Finite Group Databasebeta

← #214449

Subgroups — #214449

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2#21742C211yesmaximal, characteristic
3#13960C211yesmaximal, characteristic
4#9151C212yesmaximal, complemented
5#9143C212yesmaximal, complemented
6#9115C214yesmaximal, complemented
7#8716C212yesmaximal, complemented
8#8333C212yesmaximal, complemented
9C2×#533C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C2×#407C2212yes—
12#2848C2212yes—
13#2848C2212yes—
14#2626—21no—
15#2626—21no—
16#2580—22no—
17#2566—22no—
18#2563C412yescomplemented
19#2542C412yescomplemented
20#2538C414yescomplemented
21#2317—21no—
22#2309—21no—
23C2×#386—21no—
24#2285C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1870—22no—
28#1870—22no—
29#1817C2212yes—
30#1779C2212yes—
31C2×#328C2214yes—
32C2×#328C2214yes—
33#1441—21no—
34#1439—21no—
35#1434C2212yes—
36C22⋊#95—21no—
37C22⋊#95C2212yes—
38#1294C2214yes—
39#1294C2214yes—
40C23×D4D411yescharacteristic
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45#533—42no—
46#533—42no—
47C2×C4⋊Q8C2×C411yescharacteristic, complemented
48C2×C4⋊Q8C2×C411yescharacteristic, complemented
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2×C414yes—
52C2×C22⋊Q8C2×C414yescomplemented
53C2×C22⋊Q8C2312yes—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C412yescomplemented
56C2×C4×D4—42no—
57C2×C4×D4—42no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C412yes—
60C2×C4×D4D412yes—
61C2×C4×D4D412yes—
62C2×#117—21no—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42C2311yescharacteristic
71#407—42no—
72#407—42no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8C2314yes—
76#399—22no—
77#399—22no—
78#396—24no—
79#396—24no—
80#393—22no—
81#393—22no—
82#390—22no—
83#390—22no—
84#390—24no—
85#389—24no—
86#386—44no—
87#385—24no—
88#385—24no—
89#385—22no—
90#385—22no—
91#384—22no—
92#384C2×C412yescomplemented
93#382—22no—
94#382—21no—
95#380—21no—
96#380—42no—
97#380C2×C414yes—
98#380C2×C414yescomplemented
99#380C2312yes—
100#379—22no—
101#379—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2×C412yescomplemented
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—42no—
109C2×#95C2312yes—
110C2×#95C2312yes—
111#336—22no—
112#336—22no—
113#328—44no—
114#328—44no—
115C25C2×D411yescharacteristic
116C25C2×D411yescharacteristic
117C22×Q8—21no—
118C22×Q8C22×C412yes—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—41no—
124C22×D4—41no—
125C22×D4C22⋊C412yes—
126C22×D4C22⋊C412yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—44no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C22×C412yes—
141C23×C4C22×C412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2411yescharacteristic
149C2×M16—44no—
150C2×M16—84no—
151C22×C8—24no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154#125—44no—
155C22⋊Q8—24no—
156C22⋊Q8—24no—
157C22⋊Q8—42no—
158C22⋊Q8—42no—
159C22⋊Q8—84no—
160C22⋊Q8—84no—
161C4×Q8—44no—
162C4×Q8—44no—
163C4×D4—44no—
164C4×D4—44no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169#117—42no—
170#117—42no—
171#117—42no—
172#117—42no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C414yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8—44no—
221C22⋊C8—84no—
222#95—22no—
223#95—24no—
224#95—24no—
225#95—22no—
226#95—24no—
227#95—24no—
228#95—44no—
229#95—42no—
230#95—44no—
231#95—82no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24C2×C22⋊C412yes—
247C24C2×C22⋊C412yes—
248C24#13712yes—
249C24#13712yes—
250C24C22×D411yescharacteristic
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—44no—
254C2×Q8—82no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—44no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—48no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4—88no—
312C22×C4—88no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4AGL(1,ℤ/8)12yes—
322C22×C4#13712yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4H(5,𝔽2)12yes—
326C22×C4H(5,𝔽2)12yes—
327M16—88no—
328C2×C8—48no—
329C2×C8—44no—
330C4⋊C4—24no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—82no—
341C4⋊C4—84no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—84no—
353C22⋊C4—84no—
354C22⋊C4—84no—
355C22⋊C4—84no—
356C22⋊C4—88no—
357C22⋊C4—84no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—44no—
363C42—44no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—24no—
368C23—22no—
369C23—24no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—22no—
377C23—22no—
378C23—24no—
379C23—44no—
380C23—44no—
381C23—42no—
382C23—42no—
383C23—44no—
384C23—42no—
385C23—42no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23—82no—
398C23—82no—
399C23#41914yes—
400C23C22×C22⋊C411yescharacteristic
401C23#53411yescharacteristic
402C23#53411yescharacteristic
403C23C2×AGL(1,ℤ/8)11yescharacteristic
404C23C2×#13711yescharacteristic
405C23C2×#13711yescharacteristic
406C23C2×#13711yescharacteristic
407Q8—84no—
408Q8—84no—
409D4—48no—
410D4—48no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—816no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4#41712yes—
466C2×C4#41712yes—
467C2×C4#51812yes—
468C2×C4#53412yes—
469C8—84no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—21no—
477C22—21no—
478C22—24no—
479C22—28no—
480C22—44no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—42no—
485C22—44no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—44no—
490C22—44no—
491C22—42no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22#230811yescharacteristic
497C22#288312yes—
498C22C2×#41911yescharacteristic
499C22C2×#41711yescharacteristic
500C22#290112yes—
501C22#319311yescharacteristic
502C22#319311yescharacteristic
503C22#319511yescharacteristic
504C22#319511yescharacteristic
505C4—22no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—41no—
510C4—41no—
511C4—42no—
512C4—42no—
513C4—84no—
514C4—82no—
515C4—84no—
516C4—82no—
517C4—162no—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—21no—
522C2—44no—
523C2—82no—
524C2#1762112yes—
525C2#1763012yes—
526C2#2175011yescharacteristic
527C2#2176511yescharacteristic
528C2#2177511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9151C2semidirect
5. #9143C2semidirect
6. #9115C2semidirect
7. #8716C2semidirect
8. #8333C2semidirect
18. #2563C4semidirect
19. #2542C4semidirect
20. #2538C4semidirect
47. C2×C4⋊Q8C2×C4semidirect
48. C2×C4⋊Q8C2×C4semidirect
52. C2×C22⋊Q8C2×C4semidirect
55. C2×C4×Q8C2×C4semidirect
92. #384C2×C4semidirect
98. #380C2×C4semidirect
103. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#351722
#217422
#139602
#91512
#91432
#91152
#87162
#83332