Finite Group Databasebeta

← #453314

Subgroups — #453314

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#1921C212yesmaximal
3#13926C211yesmaximal, characteristic
4#11384C212yesmaximal
5#11380C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11219C212yesmaximal
8#11207C212yesmaximal
9#10907C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×#107C2212yes—
13#2567—28no—
14C2×#395—21no—
15C2×#395—21no—
16C2×#391—21no—
17C2×#391—21no—
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#384—21no—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1921—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1441—21no—
37#1441—21no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×Q8—21no—
42C23×Q8D411yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4×Q8—24no—
50C2×C4×Q8—24no—
51C2×C4×Q8—41no—
52C2×C4×Q8—41no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—24no—
71C2×#107—24no—
72C2×#107Q818yescomplemented
73#395—28no—
74#395—28no—
75#391—28no—
76#391—44no—
77#389—24no—
78#389—24no—
79#389—24no—
80#389D418yes—
81#384—28no—
82#384—41no—
83#384—41no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—24no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—42no—
100#340—42no—
101#336—24no—
102#336—24no—
103#336—41no—
104#336—41no—
105C22×Q8—24no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8—82no—
115C22×Q8—82no—
116C22×Q8Q1614yes—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C22×C8—22no—
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×C8C2×Q814yes—
142C4⋊Q8—48no—
143C4⋊Q8—48no—
144C4×Q8—48no—
145C4×Q8—48no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
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—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2×Q814yes—
191C2×C4⋊C4C2×Q814yes—
192C2×C4⋊C4#4114yes—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—24no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42C2×D414yes—
210C2×C42#4114yes—
211#107—216no—
212#95—24no—
213#95—28no—
214#95—28no—
215#95—44no—
216#95—42no—
217#95—44no—
218#95—82no—
219#95—82no—
220C24C22×D411yescharacteristic
221C2×Q8—28no—
222C2×Q8—28no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—44no—
228C2×Q8—48no—
229C2×Q8—88no—
230C2×Q8—88no—
231C2×Q8—84no—
232C2×Q8—84no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
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×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×C4#12012yes—
289C22×C4#12012yes—
290C22×C4#12312yes—
291C22×C4#12312yes—
292C22×C4C2×Q1612yes—
293C22×C4C2×Q1612yes—
294C22×C4C2×Q1612yes—
295C22×C4AGL(1,ℤ/8)12yes—
296C22×C4#13712yes—
297C22×C4C22×D412yes—
298C22×C4C22×Q812yes—
299C22×C4C2×#4112yes—
300C22×C4C2×#4112yes—
301C22×C4H(5,𝔽2)11yescharacteristic
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C42−1+412yes—
304C22×C42−1+412yes—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—48no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—88no—
339C4⋊C4—84no—
340C4⋊C4—88no—
341C4⋊C4—84no—
342C4⋊C4C22⋊Q818yes—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—24no—
347C42—48no—
348C42—42no—
349C42—42no—
350C42—84no—
351C23C2×#12011yescharacteristic
352C23C2×#12311yescharacteristic
353C23#53411yescharacteristic
354C23#54812yes—
355C23D4×Q812yes—
356C23C22×Q1611yescharacteristic
357C23C2×AGL(1,ℤ/8)11yescharacteristic
358C23C2×#13711yescharacteristic
359C23#57511yescharacteristic
360C23#57612yes—
361C23#57712yes—
362Q8—416no—
363Q8—816no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—28no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4#45114yes—
427C2×C4#48414yes—
428C2×C4C2×C22⋊Q812yes—
429C2×C4#53312yes—
430C2×C4#53612yes—
431C2×C4#54112yes—
432C2×C4#54212yes—
433C2×C4#54412yes—
434C2×C4#54612yes—
435C2×C4#54612yes—
436C2×C4#54812yes—
437C2×C4D4×Q812yes—
438C2×C4#55511yescharacteristic
439C2×C4#55612yes—
440C2×C4#55712yes—
441C2×C4Q8211yescharacteristic
442C2×C4#56311yescharacteristic
443C2×C4#56311yescharacteristic
444C8—44no—
445C8—82no—
446C22#240511yescharacteristic
447C22#240912yes—
448C22#244612yes—
449C22#273111yescharacteristic
450C22#273411yescharacteristic
451C22C2×#45111yescharacteristic
452C22#300611yescharacteristic
453C22#301212yes—
454C22C2×#48411yescharacteristic
455C22#309511yescharacteristic
456C22#309612yes—
457C22#318911yescharacteristic
458C22#319311yescharacteristic
459C22#319911yescharacteristic
460C22#320011yescharacteristic
461C22#336312yes—
462C22#336612yes—
463C22#337512yes—
464C22#337612yes—
465C22Q8×Q1612yes—
466C22#338412yes—
467C22#338712yes—
468C22#338912yes—
469C4—24no—
470C4—24no—
471C4—48no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—84no—
476C4—81no—
477C4—84no—
478C4—162no—
479C4#243912yes—
480C4#245112yes—
481C4#271312yes—
482C4#271612yes—
483C2#1609311yescharacteristic
484C2#2400611yescharacteristic
485C2#2401511yescharacteristic
486C2#2433312yes—
487C2#2434712yes—
488C2#2628212yes—
489C2#2629512yes—
490C2#3263111yescharacteristic
491C2#3264711yescharacteristic
492C2#3268411yescharacteristic
493C2#3269411yescharacteristic
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
72. C2×#107Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#19212
#139262
#113842
#113802
#112932
#112192
#112072
#109072