Finite Group Databasebeta

← #188043

Subgroups — #188043

Order 512.

2726 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#553C211yesmaximal, characteristic
2#21793C212yesmaximal, complemented
3#20722C212yesmaximal, complemented
4#20688C212yesmaximal, complemented
5#8543C214yesmaximal, complemented
6#8542C214yesmaximal, complemented
7C2×#553—28no—
8C2×#553—24no—
9C2×#553—24no—
10C2×#553—24no—
11C2×#553—24no—
12C22×#126C412yescomplemented
13C22×#125C411yescharacteristic, complemented
14C22×#123C2212yes—
15C22×#121C411yescharacteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#121C2212yes—
18C22×C4×D4C411yescharacteristic, complemented
19C22×C4×D4C412yescomplemented
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C411yescharacteristic, complemented
23#2280C2214yes—
24#1964—28no—
25#1820—24no—
26#1819C2218yes—
27#1819C2214yes—
28#1799—24no—
29#1781C2218yes—
30#1781C2214yes—
31C4×#99—24no—
32#1740—24no—
33C23×D4C2×C412yes—
34C23×D4C2×C411yescharacteristic
35C23×D4C2311yescharacteristic
36C23×D4C2312yes—
37C24×C4C2×C412yes—
38C24×C4C2×C412yes—
39C24×C4C2×C411yescharacteristic
40C24×C4C2311yescharacteristic
41#553—232no—
42#553—232no—
43#553—232no—
44#553—232no—
45C2×#126—24no—
46C2×#126—24no—
47C2×#126—28no—
48C2×#126—28no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—24no—
52C2×#125—24no—
53C2×#123—28no—
54C2×#123—28no—
55C2×#123—28no—
56C2×#121—24no—
57C2×#121—28no—
58C2×#121—28no—
59C2×#121—24no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—28no—
67C2×#121D418yes—
68C2×C4×D4—24no—
69C2×C4×D4—22no—
70C2×C4×D4—24no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—28no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4—28no—
78C2×C4×D4—24no—
79C2×C4×D4D414yes—
80C2×C4×D4D414yes—
81C2×#117—24no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—24no—
85C2×#117D414yes—
86C2×#117D414yes—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4C2×C412yes—
92C22×C4⋊C4C2×C412yes—
93C22×C4⋊C4C2×C412yes—
94C22×C4⋊C4C2×C411yescharacteristic
95C22×C4⋊C4C2×C411yescharacteristic
96C22×C4⋊C4C2×C411yescharacteristic
97C22×C4⋊C4C2×C411yescharacteristic
98C22×C22⋊C4C2×C412yes—
99C22×C22⋊C4C2×C412yes—
100C22×C22⋊C4C2×C411yescharacteristic
101C22×C22⋊C4C2×C412yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C42C2×C412yes—
105C22×C42C2×C411yescharacteristic
106C2×#99—28no—
107C2×#99—216no—
108#381—24no—
109#380—24no—
110#380—24no—
111#379C2318yes—
112C4×C22⋊C4—24no—
113C4×C22⋊C4—24no—
114#342—28no—
115#342—216no—
116C25C22×C411yescharacteristic
117C25C22×C412yes—
118C25C2411yescharacteristic
119C22×D4—24no—
120C22×D4—24no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—24no—
128C22×D4—21no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4—48no—
132C22×D4—84no—
133C22×D4C22⋊C414yes—
134C22×D4C2×D414yes—
135C22×D4C2×D418yes—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—84no—
148C23×C4—84no—
149C23×C4C22⋊C414yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D414yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163#126—232no—
164#126—232no—
165#125—216no—
166#125—216no—
167#123—264no—
168#121—264no—
169#121—216no—
170#121—216no—
171#121—28no—
172#121—28no—
173#121—28no—
174#121—28no—
175C4×D4—232no—
176C4×D4—232no—
177C4×D4—216no—
178C4×D4—216no—
179C4×D4—28no—
180C4×D4—28no—
181C4×D4—28no—
182C4×D4—28no—
183#117—28no—
184#117—28no—
185#117—28no—
186#117—28no—
187#117—416no—
188C2×C4⋊C4—216no—
189C2×C4⋊C4—216no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C414yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4C22⋊C412yes—
235C2×C22⋊C4C22⋊C412yes—
236C2×C22⋊C4C22⋊C418yes—
