Finite Group Databasebeta

← #215860

Subgroups — #215860

Order 512.

1424 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
1C2×#2857C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic
3#13753C211yesmaximal, characteristic
4#9908C214yesmaximal, complemented
5#9889C214yesmaximal, complemented
6#8712C214yesmaximal
7C22×C4⋊Q8C411yescharacteristic, complemented
8C22×C22⋊Q8—21no—
9C23×C42C2211yescharacteristic
10C2×#432—21no—
11#2857—21no—
12#2857—21no—
13#2857—21no—
14#2857—21no—
15#2857C412yescomplemented
16#2857C412yescomplemented
17#2857C2212yes—
18#2857C2212yes—
19C2×#391—21no—
20C2×#389—21no—
21C2×#389C412yescomplemented
22C2×#384C411yescharacteristic, complemented
23C2×#380—21no—
24C2×#380C2212yes—
25C2×C4×C22⋊C4—21no—
26C2×#376—21no—
27#1817—21no—
28#1797—21no—
29#1797C2212yes—
30#1779C2212yes—
31#1749—21no—
32#1749C2212yes—
33#1434—21no—
34#1434C2212yes—
35C22⋊#95—21no—
36C22⋊#95C2212yes—
37C4×#95C2212yes—
38#1294C22116yes—
39C23×Q8—21no—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—44no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2×C412yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—48no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C42—22no—
62C22×C42—44no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#432—44no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8C2314yes—
80C2×#97—22no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389C2×C414yes—
92#389C2×C414yes—
93#384—21no—
94#384—21no—
95#384C2×C412yes—
96#384C2×C412yes—
97#380—22no—
98#380—22no—
99#380—22no—
100#380—22no—
101#380—22no—
102#380—22no—
103#380C2×C414yes—
104#380C2314yes—
105C4×C22⋊C4—28no—
106#376—44no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95C2314yes—
114#336—22no—
115#336—22no—
116C25C2×D411yescharacteristic
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8—44no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—24no—
153C4⋊Q8—44no—
154C4⋊Q8—44no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C22⋊Q8—48no—
158C22⋊Q8—48no—
159C22⋊Q8—48no—
160C22⋊Q8—48no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4C22×C414yes—
178C2×C4⋊C4C22×C414yes—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—48no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—28no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—44no—
199C2×C42—44no—
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C4⋊C8—44no—
217C22⋊C8—28no—
218C22⋊C8—44no—
219C22⋊C8—44no—
220#97—44no—
221#95—28no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—28no—
227#95—44no—
228#95—44no—
229#95—44no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—41no—
235C24—41no—
236C24C2×C22⋊C411yescharacteristic
237C24C2×C22⋊C411yescharacteristic
238C24AGL(1,ℤ/8)14yes—
239C24C22×D411yescharacteristic
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—816no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—416no—
302C22×C4—416no—
303C22×C4—416no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4#10414yes—
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×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4#13714yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C42−1+414yes—
331C2×C8—44no—
332C2×C8—44no—
333C2×C8—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C22⋊C4—416no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—416no—
365C42#9914yes—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—28no—
381C23—28no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23#41012yes—
388C23#41812yes—
389C23#41812yes—
390C23C2×#10412yes—
391C23C22×C22⋊C411yescharacteristic
392C23#53612yes—
393C23C2×AGL(1,ℤ/8)12yes—
394C23C2×#13712yes—
395Q8—88no—
396Q8—88no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—28no—
414C2×C4—28no—
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—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—48no—
442C2×C4—48no—
443C2×C4—48no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—88no—
454C2×C4—84no—
455C2×C4—88no—
456C2×C4C2×#9912yes—
457C2×C4C2×#9912yes—
458C2×C4C2×#9912yes—
459C2×C4#41912yes—
460C2×C4#41912yes—
461C2×C4C2×#10412yes—
462C2×C4C2×#10412yes—
463C2×C4#42112yes—
464C2×C4#51912yes—
465C2×C4#53612yes—
466C8—84no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—416no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22#150514yes—
488C22#231011yescharacteristic
489C22C22×#9911yescharacteristic
490C22C2×#41011yescharacteristic
491C22C2×#41811yescharacteristic
492C22C2×#41911yescharacteristic
493C22C22×#10411yescharacteristic
494C22C2×#42111yescharacteristic
495C22#323814yes—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—88no—
508C4—84no—
509C4—84no—
510C4#150814yes—
511C2—24no—
512C2—24no—
513C2—42no—
514C2C2×#150811yescharacteristic
515C2C2×#150511yescharacteristic
516C2#1756211yescharacteristic
517C2#1757711yescharacteristic
518C2#2182511yescharacteristic
519C2#2184011yescharacteristic
520C2#2184411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9908C2semidirect
5. #9889C2semidirect
7. C22×C4⋊Q8C4semidirect
15. #2857C4semidirect
16. #2857C4semidirect
21. C2×#389C4semidirect
22. C2×#384C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#207152
#137532
#99082
#98892
#87122