Finite Group Databasebeta

← #413287

Subgroups — #413287

Order 512.

1400 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#2026C212yesmaximal
3C2×#2024C212yesmaximal
4C2×#1960C211yesmaximal, characteristic
5C2×#1956C211yesmaximal, characteristic
6#14274C211yesmaximal, characteristic
7#11052C212yesmaximal
8#11050C212yesmaximal
9#10843C211yesmaximal, characteristic
10#10842C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2660—24no—
22#2660—24no—
23C2×#398—22no—
24C2×#395—22no—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#2026—28no—
34#2024—28no—
35#1960—24no—
36#1956—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1445—22no—
44#1445—22no—
45C23×Q8—21no—
46C23×Q8C2312yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8Q814yescomplemented
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—24no—
62C2×#125—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95#398—216no—
96#395—216no—
97#391—24no—
98#391—24no—
99#391—24no—
100#391—24no—
101#384—22no—
102#384—22no—
103#384—22no—
104#384D414yes—
105#382—22no—
106#382—22no—
107#382—22no—
108#382—22no—
109#382—22no—
110#382D414yes—
111#376—22no—
112#376—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—24no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D412yes—
120C2×#95D412yes—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#336—24no—
126#336D418yes—
127C22×Q8—24no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—84no—
144C22×Q8C2×Q814yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8C2×D414yes—
169C22×C8C2×D414yes—
170C4⋊Q8—28no—
171C4⋊Q8—28no—
172C4⋊Q8—48no—
173C4⋊Q8—48no—
174C4⋊Q8—44no—
175C4⋊Q8—44no—
176#125—48no—
177#125—48no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—84no—
211C2×C4⋊C4—84no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42C2×D414yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×Q812yes—
226C2×C42#4114yes—
227C2×C42#4112yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230#103—48no—
231#103—48no—
232#103—48no—
233#103—48no—
234#97—44no—
235C4×C8—44no—
236#95—28no—
237#95—28no—
238#95—24no—
239#95—24no—
240#95—44no—
241#95—48no—
242#95—44no—
243#95—44no—
244C24C22×D411yescharacteristic
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—48no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—88no—
275C2×Q8—88no—
276C2×Q8—88no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4#12014yes—
318C22×C4C22⋊Q814yes—
319C22×C4#12414yes—
320C22×C4#12414yes—
321C22×C4#12712yes—
322C22×C4#12712yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D412yes—
326C22×C4C22×Q811yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4H(5,𝔽2)12yes—
332C22×C42−1+412yes—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—416no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—88no—
364C4⋊C4—88no—
365C42—24no—
366C42—24no—
367C42—24no—
368C42—24no—
369C42C22⋊Q814yes—
370C42C22⋊Q814yes—
371C42#12414yes—
372C42#12414yes—
373C23C2×#12011yescharacteristic
374C23C2×C22⋊Q811yescharacteristic
375C23C2×#12411yescharacteristic
376C23C2×#12411yescharacteristic
377C23C2×#12711yescharacteristic
378C23#53511yescharacteristic
379C23#53611yescharacteristic
380C23#57714yes—
381C23#57814yes—
382Q8—44no—
383Q8—44no—
384Q8—44no—
385Q8—44no—
386Q8—816no—
387Q8—82no—
388Q8—82no—
389Q8—82no—
390Q8—82no—
391C2×C4—28no—
392C2×C4—28no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—28no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4#39318yes—
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4C2×C22⋊Q811yescharacteristic
439C2×C4C2×#12411yescharacteristic
440C2×C4C2×#12411yescharacteristic
441C2×C4#53212yes—
442C2×C4#53212yes—
443C2×C4#54212yes—
444C2×C4#54212yes—
445C2×C4D4212yes—
446C2×C4D4212yes—
447C2×C4#54712yes—
448C2×C4#54812yes—
449C2×C4#54812yes—
450C2×C4D4×Q812yes—
451C2×C4#55412yes—
452C2×C4#55512yes—
453C8—44no—
454C8—44no—
455C22C2×#39311yescharacteristic
456C22#243311yescharacteristic
457C22#243611yescharacteristic
458C22#254211yescharacteristic
459C22#256311yescharacteristic
460C22#264011yescharacteristic
461C22#285811yescharacteristic
462C22#300912yes—
463C22#301112yes—
464C22#308214yes—
465C22#313714yes—
466C22#313714yes—
467C22#315412yes—
468C22#315612yes—
469C22#322212yes—
470C22#322312yes—
471C22#324214yes—
472C4—28no—
473C4—44no—
474C4—42no—
475C4—42no—
476C4—84no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4—82no—
481C4—162no—
482C4#245212yes—
483C4#245212yes—
484C4#249112yes—
485C4#249112yes—
486C2#1514411yescharacteristic
487C2#2345712yes—
488C2#2573211yescharacteristic
489C2#2573511yescharacteristic
490C2#2591412yes—
491C2#2852612yes—
492C2#2915312yes—
493C2#3021012yes—
494C2#3322912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
58. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
C2×#20262
C2×#20242
C2×#19602
C2×#19562
#142742
#110522
#110502
#108432
#108422
#106942