Finite Group Databasebeta

← #455490

Subgroups — #455490

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic
2C2×#2066C212yesmaximal
3C2×#1878C211yesmaximal, characteristic
4#15191C211yesmaximal, characteristic
5#13935C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11322C211yesmaximal, characteristic
8#11220C212yesmaximal
9#11077C212yesmaximal
10#11077C211yesmaximal, characteristic
11#11024C212yesmaximal
12C23×Q16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#475—28no—
16C2×#475—24no—
17C22×#106C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2212yes—
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#389C2212yes—
28#2066—28no—
29#1878—24no—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1446—21no—
37#1446—21no—
38#1442—22no—
39#1441—22no—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43#1434—21no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×Q8C2311yescharacteristic
48C22×Q16—22no—
49C22×Q16—22no—
50C22×Q16—42no—
51C22×Q16—42no—
52C22×Q16Q814yescomplemented
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4×Q8—22no—
61C2×C4×Q8—24no—
62C2×C4×Q8—42no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#475—232no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106D418yes—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—42no—
82C2×C4⋊C8—42no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—22no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—22no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93#391—24no—
94#391—24no—
95#391—22no—
96#391—22no—
97#391—42no—
98#391—42no—
99#389—24no—
100#389—42no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114C2×#95D412yes—
115#340—24no—
116#340—22no—
117#340—42no—
118#340—42no—
119#336—22no—
120#336—42no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—44no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—82no—
142C22×Q8C2×D412yes—
143C22×Q8C2×Q812yes—
144C22×Q8C2×Q812yes—
145C22×Q8#4112yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C2×Q16—28no—
158C2×Q16—48no—
159C2×Q16—44no—
160C2×Q16—44no—
161C2×Q16—48no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8C2×D414yes—
178C22×C8C2×Q812yes—
179C4⋊Q8—28no—
180C4⋊Q8—48no—
181C4⋊Q8—44no—
182C4×Q8—28no—
183C4×Q8—44no—
184C4×Q8—48no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—44no—
222C2×C42C2×D412yes—
223C2×C42#4112yes—
224#106—216no—
225C4⋊C8—28no—
226C4⋊C8—48no—
227#103—216no—
228#103—48no—
229#103—44no—
230#103—44no—
231#95—22no—
232#95—24no—
233#95—22no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—42no—
240#95—42no—
241#95—42no—
242#95—82no—
243#95—81no—
244#95—82no—
245C24C22×D411yescharacteristic
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—48no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—88no—
283C2×Q8—88no—
284C2×Q8—84no—
285C2×Q8—84no—
286C2×Q8C22⋊Q814yes—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4#13512yes—
343C22×C4#13714yes—
344C22×C4#13712yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D412yes—
347C22×C4C22×Q811yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4H(5,𝔽2)12yes—
351C22×C4H(5,𝔽2)12yes—
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C42−1+412yes—
354C22×C42−1+411yescharacteristic
355Q16—416no—
356C2×C8—24no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—41no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—88no—
389C4⋊C4#12014yes—
390C42—22no—
391C42—22no—
392C42—24no—
393C42—48no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—42no—
398C23C2×#12111yescharacteristic
399C23#53712yes—
400C23D4212yes—
401C23#54611yescharacteristic
402C23D4×Q811yescharacteristic
403C23C2×#13511yescharacteristic
404C23C2×#13711yescharacteristic
405C23C2×#13712yes—
406C23#57712yes—
407C23#57711yescharacteristic
408C23#57811yescharacteristic
409Q8—24no—
410Q8—24no—
411Q8—42no—
412Q8—44no—
413Q8—44no—
414Q8—42no—
415Q8—42no—
416Q8—42no—
417Q8—44no—
418Q8—816no—
419Q8—81no—
420Q8—82no—
421Q8—81no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4#46414yes—
480C2×C4C2×#12011yescharacteristic
481C2×C4C2×C22⋊Q811yescharacteristic
482C2×C4#53411yescharacteristic
483C2×C4#53511yescharacteristic
484C2×C4#53512yes—
485C2×C4#53611yescharacteristic
486C2×C4#53712yes—
487C2×C4#53711yescharacteristic
488C2×C4#53812yes—
489C2×C4#54212yes—
490C2×C4#54312yes—
491C2×C4#54612yes—
492C2×C4#54812yes—
493C2×C4D4×Q812yes—
494C2×C4#55211yescharacteristic
495C2×C4#55411yescharacteristic
496C2×C4#56212yes—
497C2×C4#56312yes—
498C8—44no—
499C8—42no—
500C8—81no—
501C22#241011yescharacteristic
502C22#241711yescharacteristic
503C22#262011yescharacteristic
504C22#267612yes—
505C22#268312yes—
506C22C2×#46411yescharacteristic
507C22#303911yescharacteristic
508C22#304311yescharacteristic
509C22#304611yescharacteristic
510C22#324812yes—
511C22#325512yes—
512C22#325812yes—
513C22#326312yes—
514C22#326612yes—
515C22#326612yes—
516C22#327612yes—
517C22#328212yes—
518C22#330411yescharacteristic
519C22#331612yes—
520C22#332811yescharacteristic
521C22#338311yescharacteristic
522C22#338712yes—
523C22#338711yescharacteristic
524C4—24no—
525C4—42no—
526C4—44no—
527C4—44no—
528C4—44no—
529C4—84no—
530C4—82no—
531C4—82no—
532C4—161no—
533C4—162no—
534C4#240911yescharacteristic
535C4#261511yescharacteristic
536C4#262611yescharacteristic
537C4#269212yes—
538C4#270212yes—
539C4#281411yescharacteristic
540C2#1626511yescharacteristic
541C2#2435711yescharacteristic
542C2#2437611yescharacteristic
543C2#2481811yescharacteristic
544C2#2483111yescharacteristic
545C2#2983111yescharacteristic
546C2#2984111yescharacteristic
547C2#3091312yes—
548C2#3092912yes—
549C2#3124212yes—
550C2#3124612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
52. C22×Q16Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
C2×#20662
C2×#18782
#151912
#139352
#113842
#113222
#112202
#110772
#110242