Finite Group Databasebeta

← #236052

Subgroups — #236052

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic, complemented
2C2×#3077C211yesmaximal, characteristic
3C2×#2542C211yesmaximal, characteristic
4#21385C211yesmaximal, characteristic, complemented
5C2×#1549C211yesmaximal, characteristic, complemented
6C2×#1478C211yesmaximal, characteristic
7#10076C214yesmaximal, complemented
8#10060C214yesmaximal, complemented
9#7901C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊Q8—22no—
13C22×C4×D4C2211yescharacteristic
14#3230—24no—
15#3230—24no—
16C2×#500C2212yescomplemented
17C2×#487C2212yescomplemented
18C2×#484—22no—
19#3077—24no—
20#3077—24no—
21C2×#477—22no—
22C2×#478—22no—
23C2×#476C2212yes—
24C2×#474C2212yes—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#428—21no—
28C2×#423C2211yescharacteristic
29C2×#418—21no—
30#2542—22no—
31#2542—22no—
32#2542—22no—
33#2542C2214yes—
34C2×#399—22no—
35C2×#390—21no—
36C2×#384—21no—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1987C2212yescomplemented
39#1981C2212yescomplemented
40#1872—22no—
41C4×#103—22no—
42#1549—24no—
43C2×#332C2214yescomplemented
44#1478—24no—
45C2×D4⋊C8C2214yes—
46#1279C2214yescomplemented
47C23×Q8—21no—
48C23×D4D411yescharacteristic
49C24×C4—21no—
50C22×M16—21no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533D414yes—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8D412yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2312yes—
65C2×#125—21no—
66C2×C22⋊Q8—28no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—416no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4D412yes—
80C2×C4×D4C2312yes—
81C2×#117—21no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D412yes—
85C22×C4⋊C4D412yes—
86C22×C4⋊C4D411yescharacteristic
87C22×C22⋊C4—24no—
88C22×C22⋊C4—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#500—24no—
92#500—24no—
93#487—24no—
94#487—24no—
95#484—216no—
96#478—216no—
97#477—216no—
98#476—24no—
99#476—24no—
100#474—24no—
101#474—24no—
102#431—22no—
103#431—22no—
104#431—22no—
105#431D414yes—
106#428—28no—
107C2×#107—22no—
108C2×#107—24no—
109C2×#106—22no—
110#423—22no—
111#423D414yes—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116#418—28no—
117C2×#103—24no—
118C2×#103—24no—
119C2×#102—22no—
120C2×#102—22no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8C2312yes—
125#399—28no—
126#399—28no—
127#390—24no—
128#390—24no—
129#384—24no—
130#384—24no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4D412yes—
134C4×C4⋊C4C2312yes—
135C2×#95—21no—
136#340—24no—
137#332—28no—
138D4⋊C8—28no—
139C25—21no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—48no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4Q1614yes—
154C22×D4C2×D412yes—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—48no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—24no—
167C2×M16—42no—
168C2×M16—42no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—22no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187C4⋊Q8—42no—
188C4⋊Q8C2×D412yes—
189C4⋊Q8C2×D412yes—
190#125—24no—
191#125—42no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C22⋊Q8—432no—
195C22⋊Q8—416no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C22⋊Q8—416no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213#117—24no—
214#117—42no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—416no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D414yes—
238C2×C22⋊C4—216no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—416no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—41no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D411yescharacteristic
256C2×C42#4112yes—
257C2×C42C2411yescharacteristic
258#107—28no—
259#107—216no—
260#106—28no—
261C4⋊C8—24no—
262C4⋊C8—24no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8C2×D414yes—
267#103—216no—
268#103—48no—
269#102—28no—
270#102—28no—
271C22⋊C8—24no—
272C22⋊C8—42no—
273C4×C8—22no—
274C4×C8—42no—
275C4×C8C2×D414yes—
276#95—22no—
277#95—22no—
278C24—22no—
279C24—24no—
280C24—44no—
281C24—44no—
282C24—42no—