237C2×C22⋊C4C2×D418yes—
238C2×C22⋊C4C2×D414yes—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—44no—
246C2×C42C22⋊C418yes—
247C2×C42C22⋊C414yes—
248#99—416no—
249#99—816no—
250#95—44no—
251#95—44no—
252C24—21no—
253C24—21no—
254C24—24no—
255C24—22no—
256C24—22no—
257C24—42no—
258C24—44no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—44no—
264C24—44no—
265C24C2×C22⋊C414yes—
266C24C2×C22⋊C412yes—
267C24C23×C411yescharacteristic
268C24C22×D412yes—
269C24C22×D412yes—
270C24C22×D412yes—
271C2×D4—216no—
272C2×D4—28no—
273C2×D4—28no—
274C2×D4—216no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—416no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—416no—
285C2×D4—48no—
286C2×D4—416no—
287C2×D4—48no—
288C2×D4—44no—
289C2×D4—88no—
290C2×D4—88no—
291C22×C4—28no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—216no—
306C22×C4—216no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—44no—
310C22×C4—48no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—48no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C414yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C414yes—
338C22×C4C2×C22⋊C414yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4#12014yes—
346C22×C4#12014yes—
347C22×C4#12014yes—
348C22×C4#12014yes—
349C4⋊C4—216no—
350C4⋊C4—216no—
351C4⋊C4—216no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—88no—
361C22⋊C4—216no—
362C22⋊C4—216no—
363C22⋊C4—216no—
364C22⋊C4—28no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—28no—
368C22⋊C4—28no—
369C22⋊C4—416no—
370C22⋊C4—416no—
371C22⋊C4—416no—
372C42—28no—
373C42—28no—
374C42—24no—
375C42—24no—
376C42—88no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—24no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—24no—
386C23—24no—
387C23—24no—
388C23—28no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—416no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—88no—
411C23—816no—
412C23#41014yes—
413C23C22×C22⋊C411yescharacteristic
414C23C22×C22⋊C411yescharacteristic
415C23C22×C22⋊C411yescharacteristic
416C23C2×#12014yes—
417D4—48no—
418D4—416no—
419D4—432no—
420D4—48no—
421D4—816no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—28no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—48no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—816no—
460C2×C4—816no—
461C2×C4—88no—
462C2×C4—88no—
463C2×C4—88no—
464C2×C4#37912yes—
465C2×C4#37912yes—
466C2×C4#37912yes—
467C2×C4#37912yes—
468C2×C4#41014yes—
469C22—28no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—24no—
477C22—24no—
478C22—24no—
479C22—24no—
480C22—44no—
481C22—48no—
482C22—44no—
483C22—48no—
484C22—42no—
485C22—42no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—84no—
490C22—84no—
491C22—84no—
492C22—88no—
493C22—88no—
494C22—88no—
495C22C2×#37911yescharacteristic
496C22C2×#41012yes—
497C22C2×#41012yes—
498C22#288312yes—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—44no—
505C4—44no—
506C4—44no—
507C4—42no—
508C4—168no—
509C4#178214yes—
510C2—22no—
511C2—22no—
512C2—22no—
513C2—44no—
514C2—44no—
515C2—84no—
516C2C2×#178211yescharacteristic
517C2#2058411yescharacteristic
518C2#2058611yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#553C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21793C2semidirect
3. #20722C2semidirect
4. #20688C2semidirect
5. #8543C2semidirect
6. #8542C2semidirect
12. C22×#126C4semidirect
13. C22×#125C4semidirect
15. C22×#121C4semidirect
18. C22×C4×D4C4semidirect
19. C22×C4×D4C4semidirect
22. C23×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5532
#217932
#207222
#206882
#85432
#85422