41C2×#334C2212yes—
42#1549—24no—
43C2×#332C2214yes—
44#1296C2214yes—
45C23×Q8—21no—
46C22×Q16—24no—
47C22×M16—21no—
48C23×C8D411yescharacteristic
49C23×C8D411yescharacteristic
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D414yes—
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8D412yes—
67C2×C4×Q8—21no—
68C2×C4×Q8C2312yes—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#494—28no—
77#487—28no—
78#462—28no—
79#462—28no—
80#451—216no—
81#431—22no—
82#431—22no—
83#431—22no—
84#431D414yes—
85#423—22no—
86#423—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×#103—24no—
92C2×#103—24no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C4×C8—22no—
96C2×C4×C8—21no—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8D411yescharacteristic
100C2×C4×C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#384—21no—
104#384—21no—
105#384—21no—
106#384C2312yes—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—24no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#335—24no—
117#334—24no—
118#332—28no—
119C22×Q8—21no—
120C22×Q8—24no—
121C22×Q8—22no—
122C22×Q8—41no—
123C22×Q8—42no—
124C22×Q8—82no—
125C22×Q8—82no—
126C22×Q8C2×D412yes—
127C23×C4—41no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4#4111yescharacteristic
132C2×Q16—216no—
133C2×Q16—84no—
134C2×Q16—84no—
135C2×M16—42no—
136C2×M16—42no—
137C2×M16—42no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8Q1614yescomplemented
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C4⋊Q8—22no—
152C4⋊Q8—24no—
153C4⋊Q8—22no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8C2×D414yes—
165#125—42no—
166#125—42no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—42no—
176C4×Q8—42no—
177#117—42no—
178#117—42no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D412yes—
203C2×C22⋊C4—22no—
204C2×C42—22no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42#4111yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—42no—
219C4⋊C8C2×D414yes—
220#103—216no—
221#103—84no—
222C22⋊C8—24no—
223C22⋊C8—42no—
224C4×C8—24no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—42no—
232C24#12111yescharacteristic
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—88no—
246C2×Q8—82no—
247C2×Q8—82no—
248C2×Q8—88no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4#12012yes—
264C22×C4#12111yescharacteristic
265C22×C4#12112yes—
266C22×C4#12112yes—
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12312yes—
270C22×C4#12312yes—
271C22×C4#12412yes—
272C22×C4C2×Q1612yes—
273C22×C4#13512yes—
274C22×C4#13712yes—
275C22×C4#13712yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C2×#4111yescharacteristic
279Q16—816no—
280M16—84no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—28no—
286C2×C8—24no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—44no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8C2×Q1612yes—
303C2×C8C2×Q1612yes—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C22⋊C4—44no—
326C42—42no—
327C42—42no—
328C42#12012yes—
329C42#12112yes—
330C42#12112yes—
331C42#12312yes—
332C42#12312yes—
333C42#12412yes—
334C42C22×D411yescharacteristic
335C42C22×D411yescharacteristic
336C42C2×#4111yescharacteristic
337C42C2×#4111yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23#39412yes—
343C23#46212yes—
344C23#46412yes—
345C23C2×#12111yescharacteristic
346Q8—44no—
347Q8—44no—
348Q8—48no—
349Q8—88no—
350Q8—84no—
351Q8—84no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—88no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—164no—
374C2×C4#39412yes—
375C2×C4#39412yes—
376C2×C4#39412yes—
377C2×C4#46212yes—
378C2×C4#46214yes—
379C2×C4#46412yes—
380C2×C4#46414yes—
381C2×C4#46714yes—
382C2×C4#47014yes—
383C2×C4#47112yes—
384C2×C4#47312yes—
385C2×C4C2×#12011yescharacteristic
386C2×C4C2×#12111yescharacteristic
387C2×C4C2×#12111yescharacteristic
388C2×C4C2×#12311yescharacteristic
389C2×C4C2×#12311yescharacteristic
390C2×C4C2×#12411yescharacteristic
391C2×C4C22×Q1611yescharacteristic
392C2×C4C2×#13511yescharacteristic
393C2×C4C2×#13711yescharacteristic
394C2×C4C2×#13711yescharacteristic
395C2×C4#57612yes—
396C2×C4#57712yes—
397C8—22no—
398C8—22no—
399C8—44no—
400C8—41no—
401C8—41no—
402C8—82no—
403C8—82no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22#205614yes—
411C22C2×#39411yescharacteristic
412C22C2×#46211yescharacteristic
413C22C2×#46411yescharacteristic
414C22#305711yescharacteristic
415C22#305811yescharacteristic
416C22C2×#47111yescharacteristic
417C22C2×#47311yescharacteristic
418C4—22no—
419C4—21no—
420C4—21no—
421C4—82no—
422C4—84no—
423C4—82no—
424C4—162no—
425C4#205214yes—
426C4#205614yes—
427C4#301212yes—
428C4C2×#46211yescharacteristic
429C4C2×#46411yescharacteristic
430C4C2×#46711yescharacteristic
431C4C2×#47011yescharacteristic
432C4#309612yes—
433C2—24no—
434C2#1920711yescharacteristic
435C2#1920811yescharacteristic
436C2#1921311yescharacteristic
437C2#1921411yescharacteristic
438C2#2348611yescharacteristic
439C2C2×#205211yescharacteristic
440C2C2×#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. #10479C2semidirect
8. #10478C2semidirect
146. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#218602
C2×#19902
#213002
C2×#15642
C2×#15492
#104792
#104782
#80432