Finite Group Databasebeta

← #382456

Subgroups — #382456

Order 512.

2466 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1825C211yesmaximal, characteristic
4C2×#1825C211yesmaximal, characteristic
5C2×#1823C211yesmaximal, characteristic
6C2×#1818C211yesmaximal, characteristic
7#14343C211yesmaximal, characteristic
8#11385C214yesmaximal, complemented
9#11384C214yesmaximal, complemented
10C2×#544—24no—
11C2×#544—24no—
12C2×#544—24no—
13C2×#544—24no—
14C2×#544—24no—
15C2×#544C418yescomplemented
16C22×C4⋊Q8—21no—
17C22×C4⋊Q8C2212yes—
18C22×#125—21no—
19C22×#125C2212yes—
20C22×C22⋊Q8—22no—
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28#1825—24no—
29#1825—24no—
30#1824—24no—
31#1823—24no—
32#1818—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2214yes—
36C2×#340C2214yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2214yes—
40#1445—21no—
41#1445—21no—
42#1440—22no—
43C4⋊#95C2214yes—
44C23×Q8—22no—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49#544—232no—
50#544—232no—
51#544—232no—
52#544—232no—
53C2×C4⋊Q8—28no—
54C2×C4⋊Q8—28no—
55C2×C4⋊Q8—28no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8C2×C418yes—
62C2×#125—28no—
63C2×#125—28no—
64C2×#125—28no—
65C2×#125—24no—
66C2×#125—24no—
67C2×#125—24no—
68C2×#125—24no—
69C2×#125—24no—
70C2×#125C2×C418yes—
71C2×C22⋊Q8—216no—
72C2×C22⋊Q8—216no—
73C2×C22⋊Q8—216no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117—22no—
81C2×#117C2×C414yes—
82C2×#117C2×C414yes—
83C2×#117D414yes—
84C2×#117D414yes—
85C2×#117D414yes—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4C2×C414yes—
94C22×C4⋊C4D411yescharacteristic
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4D414yes—
97C22×C4⋊C4D414yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4D412yes—
107C22×C42—21no—
108C22×C42C2312yes—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—24no—
122C2×#95—24no—
123#340—28no—
124#340—28no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#336—28no—
130C25C2×D411yescharacteristic
131C22×Q8—28no—
132C22×Q8—28no—
133C22×Q8—28no—
134C22×Q8—84no—
135C22×Q8—84no—
136C23×C4—28no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C22×C412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—28no—
164C22×C8—28no—
165C4⋊Q8—216no—
166C4⋊Q8—216no—
167C4⋊Q8—264no—
168C4⋊Q8—216no—
169C4⋊Q8—216no—
170#125—216no—
171#125—216no—
172#125—264no—
173#125—216no—
174#125—216no—
175C22⋊Q8—2128no—
176#117—28no—
177#117—28no—
178#117—28no—
179#117—28no—
180#117—28no—
181#117—28no—
182#117—28no—
183#117C22⋊C4116yes—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4C22⋊C414yes—
219C2×C4⋊C4C22⋊C414yes—
220C2×C4⋊C4C22⋊C4116yes—
221C2×C4⋊C4C22×C414yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4C22⋊C418yes—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42C22×C414yes—
248C2×C42C2×D414yes—
249C22⋊C8—44no—
250C22⋊C8—44no—
251#95—42no—
252#95—42no—
253#95—84no—
254#95—84no—
255C24—24no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24C2×C22⋊C412yes—
261C24#12012yes—
262C24#12012yes—
263C24C22×D411yescharacteristic
264C2×Q8—232no—
265C2×Q8—816no—
266C2×Q8—88no—
267C2×Q8—88no—
268C2×Q8—816no—
269C22×C4—24no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—216no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—88no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—88no—
314C22×C4C2×C22⋊C414yes—
315C22×C4C2×C22⋊C414yes—
316C22×C4C2×C22⋊C414yes—
317C22×C4C2×C22⋊C414yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4#12014yes—
320C22×C4#12014yes—
321C22×C4#12012yes—
322C22×C4#12012yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C42−1+418yes—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C4⋊C4—232no—
339C4⋊C4—232no—
340C4⋊C4—232no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—216no—
351C4⋊C4—28no—
352C4⋊C4—28no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4C2×C22⋊C418yes—
358C4⋊C4C2×C22⋊C414yes—
359C4⋊C4C2×C22⋊C414yes—
360C22⋊C4—216no—
361C22⋊C4—216no—
362C22⋊C4—84no—
363C22⋊C4—84no—
364C42—24no—
365C42—24no—
366C42—24no—
367C42—24no—
368C42—216no—
369C23—24no—
370C23—24no—
371C23—24no—
372C23—24no—
373C23—24no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—24no—
379C23—24no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23#37914yes—
385C23C22×C22⋊C411yescharacteristic
386C23C2×#12011yescharacteristic
387C23C2×#12011yescharacteristic
388C23#53614yes—
389C23#57612yes—
390C23#57712yes—
391C23#57712yes—
392C23#57812yes—
393Q8—832no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—28no—
408C2×C4—48no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—48no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—816no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4#37914yes—
438C2×C4C22×C22⋊C411yescharacteristic
439C2×C4C22×C22⋊C411yescharacteristic
440C2×C4#51914yes—
441C2×C4C2×#12011yescharacteristic
442C2×C4C2×#12011yescharacteristic
443C2×C4#53614yes—
444C2×C4#53614yes—
445C2×C4#53614yes—
446C2×C4#53614yes—
447C2×C4#53614yes—
448C8—82no—
449C8—82no—
450C22—24no—
451C22—24no—
452C22—24no—
453C22—48no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22C2×#37911yescharacteristic
459C22#231012yes—
460C22#261412yes—
461C22#261412yes—
462C22#289712yes—
463C22#289812yes—
464C22#300811yescharacteristic
465C22#300911yescharacteristic
466C22#301011yescharacteristic
467C22#301111yescharacteristic
468C22#301212yes—
469C22#301312yes—
470C22#324114yes—
471C22#324214yes—
472C22#324314yes—
473C22#324414yes—
474C4—24no—
475C4—44no—
476C4—48no—
477C4—44no—
478C4—44no—
479C4—44no—
480C4—162no—
481C4—162no—
482C4#231012yes—
483C4#231012yes—
484C4#261412yes—
485C4#261412yes—
486C2—44no—
487C2#1433611yescharacteristic
488C2#2060211yescharacteristic
489C2#2060311yescharacteristic
490C2#2184314yes—
491C2#2964612yes—
492C2#2964712yes—
493C2#2964812yes—
494C2#2964912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11385C2semidirect
9. #11384C2semidirect
15. C2×#544C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#18242
C2×#18252
C2×#18232
C2×#18182
#143432
#113852
#113842