Finite Group Databasebeta

← #468172

Subgroups — #468172

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic
3C2×#1918C211yesmaximal, characteristic
4#15158C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11380C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11328C212yesmaximal
9#11292C212yesmaximal
10#11219C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2212yes—
17#2546—24no—
18#2546—24no—
19C2×#398—22no—
20C2×#395—21no—
21C2×#391—21no—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#2057—24no—
28#1918—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1441—22no—
42#1441—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8D411yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—28no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8—44no—
55C2×C4⋊Q8—44no—
56C2×C4⋊Q8D414yes—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—42no—
60C2×C4×Q8—42no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—24no—
80C2×#107—42no—
81C2×#107—42no—
82C2×#107Q814yescomplemented
83#398—216no—
84#395—44no—
85#391—28no—
86#389—24no—
87#389—24no—
88#389—24no—
89#389—24no—
90#389—22no—
91#389—22no—
92#389—42no—
93#389—42no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4D414yes—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105#340—24no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—24no—
110#340—42no—
111#340—42no—
112#340Q818yescomplemented
113#336—22no—
114#336—22no—
115#336—42no—
116#336—42no—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—44no—
123C22×Q8—44no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8D814yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2411yescharacteristic
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8C2×Q814yes—
154C22×C8C2×Q812yes—
155C4⋊Q8—48no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C4⋊Q8—432no—
159C4×Q8—48no—
160C4×Q8—44no—
161C4×Q8—44no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×Q814yes—
209C2×C4⋊C4C2×Q812yes—
210C2×C4⋊C4C2×Q812yes—
211C2×C4⋊C4C2×Q814yes—
212C2×C4⋊C4#4112yes—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226#107—28no—
227#107—48no—
228#95—28no—
229#95—28no—
230#95—24no—
231#95—44no—
232#95—42no—
233#95—44no—
234#95—42no—
235#95—81no—
236#95—82no—
237C24C22×D411yescharacteristic
238C2×Q8—28no—
239C2×Q8—48no—
240C2×Q8—44no—
241C2×Q8—48no—
242C2×Q8—44no—
243C2×Q8—48no—
244C2×Q8—48no—
245C2×Q8—48no—
246C2×Q8—88no—
247C2×Q8—88no—
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—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4#12012yes—
306C22×C4#12012yes—
307C22×C4C2×D814yes—
308C22×C4C2×D812yes—
309C22×C4AGL(1,ℤ/8)12yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×Q811yescharacteristic
313C22×C4C22×Q812yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4H(5,𝔽2)12yes—
316C22×C4H(5,𝔽2)12yes—
317C22×C42−1+412yes—
318C22×C42−1+411yescharacteristic
319C22×C42−1+411yescharacteristic
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C4⋊C4—416no—
352C4⋊C4—416no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—88no—
363C4⋊C4—84no—
364C4⋊C4C22⋊Q814yes—
365C4⋊C4C4⋊Q814yes—
366C42—24no—
367C42—22no—
368C42—22no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—48no—
374C23C2×#12011yescharacteristic
375C23#53412yes—
376C23#54811yescharacteristic
377C23D4×Q811yescharacteristic
378C23D4×Q812yes—
379C23C22×D811yescharacteristic
380C23C2×AGL(1,ℤ/8)11yescharacteristic
381C23#57512yes—
382C23#57711yescharacteristic
383C23#57811yescharacteristic
384C23#57812yes—
385Q8—416no—
386Q8—816no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
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—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4#44714yes—
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4C2×C4⋊Q811yescharacteristic
451C2×C4#53412yes—
452C2×C4#53511yescharacteristic
453C2×C4#53511yescharacteristic
454C2×C4#53611yescharacteristic
455C2×C4#53611yescharacteristic
456C2×C4#54212yes—
457C2×C4#54412yes—
458C2×C4#54412yes—
459C2×C4#54412yes—
460C2×C4#54811yescharacteristic
461C2×C4D4×Q812yes—
462C2×C4D4×Q811yescharacteristic
463C2×C4#55412yes—
464C2×C4Q8212yes—
465C2×C4#56312yes—
466C2×C4#56312yes—
467C8—42no—
468C8—44no—
469C8—81no—
470C22#240911yescharacteristic
471C22#243411yescharacteristic
472C22#261311yescharacteristic
473C22#272912yes—
474C22#273412yes—
475C22C2×#44711yescharacteristic
476C22#300511yescharacteristic
477C22#301111yescharacteristic
478C22#301311yescharacteristic
479C22#318612yes—
480C22#319412yes—
481C22#320112yes—
482C22#320212yes—
483C22#335811yescharacteristic
484C22#336711yescharacteristic
485C22#337712yes—
486C22Q8×D811yescharacteristic
487C22Q8×D812yes—
488C22#338411yescharacteristic
489C22#338612yes—
490C22#338812yes—
491C22#339012yes—
492C22#339012yes—
493C4—24no—
494C4—44no—
495C4—44no—
496C4—42no—
497C4—44no—
498C4—82no—
499C4—84no—
500C4—84no—
501C4—82no—
502C4—161no—
503C4#256711yescharacteristic
504C4#262811yescharacteristic
505C4#266011yescharacteristic
506C4#272912yes—
507C4#273412yes—
508C4#282111yescharacteristic
509C2#1660311yescharacteristic
510C2#2432611yescharacteristic
511C2#2434411yescharacteristic
512C2#2583811yescharacteristic
513C2#2584611yescharacteristic
514C2#2959211yescharacteristic
515C2#2960511yescharacteristic
516C2#3256412yes—
517C2#3258612yes—
518C2#3268212yes—
519C2#3269012yes—
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
82. C2×#107Q8semidirect
112. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#20572
C2×#19182
#151582
#113842
#113802
#113282
#112922
#112192
#108062