Finite Group Databasebeta

← #196457

Subgroups — #196457

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic, complemented
2#35178C211yesmaximal, characteristic
3C2×#1514C211yesmaximal, characteristic
4C2×#1505C211yesmaximal, characteristic
5#13961C211yesmaximal, characteristic, complemented
6#9118C214yesmaximal, complemented
7#9114C214yesmaximal, complemented
8#8022C211yesmaximal, characteristic, complemented
9#8018C211yesmaximal, characteristic, complemented
10C2×#536—21no—
11C2×#532C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#2983—24no—
14#2983—24no—
15C2×#446—22no—
16C2×#446—22no—
17C2×#432C2212yes—
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2212yes—
21#2854C2212yes—
22#2854C2212yes—
23#2618—21no—
24#2618—21no—
25#2548C412yescomplemented
26#2546C412yescomplemented
27#2545—22no—
28#2538C414yescomplemented
29#2536—22no—
30#2311—21no—
31#2285C2211yescharacteristic
32C2×#383C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1514—24no—
35#1505—24no—
36C2×#327C2214yes—
37C2×#327C2214yes—
38#1446—21no—
39#1445—21no—
40#1436—21no—
41#1436—21no—
42#1434C2212yes—
43#1434C2212yes—
44#1294C2214yes—
45#1294C2214yes—
46C22×#41—21no—
47C23×Q8D411yescharacteristic
48#536—48no—
49#532—22no—
50#532—22no—
51#532—22no—
52#532—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#127C2×C411yescharacteristic
56C2×#124—22no—
57C2×#124—21no—
58C2×#123—22no—
59C2×#123—22no—
60C2×C22⋊Q8—24no—
61C2×#121—21no—
62C2×#121C2312yes—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8D412yes—
71C2×C4×D4—21no—
72C2×C4×D4C2×C412yes—
73C2×#117—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#446—216no—
82#446—216no—
83#432—22no—
84#432—22no—
85#432—24no—
86#432—24no—
87#432D414yes—
88#432D414yes—
89#431—24no—
90#431—24no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2312yes—
97C2×#97—22no—
98C2×#97—21no—
99C2×#97—21no—
100C2×C4×C8—22no—
101#398—22no—
102#398—22no—
103#396—24no—
104#396—24no—
105#394—22no—
106#394—22no—
107#391—22no—
108#390—22no—
109#390C2×C414yes—
110#390C2×C414yes—
111#389C2×C414yes—
112#389C2×C414yes—
113#388—24no—
114#388—24no—
115#386—22no—
116#386—22no—
117#384C2×C412yes—
118#383—22no—
119#383—22no—
120#382—21no—
121#380—21no—
122#380—24no—
123#380C2312yes—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2×C412yes—
126#376—22no—
127#376—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95C2312yes—
136C2×#95C2312yes—
137#327—28no—
138#327—28no—
139C2×#41—44no—
140C2×#41—44no—
141C2×#41—42no—
142C2×#41—42no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—24no—
147C22×Q8—42no—
148C22×Q8C22⋊C412yes—
149C22×Q8C22⋊C412yes—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22×C412yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C22×C8—22no—
165C22×C8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—48no—
169#127—22no—
170#127—22no—
171#124—48no—
172#124—42no—
173#124—42no—
174#123—48no—
175#123—48no—
176C22⋊Q8—416no—
177#121—22no—
178#121—22no—
179#121—22no—
180#121—22no—
181#121—42no—
182#121—42no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—24no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—42no—
194C4×D4—42no—
195#117—44no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C414yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
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—22no—
231C2×C42—44no—
232C2×C42—42no—
233C2×C42C22×C412yes—
234C2×C42C22×C412yes—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4112yes—
244C2×C42#4112yes—
245C4⋊C8—28no—
246C4⋊C8—28no—
247C4⋊C8—24no—
248C4⋊C8—24no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8C2×D414yes—
252C22⋊C8C2×D414yes—
253#97—28no—
254#97—24no—
255#97—24no—
256C4×C8—28no—
257#95—22no—
258#95—24no—
259#95—24no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—44no—
272C24—22no—
273C24C22×D411yescharacteristic
274#41—88no—
275#41—88no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—48no—
283C2×Q8—48no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—44no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—82no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C412yes—
343C22×C4#12112yes—
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4AGL(1,ℤ/8)14yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—44no—
357C2×C8—44no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—48no—
369C4⋊C4—42no—
370C4⋊C4—48no—
371C4⋊C4—82no—
372C22⋊C4—48no—
373C22⋊C4—44no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—48no—
378C22⋊C4—84no—
379C22⋊C4—84no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—44no—
389C42—48no—
390C42C4×D412yes—
391C42C4×D412yes—
392C42C4×D412yes—
393C42C4×D412yes—
394C42#12012yes—
395C42#12012yes—
396C42#12312yes—
397C42#12312yes—
398C23—22no—
399C23—22no—
400C23—44no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—42no—
405C23C22×C22⋊C411yescharacteristic
406C23C2×#12111yescharacteristic
407C23C2×#12111yescharacteristic
408C23C2×AGL(1,ℤ/8)12yes—
409C23#57512yes—
410Q8—48no—
411Q8—48no—
412Q8—84no—
413D4—82no—
414D4—82no—
415D4—84no—
416D4—84no—
417D4—84no—
418D4—84no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—88no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4#38612yes—
474C2×C4#38612yes—
475C2×C4#38612yes—
476C2×C4#38612yes—
477C2×C4#41814yes—
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×#12011yescharacteristic
481C2×C4C2×#12311yescharacteristic
482C8—44no—
483C8—44no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—21no—
491C22—21no—
492C22—82no—
493C22—84no—
494C22—84no—
495C22—82no—
496C22—82no—
497C22C2×#38611yescharacteristic
498C22#288412yes—
499C22C2×#41811yescharacteristic
500C22#289611yescharacteristic
501C22#290112yes—
502C22#305512yes—
503C22#305512yes—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—24no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—84no—
517C4—84no—
518C4—82no—
519C4—162no—
520C2—22no—
521C2—22no—
522C2—162no—
523C2#1762412yes—
524C2#1763112yes—
525C2#2090311yescharacteristic
526C2#2097311yescharacteristic
527C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2983C2semidirect
5. #13961C2semidirect
6. #9118C2semidirect
7. #9114C2semidirect
8. #8022C2semidirect
9. #8018C2semidirect
25. #2548C4semidirect
26. #2546C4semidirect
28. #2538C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#351782
C2×#15142
C2×#15052
#139612
#91182
#91142
#80222
#80182