Finite Group Databasebeta

← #455428

Subgroups — #455428

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×#2085C212yesmaximal
3C2×#1865C211yesmaximal, characteristic
4#15230C211yesmaximal, characteristic
5#13935C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11255C211yesmaximal, characteristic
8#11227C211yesmaximal, characteristic
9#11219C212yesmaximal
10#11190C212yesmaximal
11#11052C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#477—28no—
15C2×#477—24no—
16C22×#107C2212yes—
17C22×#106C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C2×#391C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#382C2212yes—
26#2085—28no—
27#1865—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1446—22no—
37#1441—22no—
38#1441—21no—
39#1441—21no—
40C4⋊#95—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43#1434—21no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—42no—
57C2×C4×Q8D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70#477—232no—
71C2×#107—24no—
72C2×#107—24no—
73C2×#107D418yes—
74C2×#106—24no—
75C2×#106—22no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—22no—
78C2×#103—24no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—22no—
84C2×#103—42no—
85C2×#103Q814yescomplemented
86#391—24no—
87#391—42no—
88#389—24no—
89#389—22no—
90#389—22no—
91#389—42no—
92#382—24no—
93#382—42no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95D411yescharacteristic
105C2×#95D411yescharacteristic
106C2×#95D412yes—
107#340—24no—
108#340—24no—
109#340—22no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—22no—
114#336—42no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8—44no—
124C22×Q8—42no—
125C22×Q8—81no—
126C22×Q8—81no—
127C22×Q8C2×D412yes—
128C22×Q8C2×Q812yes—
129C22×Q8#4112yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8C2×D414yes—
157C22×C8C2×Q812yes—
158C4⋊Q8—28no—
159C4⋊Q8—48no—
160C4⋊Q8—44no—
161C4×Q8—28no—
162C4×Q8—44no—
163C4×Q8—48no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×Q812yes—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—44no—
210C2×C42C2×D412yes—
211C2×C42#4112yes—
212#107—216no—
213#106—44no—
214C4⋊C8—28no—
215#103—216no—
216#103—28no—
217#103—44no—
218#95—22no—
219#95—22no—
220#95—42no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—42no—
227#95—42no—
228#95—81no—
229#95—82no—
230#95—82no—
231C24C22×D411yescharacteristic
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—48no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—84no—
251C2×Q8—84no—
252C2×Q8—88no—
253C2×Q8—88no—
254C2×Q8C22⋊Q814yes—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
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—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#13512yes—
314C22×C4#13712yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D412yes—
317C22×C4C22×Q811yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4H(5,𝔽2)12yes—
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C42−1+412yes—
323C22×C42−1+411yescharacteristic
324C22×C42−1+412yes—
325C2×C8—24no—
326C2×C8—22no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—88no—
362C4⋊C4—84no—
363C4⋊C4#12014yes—
364C42—22no—
365C42—22no—
366C42—24no—
367C42—48no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—42no—
372C23C2×#12111yescharacteristic
373C23#53612yes—
374C23D4212yes—
375C23#54611yescharacteristic
376C23D4×Q811yescharacteristic
377C23C2×#13511yescharacteristic
378C23C2×#13711yescharacteristic
379C23#57512yes—
380C23#57611yescharacteristic
381C23#57711yescharacteristic
382C23#57812yes—
383Q8—24no—
384Q8—24no—
385Q8—42no—
386Q8—44no—
387Q8—44no—
388Q8—42no—
389Q8—816no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4#46414yes—
450C2×C4C2×#12011yescharacteristic
451C2×C4C2×C22⋊Q811yescharacteristic
452C2×C4#53411yescharacteristic
453C2×C4#53611yescharacteristic
454C2×C4#53612yes—
455C2×C4#53611yescharacteristic
456C2×C4#53712yes—
457C2×C4#53711yescharacteristic
458C2×C4#53912yes—
459C2×C4#54112yes—
460C2×C4#54412yes—
461C2×C4#54612yes—
462C2×C4#54812yes—
463C2×C4D4×Q812yes—
464C2×C4#55211yescharacteristic
465C2×C4#55411yescharacteristic
466C2×C4#56212yes—
467C2×C4#56312yes—
468C8—44no—
469C8—42no—
470C8—81no—
471C22#241011yescharacteristic
472C22#241711yescharacteristic
473C22#262111yescharacteristic
474C22#267712yes—
475C22#268112yes—
476C22C2×#46411yescharacteristic
477C22#304011yescharacteristic
478C22#304411yescharacteristic
479C22#304511yescharacteristic
480C22#323312yes—
481C22#323712yes—
482C22#324212yes—
483C22#324312yes—
484C22#326712yes—
485C22#327012yes—
486C22#327312yes—
487C22#328512yes—
488C22#330511yescharacteristic
489C22#331312yes—
490C22#332911yescharacteristic
491C22#338111yescharacteristic
492C22#338511yescharacteristic
493C22#339012yes—
494C4—24no—
495C4—42no—
496C4—44no—
497C4—44no—
498C4—84no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—161no—
503C4—162no—
504C4#240911yescharacteristic
505C4#261611yescharacteristic
506C4#262711yescharacteristic
507C4#269012yes—
508C4#270412yes—
509C4#281411yescharacteristic
510C2#1626411yescharacteristic
511C2#2436111yescharacteristic
512C2#2437311yescharacteristic
513C2#2481411yescharacteristic
514C2#2483211yescharacteristic
515C2#2986711yescharacteristic
516C2#2987511yescharacteristic
517C2#3096112yes—
518C2#3097712yes—
519C2#3112912yes—
520C2#3114312yes—
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
85. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#20852
C2×#18652
#152302
#139352
#113842
#112552
#112272
#112192
#111902
#110522