Finite Group Databasebeta

← #482384

Subgroups — #482384

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#11883C214yesmaximal, complemented
5#11807C214yesmaximal, complemented
6#11780C214yesmaximal, complemented
7C2×#536—21no—
8C22×C4⋊Q8C411yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2920—21no—
11#2920—21no—
12#2857—21no—
13#2857—21no—
14#2621C412yes—
15#2621C2212yes—
16#2547—21no—
17#2547C412yes—
18#2546—21no—
19#2546C2212yes—
20#2486—21no—
21#2486—21no—
22#2486C412yes—
23#2486C2212yes—
24#2310—21no—
25#2285C411yescharacteristic
26#1822C2212yes—
27#1801C2212yes—
28#1800—21no—
29#1785—21no—
30C4×#101—21no—
31C4×#100C2212yes—
32#1465—21no—
33#1465C2212yes—
34#1464—21no—
35#1464C2212yes—
36#1461C2212yes—
37#1460—21no—
38#1346C22116yes—
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×Q8D411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
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—41no—
79#432—41no—
80#432—41no—
81#431—22no—
82#431—41no—
83#431—41no—
84#431—41no—
85#423—22no—
86#423—41no—
87C2×#101—22no—
88C2×#100—22no—
89#407—22no—
90#407—22no—
91C4×M16—22no—
92C4×M16—41no—
93#398—22no—
94#398—22no—
95#398C2×C414yes—
96#398C2314yes—
97#397—22no—
98#397—22no—
99#397C2×C414yes—
100#397C2×C414yes—
101#391—22no—
102#391—22no—
103#391—22no—
104#391—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#384—21no—
110#382C2×C412yes—
111#381—22no—
112#381—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—21no—
117#380—22no—
118#380C2×C412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121#343—22no—
122#343C2314yes—
123#324—22no—
124#324C2314yes—
125#323—22no—
126#323—22no—
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×Q8C2×D411yescharacteristic
138C22×D4C22⋊C411yescharacteristic
139C22×D4C22⋊C411yescharacteristic
140C22×D4C2×D411yescharacteristic
141C23×C4—22no—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—24no—
152C2×M16—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—48no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164#124—48no—
165#123—48no—
166#123—48no—
167C22⋊Q8—48no—
168C22⋊Q8—48no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×Q8—41no—
172C4×Q8—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—
184C4×D4—41no—
185#117—41no—
186#117—41no—
187#117—41no—
188#117—41no—
189#117—41no—
190#117—41no—
191#117—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
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⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2411yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4C22⋊C411yescharacteristic
215C2×C22⋊C4C22⋊C411yescharacteristic
216C2×C22⋊C4C22×C411yescharacteristic
217C2×C22⋊C4C22×C411yescharacteristic
218C2×C22⋊C4C2×D411yescharacteristic
219C2×C22⋊C4C2×D411yescharacteristic
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—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247#101—44no—
248#100—44no—
249C22⋊C8—24no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253#97—42no—
254#97—42no—
255C4×C8—42no—
256C4×C8—42no—
257#95—22no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95C22×C414yes—
263C24C2×C22⋊C411yescharacteristic
264C24C2×C22⋊C411yescharacteristic
265C24C2×C22⋊C411yescharacteristic
266#41—48no—
267#41—48no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—44no—
277C2×Q8—88no—
278C2×Q8—88no—
279C2×Q8AGL(1,ℤ/8)14yes—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
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—22no—
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×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336M16—44no—
337M16—44no—
338M16—84no—
339C2×C8—44no—
340C2×C8—44no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—48no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—44no—
365C22⋊C4—48no—
366C22⋊C4—48no—
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—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—44no—
386C42#9911yescharacteristic
387C42#9911yescharacteristic
388C42#9911yescharacteristic
389C42#9911yescharacteristic
390C23—21no—
391C23—21no—
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#42112yes—
405C23C22×C22⋊C411yescharacteristic
406Q8—44no—
407Q8—84no—
408Q8—84no—
409Q8—88no—
410D4—44no—
411D4—44no—
412D4—44no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
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—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—88no—
460C2×C4C2×#9911yescharacteristic
461C2×C4C2×#9911yescharacteristic
462C2×C4C2×#9911yescharacteristic
463C2×C4C2×#9911yescharacteristic
464C2×C4C2×#9911yescharacteristic
465C2×C4C2×#9911yescharacteristic
466C2×C4#41812yes—
467C2×C4#41812yes—
468C2×C4C2×AGL(1,ℤ/8)12yes—
469C8—84no—
470C8—84no—
471C22—21no—
472C22—21no—
473C22—22no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—81no—
487C22—81no—
488C22C22×#9911yescharacteristic
489C22C2×#41811yescharacteristic
490C22C2×#42111yescharacteristic
491C4—21no—
492C4—21no—
493C4—22no—
494C4—21no—
495C4—21no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—84no—
505C4—84no—
506C4#212312yes—
507C4#213512yes—
508C2—21no—
509C2—21no—
510C2—42no—
511C2—81no—
512C2—81no—
513C2#1757711yescharacteristic
514C2C2×#212311yescharacteristic
515C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #11883C2semidirect
5. #11807C2semidirect
6. #11780C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#207282
#173662
#118832
#118072
#117802