Finite Group Databasebeta

← #204632

Subgroups — #204632

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35019C211yesmaximal, characteristic
2#20526C211yesmaximal, characteristic, complemented
3C2×#1520C211yesmaximal, characteristic, complemented
4C2×#1519C211yesmaximal, characteristic
5C22×#328C211yesmaximal, characteristic, complemented
6C22×#327C211yesmaximal, characteristic
7#13965C211yesmaximal, characteristic, complemented
8#9682C218yesmaximal, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2850C2214yes—
14#2639—22no—
15#2591—22no—
16#2561—22no—
17#2504C412yescomplemented
18#2503C412yescomplemented
19#2436C412yescomplemented
20#2435C412yescomplemented
21#2308—22no—
22C2×#385—22no—
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C22×#95C2211yescharacteristic
26#1752—22no—
27#1752—22no—
28#1749C2214yescomplemented
29#1520—24no—
30#1519—24no—
31C2×#328—24no—
32C2×#328C2214yescomplemented
33C2×#328C2214yescomplemented
34C2×#327—24no—
35C2×#327C2214yes—
36C2×#327C2214yes—
37#1444—22no—
38C4⋊#95C2214yescomplemented
39C23×Q8C2×C411yescharacteristic
40C23×D4—21no—
41C23×D4C2×C411yescharacteristic
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8—21no—
45C23×C8—21no—
46C2×#124C2×C414yescomplemented
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8C2×C412yes—
54C2×C22⋊Q8C2312yes—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121C2×C412yes—
62C2×#121C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4D412yes—
72C22×C42C2312yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8D414yes—
78C2×C22⋊C8D414yes—
79C2×C22⋊C8C2312yes—
80C2×C22⋊C8C2312yes—
81C2×#97—24no—
82#400—24no—
83#397—24no—
84#394—24no—
85#393—24no—
86#391C2×C414yescomplemented
87#390C2×C414yescomplemented
88#388—28no—
89#386—28no—
90#386—24no—
91#386—24no—
92#385—48no—
93#382C2×C414yescomplemented
94#381—24no—
95#380—22no—
96#380—22no—
97#379—24no—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4C2×C412yes—
100C4×C22⋊C4D412yes—
101C4×C22⋊C4D412yes—
102C2×#95—22no—
103C2×#95—22no—
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—
114C25—21no—
115C25C2×D411yescharacteristic
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—82no—
120C22×Q8C22×C412yes—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—42no—
127C22×D4—42no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—82no—
131C22×D4C22×C412yes—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—44no—
138C23×C4—42no—
139C23×C4—44no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D414yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—24no—
156C22×C8—24no—
157C22×C8—82no—
158C22×C8—82no—
159#124—84no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170#121—24no—
171#121—24no—
172#121—22no—
173#121—22no—
174#121—22no—
175#121—22no—
176#121—42no—
177#121—42no—
178#121—42no—
179#121—42no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—84no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4C22⋊C414yes—
212C2×C22⋊C4C22⋊C414yes—
213C2×C42—82no—
214C2×C42C22×C414yes—
215C2×C42C2×D414yes—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8C2×D414yes—
221C22⋊C8C2×D414yes—
222#97—84no—
223#95—28no—
224#95—28no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—48no—
230#95—42no—
231#95C22⋊C414yes—
232#95C22⋊C414yes—
233C24—22no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24—44no—
238C24—82no—
239C24—82no—
240C24C2×C22⋊C411yescharacteristic
241C24C2×C22⋊C411yescharacteristic
242C24#12412yes—
243C24#12712yes—
244C24AGL(1,ℤ/8)14yes—
245C24#13714yes—
246C24C22×D411yescharacteristic
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—44no—
250C2×Q8—42no—
251C2×Q8—82no—
252C2×Q8—84no—
253C2×Q8—84no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—44no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—42no—
264C2×D4—82no—
265C2×D4—84no—
266C2×D4—84no—
267C2×D4—88no—
268C2×D4—84no—
269C2×D4—84no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—48no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—816no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4C2×C22⋊C418yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4C4×D412yes—
313C22×C4C4×D412yes—
314C22×C4#12112yes—
315C22×C4#12112yes—
316C22×C4#12112yes—
317C22×C4#12112yes—
318C22×C4#12412yes—
319C22×C4#12712yes—
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—44no—
329C2×C8—44no—
330C2×C8—84no—
331C2×C8—84no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—816no—
342C4⋊C4—84no—
343C22⋊C4—28no—
344C22⋊C4—28no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—88no—
352C22⋊C4—88no—
353C22⋊C4—84no—
354C42—44no—
355C42—44no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—48no—
366C23—42no—
367C23—48no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—42no—
372C23—84no—
373C23—84no—
374C23—84no—
375C23—168no—
376C23#39012yes—
377C23#39012yes—
378C23#41714yes—
379C23#41714yes—
380C23#48918yes—
381C23#49614yes—
382C23#49714yes—
383C23C22×C22⋊C411yescharacteristic
384C23C2×#12411yescharacteristic
385C23C2×#12711yescharacteristic
386C23C2×AGL(1,ℤ/8)12yes—
387C23C2×#13712yes—
388Q8—82no—
389Q8—82no—
390Q8—84no—
391D4—82no—
392D4—82no—
393D4—84no—
394D4—84no—
395D4—82no—
396D4—82no—
397D4—88no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—84no—
430C2×C4—82no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—816no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—84no—
440C2×C4—82no—
441C2×C4#39012yes—
442C2×C4#39012yes—
443C2×C4C2×C4×D411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×#12111yescharacteristic
446C2×C4C2×#12111yescharacteristic
447C8—44no—
448C8—44no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—21no—
453C22—24no—
454C22—24no—
455C22—28no—
456C22—28no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—21no—
461C22—44no—
462C22—48no—
463C22—44no—
464C22—82no—
465C22—82no—
466C22—88no—
467C22—82no—
468C22—168no—
469C22#195914yes—
470C22#195914yes—
471C22C2×#39011yescharacteristic
472C22#288312yes—
473C22C2×#41712yes—
474C22#290112yes—
475C22C2×#48912yes—
476C22C2×#49611yescharacteristic
477C22C2×#49711yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—22no—
483C4—22no—
484C4—48no—
485C4—84no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4—162no—
490C2—21no—
491C2—24no—
492C2—22no—
493C2—21no—
494C2—24no—
495C2—162no—
496C2#1762214yes—
497C2C2×#195911yescharacteristic
498C2#2121311yescharacteristic
499C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20526C2semidirect
3. C2×#1520C2semidirect
5. C22×#328C2semidirect
7. #13965C2semidirect
8. #9682C2semidirect
17. #2504C4semidirect
18. #2503C4semidirect
19. #2436C4semidirect
20. #2435C4semidirect
28. #1749C22semidirect
32. C2×#328C22semidirect
33. C2×#328C22semidirect
38. C4⋊#95C22semidirect
46. C2×#124C2×C4semidirect
86. #391C2×C4semidirect
87. #390C2×C4semidirect
93. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350192
#205262
C2×#15202
C2×#15192
C22×#3282
C22×#3272
#139652
#96822