Finite Group Databasebeta

← #192128

Subgroups — #192128

Order 512.

1594 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2312C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic, complemented
3#21938C211yesmaximal, characteristic, complemented
4C22×#323C211yesmaximal, characteristic
5#14338C211yesmaximal, characteristic
6#8858C218yesmaximal, complemented
7#8023C211yesmaximal, characteristic, complemented
8#8023C211yesmaximal, characteristic, complemented
9C23×C22⋊C4C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2312—22no—
13#2312—22no—
14#2312—22no—
15#2312C2214yes—
16C2×#388—22no—
17C2×#388—22no—
18C2×#380C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#379—21no—
23C2×#379—21no—
24C2×C4×C22⋊C4—21no—
25#1853—22no—
26#1853C2214yescomplemented
27#1852—22no—
28#1852C2214yescomplemented
29#1808—22no—
30#1807—22no—
31C2×#323C418yescomplemented
32C2×#323C2214yes—
33#1439—22no—
34#1439C2214yes—
35C22⋊#95—21no—
36#1294C2214yescomplemented
37#1294C2214yescomplemented
38C26D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C23×C8—21no—
42C23×C8—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4C2312yes—
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—24no—
60C22×C22⋊C4C2×C414yes—
61C22×C22⋊C4D412yes—
62C22×C22⋊C4Q814yes—
63C22×C22⋊C4C2312yes—
64C22×C42—22no—
65C2×#107—24no—
66C2×#106—24no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8C2×C414yes—
72C2×C22⋊C8C2×C414yes—
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2312yes—
75#388—28no—
76#388—28no—
77#388—28no—
78#388—28no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380—22no—
84#380—22no—
85#380—21no—
86#380—21no—
87#380—21no—
88#380—21no—
89#380D412yes—
90#380D412yes—
91#380C2312yes—
92#380C2312yes—
93#379—24no—
94#379—24no—
95#379—44no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—24no—
103C2×#95C2312yes—
104#340—24no—
105#340—24no—
106#323C2×C4116yescomplemented
107C25—21no—
108C25—21no—
109C25—22no—
110C25—22no—
111C25—22no—
112C25—22no—
113C25—22no—
114C25—82no—
115C25—82no—
116C25C4⋊C414yes—
117C25C2×D412yes—
118C25C2×D411yescharacteristic
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—82no—
127C23×C4—82no—
128C23×C4C4⋊C414yes—
129C23×C4C22×C412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×Q812yes—
137C23×C4#4112yes—
138C23×C4#4112yes—
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—24no—
143C22×C8—24no—
144C22×C8—82no—
145C22×C8—82no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4C2×D414yes—
156C2×C4⋊C4C2×D414yes—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—48no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—84no—
191C2×C22⋊C4—84no—
192C2×C22⋊C4—84no—
193C2×C22⋊C4—84no—
194C2×C22⋊C4C4⋊C418yes—
195C2×C22⋊C4C22×C414yes—
196C2×C22⋊C4C2×Q814yes—
197C2×C42—28no—
198C2×C42—84no—
199#107—84no—
200#106—84no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8C22×C414yes—
206C22⋊C8C22×C414yes—
207#95—28no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—41no—
215#95—41no—
216#95—84no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—24no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—24no—
225C24—28no—
226C24—28no—
227C24—42no—
228C24—42no—
229C24—44no—
230C24—44no—
231C24—44no—
232C24—44no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24—44no—
238C24—44no—
239C24—82no—
240C24—82no—
241C24—84no—
242C24—84no—
243C24—84no—
244C24—84no—
245C24—88no—
246C24—88no—
247C24—88no—
248C24—88no—
249C24C2×C4⋊C414yes—
250C24C2×C4⋊C412yes—
251C24#12011yescharacteristic
252C24#12011yescharacteristic
253C24#12312yes—
254C24#13512yes—
255C24#13512yes—
256C24AGL(1,ℤ/8)12yes—
257C24#13712yes—
258C24C22×D411yescharacteristic
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—84no—
294C22×C4—82no—
295C22×C4—84no—
296C22×C4C2×C4⋊C414yes—
297C22×C4C2×C4⋊C412yes—
298C22×C4C4×D412yes—
299C22×C4C4×D412yes—
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4C22⋊Q812yes—
305C22×C4C22⋊Q812yes—
306C22×C4C22⋊Q812yes—
307C22×C4C22⋊Q812yes—
308C22×C4#12312yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×Q811yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—84no—
320C2×C8—84no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—84no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—28no—
333C22⋊C4—28no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—88no—
341C22⋊C4—88no—
342C22⋊C4—816no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—84no—
346C42—88no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—48no—
367C23—48no—
368C23—48no—
369C23—48no—
370C23—42no—
371C23—42no—
372C23—48no—
373C23—48no—
374C23—42no—
375C23—42no—
376C23—44no—
377C23—44no—
378C23—82no—
379C23—82no—
380C23—84no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23—88no—
385C23—88no—
386C23—88no—
387C23—88no—
388C23—816no—
389C23—816no—
390C23—816no—
391C23—816no—
392C23#38512yes—
393C23#38512yes—
394C23#42714yes—
395C23#42814yes—
396C23#44912yes—
397C23#44912yes—
398C23#45212yes—
399C23#45212yes—
400C23#48214yes—
401C23#48514yes—
402C23C22×C4⋊C411yescharacteristic
403C23C2×#12011yescharacteristic
404C23C2×#12311yescharacteristic
405C23C2×#13511yescharacteristic
406C23C2×#13511yescharacteristic
407C23C2×AGL(1,ℤ/8)11yescharacteristic
408C23C2×#13711yescharacteristic
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—48no—
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—82no—
438C2×C4—84no—
439C2×C4—88no—
440C2×C4—88no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4#38512yes—
452C2×C4#38512yes—
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C2×C22⋊Q811yescharacteristic
456C2×C4C2×C22⋊Q811yescharacteristic
457C8—44no—
458C8—44no—
459C22—22no—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—21no—
464C22—22no—
465C22—21no—
466C22—22no—
467C22—22no—
468C22—24no—
469C22—22no—
470C22—24no—
471C22—21no—
472C22—21no—
473C22—22no—
474C22—22no—
475C22—44no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—48no—
480C22—44no—
481C22—44no—
482C22—44no—
483C22—84no—
484C22—84no—
485C22—84no—
486C22—816no—
487C22—816no—
488C22—84no—
489C22—84no—
490C22#185414yes—
491C22#185414yes—
492C22C2×#38511yescharacteristic
493C22C2×#42711yescharacteristic
494C22C2×#42811yescharacteristic
495C22C2×#44911yescharacteristic
496C22C2×#45211yescharacteristic
497C22#301912yes—
498C22#302612yes—
499C22C2×#48211yescharacteristic
500C22C2×#48511yescharacteristic
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—22no—
506C4—22no—
507C4—44no—
508C4—84no—
509C4—84no—
510C4—162no—
511C4—162no—
512C2—21no—
513C2—21no—
514C2—21no—
515C2—24no—
516C2—24no—
517C2—21no—
518C2—84no—
519C2#1829612yes—
520C2#1829712yes—
521C2C2×#185411yescharacteristic
522C2#2085611yescharacteristic
523C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21940C2semidirect
3. #21938C2semidirect
6. #8858C2semidirect
7. #8023C2semidirect
8. #8023C2semidirect
26. #1853C22semidirect
28. #1852C22semidirect
31. C2×#323C4semidirect
36. #1294C22semidirect
37. #1294C22semidirect
106. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23122
#219402
#219382
C22×#3232
#143382
#88582
#80232