Finite Group Databasebeta

← #370254

Subgroups — #370254

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C212yesmaximal
2C2×#1947C211yesmaximal, characteristic
3C2×#1940C211yesmaximal, characteristic
4C2×#1776C211yesmaximal, characteristic
5#13911C211yesmaximal, characteristic
6#13804C211yesmaximal, characteristic
7#11268C212yesmaximal
8#11232C212yesmaximal
9#10907C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10843C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2212yes—
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1949—28no—
26#1947—24no—
27#1940C418yescomplemented
28#1776—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1442—21no—
38#1442—21no—
39#1436—22no—
40C4⋊#95—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43#1434—21no—
44C4×#95—21no—
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C22×C42—22no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C2×#107—22no—
63C2×#107—22no—
64C2×#107—22no—
65C2×#107—24no—
66C2×#107—24no—
67C2×#107—24no—
68C2×#106—22no—
69C2×#106—22no—
70C2×#106D414yes—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2×C418yes—
76C2×C4⋊C8C2×C414yes—
77C2×#97—22no—
78C2×#97—22no—
79C2×#97C2×C414yes—
80#384—22no—
81#384—22no—
82#382—22no—
83#382—24no—
84#382—42no—
85#382Q814yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—42no—
91C4×C4⋊C4C2×C414yes—
92C4×C4⋊C4D414yes—
93C4×C4⋊C4Q814yes—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D411yescharacteristic
106C2×#95D411yescharacteristic
107#340—24no—
108#340—24no—
109#340—24no—
110#340—22no—
111#340—22no—
112#340—24no—
113#340—42no—
114#336C2×C418yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8C22×C412yes—
143C22×C8C22×C414yes—
144C22×C8C22×C412yes—
145C22×C8C2×D412yes—
146C22×C8#4112yes—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
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—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C4⋊C418yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×Q814yes—
188C2×C4⋊C4#4112yes—
189C2×C4⋊C4#4112yes—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—44no—
196C2×C42—44no—
197C2×C42—44no—
198C2×C42—42no—
199C2×C42—82no—
200C2×C42—82no—
201C2×C42C22×C414yes—
202C2×C42C22×C412yes—
203C2×C42C2×D412yes—
204C2×C42C2×Q814yes—
205C2×C42C2×Q812yes—
206C2×C42C2×Q812yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209#107—28no—
210#107—48no—
211#106—28no—
212C4⋊C8—216no—
213C4⋊C8—28no—
214#97—28no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—42no—
223#95—82no—
224#95—82no—
225#95—82no—
226#95C4⋊C414yes—
227#95C4⋊C414yes—
228C24C22×D411yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4C2×C4⋊C414yes—
280C22×C4C2×C4⋊C414yes—
281C22×C4C2×C4⋊C412yes—
282C22×C4C2×C4⋊C412yes—
283C22×C4#12112yes—
284C22×C4#12112yes—
285C22×C4C22⋊Q812yes—
286C22×C4C22⋊Q812yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×Q811yescharacteristic
291C22×C4C22×Q811yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4H(5,𝔽2)11yescharacteristic
298C22×C42−1+411yescharacteristic
299C22×C42−1+411yescharacteristic
300C22×C42−1+411yescharacteristic
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8C4×D414yes—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—48no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4#12114yes—
349C42—24no—
350C42—24no—
351C42—24no—
352C42—24no—
353C42—84no—
354C42—84no—
355C42—88no—
356C42C4×D414yes—
357C42C4×Q814yes—
358C42#12514yes—
359C42C4⋊Q814yes—
360C23C22×C4⋊C411yescharacteristic
361C23C2×#12111yescharacteristic
362C23C2×C22⋊Q811yescharacteristic
363C23D4211yescharacteristic
364C23#54711yescharacteristic
365C23#54811yescharacteristic
366C23#54811yescharacteristic
367C23#57512yes—
368C23#57512yes—
369C23#57612yes—
370C23#57812yes—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4#38914yes—
428C2×C4#38914yes—
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×C4×Q811yescharacteristic
432C2×C4C4×#4111yescharacteristic
433C2×C4#51911yescharacteristic
434C2×C4#52011yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C2×C4#52511yescharacteristic
437C2×C4C2×#12511yescharacteristic
438C2×C4#52911yescharacteristic
439C2×C4C2×C4⋊Q811yescharacteristic
440C2×C4#53211yescharacteristic
441C2×C4#53611yescharacteristic
442C2×C4#53711yescharacteristic
443C2×C4#54111yescharacteristic
444C2×C4#54211yescharacteristic
445C2×C4#54711yescharacteristic
446C2×C4D4×Q811yescharacteristic
447C2×C4D4×Q811yescharacteristic
448C2×C4#55211yescharacteristic
449C2×C4#55311yescharacteristic
450C2×C4#55411yescharacteristic
451C2×C4#55411yescharacteristic
452C2×C4#55611yescharacteristic
453C2×C4#55711yescharacteristic
454C2×C4#55711yescharacteristic
455C2×C4#55711yescharacteristic
456C2×C4#55711yescharacteristic
457C8—24no—
458C8—42no—
459C8—44no—
460C22C2×#38911yescharacteristic
461C22D4×C4⋊C411yescharacteristic
462C22#235111yescharacteristic
463C22#241511yescharacteristic
464C22#241611yescharacteristic
465C22#245211yescharacteristic
466C22#245311yescharacteristic
467C22#291112yes—
468C22#291312yes—
469C22#304011yescharacteristic
470C22#304011yescharacteristic
471C22#304111yescharacteristic
472C22#304411yescharacteristic
473C22#308012yes—
474C22#308312yes—
475C22#326812yes—
476C22#326912yes—
477C22#334712yes—
478C22#334912yes—
479C22#337012yes—
480C22#337012yes—
481C22#337712yes—
482C22#337712yes—
483C4—24no—
484C4—24no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4—84no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—162no—
493C4C4×#12111yescharacteristic
494C4#232311yescharacteristic
495C4#236511yescharacteristic
496C4#237011yescharacteristic
497C4#250611yescharacteristic
498C4#250911yescharacteristic
499C4#254611yescharacteristic
500C4#258411yescharacteristic
501C2#1404611yescharacteristic
502C2#2117211yescharacteristic
503C2#2117311yescharacteristic
504C2#2266012yes—
505C2#2274412yes—
506C2#2469311yescharacteristic
507C2#2469911yescharacteristic
508C2#2474311yescharacteristic
509C2#2474911yescharacteristic
510C2#2647712yes—
511C2#2652312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1940C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19492
C2×#19472
C2×#19402
C2×#17762
#139112
#138042
#112682
#112322
#109072
#108502
#108432