Finite Group Databasebeta

← #481642

Subgroups — #481642

Order 512.

1013 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2C4×#410C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#11787C214yesmaximal, complemented
5#11785C214yesmaximal, complemented
6#11717C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic, complemented
8C2×C4×#41C2211yescharacteristic
9#2920—21no—
10C2×#410—21no—
11#2857—21no—
12#2857—21no—
13#2628C412yescomplemented
14#2628C2212yes—
15#2568—21no—
16#2567C412yescomplemented
17#2563C2212yes—
18#2542—21no—
19#2487—21no—
20#2487—21no—
21#2409C412yescomplemented
22#2409C2212yes—
23#2317—21no—
24#2311—21no—
25#2285C411yescharacteristic, complemented
26C4×#117—21no—
27C4×#99—21no—
28C4×#99—21no—
29C4×#99C2212yes—
30C4×#99C2212yes—
31#1740C2212yes—
32#1467—21no—
33#1466C2212yes—
34#1465—21no—
35#1465C2212yes—
36#1458—21no—
37#1458C2212yes—
38#1351C22116yes—
39C22×#41D411yescharacteristic
40C23×Q8C2×C412yes—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—41no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C4×#41—41no—
52C4×#41—41no—
53C4×#41—41no—
54C4×#41—41no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×#117—21no—
64C2×#117—22no—
65C2×#117C2×C411yescharacteristic
66C2×#117D411yescharacteristic
67C2×#117C2311yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C42C2×C411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73#432—22no—
74#432—41no—
75#431—22no—
76#431—41no—
77#423—41no—
78#410—28no—
79C2×#99—22no—
80C2×#99—22no—
81#407—22no—
82C4×M16—41no—
83#400—22no—
84#400—22no—
85#399—22no—
86#399—22no—
87#395C2×C414yes—
88#395C2×C414yes—
89#393C2×C414yes—
90#393C2314yes—
91#391—22no—
92#391—22no—
93#391—22no—
94#390—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#386—22no—
100#386—22no—
101#385—22no—
102#385—22no—
103#384—22no—
104#384—21no—
105#382—22no—
106#382C2×C412yes—
107#380—22no—
108#380—21no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380C2×C412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—41no—
115C4×C4⋊C4—41no—
116C4×C4⋊C4C2×C412yes—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119#376—41no—
120C43—41no—
121#342—22no—
122#342C2314yes—
123#324—22no—
124#324—22no—
125#323—22no—
126#323C2314yes—
127C2×#41—24no—
128C2×#41—24no—
129C2×#41—41no—
130C2×#41—41no—
131C22×Q8—22no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—84no—
135C22×Q8C22⋊C411yescharacteristic
136C22×Q8C22×C414yes—
137C22×D4C22⋊C411yescharacteristic
138C22×D4C2×D411yescharacteristic
139C22×D4C2×D411yescharacteristic
140C23×C4—22no—
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C22⋊Q8—84no—
162C22⋊Q8—84no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×D4—41no—
168C4×D4—41no—
169C4×D4—41no—
170C4×D4—41no—
171C4×D4—41no—
172C4×D4—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178C4×D4—41no—
179#117—28no—
180#117—41no—
181#117—41no—
182#117—41no—
183#117—41no—
184#117—41no—
185#117—41no—
186#117—44no—
187#117—44no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4C22⋊C411yescharacteristic
213C2×C22⋊C4C22⋊C411yescharacteristic
214C2×C22⋊C4C22×C411yescharacteristic
215C2×C22⋊C4C2×D411yescharacteristic
216C2×C22⋊C4C2×D411yescharacteristic
217C2×C22⋊C4C2411yescharacteristic
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C4⋊C8—42no—
245C4⋊C8—42no—
246#99—28no—
247#99—28no—
248C22⋊C8—24no—
249C22⋊C8—42no—
250#97—42no—
251C4×C8—42no—
252#95—22no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—42no—
258#95—42no—
259#95C22×C414yes—
260C24C2×C22⋊C411yescharacteristic
261C24C2×C22⋊C411yescharacteristic
262C24C22×D411yescharacteristic
263#41—48no—
264#41—48no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—84no—
274C2×Q8—88no—
275C2×Q8—88no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—84no—
318C22×C4#10414yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4#13714yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332M16—84no—
333C2×C8—44no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—48no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C22⋊C4—28no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—84no—
362C22⋊C4—84no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—48no—
382C42—44no—
383C42—84no—
384C42—84no—
385C42#9911yescharacteristic
386C42#9911yescharacteristic
387C42#9911yescharacteristic
388C42#9911yescharacteristic
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23C2×#10412yes—
403C23C22×C22⋊C411yescharacteristic
404C23C2×#13712yes—
405Q8—44no—
406Q8—84no—
407Q8—88no—
408Q8—84no—
409D4—44no—
410D4—44no—
411D4—44no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
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—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—88no—
461C2×C4C2×#9911yescharacteristic
462C2×C4C2×#9911yescharacteristic
463C2×C4C2×#9911yescharacteristic
464C2×C4C2×#9911yescharacteristic
465C2×C4C2×#9911yescharacteristic
466C2×C4C2×#9911yescharacteristic
467C2×C4#41912yes—
468C2×C4#41912yes—
469C2×C4C2×#10412yes—
470C2×C4C2×#10412yes—
471C8—84no—
472C22—21no—
473C22—21no—
474C22—22no—
475C22—22no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—81no—
488C22—81no—
489C22C22×#9911yescharacteristic
490C22C2×#41911yescharacteristic
491C22C22×#10411yescharacteristic
492C4—21no—
493C4—21no—
494C4—22no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—84no—
504C4—88no—
505C4—84no—
506C4#150812yes—
507C4#150812yes—
508C4#212212yes—
509C4#212912yes—
510C2—21no—
511C2—21no—
512C2—42no—
513C2—42no—
514C2—81no—
515C2C2×#150811yescharacteristic
516C2C2×#212211yescharacteristic
517C2C2×#212911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11787C2semidirect
5. #11785C2semidirect
6. #11717C2semidirect
7. C22×C4⋊Q8C4semidirect
13. #2628C4semidirect
16. #2567C4semidirect
21. #2409C4semidirect
25. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#352792
C4×#4102
#173612
#117872
#117852
#117172