Finite Group Databasebeta

← #468152

Subgroups — #468152

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic
3C2×#1928C211yesmaximal, characteristic
4#15201C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11380C211yesmaximal, characteristic
7#11293C212yesmaximal
8#11219C212yesmaximal
9#11067C212yesmaximal
10#11067C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18#2567—24no—
19#2567—24no—
20C2×#395—22no—
21C2×#391—21no—
22C2×#391C2211yescharacteristic
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#384—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#2024—24no—
29#1928—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1446—21no—
41#1442—22no—
42#1442—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8D411yescharacteristic
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8D414yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—42no—
58C2×C4×Q8—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×#107—22no—
73C2×#107—24no—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103Q814yescomplemented
82#395—216no—
83#391—28no—
84#391—22no—
85#391—22no—
86#391—42no—
87#391—42no—
88#389—24no—
89#389—24no—
90#389—24no—
91#389—24no—
92#384—28no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106#340—22no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—22no—
114#336—42no—
115#336—42no—
116C22×Q8—22no—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—44no—
131C22×Q8—81no—
132C22×Q8—82no—
133C22×Q8—81no—
134C22×Q8Q1614yes—
135C22×Q8C2×Q812yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8C2×Q812yes—
164C4⋊Q8—48no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4×Q8—48no—
168C4×Q8—44no—
169C4×Q8—44no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C4⋊C4#4112yes—
208C2×C4⋊C4#4114yes—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—44no—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221#107—44no—
222#103—28no—
223#103—48no—
224#103—44no—
225#95—28no—
226#95—28no—
227#95—24no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—81no—
235#95—82no—
236C24C22×D411yescharacteristic
237C2×Q8—28no—
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—22no—
241C2×Q8—24no—
242C2×Q8—48no—
243C2×Q8—44no—
244C2×Q8—48no—
245C2×Q8—44no—
246C2×Q8—48no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—84no—
262C2×Q8—84no—
263C2×Q8—88no—
264C2×Q8—88no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4#12012yes—
319C22×C4#12012yes—
320C22×C4C2×Q1614yes—
321C22×C4C2×Q1612yes—
322C22×C4#13712yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×Q811yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4H(5,𝔽2)12yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+411yescharacteristic
331C22×C42−1+411yescharacteristic
332C22×C42−1+412yes—
333C2×C8—22no—
334C2×C8—24no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—88no—
370C4⋊C4—88no—
371C4⋊C4C22⋊Q814yes—
372C4⋊C4#12414yes—
373C42—24no—
374C42—22no—
375C42—22no—
376C42—48no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C23C2×#12011yescharacteristic
382C23#53412yes—
383C23#54812yes—
384C23#54811yescharacteristic
385C23D4×Q811yescharacteristic
386C23C22×Q1611yescharacteristic
387C23C2×#13711yescharacteristic
388C23#57512yes—
389C23#57611yescharacteristic
390C23#57712yes—
391C23#57711yescharacteristic
392Q8—416no—
393Q8—42no—
394Q8—42no—
395Q8—44no—
396Q8—816no—
397Q8—81no—
398Q8—81no—
399Q8—82no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—81no—
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×C4—81no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4#45114yes—
458C2×C4C2×C22⋊Q811yescharacteristic
459C2×C4C2×#12411yescharacteristic
460C2×C4#53412yes—
461C2×C4#53511yescharacteristic
462C2×C4#53511yescharacteristic
463C2×C4#53611yescharacteristic
464C2×C4#53611yescharacteristic
465C2×C4#54212yes—
466C2×C4#54212yes—
467C2×C4#54212yes—
468C2×C4#54412yes—
469C2×C4#54812yes—
470C2×C4#54811yescharacteristic
471C2×C4D4×Q811yescharacteristic
472C2×C4#55412yes—
473C2×C4#55512yes—
474C2×C4#56312yes—
475C2×C4#56312yes—
476C8—42no—
477C8—81no—
478C8—82no—
479C22#240911yescharacteristic
480C22#243611yescharacteristic
481C22#261311yescharacteristic
482C22#272912yes—
483C22#273112yes—
484C22C2×#45111yescharacteristic
485C22#300511yescharacteristic
486C22#300911yescharacteristic
487C22#301211yescharacteristic
488C22#318912yes—
489C22#319612yes—
490C22#319812yes—
491C22#320012yes—
492C22#336311yescharacteristic
493C22#336312yes—
494C22#336912yes—
495C22#337012yes—
496C22#337511yescharacteristic
497C22#337612yes—
498C22#337612yes—
499C22Q8×Q1611yescharacteristic
500C22#338711yescharacteristic
501C22#338812yes—
502C4—24no—
503C4—44no—
504C4—44no—
505C4—44no—
506C4—42no—
507C4—82no—
508C4—84no—
509C4—82no—
510C4—161no—
511C4—162no—
512C4#256311yescharacteristic
513C4#262811yescharacteristic
514C4#264011yescharacteristic
515C4#272912yes—
516C4#273112yes—
517C4#282111yescharacteristic
518C2#1660211yescharacteristic
519C2#2433311yescharacteristic
520C2#2434311yescharacteristic
521C2#2593611yescharacteristic
522C2#2594011yescharacteristic
523C2#2959711yescharacteristic
524C2#2960411yescharacteristic
525C2#3257012yes—
526C2#3258212yes—
527C2#3263112yes—
528C2#3263812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
81. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#20242
C2×#19282
#152012
#113842
#113802
#112932
#112192
#110672
#108072