283C24—42no—
284C24#12011yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—48no—
288C2×Q8—44no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—42no—
293C2×Q8—48no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—88no—
297C2×Q8—88no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—28no—
304C2×D4—24no—
305C2×D4—28no—
306C2×D4—42no—
307C2×D4—44no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4C2×Q1614yes—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—48no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—48no—
321C22×C4—44no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—48no—
329C22×C4—44no—
330C22×C4—88no—
331C22×C4—88no—
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12012yes—
335C22×C4#12012yes—
336C22×C4#12011yescharacteristic
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#12712yes—
341C22×C4C2×Q1612yes—
342C22×C4#13512yes—
343C22×C4#13712yes—
344C22×C4#13712yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348M16—48no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—44no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—44no—
356C4⋊C4—28no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—432no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4AGL(1,ℤ/8)18yes—
379C22⋊C4—432no—
380C22⋊C4—44no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—44no—
384C42—42no—
385C42—42no—
386C42—44no—
387C42—44no—
388C42#12012yes—
389C42#12012yes—
390C42#12112yes—
391C42#12112yes—
392C42#12112yes—
393C42#12712yes—
394C42C22×D411yescharacteristic
395C42C22×D411yescharacteristic
396C42C22×D411yescharacteristic
397C42C2×#4111yescharacteristic
398C23—22no—
399C23—22no—
400C23—24no—
401C23—416no—
402C23—42no—
403C23—42no—
404C23—44no—
405C23—44no—
406C23—48no—
407C23—84no—
408C23—88no—
409C23—84no—
410C23#39212yes—
411C23#45112yes—
412C23#45212yes—
413C23C2×#12011yescharacteristic
414Q8—44no—
415Q8—44no—
416Q8—82no—
417Q8—88no—
418Q8—82no—
419Q8—84no—
420Q8—84no—
421D4—24no—
422D4—24no—
423D4—42no—
424D4—48no—
425D4—42no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—28no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—416no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—48no—
442C2×C4—44no—
443C2×C4—88no—
444C2×C4—84no—
445C2×C4—88no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4#39212yes—
451C2×C4#39212yes—
452C2×C4#39212yes—
453C2×C4#45114yes—
454C2×C4#45112yes—
455C2×C4#45214yes—
456C2×C4#45212yes—
457C2×C4#46714yes—
458C2×C4#47014yes—
459C2×C4C2×#12011yescharacteristic
460C2×C4C2×#12011yescharacteristic
461C2×C4C2×#12111yescharacteristic
462C2×C4C2×#12111yescharacteristic
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×#12711yescharacteristic
465C2×C4C22×Q1611yescharacteristic
466C2×C4C2×#13511yescharacteristic
467C2×C4C2×AGL(1,ℤ/8)14yes—
468C2×C4C2×#13711yescharacteristic
469C2×C4C2×#13711yescharacteristic
470C8—44no—
471C8—44no—
472C8—82no—
473C22—24no—
474C22—24no—
475C22—22no—
476C22—22no—
477C22—48no—
478C22—44no—
479C22—816no—
480C22—84no—
481C22—82no—
482C22—82no—
483C22#200114yes—
484C22C2×#39211yescharacteristic
485C22C2×#45111yescharacteristic
486C22C2×#45211yescharacteristic
487C22#300312yes—
488C22#302012yes—
489C4—21no—
490C4—21no—
491C4—22no—
492C4—22no—
493C4—48no—
494C4—82no—
495C4—82no—
496C4—84no—
497C4—82no—
498C4#200114yes—
499C4C2×#45111yescharacteristic
500C4C2×#45211yescharacteristic
501C4#303812yes—
502C4C2×#46711yescharacteristic
503C4C2×#47011yescharacteristic
504C4#315112yes—
505C2—24no—
506C2—84no—
507C2#1875912yes—
508C2#1878312yes—
509C2C2×#200111yescharacteristic
510C2#2340311yescharacteristic
511C2#2343011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3230C2semidirect
4. #21385C2semidirect
5. C2×#1549C2semidirect
7. #10076C2semidirect
8. #10060C2semidirect
9. #7901C2semidirect
16. C2×#500C22semidirect
17. C2×#487C22semidirect
38. #1987C22semidirect
39. #1981C22semidirect
43. C2×#332C22semidirect
46. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C2×#30772
C2×#25422
#213852
C2×#15492
C2×#14782
#100762
#100602
#79012