Finite Group Databasebeta

← #240532

Subgroups — #240532

Order 512.

968 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
1C2×#3230C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4#21300C211yesmaximal, characteristic
5C2×#1564C211yesmaximal, characteristic
6C2×#1548C211yesmaximal, characteristic
7#10483C214yesmaximal, complemented
8#10482C214yesmaximal, complemented
9#8043C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3230—24no—
13#3230—24no—
14C2×#500—21no—
15C2×#487—21no—
16C2×#484—22no—
17C2×#477C2214yes—
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#1548—24no—
43C2×#333C2214yes—
44#1296C2214yes—
45C23×Q8—21no—
46C22×M16—21no—
47C23×C8D411yescharacteristic
48C23×C8D411yescharacteristic
49#533—22no—
50#533—22no—
51#533—22no—
52#533—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D414yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8C2312yes—
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8D412yes—
66C2×C4×Q8—21no—
67C2×C4×Q8C2312yes—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#500—28no—
76#487—28no—
77#484—216no—
78#477—28no—
79#477—28no—
80#431—22no—
81#431—22no—
82#431—22no—
83#431D414yes—
84C2×#107—24no—
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#333—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×M16—42no—
133C2×M16—42no—
134C2×M16—42no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8Q1614yes—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C4⋊Q8—24no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—24no—
152C4⋊Q8—22no—
153C4⋊Q8—24no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8C2×D414yes—
162#125—42no—
163#125—42no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—42no—
173C4×Q8—42no—
174#117—42no—
175#117—42no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—22no—
201C2×C42—22no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42#4111yescharacteristic
209C2×C42C2411yescharacteristic
210#107—216no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—42no—
217C4⋊C8C2×D414yes—
218#103—216no—
219#103—84no—
220C22⋊C8—24no—
221C22⋊C8—42no—
222C4×C8—24no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—42no—
230C24#12111yescharacteristic
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—44no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—88no—
242C2×Q8—88no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×Q8—82no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4#12012yes—
262C22×C4#12111yescharacteristic
263C22×C4#12112yes—
264C22×C4#12112yes—
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12312yes—
268C22×C4#12312yes—
269C22×C4#12412yes—
270C22×C4C2×Q1612yes—
271C22×C4#13512yes—
272C22×C4#13712yes—
273C22×C4#13712yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C2×#4111yescharacteristic
277M16—84no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—28no—
283C2×C8—24no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—44no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8C2×Q1612yes—
300C2×C8C2×Q1612yes—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—44no—
323C42—42no—
324C42—42no—
325C42#12012yes—
326C42#12112yes—
327C42#12112yes—
328C42#12312yes—
329C42#12312yes—
330C42#12412yes—
331C42C22×D411yescharacteristic
332C42C22×D411yescharacteristic
333C42C2×#4111yescharacteristic
334C42C2×#4111yescharacteristic
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23#39412yes—
340C23#46212yes—
341C23#46412yes—
342C23C2×#12111yescharacteristic
343Q8—44no—
344Q8—44no—
345Q8—84no—
346Q8—84no—
347Q8—84no—
348Q8—88no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—88no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—164no—
371C2×C4#39412yes—
372C2×C4#39412yes—
373C2×C4#39412yes—
374C2×C4#46212yes—
375C2×C4#46214yes—
376C2×C4#46412yes—
377C2×C4#46414yes—
378C2×C4#46714yes—
379C2×C4#47014yes—
380C2×C4#47112yes—
381C2×C4#47212yes—
382C2×C4C2×#12011yescharacteristic
383C2×C4C2×#12111yescharacteristic
384C2×C4C2×#12111yescharacteristic
385C2×C4C2×#12311yescharacteristic
386C2×C4C2×#12311yescharacteristic
387C2×C4C2×#12411yescharacteristic
388C2×C4C22×Q1611yescharacteristic
389C2×C4C2×#13511yescharacteristic
390C2×C4C2×#13711yescharacteristic
391C2×C4C2×#13711yescharacteristic
392C2×C4#57712yes—
393C2×C4#57812yes—
394C8—22no—
395C8—22no—
396C8—44no—
397C8—41no—
398C8—41no—
399C8—82no—
400C8—82no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22#205614yes—
408C22C2×#39411yescharacteristic
409C22C2×#46211yescharacteristic
410C22C2×#46411yescharacteristic
411C22#305711yescharacteristic
412C22#305911yescharacteristic
413C22C2×#47111yescharacteristic
414C22C2×#47211yescharacteristic
415C4—22no—
416C4—21no—
417C4—21no—
418C4—82no—
419C4—82no—
420C4—84no—
421C4—162no—
422C4#205114yes—
423C4#205614yes—
424C4#301312yes—
425C4C2×#46211yescharacteristic
426C4C2×#46411yescharacteristic
427C4C2×#46711yescharacteristic
428C4C2×#47011yescharacteristic
429C4#309712yes—
430C2—24no—
431C2#1920711yescharacteristic
432C2#1920911yescharacteristic
433C2#1921211yescharacteristic
434C2#1921511yescharacteristic
435C2#2348711yescharacteristic
436C2C2×#205111yescharacteristic
437C2C2×#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
#218602
C2×#19902
#213002
C2×#15642
C2×#15482
#104832
#104822
#80432