Finite Group Databasebeta

← #214463

Subgroups — #214463

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2#21790C211yesmaximal, characteristic
3#13960C211yesmaximal, characteristic
4#9151C212yesmaximal, complemented
5#9143C212yesmaximal, complemented
6#9116C214yesmaximal, complemented
7#8712C212yesmaximal, complemented
8#8329C212yesmaximal, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C2×#432C2212yes—
12#2855C2212yes—
13#2848C2212yes—
14#2627—21no—
15#2626—21no—
16#2579—22no—
17#2564—22no—
18#2487C412yescomplemented
19#2417C414yescomplemented
20#2409C412yescomplemented
21#2315—21no—
22#2308—21no—
23C2×#386—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1955—22no—
28#1955—22no—
29#1779C2212yes—
30#1779C2212yes—
31C2×#328C2214yes—
32C2×#328C2214yes—
33#1441—21no—
34#1439—21no—
35#1434C2212yes—
36C22⋊#95—21no—
37C22⋊#95C2212yes—
38#1294C2214yes—
39#1294C2214yes—
40C23×Q8C2×C411yescharacteristic, complemented
41C23×D4D411yescharacteristic
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—42no—
48C2×C22⋊Q8—42no—
49C2×C22⋊Q8C2×C414yes—
50C2×C22⋊Q8C2×C412yescomplemented
51C2×C22⋊Q8C2312yes—
52C2×C4×D4—42no—
53C2×C4×D4—42no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C412yes—
56C2×C4×D4D412yes—
57C2×C4×D4D412yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C411yescharacteristic, complemented
61C22×C4⋊C4C2×C412yescomplemented
62C22×C4⋊C4D411yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4D411yescharacteristic
69C22×C42C2311yescharacteristic
70#432—42no—
71#432—42no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8C2314yes—
77C2×#97—22no—
78#400—22no—
79#399—22no—
80#398—24no—
81#396—24no—
82#395—22no—
83#393—22no—
84#391C2×C414yescomplemented
85#389C2×C414yescomplemented
86#387—24no—
87#386—24no—
88#386—22no—
89#386—22no—
90#386—44no—
91#385—24no—
92#385—22no—
93#385—22no—
94#385—22no—
95#385—22no—
96#381—22no—
97#380—21no—
98#380—21no—
99#380—24no—
100#380—42no—
101#380C2×C414yes—
102#380C2×C412yescomplemented
103#380C2312yes—
104#379—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—42no—
111C2×#95C2312yes—
112C2×#95C2312yes—
113#328—44no—
114#328—44no—
115C25C2×D411yescharacteristic
116C25C2×D411yescharacteristic
117C22×Q8—21no—
118C22×Q8—42no—
119C22×Q8—82no—
120C22×Q8C22×C412yes—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—41no—
126C22×D4—41no—
127C22×D4C22⋊C412yes—
128C22×D4C22⋊C412yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—44no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2411yescharacteristic
156C22×C8—24no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—44no—
160C22⋊Q8—44no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C22⋊Q8—84no—
164C4×D4—44no—
165C4×D4—44no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—42no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C414yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42C22×C412yes—
214C2×C42C2×D412yes—
215C4⋊C8—44no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—84no—
219C22⋊C8—82no—
220C22⋊C8—82no—
221#97—44no—
222#95—22no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—22no—
228#95—24no—
229#95—24no—
230#95—44no—
231#95—42no—
232#95—44no—
233#95—82no—
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⋊C411yescharacteristic
247C24C2×C22⋊C411yescharacteristic
248C24C2×C22⋊C412yes—
249C24AGL(1,ℤ/8)12yes—
250C24#13712yes—
251C24#13712yes—
252C24#13712yes—
253C24C22×D411yescharacteristic
254C2×Q8—24no—
255C2×Q8—42no—
256C2×Q8—44no—
257C2×Q8—82no—
258C2×Q8—88no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—48no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4H(5,𝔽2)12yes—
339C22×C4H(5,𝔽2)12yes—
340C2×C8—44no—
341C2×C8—44no—
342C2×C8—44no—
343C4⋊C4—24no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—84no—
351C4⋊C4—82no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—44no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—44no—
364C22⋊C4—84no—
365C22⋊C4—84no—
366C22⋊C4—88no—
367C22⋊C4—84no—
368C42—24no—
369C42—22no—
370C42—22no—
371C23—22no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—22no—
376C23—24no—
377C23—22no—
378C23—22no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—24no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—42no—
398C23—42no—
399C23—44no—
400C23—42no—
401C23—42no—
402C23—44no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—82no—
408C23—82no—
409C23#41712yes—
410C23#41712yes—
411C23#41914yes—
412C23C22×C22⋊C411yescharacteristic
413C23#53411yescharacteristic
414C23#53411yescharacteristic
415C23C2×AGL(1,ℤ/8)11yescharacteristic
416C23C2×#13711yescharacteristic
417C23C2×#13711yescharacteristic
418C23C2×#13711yescharacteristic
419Q8—84no—
420Q8—84no—
421D4—48no—
422D4—48no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—84no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—84no—
469C2×C4—816no—
470C2×C4—84no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—164no—
480C2×C4#51812yes—
481C2×C4#53412yes—
482C8—84no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—22no—
493C22—24no—
494C22—28no—
495C22—24no—
496C22—24no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—41no—
502C22—42no—
503C22—41no—
504C22—42no—
505C22—42no—
506C22—44no—
507C22—41no—
508C22—41no—
509C22—42no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22#230811yescharacteristic
515C22#288312yes—
516C22C2×#41911yescharacteristic
517C22C2×#41711yescharacteristic
518C22#290112yes—
519C22#319311yescharacteristic
520C22#319311yescharacteristic
521C22#319511yescharacteristic
522C22#319511yescharacteristic
523C4—22no—
524C4—22no—
525C4—42no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—41no—
530C4—84no—
531C4—82no—
532C4—84no—
533C4—82no—
534C4—162no—
535C2—21no—
536C2—21no—
537C2—21no—
538C2—21no—
539C2—44no—
540C2—42no—
541C2—42no—
542C2#1762112yes—
543C2#1762212yes—
544C2#2174911yescharacteristic
545C2#2176511yescharacteristic
546C2#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. #9116C2semidirect
7. #8712C2semidirect
8. #8329C2semidirect
18. #2487C4semidirect
19. #2417C4semidirect
20. #2409C4semidirect
40. C23×Q8C2×C4semidirect
50. C2×C22⋊Q8C2×C4semidirect
60. C22×C4⋊C4C2×C4semidirect
61. C22×C4⋊C4C2×C4semidirect
84. #391C2×C4semidirect
85. #389C2×C4semidirect
102. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350102
#217902
#139602
#91512
#91432
#91162
#87122
#83292