Finite Group Databasebeta

← #207282

Subgroups — #207282

Order 512.

1070 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#35079C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic
3#13796C211yesmaximal, characteristic
4#9913C212yesmaximal, complemented
5#9913C212yesmaximal, complemented
6#9912C212yesmaximal, complemented
7#9911C212yesmaximal, complemented
8#8020C211yesmaximal, characteristic
9#8019C211yesmaximal, characteristic
10#8012C211yesmaximal, characteristic
11C2×#1294C211yesmaximal, characteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2857C2212yes—
15#2851C2212yes—
16#2667—21no—
17#2665—21no—
18#2569—21no—
19#2549—21no—
20#2442C412yescomplemented
21#2441C412yescomplemented
22#2440C412yescomplemented
23#2439C412yescomplemented
24#2437—22no—
25#2329—22no—
26#2280C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C22×#95C2211yescharacteristic
31#1851—21no—
32#1851—21no—
33#1850—22no—
34#1797C2212yes—
35#1797C2212yes—
36#1434—21no—
37#1434—21no—
38#1434C2212yes—
39#1434C2212yes—
40C22⋊#95—22no—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2211yescharacteristic
44#1294C2211yescharacteristic
45#1294C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8C2×C412yescomplemented
56C2×#125C2×C412yescomplemented
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×#117C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D412yes—
64C22×C42—21no—
65C22×C42C2312yes—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68#407—22no—
69#407—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8C2311yescharacteristic
75C2×C22⋊C8C2311yescharacteristic
76C2×C22⋊C8C2312yes—
77#400—22no—
78#399—22no—
79#398—22no—
80#398—22no—
81#396—22no—
82#396—22no—
83#395—22no—
84#393—22no—
85#391—22no—
86#391—22no—
87#389—22no—
88#389—22no—
89#389C2×C412yescomplemented
90#389C2×C412yescomplemented
91#389C2×C412yescomplemented
92#389C2×C412yescomplemented
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#384C2×C412yescomplemented
102#382C2×C412yescomplemented
103#381—21no—
104#380—22no—
105#380—21no—
106#380—21no—
107#380—22no—
108#380C2×C411yescharacteristic
109#380C2×C412yes—
110#380C2×C411yescharacteristic
111#380C2311yescharacteristic
112#380C2312yes—
113#380C2311yescharacteristic
114#379—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4C2×C411yescharacteristic
120C4×C22⋊C4C2×C411yescharacteristic
121C4×C22⋊C4D412yes—
122C4×C22⋊C4D411yescharacteristic
123C4×C22⋊C4D411yescharacteristic
124C4×C22⋊C4Q814yescomplemented
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95D411yescharacteristic
133C2×#95D411yescharacteristic
134C2×#95Q814yescomplemented
135C2×#95C2312yes—
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138#336—22no—
139#336—22no—
140C25C2×D411yescharacteristic
141C22×Q8—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—44no—
148C23×C4—82no—
149C23×C4—81no—
150C23×C4—81no—
151C23×C4—82no—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×Q812yes—
160C23×C4C2×Q812yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4C2411yescharacteristic
164C2×M16—42no—
165C2×M16—82no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—42no—
170C22×C8—82no—
171C4⋊Q8—82no—
172#125—82no—
173C22⋊Q8—48no—
174C22⋊Q8—82no—
175C22⋊Q8—82no—
176#117—82no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—81no—
203C2×C22⋊C4—81no—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—82no—
211C2×C42—82no—
212C2×C42C22×C412yes—
213C2×C42C22×C412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C4⋊C8—82no—
217C4⋊C8—82no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—42no—
225C22⋊C8—42no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—24no—
236#95—22no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95—82no—
243#95—82no—
244#95—82no—
245#95—82no—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248#95C22⋊C411yescharacteristic
249#95C22⋊C411yescharacteristic
250#95C22⋊C411yescharacteristic
251#95C22⋊C411yescharacteristic
252#95C2×Q814yes—
253#95C2×Q812yes—
254#95C2×Q812yes—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—24no—
260C24—42no—
261C24C2×C22⋊C411yescharacteristic
262C24C2×C22⋊C411yescharacteristic
263C24C22⋊Q812yes—
264C24C22⋊Q812yes—
265C24C22×D411yescharacteristic
266C2×Q8—44no—
267C2×Q8—82no—
268C2×Q8—82no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—48no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—44no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—88no—
315C22×C4—88no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C4×Q812yes—
323C22×C4C4×Q812yes—
324C22×C4C4×Q812yes—
325C22×C4C4×Q812yes—
326C22×C4C22⋊Q812yes—
327C22×C4C22⋊Q812yes—
328C22×C4#12412yes—
329C22×C4#12412yes—
330C22×C4C4⋊Q812yes—
331C22×C4C4⋊Q812yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×Q811yescharacteristic
335C22×C4C22×Q811yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338M16—84no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—84no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—48no—
366C22⋊C4—42no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—42no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—44no—
373C22⋊C4—81no—
374C22⋊C4—81no—
375C22⋊C4—82no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—82no—
381C42—84no—
382C23—22no—
383C23—22no—
384C23—21no—
385C23—21no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—21no—
393C23—21no—
394C23—22no—
395C23—24no—
396C23—24no—
397C23—42no—
398C23—42no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23#39112yes—
404C23#39112yes—
405C23#42114yes—
406C23C22×C22⋊C411yescharacteristic
407C23C2×C22⋊Q811yescharacteristic
408C23C2×C22⋊Q811yescharacteristic
409C23#57611yescharacteristic
410C23#57711yescharacteristic
411C23#57711yescharacteristic
412C23#57811yescharacteristic
413Q8—164no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—816no—
478C2×C4—164no—
479C2×C4#39112yes—
480C2×C4#39112yes—
481C2×C4C2×C4×Q811yescharacteristic
482C2×C4C2×C4×Q811yescharacteristic
483C2×C4C2×#12411yescharacteristic
484C2×C4C2×C4⋊Q811yescharacteristic
485C8—81no—
486C8—81no—
487C8—82no—
488C22—22no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—24no—
497C22—22no—
498C22—22no—
499C22—24no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—44no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22#199514yes—
510C22C2×#39111yescharacteristic
511C22#288312yes—
512C22#289711yescharacteristic
513C22#289811yescharacteristic
514C22C2×#42112yes—
515C22#308111yescharacteristic
516C22#308211yescharacteristic
517C22#308311yescharacteristic
518C22#308411yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—82no—
530C4—82no—
531C4—84no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—82no—
536C2—21no—
537C2—21no—
538C2—21no—
539C2—21no—
540C2—24no—
541C2—42no—
542C2#1759712yes—
543C2#1759812yes—
544C2#2131511yescharacteristic
545C2C2×#199511yescharacteristic
546C2#2133311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
4. #9913C2semidirect
5. #9913C2semidirect
6. #9912C2semidirect
7. #9911C2semidirect
20. #2442C4semidirect
21. #2441C4semidirect
22. #2440C4semidirect
23. #2439C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
56. C2×#125C2×C4semidirect
89. #389C2×C4semidirect
90. #389C2×C4semidirect
91. #389C2×C4semidirect
92. #389C2×C4semidirect
101. #384C2×C4semidirect
102. #382C2×C4semidirect
124. C4×C22⋊C4Q8semidirect
134. C2×#95Q8semidirect

Maximal subgroups

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

subgroupindex
#350792
#219322
#137962
#99132
#99122
#99112
#80202
#80192
#80122
C2×#12942