Finite Group Databasebeta

← #216218

Subgroups — #216218

Order 512.

1282 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2293C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic, complemented
3#20777C211yesmaximal, characteristic, complemented
4#9879C212yesmaximal, complemented
5#9649C212yesmaximal
6#9084C214yesmaximal, complemented
7#8559C214yesmaximal, complemented
8C22×C4×D4C411yescharacteristic
9C22×#117C412yes—
10C2×#432C2212yes—
11#2293—21no—
12#2293—21no—
13#2293—21no—
14#2293—21no—
15#2293—21no—
16#2293—21no—
17#2293—21no—
18#2293—21no—
19C2×#390C411yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C411yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388C412yes—
24C2×#388C2212yes—
25C2×C4×C4⋊C4C411yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27C2×C43C2211yescharacteristic
28#1842—21no—
29#1839—21no—
30#1831—21no—
31#1831—21no—
32#1817C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1797C2212yes—
36#1767—21no—
37#1767—21no—
38C4×C4⋊C8—21no—
39C4×C22⋊C8C2212yes—
40#1744—21no—
41C2×#323C2218yes—
42#1280C2218yes—
43C23×D4C2×C411yescharacteristic
44C24×C4C2×C412yes—
45C2×C4×D4—24no—
46C2×C4×D4—24no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×C4×D4—21no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—48no—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2312yes—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C412yes—
70C22×C42C2×C411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#432—44no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C22⋊C8C2314yes—
80C2×C22⋊C8C2314yes—
81C2×#97—22no—
82C2×C4×C8—22no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2312yes—
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#323—48no—
133C25C22×C412yes—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4M1614yes—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—24no—
157C22×C8—24no—
158C4×D4—28no—
159C4×D4—28no—
160C4×D4—28no—
161C4×D4—28no—
162#117—48no—
163#117—48no—
164#117—48no—
165#117—48no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4M1614yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—44no—
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22⋊C411yescharacteristic
224C2×C42C22⋊C411yescharacteristic
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22⋊C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C4⋊C8—44no—
240C4⋊C8—44no—
241C4⋊C8—44no—
242C4⋊C8—44no—
243C22⋊C8—24no—
244C22⋊C8—28no—
245C22⋊C8—24no—
246#97—44no—
247C4×C8—44no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254C24—22no—
255C24—22no—
256C24—24no—
257C24—24no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24C2×M1614yes—
263C24C23×C411yescharacteristic
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—48no—
269C2×D4—48no—
270C2×D4—42no—
271C2×D4—42no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—28no—
279C22×C4—28no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—44no—
301C22×C4—416no—
302C22×C4—416no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×M1612yes—
325C22×C4C2×M1612yes—
326C22×C4C2×M1614yes—
327C22×C4#13114yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4H(5,𝔽2)12yes—
331C22×C42−1+412yes—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C22⋊C4—28no—
349C22⋊C4—28no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—44no—
389C42—44no—
390C42#10014yes—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—24no—
396C23—28no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—84no—
409C23—84no—
410C23—84no—
411C23—84no—
412C23C22×C22⋊C411yescharacteristic
413C23#52012yes—
414C23C22×M1611yescharacteristic
415C23C22×M1611yescharacteristic
416C23C2×#13112yes—
417D4—48no—
418D4—48no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—21no—
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—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—416no—
460C2×C4—416no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4#40712yes—
476C2×C4#40712yes—
477C2×C4#40812yes—
478C2×C4#40812yes—
479C2×C4#41012yes—
480C2×C4#41012yes—
481C2×C4C2×#10012yes—
482C2×C4C2×#10012yes—
483C2×C4C2×#10012yes—
484C2×C4#41312yes—
485C2×C4#53511yescharacteristic
486C2×C4#53611yescharacteristic
487C2×C4#53712yes—
488C8—84no—
489C8—84no—
490C22—22no—
491C22—22no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—88no—
502C22—88no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22#231111yescharacteristic
508C22C2×#40711yescharacteristic
509C22C2×#40811yescharacteristic
510C22C2×#41011yescharacteristic
511C22C2×#41011yescharacteristic
512C22C22×#10011yescharacteristic
513C22C2×#41311yescharacteristic
514C22#299012yes—
515C22#299112yes—
516C4—21no—
517C4—22no—
518C4—21no—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—44no—
530C4—41no—
531C4—82no—
532C4—82no—
533C4#146414yes—
534C2—22no—
535C2—22no—
536C2—84no—
537C2C2×#146411yescharacteristic
538C2#1736411yescharacteristic
539C2#1738211yescharacteristic
540C2#1738411yescharacteristic
541C2#2184911yescharacteristic
542C2#2187511yescharacteristic
543C2#2187811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20810C2semidirect
3. #20777C2semidirect
4. #9879C2semidirect
6. #9084C2semidirect
7. #8559C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22932
#208102
#207772
#98792
#96492
#90842
#85592