Finite Group Databasebeta

← #375896

Subgroups — #375896

Order 512.

1322 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic
2C2×#2026C211yesmaximal, characteristic
3C2×#1863C212yesmaximal
4#13786C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic
6#11324C211yesmaximal, characteristic
7#11050C211yesmaximal, characteristic
8#10932C212yesmaximal
9#10907C212yesmaximal
10#10884C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17#2330—28no—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#389—22no—
21C2×#389C2212yes—
22C2×#387—22no—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#2026C418yescomplemented
28#1863—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1436—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41#1434—22no—
42C23×Q8C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#125—24no—
46C2×#125—24no—
47C2×#125—22no—
48C2×#125C2×C414yes—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—24no—
52C2×C4×Q8—42no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C42—21no—
63C22×C42—22no—
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#107—24no—
69C2×#107—24no—
70C2×#107—42no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—42no—
76C2×#103C2×C418yes—
77#391—22no—
78#391—22no—
79#391C2×C414yes—
80#391Q814yes—
81#389—216no—
82#389—24no—
83#389D418yes—
84#387—216no—
85#384—24no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—42no—
89#376—24no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98C2×#95D412yes—
99#340—24no—
100#340—24no—
101#340—42no—
102#340—42no—
103#340C2×C418yes—
104#336—24no—
105#336—22no—
106#336—24no—
107#336C2×C414yes—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8C22×C412yes—
116C22×Q8C2×Q812yes—
117C22×Q8#4112yes—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2411yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8C22×C414yes—
143#125—216no—
144#125—216no—
145C4×Q8—48no—
146C4×Q8—48no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C414yes—
178C2×C4⋊C4C2×D414yes—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4C2×Q812yes—
181C2×C4⋊C4#4112yes—
182C2×C4⋊C4#4114yes—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—28no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—82no—
196C2×C42—82no—
197C2×C42—82no—
198C2×C42C22×C412yes—
199C2×C42C2×D414yes—
200C2×C42C2×Q812yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203#107—48no—
204#103—216no—
205#103—48no—
206#95—28no—
207#95—28no—
208#95—28no—
209#95—24no—
210#95—24no—
211#95—28no—
212#95—42no—
213#95—82no—
214#95—82no—
215C24C22×D411yescharacteristic
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—24no—
220C2×Q8—24no—
221C2×Q8—24no—
222C2×Q8—24no—
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8C4×Q814yes—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
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—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C4×D414yes—
289C22×C4#12012yes—
290C22×C4#12012yes—
291C22×C4C22⋊Q814yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D412yes—
294C22×C4C22×Q811yescharacteristic
295C22×C4C2×#4112yes—
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4H(5,𝔽2)12yes—
301C22×C42−1+412yes—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—232no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—48no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—88no—
332C4⋊C4—88no—
333C4⋊C4C4×D418yes—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—48no—
341C42—88no—
342C42—84no—
343C23C2×C4×D411yescharacteristic
344C23C2×#12011yescharacteristic
345C23C2×C22⋊Q811yescharacteristic
346C23#52511yescharacteristic
347C23#52511yescharacteristic
348C23#53511yescharacteristic
349C23#53611yescharacteristic
350C23#57612yes—
351C23#57712yes—
352C23#57812yes—
353C23#57812yes—
354Q8—28no—
355Q8—44no—
356Q8—44no—
357Q8—44no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—28no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4C2×C4×D412yes—
419C2×C4C2×C4×Q811yescharacteristic
420C2×C4C4×#4111yescharacteristic
421C2×C4#52012yes—
422C2×C4#52512yes—
423C2×C4#52911yescharacteristic
424C2×C4#53311yescharacteristic
425C2×C4#54212yes—
426C2×C4#54612yes—
427C2×C4#54612yes—
428C2×C4#54812yes—
429C2×C4D4×Q812yes—
430C2×C4#55212yes—
431C2×C4#55412yes—
432C2×C4#55612yes—
433C2×C4#55912yes—
434C8—44no—
435C8—82no—
436C22C4×#12011yescharacteristic
437C22C4×C22⋊Q811yescharacteristic
438C22#232611yescharacteristic
439C22#240411yescharacteristic
440C22#240911yescharacteristic
441C22#258011yescharacteristic
442C22#259911yescharacteristic
443C22#294712yes—
444C22#295112yes—
445C22#301112yes—
446C22#301212yes—
447C22#308212yes—
448C22#308312yes—
449C22#311912yes—
450C22#312012yes—
451C22#312412yes—
452C22#312712yes—
453C22#322011yescharacteristic
454C22#322111yescharacteristic
455C22#322311yescharacteristic
456C22#322311yescharacteristic
457C22#324212yes—
458C22#324312yes—
459C4—44no—
460C4—42no—
461C4—42no—
462C4—44no—
463C4—48no—
464C4—44no—
465C4—82no—
466C4—82no—
467C4—84no—
468C4—82no—
469C4#236012yes—
470C4#236712yes—
471C4#245812yes—
472C4#246512yes—
473C2#1417211yescharacteristic
474C2#2146512yes—
475C2#2162112yes—
476C2#2222811yescharacteristic
477C2#2223111yescharacteristic
478C2#2394111yescharacteristic
479C2#2394311yescharacteristic
480C2#2431112yes—
481C2#2926611yescharacteristic
482C2#2926811yescharacteristic
483C2#2940712yes—
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
27. #2026C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23302
C2×#20262
C2×#18632
#137862
#113852
#113242
#110502
#109322
#109072
#108842
#107962