Finite Group Databasebeta

← #403645

Subgroups — #403645

Order 512.

1322 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#1921C211yesmaximal, characteristic
4#15194C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11220C212yesmaximal
9#10908C212yesmaximal
10#10884C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#107C2211yescharacteristic
15C22×#106C2212yes—
16#2365—24no—
17#2365—24no—
18C2×#391—21no—
19C2×#389C2211yescharacteristic
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#387—22no—
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#2058—24no—
27#1921—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1445—21no—
41#1444—21no—
42#1441—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8—21no—
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8D414yes—
53C2×C4×Q8—28no—
54C2×C4×Q8—48no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C42—22no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—24no—
77#391—28no—
78#389—24no—
79#389—24no—
80#389—24no—
81#389—22no—
82#389—22no—
83#389—42no—
84#387—216no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4D414yes—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95D412yes—
101#340—24no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340Q818yescomplemented
109#336—24no—
110#336—22no—
111#336—22no—
112#336—42no—
113C22×Q8—28no—
114C22×Q8—24no—
115C22×Q8—44no—
116C22×Q8—42no—
117C22×Q8—42no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8C2×Q814yes—
145C22×C8#4112yes—
146C4⋊Q8—28no—
147C4⋊Q8—28no—
148C4⋊Q8—216no—
149C4×Q8—432no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×Q814yes—
193C2×C4⋊C4C2×Q814yes—
194C2×C4⋊C4#4112yes—
195C2×C4⋊C4#4114yes—
196C2×C4⋊C4#4114yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—28no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—44no—
208C2×C42—44no—
209C2×C42—44no—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212#107—28no—
213#106—48no—
214#95—28no—
215#95—28no—
216#95—24no—
217#95—24no—
218#95—44no—
219#95—42no—
220#95—42no—
221#95—81no—
222#95—82no—
223#95—82no—
224C24C22×D411yescharacteristic
225C2×Q8—216no—
226C2×Q8—48no—
227C2×Q8—416no—
228C2×Q8—416no—
229C2×Q8—48no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
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—41no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4#12012yes—
284C22×C4#12012yes—
285C22×C4#13512yes—
286C22×C4AGL(1,ℤ/8)14yes—
287C22×C4#13712yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×Q812yes—
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4H(5,𝔽2)11yescharacteristic
296C22×C42−1+412yes—
297C22×C42−1+411yescharacteristic
298C2×C8—22no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—42no—
315C4⋊C4—28no—
316C4⋊C4—216no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—416no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—88no—
340C4⋊C4—84no—
341C4⋊C4#12318yes—
342C4⋊C4C4⋊Q814yes—
343C42—24no—
344C42—22no—
345C42—22no—
346C42—48no—
347C42—416no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C23C2×#12011yescharacteristic
353C23#52512yes—
354C23#54611yescharacteristic
355C23#54711yescharacteristic
356C23D4×Q812yes—
357C23C2×#13511yescharacteristic
358C23C2×AGL(1,ℤ/8)12yes—
359C23C2×#13711yescharacteristic
360C23#57712yes—
361C23#57711yescharacteristic
362C23#57811yescharacteristic
363Q8—432no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—28no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4#45214yes—
422C2×C4C2×#12312yes—
423C2×C4#52511yescharacteristic
424C2×C4#52511yescharacteristic
425C2×C4#52511yescharacteristic
426C2×C4C2×C4⋊Q811yescharacteristic
427C2×C4#53312yes—
428C2×C4#53312yes—
429C2×C4#53611yescharacteristic
430C2×C4#53711yescharacteristic
431C2×C4#54112yes—
432C2×C4#54212yes—
433C2×C4#54811yescharacteristic
434C2×C4D4×Q812yes—
435C2×C4#55011yescharacteristic
436C2×C4#55212yes—
437C2×C4#55412yes—
438C2×C4#55612yes—
439C2×C4#55712yes—
440C8—42no—
441C8—44no—
442C8—81no—
443C22#240711yescharacteristic
444C22#243411yescharacteristic
445C22#246212yes—
446C22#246512yes—
447C22#284911yescharacteristic
448C22C2×#45211yescharacteristic
449C22#300311yescharacteristic
450C22#301011yescharacteristic
451C22#301311yescharacteristic
452C22#310512yes—
453C22#311012yes—
454C22#312112yes—
455C22#312612yes—
456C22#330411yescharacteristic
457C22#330612yes—
458C22#332811yescharacteristic
459C22#334211yescharacteristic
460C22#334512yes—
461C22#335611yescharacteristic
462C22#338112yes—
463C22#338412yes—
464C22#338612yes—
465C22#338712yes—
466C4—24no—
467C4—44no—
468C4—48no—
469C4—44no—
470C4—42no—
471C4—82no—
472C4—82no—
473C4—82no—
474C4—84no—
475C4—161no—
476C4#239911yescharacteristic
477C4#242312yes—
478C4#244612yes—
479C4#258411yescharacteristic
480C4#259911yescharacteristic
481C4#263711yescharacteristic
482C2#1487911yescharacteristic
483C2#2412311yescharacteristic
484C2#2412811yescharacteristic
485C2#2583611yescharacteristic
486C2#2584411yescharacteristic
487C2#2685912yes—
488C2#2687512yes—
489C2#2699912yes—
490C2#2701712yes—
491C2#3300911yescharacteristic
492C2#3301511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
108. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#20582
C2×#19212
#151942
#113842
#113462
#113282
#112202
#109082
#108842
#108072