Finite Group Databasebeta

← #482364

Subgroups — #482364

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#11878C214yesmaximal, complemented
5#11801C214yesmaximal, complemented
6#11773C214yesmaximal, complemented
7C2×#536—21no—
8C22×C4⋊Q8C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10C2×#432—21no—
11#2857—21no—
12#2857—21no—
13#2621C412yescomplemented
14#2621C2212yes—
15#2547—21no—
16#2547C2212yes—
17#2546—21no—
18#2546C412yescomplemented
19#2486—21no—
20#2486—21no—
21#2486C412yescomplemented
22#2486C2212yes—
23#2310—21no—
24#2285C411yescharacteristic, complemented
25#2283—21no—
26#1820C2212yes—
27#1799—21no—
28#1799C2212yes—
29#1782—21no—
30C4×#99—21no—
31C4×#99C2212yes—
32#1467—21no—
33#1466C2212yes—
34#1464—21no—
35#1464C2212yes—
36#1459—21no—
37#1459C2212yes—
38#1344C22116yes—
39C22×#41D411yescharacteristic
40C23×Q8—21no—
41#536—48no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2×C412yes—
51C2×#124—22no—
52C2×#123—22no—
53C2×#123—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C4×#41—41no—
57C4×#41—41no—
58C4×#41—41no—
59C4×#41—41no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—21no—
69C2×#117C2×C411yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117C2311yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C412yes—
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77#432—22no—
78#432—22no—
79#432—41no—
80#432—41no—
81C2×C4⋊C8—41no—
82C2×#99—22no—
83C2×#99—22no—
84C2×C22⋊C8—22no—
85C2×#97—41no—
86#398—22no—
87#398—22no—
88#398C2×C414yes—
89#398C2×C414yes—
90#397—22no—
91#397—22no—
92#397C2×C414yes—
93#397C2314yes—
94#391—22no—
95#391—22no—
96#391—22no—
97#391—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#384—21no—
103#383—41no—
104#382—41no—
105#382C2×C412yes—
106#381—22no—
107#381—22no—
108#381—22no—
109#380—22no—
110#380—21no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380—22no—
115#380C2×C412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—41no—
118C4×C4⋊C4C2×C412yes—
119C4×C22⋊C4—22no—
120#376—41no—
121#342—22no—
122#342C2314yes—
123#324—22no—
124#324—22no—
125#323—22no—
126#323C2314yes—
127C2×#41—24no—
128C2×#41—41no—
129C2×#41—41no—
130C2×#41—42no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8C22⋊C411yescharacteristic
138C22×D4C22⋊C411yescharacteristic
139C22×D4C2×D411yescharacteristic
140C22×D4C2×D411yescharacteristic
141C23×C4—22no—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C22×C8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—48no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163#124—48no—
164#123—48no—
165#123—48no—
166C22⋊Q8—48no—
167C22⋊Q8—48no—
168C4×Q8—41no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×Q8—41no—
172C4×D4—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184#117—41no—
185#117—41no—
186#117—41no—
187#117—41no—
188#117—41no—
189#117—41no—
190#117—44no—
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—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4C22⋊C411yescharacteristic
216C2×C22⋊C4C22⋊C411yescharacteristic
217C2×C22⋊C4C22×C411yescharacteristic
218C2×C22⋊C4C2×D411yescharacteristic
219C2×C22⋊C4C2×D411yescharacteristic
220C2×C22⋊C4C2411yescharacteristic
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—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C4⋊C8—42no—
246C4⋊C8—42no—
247#99—44no—
248#99—44no—
249C22⋊C8—24no—
250C22⋊C8—42no—
251#97—42no—
252#97—42no—
253#95—22no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95C22×C414yes—
263C24C2×C22⋊C411yescharacteristic
264C24C2×C22⋊C411yescharacteristic
265C24C22×D411yescharacteristic
266#41—48no—
267#41—48no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—44no—
278C2×Q8—88no—
279C2×Q8—88no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4AGL(1,ℤ/8)14yes—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C2×C8—44no—
338C2×C8—84no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—48no—
351C4⋊C4—48no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—84no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—44no—
364C22⋊C4—48no—
365C22⋊C4—48no—
366C22⋊C4—44no—
367C22⋊C4—44no—
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—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—44no—
387C42—84no—
388C42#9911yescharacteristic
389C42#9911yescharacteristic
390C42#9911yescharacteristic
391C42#9911yescharacteristic
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23#41812yes—
405C23#42112yes—
406C23C22×C22⋊C411yescharacteristic
407C23C2×AGL(1,ℤ/8)12yes—
408Q8—44no—
409Q8—84no—
410Q8—84no—
411Q8—88no—
412D4—44no—
413D4—44no—
414D4—44no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
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—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—88no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4C2×#9911yescharacteristic
468C2×C4C2×#9911yescharacteristic
469C2×C4C2×#9911yescharacteristic
470C2×C4C2×#9911yescharacteristic
471C2×C4C2×#9911yescharacteristic
472C2×C4C2×#9911yescharacteristic
473C2×C4#41812yes—
474C8—84no—
475C22—21no—
476C22—21no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—81no—
491C22—81no—
492C22C22×#9911yescharacteristic
493C22C2×#41811yescharacteristic
494C22C2×#42111yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—22no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—84no—
509C4—84no—
510C4—164no—
511C4#212312yes—
512C4#212912yes—
513C2—21no—
514C2—21no—
515C2—42no—
516C2—42no—
517C2—42no—
518C2#1757711yescharacteristic
519C2C2×#212311yescharacteristic
520C2C2×#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. #11878C2semidirect
5. #11801C2semidirect
6. #11773C2semidirect
8. C22×C4⋊Q8C4semidirect
13. #2621C4semidirect
18. #2546C4semidirect
21. #2486C4semidirect
24. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#353162
#207232
#173632
#118782
#118012
#117732