Finite Group Databasebeta

← #403415

Subgroups — #403415

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic
3C2×#1927C211yesmaximal, characteristic
4#15201C211yesmaximal, characteristic
5#11380C211yesmaximal, characteristic
6#11357C211yesmaximal, characteristic
7#11325C212yesmaximal
8#11052C212yesmaximal
9#10932C212yesmaximal
10#10907C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18#2355—24no—
19#2355—24no—
20C2×#391—21no—
21C2×#391C2212yes—
22C2×#391C2211yescharacteristic
23C2×#387—22no—
24C2×#384C2211yescharacteristic
25C2×#382C2212yes—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2095—24no—
30#1927—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40#1446—21no—
41#1442—21no—
42#1442—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C42—21no—
64C22×C42—22no—
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C2×#107—24no—
68C2×#107—24no—
69C2×#107—42no—
70C2×#106—22no—
71C2×#106—42no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—22no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103Q814yescomplemented
80#391—28no—
81#391—22no—
82#391—22no—
83#391—24no—
84#391—24no—
85#387—216no—
86#384—22no—
87#384D414yes—
88#382—24no—
89#382—24no—
90C4×C4⋊C4—28no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107#340—24no—
108#340—22no—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—22no—
114#336—24no—
115#336—24no—
116C22×Q8—22no—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—42no—
130C22×Q8C2×Q812yes—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—21no—
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—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8C2×Q812yes—
158C22×C8#4114yes—
159C4×Q8—28no—
160C4×Q8—28no—
161C4×Q8—216no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
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—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—81no—
192C2×C4⋊C4—81no—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×Q812yes—
197C2×C4⋊C4#4112yes—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42—28no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—82no—
209C2×C42—82no—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42#4114yes—
213C2×C42#4114yes—
214#107—48no—
215#106—44no—
216#103—28no—
217#103—48no—
218#103—44no—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—24no—
223#95—44no—
224#95—42no—
225#95—42no—
226#95—44no—
227#95—44no—
228#95—42no—
229#95—82no—
230#95—81no—
231C24C22×D411yescharacteristic
232C2×Q8—28no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—48no—
238C2×Q8—48no—
239C2×Q8—44no—
240C2×Q8—41no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—42no—
244C2×Q8—44no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4#12012yes—
310C22×C4#12012yes—
311C22×C4#12314yes—
312C22×C4#12314yes—
313C22×C4#13512yes—
314C22×C4#13712yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×Q811yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4112yes—
320C22×C4C2×#4112yes—
321C22×C4C2×#4112yes—
322C22×C4H(5,𝔽2)12yes—
323C22×C42−1+411yescharacteristic
324C22×C42−1+411yescharacteristic
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—42no—
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—28no—
343C4⋊C4—216no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—48no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—84no—
363C4⋊C4C22⋊Q814yes—
364C42—22no—
365C42—22no—
366C42—24no—
367C42—24no—
368C42—24no—
369C42—24no—
370C42—24no—
371C42—48no—
372C42—88no—
373C23C2×#12011yescharacteristic
374C23C2×#12312yes—
375C23#54811yescharacteristic
376C23D4×Q811yescharacteristic
377C23#55012yes—
378C23C2×#13511yescharacteristic
379C23C2×#13711yescharacteristic
380C23#57512yes—
381C23#57611yescharacteristic
382C23#57711yescharacteristic
383C23#57812yes—
384Q8—416no—
385Q8—42no—
386Q8—42no—
387Q8—44no—
388Q8—81no—
389Q8—81no—
390Q8—82no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
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—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4#45214yes—
451C2×C4C2×C22⋊Q811yescharacteristic
452C2×C4#52511yescharacteristic
453C2×C4#52511yescharacteristic
454C2×C4#52512yes—
455C2×C4#52912yes—
456C2×C4#53211yescharacteristic
457C2×C4#53312yes—
458C2×C4#53511yescharacteristic
459C2×C4#53611yescharacteristic
460C2×C4#54212yes—
461C2×C4#54212yes—
462C2×C4#54612yes—
463C2×C4#54811yescharacteristic
464C2×C4D4×Q811yescharacteristic
465C2×C4#55212yes—
466C2×C4#55412yes—
467C2×C4#55612yes—
468C2×C4#55912yes—
469C8—42no—
470C8—44no—
471C8—81no—
472C22#240911yescharacteristic
473C22#242312yes—
474C22#243111yescharacteristic
475C22#244812yes—
476C22#284911yescharacteristic
477C22C2×#45211yescharacteristic
478C22#300511yescharacteristic
479C22#301111yescharacteristic
480C22#301211yescharacteristic
481C22#309112yes—
482C22#309512yes—
483C22#309712yes—
484C22#309812yes—
485C22#336111yescharacteristic
486C22#337211yescharacteristic
487C22#337712yes—
488C22#338111yescharacteristic
489C22#338511yescharacteristic
490C22#339012yes—
491C22#339312yes—
492C22#339812yes—
493C22#340112yes—
494C22#340312yes—
495C4—24no—
496C4—44no—
497C4—44no—
498C4—44no—
499C4—42no—
500C4—82no—
501C4—82no—
502C4—84no—
503C4—161no—
504C4—162no—
505C4#239611yescharacteristic
506C4#245812yes—
507C4#246512yes—
508C4#258011yescharacteristic
509C4#260411yescharacteristic
510C4#263211yescharacteristic
511C2#1487511yescharacteristic
512C2#2433111yescharacteristic
513C2#2434411yescharacteristic
514C2#2506712yes—
515C2#2507512yes—
516C2#2563211yescharacteristic
517C2#2563911yescharacteristic
518C2#2632512yes—
519C2#2633312yes—
520C2#3298411yescharacteristic
521C2#3298911yescharacteristic
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
79. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#20952
C2×#19272
#152012
#113802
#113572
#113252
#110522
#109322
#109072
#108062