Finite Group Databasebeta

← #481647

Subgroups — #481647

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34947C211yesmaximal, characteristic
2C4×#413C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#11795C212yesmaximal, complemented
5#11794C212yesmaximal, complemented
6#11793C214yesmaximal, complemented
7#11721C214yesmaximal, complemented
8C2×#533C411yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2920—21no—
11C2×#413—21no—
12C2×C4×M16—21no—
13#2568C2211yescharacteristic
14#2567C2211yescharacteristic
15#2566—21no—
16#2565—21no—
17#2564C412yes—
18#2564C2212yes—
19#2563C411yescharacteristic
20#2542C411yescharacteristic
21#2417—21no—
22#2417—21no—
23#2396C412yes—
24#2396C2212yes—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27#2297—21no—
28#2285—21no—
29C2×C4×C4⋊C4C411yescharacteristic
30C4×#101—21no—
31C4×#101C2212yes—
32C4×#100—21no—
33C4×#100C2212yes—
34#1742C2212yes—
35#1467C2212yes—
36#1466—21no—
37#1465—21no—
38#1465C2212yes—
39#1458—21no—
40#1458C2212yes—
41#1354C2218yes—
42#1353C2218yes—
43C22×#41D411yescharacteristic
44C22×M16—22no—
45#533—41no—
46#533—41no—
47#533—41no—
48#533—41no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C412yes—
56C4×#41—41no—
57C4×#41—41no—
58C4×#41—41no—
59C4×#41—41no—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117C2×C411yescharacteristic
72C2×#117D411yescharacteristic
73C2×#117C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C42—21no—
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79#432—22no—
80#432—41no—
81#431—22no—
82#431—41no—
83#423—41no—
84#413—28no—
85C2×#101—22no—
86C2×#100—22no—
87#407—22no—
88C4×M16—22no—
89C4×M16—22no—
90C4×M16—41no—
91C4×M16—41no—
92C4×M16—41no—
93C2×#97—41no—
94C2×C4×C8—41no—
95#400C2×C412yes—
96#400C2312yes—
97#399C2×C412yes—
98#399C2×C412yes—
99#398—22no—
100#398—22no—
101#396—22no—
102#396—22no—
103#395C2×C412yes—
104#395C2312yes—
105#393C2×C412yes—
106#393C2×C412yes—
107#391—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#390—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#387—22no—
116#387—22no—
117#386—22no—
118#386—22no—
119#385—22no—
120#385—22no—
121#384—21no—
122#384—21no—
123#382—21no—
124#382—21no—
125#380—21no—
126#380—21no—
127#380—22no—
128#380—22no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C412yes—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C2×#95C2×C412yes—
139#343—22no—
140#343C2314yes—
141#324—22no—
142#324C2314yes—
143#323—22no—
144#323—22no—
145C2×#41—24no—
146C2×#41—24no—
147C2×#41—41no—
148C2×#41—41no—
149C22×Q8—21no—
150C22×Q8C22×C412yes—
151C22×Q8C2×D411yescharacteristic
152C22×D4C22⋊C411yescharacteristic
153C22×D4C22⋊C411yescharacteristic
154C22×D4C2×D411yescharacteristic
155C23×C4—22no—
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C2×M16—28no—
166C2×M16—24no—
167C2×M16—42no—
168C2×M16—44no—
169C2×M16—42no—
170C22×C8—42no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175#125—42no—
176#125—42no—
177C22⋊Q8—44no—
178C22⋊Q8—44no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205#117—41no—
206#117—41no—
207#117—41no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—42no—
212#117—42no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2411yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4C22⋊C411yescharacteristic
239C2×C22⋊C4C22⋊C411yescharacteristic
240C2×C22⋊C4C22×C411yescharacteristic
241C2×C22⋊C4C22×C411yescharacteristic
242C2×C22⋊C4C2×D411yescharacteristic
243C2×C22⋊C4C2×D411yescharacteristic
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—44no—
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C4⋊C8—42no—
272C4⋊C8—42no—
273#101—28no—
274#100—28no—
275C22⋊C8—24no—
276C22⋊C8—42no—
277#97—42no—
278#97—42no—
279#97—42no—
280C4×C8—42no—
281C4×C8—42no—
282C4×C8—42no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95C22×C412yes—
293#95C22×C412yes—
294C24C2×C22⋊C411yescharacteristic
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C411yescharacteristic
297#41—48no—
298#41—48no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—82no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—82no—
351C22×C4—81no—
352C22×C4#10414yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4#13714yes—
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366M16—28no—
367M16—48no—
368M16—44no—
369M16—84no—
370C2×C8—44no—
371C2×C8—48no—
372C2×C8—44no—
373C4⋊C4—24no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—44no—
427C42—42no—
428C42#9911yescharacteristic
429C42#9911yescharacteristic
430C42#9911yescharacteristic
431C42#9911yescharacteristic
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23#41912yes—
446C23C2×#10412yes—
447C23C22×C22⋊C411yescharacteristic
448Q8—44no—
449Q8—82no—
450Q8—84no—
451Q8—82no—
452D4—44no—
453D4—44no—
454D4—44no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—88no—
507C2×C4C2×#9911yescharacteristic
508C2×C4C2×#9911yescharacteristic
509C2×C4C2×#9911yescharacteristic
510C2×C4C2×#9911yescharacteristic
511C2×C4C2×#9911yescharacteristic
512C2×C4C2×#9911yescharacteristic
513C2×C4#41912yes—
514C2×C4C2×#10412yes—
515C2×C4C2×#10412yes—
516C2×C4C2×#13712yes—
517C8—48no—
518C8—84no—
519C22—21no—
520C22—21no—
521C22—22no—
522C22—22no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—81no—
535C22—81no—
536C22C22×#9911yescharacteristic
537C22C2×#41911yescharacteristic
538C22C22×#10411yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—22no—
543C4—22no—
544C4—44no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4#150812yes—
556C4#150812yes—
557C4#213212yes—
558C2—21no—
559C2—21no—
560C2—42no—
561C2—42no—
562C2—81no—
563C2C2×#150811yescharacteristic
564C2#3334711yescharacteristic
565C2C2×#213211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
socleC22C22×#99
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11795C2semidirect
5. #11794C2semidirect
6. #11793C2semidirect
7. #11721C2semidirect

Maximal subgroups

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

subgroupindex
#349472
C4×#4132
#173612
#117952
#117942
#117932
#117212