Finite Group Databasebeta

← #381437

Subgroups — #381437

Order 512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2327C211yesmaximal, characteristic
2C2×#2016C211yesmaximal, characteristic
3C2×#1878C212yesmaximal
4#14261C211yesmaximal, characteristic
5#11268C211yesmaximal, characteristic
6#11067C212yesmaximal
7#11052C211yesmaximal, characteristic
8#11024C212yesmaximal
9#11010C211yesmaximal, characteristic
10#10796C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19#2327—28no—
20C2×#391C2212yes—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#387—22no—
26C2×#387—22no—
27C2×#382C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#2016C418yescomplemented
30#1878—28no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1443—22no—
39#1442—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C23×Q8C2311yescharacteristic
43C23×Q8C2312yes—
44C22×Q16—24no—
45C22×Q16—24no—
46C22×Q16—42no—
47C22×Q16—42no—
48C22×Q16—42no—
49C22×Q16—42no—
50C22×Q16C2×C418yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C2×C4×Q8—42no—
60C2×C4×Q8—42no—
61C2×C4×Q8—42no—
62C2×C4×Q8D418yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C42—22no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103C2×C418yes—
86#391—24no—
87#391—24no—
88#391—22no—
89#391—24no—
90#391—42no—
91#391C2×C414yes—
92#389—42no—
93#387—216no—
94#387—216no—
95#382—24no—
96#376—24no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95D412yes—
108C2×#95D412yes—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—22no—
114#336—42no—
115#336C2×C414yes—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8C22×C412yes—
137C22×Q8C22×C414yes—
138C22×Q8C2×D414yes—
139C22×Q8#4114yes—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2411yescharacteristic
151C2×Q16—216no—
152C2×Q16—48no—
153C2×Q16—48no—
154C2×Q16—48no—
155C2×Q16—48no—
156C2×Q16—48no—
157C2×Q16—48no—
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×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C22×C414yes—
172C4⋊Q8—216no—
173C4⋊Q8—216no—
174C4×Q8—48no—
175C4×Q8—48no—
176C4×Q8—48no—
177C4×Q8—48no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
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—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C2×D414yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—28no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—44no—
215C2×C42—44no—
216C2×C42C22×C412yes—
217C2×C42C2×D414yes—
218C2×C42#4112yes—
219#103—48no—
220#103—48no—
221#103—48no—
222#103—48no—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—28no—
227#95—24no—
228#95—24no—
229#95—42no—
230#95—42no—
231#95—42no—
232#95—44no—
233#95—82no—
234#95—82no—
235C24C22×D411yescharacteristic
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
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—41no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8C4×D418yes—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4C4×D414yes—
339C22×C4#12014yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D412yes—
342C22×C4C2×#4112yes—
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)12yes—
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)12yes—
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C42−1+411yescharacteristic
350Q16—432no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C4⋊C4—232no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—416no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—84no—
378C42—24no—
379C42—24no—
380C42—24no—
381C42—24no—
382C42—416no—
383C42—42no—
384C42—42no—
385C23C2×C4×D411yescharacteristic
386C23C2×#12011yescharacteristic
387C23#52511yescharacteristic
388C23#53511yescharacteristic
389C23#53511yescharacteristic
390C23#53511yescharacteristic
391C23#53711yescharacteristic
392C23#57712yes—
393C23#57812yes—
394C23#57812yes—
395C23#57812yes—
396Q8—28no—
397Q8—28no—
398Q8—44no—
399Q8—44no—
400Q8—44no—
401Q8—44no—
402Q8—44no—
403Q8—44no—
404Q8—44no—
405Q8—44no—
406Q8—82no—
407Q8—82no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—44no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4C2×C4×D412yes—
469C2×C4#51811yescharacteristic
470C2×C4#51812yes—
471C2×C4#52011yescharacteristic
472C2×C4#52512yes—
473C2×C4#53412yes—
474C2×C4#53412yes—
475C2×C4#53512yes—
476C2×C4#53712yes—
477C2×C4#53812yes—
478C2×C4#53812yes—
479C2×C4#53812yes—
480C2×C4#53912yes—
481C2×C4#54012yes—
482C2×C4#54011yescharacteristic
483C2×C4#54211yescharacteristic
484C8—44no—
485C8—82no—
486C22C4×#12011yescharacteristic
487C22#232411yescharacteristic
488C22#232811yescharacteristic
489C22#261311yescharacteristic
490C22#261511yescharacteristic
491C22#263111yescharacteristic
492C22#263611yescharacteristic
493C22#294712yes—
494C22#295012yes—
495C22#301112yes—
496C22#301312yes—
497C22#312112yes—
498C22#312712yes—
499C22#322211yescharacteristic
500C22#322311yescharacteristic
501C22#322311yescharacteristic
502C22#322312yes—
503C22#322311yescharacteristic
504C22#322312yes—
505C22#322512yes—
506C22#322512yes—
507C22#325812yes—
508C22#326112yes—
509C4—44no—
510C4—44no—
511C4—48no—
512C4—82no—
513C4—84no—
514C4—82no—
515C4—81no—
516C4—81no—
517C4—161no—
518C4—161no—
519C4#232112yes—
520C4#232512yes—
521C4#263012yes—
522C4#263512yes—
523C2#1429911yescharacteristic
524C2#2146312yes—
525C2#2216412yes—
526C2#2227711yescharacteristic
527C2#2227811yescharacteristic
528C2#2963312yes—
529C2#2967411yescharacteristic
530C2#2967611yescharacteristic
531C2#3005011yescharacteristic
532C2#3005211yescharacteristic
533C2#3014012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
29. #2016C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23272
C2×#20162
C2×#18782
#142612
#112682
#110672
#110522
#110242
#110102
#107962
#107892