Finite Group Databasebeta

← #444561

Subgroups — #444561

Order 512.

1190 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2550C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic
4#15151C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic
6#11384C212yesmaximal
7#11292C212yesmaximal
8#11208C212yesmaximal
9#10885C212yesmaximal
10#10881C211yesmaximal, characteristic
11#10875C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#97C2211yescharacteristic
15#2550—24no—
16#2550—24no—
17C2×#398—21no—
18C2×#398—21no—
19C2×#398—21no—
20C2×#395—21no—
21C2×#391—21no—
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2100—24no—
31#1992—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—42no—
54C2×#125—24no—
55C2×#125—22no—
56C2×#125—42no—
57C2×#125D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#97—22no—
79C2×#97—42no—
80#398—28no—
81#398—44no—
82#398—44no—
83#395—28no—
84#391—28no—
85#389—28no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—24no—
90#389—24no—
91#389D414yes—
92#384—22no—
93#384—22no—
94#382—22no—
95#382—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#340Q818yescomplemented
120#336—22no—
121#336—24no—
122#336—42no—
123C22×Q8—24no—
124C22×Q8—44no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—82no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8C2×Q814yes—
155C4⋊Q8—48no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158#125—28no—
159#125—44no—
160#125—48no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×Q814yes—
213C2×C4⋊C4C2×Q814yes—
214C2×C4⋊C4#4114yes—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—82no—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42#4114yes—
229#97—44no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—22no—
236#95—22no—
237#95—42no—
238#95—42no—
239#95—44no—
240#95—44no—
241#95—44no—
242C24C22×D411yescharacteristic
243C2×Q8—48no—
244C2×Q8—48no—
245C2×Q8—88no—
246C2×Q8—84no—
247C2×Q8—88no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4#12012yes—
309C22×C4#12012yes—
310C22×C4#12412yes—
311C22×C4#12412yes—
312C22×C4AGL(1,ℤ/8)12yes—
313C22×C4AGL(1,ℤ/8)12yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×Q812yes—
317C22×C4C2×#4112yes—
318C22×C4C2×#4112yes—
319C22×C4H(5,𝔽2)12yes—
320C22×C42−1+412yes—
321C22×C42−1+412yes—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—84no—
361C4⋊C4—88no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4#12314yes—
365C4⋊C4C4⋊Q814yes—
366C42—22no—
367C42—22no—
368C42—24no—
369C42—24no—
370C42—24no—
371C42—41no—
372C42—41no—
373C42—42no—
374C42—84no—
375C23C2×#12011yescharacteristic
376C23C2×#12411yescharacteristic
377C23#53611yescharacteristic
378C23#54612yes—
379C23D4×Q812yes—
380C23C2×AGL(1,ℤ/8)11yescharacteristic
381C23C2×AGL(1,ℤ/8)11yescharacteristic
382C23#57511yescharacteristic
383C23#57511yescharacteristic
384C23#57712yes—
385C23#57812yes—
386Q8—816no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4C2×#12311yescharacteristic
454C2×C4C2×C4⋊Q811yescharacteristic
455C2×C4#53312yes—
456C2×C4#53411yescharacteristic
457C2×C4#53611yescharacteristic
458C2×C4#54112yes—
459C2×C4#54212yes—
460C2×C4#54412yes—
461C2×C4#54612yes—
462C2×C4#54812yes—
463C2×C4#54812yes—
464C2×C4D4×Q812yes—
465C2×C4#55412yes—
466C2×C4#55612yes—
467C2×C4#55712yes—
468C2×C4#56312yes—
469C8—44no—
470C8—82no—
471C22#240511yescharacteristic
472C22#243411yescharacteristic
473C22#243611yescharacteristic
474C22#249212yes—
475C22#271312yes—
476C22#300611yescharacteristic
477C22#300611yescharacteristic
478C22#300911yescharacteristic
479C22#301111yescharacteristic
480C22#313511yescharacteristic
481C22#313712yes—
482C22#313811yescharacteristic
483C22#323511yescharacteristic
484C22#323511yescharacteristic
485C22#324212yes—
486C22#330612yes—
487C22#330812yes—
488C22#332312yes—
489C22#332712yes—
490C22#338412yes—
491C22#338612yes—
492C22#338812yes—
493C22#339012yes—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—44no—
499C4—42no—
500C4—42no—
501C4—44no—
502C4—84no—
503C4—82no—
504C4—82no—
505C4—162no—
506C4#244612yes—
507C4#255012yes—
508C4#271612yes—
509C4#272912yes—
510C2#1578311yescharacteristic
511C2#2404811yescharacteristic
512C2#2405111yescharacteristic
513C2#2584911yescharacteristic
514C2#2585111yescharacteristic
515C2#2591111yescharacteristic
516C2#2591411yescharacteristic
517C2#2782012yes—
518C2#2783012yes—
519C2#3215612yes—
520C2#3216012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
119. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25502
C2×#21002
C2×#19922
#151512
#113852
#113842
#112922
#112082
#108852
#108812
#108752