Finite Group Databasebeta

← #464521

Subgroups — #464521

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic
3C2×#2058C212yesmaximal
4C2×#1920C211yesmaximal, characteristic
5#15194C211yesmaximal, characteristic
6#11384C212yesmaximal
7#11373C211yesmaximal, characteristic
8#11346C211yesmaximal, characteristic
9#11327C212yesmaximal
10#11293C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2212yes—
14C22×#106C2212yes—
15C22×#106C2211yescharacteristic
16#2551—24no—
17#2551—24no—
18C2×#395—22no—
19C2×#389—21no—
20C2×#389C2212yes—
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#2084—24no—
27#2058—28no—
28#1920—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1445—21no—
42#1444—21no—
43#1441—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2211yescharacteristic
46C23×Q8—22no—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4⋊Q8—28no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8—44no—
58C2×C4⋊Q8—44no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—24no—
79C2×#106—24no—
80C2×#106—42no—
81C2×#106D414yes—
82#395—216no—
83#389—28no—
84#389—24no—
85#389—24no—
86#389—24no—
87#389—22no—
88#389—22no—
89#389—42no—
90#384—22no—
91#384D414yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4D414yes—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95D412yes—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—42no—
112#340Q818yescomplemented
113#336—24no—
114#336—22no—
115#336—22no—
116#336—42no—
117C22×Q8—28no—
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8—82no—
123C22×Q8—82no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8C2×D412yes—
151C22×C8C2×Q814yes—
152C4⋊Q8—48no—
153C4⋊Q8—48no—
154C4⋊Q8—48no—
155C4⋊Q8—432no—
156C4⋊Q8—48no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4SD1614yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×Q814yes—
209C2×C4⋊C4C2×Q814yes—
210C2×C4⋊C4#4112yes—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224#106—28no—
225#106—48no—
226#95—28no—
227#95—24no—
228#95—24no—
229#95—44no—
230#95—42no—
231#95—42no—
232#95—81no—
233#95—82no—
234#95—82no—
235C24C22×D411yescharacteristic
236C2×Q8—416no—
237C2×Q8—416no—
238C2×Q8—88no—
239C2×Q8—88no—
240C2×Q8—88no—
241C2×Q8—88no—
242C2×Q8—88no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4#12012yes—
298C22×C4#12012yes—
299C22×C4C2×SD1614yes—
300C22×C4C2×SD1612yes—
301C22×C4AGL(1,ℤ/8)12yes—
302C22×C4#13714yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q812yes—
307C22×C4C2×#4111yescharacteristic
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C4H(5,𝔽2)11yescharacteristic
310C22×C42−1+412yes—
311C22×C42−1+412yes—
312C22×C42−1+412yes—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—28no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—416no—
344C4⋊C4—416no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—48no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—84no—
356C4⋊C4—88no—
357C4⋊C4—84no—
358C4⋊C4#12114yes—
359C4⋊C4C4⋊Q814yes—
360C42—22no—
361C42—22no—
362C42—24no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—48no—
368C23C2×#12011yescharacteristic
369C23#53612yes—
370C23D4211yescharacteristic
371C23#54611yescharacteristic
372C23D4×Q812yes—
373C23C22×SD1611yescharacteristic
374C23C2×AGL(1,ℤ/8)11yescharacteristic
375C23C2×#13712yes—
376C23#57611yescharacteristic
377C23#57712yes—
378C23#57711yescharacteristic
379Q8—832no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4#45014yes—
440C2×C4C2×#12111yescharacteristic
441C2×C4C2×C4⋊Q811yescharacteristic
442C2×C4#53511yescharacteristic
443C2×C4#53611yescharacteristic
444C2×C4#53611yescharacteristic
445C2×C4#53611yescharacteristic
446C2×C4#53612yes—
447C2×C4#53612yes—
448C2×C4#54112yes—
449C2×C4#54412yes—
450C2×C4#54412yes—
451C2×C4D4211yescharacteristic
452C2×C4#54611yescharacteristic
453C2×C4D4×Q812yes—
454C2×C4D4×Q812yes—
455C2×C4#55712yes—
456C2×C4#56312yes—
457C2×C4#56312yes—
458C8—42no—
459C8—44no—
460C8—81no—
461C22#240811yescharacteristic
462C22#243411yescharacteristic
463C22#261411yescharacteristic
464C22#267712yes—
465C22#271312yes—
466C22C2×#45011yescharacteristic
467C22#300411yescharacteristic
468C22#301011yescharacteristic
469C22#301211yescharacteristic
470C22#322912yes—
471C22#323812yes—
472C22#324112yes—
473C22#324412yes—
474C22#326612yes—
475C22#327411yescharacteristic
476C22D4×SD1611yescharacteristic
477C22#329511yescharacteristic
478C22#332012yes—
479C22#332411yescharacteristic
480C22Q8×SD1612yes—
481C22#338412yes—
482C22#338512yes—
483C22#338712yes—
484C4—24no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4—42no—
489C4—44no—
490C4—84no—
491C4—82no—
492C4—84no—
493C4—161no—
494C4#254611yescharacteristic
495C4#262111yescharacteristic
496C4#266011yescharacteristic
497C4#267712yes—
498C4#271312yes—
499C4#282111yescharacteristic
500C2#1650711yescharacteristic
501C2#2424011yescharacteristic
502C2#2425111yescharacteristic
503C2#2583411yescharacteristic
504C2#2584711yescharacteristic
505C2#2964111yescharacteristic
506C2#2964711yescharacteristic
507C2#3094912yes—
508C2#3097012yes—
509C2#3213212yes—
510C2#3215012yes—
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
112. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#20842
C2×#20582
C2×#19202
#151942
#113842
#113732
#113462
#113272
#112932
#108062