Finite Group Databasebeta

← #204636

Subgroups — #204636

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35180C211yesmaximal, characteristic
2#20536C211yesmaximal, characteristic, complemented
3C2×#1518C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic, complemented
6C2×#1502C211yesmaximal, characteristic
7#13954C211yesmaximal, characteristic, complemented
8#9682C218yesmaximal, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2850C2214yes—
14#2639—22no—
15#2592—22no—
16#2568C412yescomplemented
17#2563C412yescomplemented
18#2562—22no—
19#2548C412yescomplemented
20#2547C412yescomplemented
21#2311—22no—
22C2×#389—22no—
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1770—22no—
27#1770—22no—
28#1749C2214yes—
29#1518—24no—
30#1517—24no—
31#1503—24no—
32#1502—24no—
33C2×#328C2214yes—
34C2×#328C2214yes—
35C2×#327C2214yes—
36C2×#327C2214yes—
37#1444—22no—
38C4⋊#95C2214yes—
39C22×#41—21no—
40#533—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44#532—22no—
45#532—22no—
46#532—22no—
47#532—22no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#125—21no—
53C2×#124—21no—
54C2×#124C2×C414yes—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2312yes—
57C2×#121—21no—
58C2×#121C2312yes—
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C412yes—
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4D412yes—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71#431—22no—
72#431—22no—
73#431D414yes—
74#431D414yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79C2×#97—24no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#400—24no—
83#397—24no—
84#394—24no—
85#393—24no—
86#391—24no—
87#391C2×C414yes—
88#390—24no—
89#390C2×C414yes—
90#389—48no—
91#388—28no—
92#386—28no—
93#382—22no—
94#382—22no—
95#382C2×C414yes—
96#380—24no—
97#380—24no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C2×#95—24no—
105C2×#95—21no—
106C2×#95D412yes—
107C2×#95C2312yes—
108C2×#95C2312yes—
109#336—24no—
110#328—28no—
111#328—28no—
112#327—28no—
113#327—28no—
114C2×#41—42no—
115C2×#41—42no—
116C2×#41—44no—
117C2×#41—44no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8C22×C412yes—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4C22×C412yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4C2×D414yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140#127—22no—
141#127—22no—
142#125—42no—
143#125—42no—
144#124—42no—
145#124—42no—
146#124—84no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153#121—22no—
154#121—22no—
155#121—22no—
156#121—22no—
157#121—42no—
158#121—42no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—42no—
166C4×D4—42no—
167#117—42no—
168#117—42no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—84no—
184C2×C4⋊C4C22⋊C414yes—
185C2×C4⋊C4C22⋊C414yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C42—22no—
194C2×C42—42no—
195C2×C42—44no—
196C2×C42—82no—
197C2×C42C22×C414yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D414yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C4⋊C8—24no—
208C4⋊C8—24no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8C2×D414yes—
212C22⋊C8C2×D414yes—
213#97—84no—
214C4×C8—24no—
215C4×C8—24no—
216#95—28no—
217#95—28no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—48no—
223#95—42no—
224#95C22⋊C414yes—
225#95C22⋊C414yes—
226C24—22no—
227C24C22×D411yescharacteristic
228#41—88no—
229#41—84no—
230#41—84no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—44no—
238C2×Q8—82no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—44no—
250C2×D4—82no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—48no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4C2×C22⋊C418yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4#12412yes—
281C22×C4#12412yes—
282C22×C4#12712yes—
283C22×C4#12712yes—
284C22×C4AGL(1,ℤ/8)14yes—
285C22×C4#13714yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C4⋊C4—28no—
299C4⋊C4—28no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—48no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—816no—
313C4⋊C4—84no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—88no—
320C22⋊C4—88no—
321C42—44no—
322C42—44no—
323C42—44no—
324C42—84no—
325C42C4×D412yes—
326C42C4×D412yes—
327C42C4×D412yes—
328C42C4×D412yes—
329C42#12112yes—
330C42#12112yes—
331C42#12112yes—
332C42#12112yes—
333C23—21no—
334C23—21no—
335C23—22no—
336C23—42no—
337C23—42no—
338C23—44no—
339C23—44no—
340C23—42no—
341C23C22×C22⋊C411yescharacteristic
342C23C2×#12411yescharacteristic
343C23C2×#12711yescharacteristic
344C23C2×AGL(1,ℤ/8)12yes—
345C23C2×#13712yes—
346Q8—82no—
347Q8—82no—
348Q8—84no—
349Q8—84no—
350D4—82no—
351D4—82no—
352D4—84no—
353D4—84no—
354D4—82no—
355D4—82no—
356D4—84no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—28no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—88no—
389C2×C4—84no—
390C2×C4—82no—
391C2×C4—84no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#39012yes—
395C2×C4#39012yes—
396C2×C4#39012yes—
397C2×C4#39012yes—
398C2×C4#41714yes—
399C2×C4#41714yes—
400C2×C4#48918yes—
401C2×C4#49614yes—
402C2×C4#49714yes—
403C2×C4C2×C4×D411yescharacteristic
404C2×C4C2×C4×D411yescharacteristic
405C2×C4C2×#12111yescharacteristic
406C2×C4C2×#12111yescharacteristic
407C8—44no—
408C8—44no—
409C22—21no—
410C22—21no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—21no—
416C22—21no—
417C22—82no—
418C22—82no—
419C22—84no—
420C22—84no—
421C22—82no—
422C22C2×#39011yescharacteristic
423C22#288311yescharacteristic
424C22#288411yescharacteristic
425C22C2×#41712yes—
426C22#290112yes—
427C22C2×#48912yes—
428C22C2×#49611yescharacteristic
429C22C2×#49711yescharacteristic
430C4—22no—
431C4—22no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—48no—
437C4—84no—
438C4—82no—
439C4—82no—
440C4—82no—
441C4—162no—
442C4#195914yes—
443C4#195914yes—
444C2—21no—
445C2—22no—
446C2—21no—
447C2—162no—
448C2#1762812yes—
449C2#1763012yes—
450C2C2×#195911yescharacteristic
451C2#2121411yescharacteristic
452C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20536C2semidirect
3. C2×#1518C2semidirect
5. C2×#1503C2semidirect
7. #13954C2semidirect
8. #9682C2semidirect
16. #2568C4semidirect
17. #2563C4semidirect
19. #2548C4semidirect
20. #2547C4semidirect

Maximal subgroups

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

subgroupindex
#351802
#205362
C2×#15182
C2×#15172
C2×#15032
C2×#15022
#139542
#96822