Finite Group Databasebeta

← #6482128

Subgroups — #6482128

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3236C212yesmaximal
2#35173C211yesmaximal, characteristic
3C2×#2370C211yesmaximal, characteristic
4#33109C212yesmaximal
5#33108C212yesmaximal
6#29757C212yesmaximal
7#29756C212yesmaximal
8#25467C218yesmaximal, complemented
9#25290C218yesmaximal, complemented
10#21955C211yesmaximal, characteristic
11#21394C212yesmaximal
12C2×#533C2212yes—
13C22×#125C2211yescharacteristic
14#3236—216no—
15C2×#501C2214yes—
16C2×#491C2214yes—
17C2×#484C2214yes—
18C2×#483C2214yes—
19C2×#479C2218yes—
20C2×#432C2212yes—
21#2857C2212yes—
22#2857C2212yes—
23#2659—21no—
24#2659—21no—
25#2584—24no—
26#2567C2214yes—
27#2546C2214yes—
28#2370—24no—
29#2370C2218yes—
30#2316—21no—
31#2316—21no—
32C2×#389—21no—
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#387—21no—
36C2×#387—21no—
37C2×C4×C4⋊C4—21no—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42#2084C2214yes—
43#2083C2214yes—
44#2068C2218yes—
45#2066C2218yes—
46#2043C2214yes—
47#2043C2214yes—
48#1988C2214yes—
49#1981C2214yes—
50#1981C2214yes—
51#1981C2214yes—
52#1948C2214yes—
53#1947C2214yes—
54#1940C2218yes—
55#1928C2218yes—
56#1927C2218yes—
57#1872C2214yes—
58#1872C2214yes—
59#1797C2212yes—
60#1797C2212yes—
61#1763C2218yes—
62#533—28no—
63#533—28no—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×C4⋊Q8C2312yes—
68C2×#125—22no—
69C2×#125—22no—
70C2×#125—24no—
71C2×#125D418yes—
72C2×#125C2314yes—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8C2314yes—
75C2×C4×Q8—22no—
76C2×C4×Q8C2314yes—
77C2×#117—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C42—21no—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#501—216no—
95#491—216no—
96#484—216no—
97#483—216no—
98#479—232no—
99#432—28no—
100C2×#107C2318yes—
101C2×#106C2318yes—
102C2×C4⋊C8C2314yes—
103C2×C4⋊C8C2314yes—
104C2×C4⋊C8C2314yes—
105C2×#103C2318yes—
106C2×#103C2318yes—
107C2×#103C2318yes—
108C2×#103C2318yes—
109C2×C22⋊C8C2314yes—
110C2×#97C2314yes—
111#398—24no—
112#398—24no—
113#395—24no—
114#395—24no—
115#391C2314yes—
116#391C2314yes—
117#389—24no—
118#389—24no—
119#389—24no—
120#389—24no—
121#389—28no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389C2314yes—
129#389C2314yes—
130#387—28no—
131#387—28no—
132#384C2312yes—
133#384C2312yes—
134#380—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4D414yes—
143C4×C4⋊C4C2314yes—
144C4×C4⋊C4C2314yes—
145C4×C4⋊C4C2314yes—
146C4×C4⋊C4C2314yes—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95—21no—
152C2×#95D411yescharacteristic
153C2×#95D411yescharacteristic
154#340C2318yes—
155#340C2318yes—
156#336C2314yes—
157#336C2314yes—
158C22×Q8—22no—
159C22×Q8C2414yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8C2418yes—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—24no—
176C4⋊Q8—44no—
177C4⋊Q8—44no—
178#125—216no—
179#125—28no—
180#125—48no—
181#125C2×D4116yes—
182C22⋊Q8—28no—
183C22⋊Q8—28no—
184C22⋊Q8—48no—
185C4×Q8—28no—
186C4×Q8—28no—
187C4×Q8—48no—
188#117—48no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2412yes—
225C2×C4⋊C4C2414yes—
226C2×C4⋊C4C2412yes—
227C2×C4⋊C4C2412yes—
228C2×C4⋊C4C2412yes—
229C2×C22⋊C4—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—44no—
237C2×C42—82no—
238C2×C42—82no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D414yes—
243C2×C42#4114yes—
244C2×C42C2412yes—
245C2×C42C2412yes—
246C2×C42C2412yes—
247#107—216no—
248#106—216no—
249C4⋊C8—28no—
250C4⋊C8—44no—
251C4⋊C8—44no—
252#103—216no—
253#103—216no—
254#103—216no—
255#103—48no—
256C22⋊C8—28no—
257#97—28no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95C2×D414yes—
268#95C2×D414yes—
269C24C22×D411yescharacteristic
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—28no—
273C2×Q8—28no—
274C2×Q8—44no—
275C2×Q8—82no—
276C2×Q8—82no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4#12114yes—
312C22×C4#12114yes—
313C22×C4#13714yes—
314C22×C4#13714yes—
315C22×C4C22×D412yes—
316C22×C4C22×D412yes—
317C22×C4C22×D412yes—
318C22×C4C22×D412yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D412yes—
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C2×#4112yes—
324C22×C4C2511yescharacteristic
325C2×C8—28no—
326C2×C8—28no—
327C2×C8—28no—
328C4⋊C4—24no—
329C4⋊C4—216no—
330C4⋊C4—216no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—84no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4C22×D414yes—
358C4⋊C4C22×D414yes—
359C4⋊C42−1+414yes—
360C22⋊C4—48no—
361C42—24no—
362C42—24no—
363C42—42no—
364C42—42no—
365C42—48no—
366C42—48no—
367C42—84no—
368C42—84no—
369C42#12118yes—
370C42C22×D414yes—
371C42C2×#4114yes—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23C2×#12112yes—
377C23C2×#13712yes—
378C23C2×#13712yes—
379C23C23×D411yescharacteristic
380Q8—48no—
381Q8—48no—
382Q8—88no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—28no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—48no—
409C2×C4—48no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—84no—
429C2×C4C2×#12114yes—
430C2×C4C2×#12112yes—
431C2×C4C2×#12114yes—
432C2×C4C2×#13714yes—
433C2×C4C2×#13714yes—
434C2×C4C23×D411yescharacteristic
435C2×C4C22×#4111yescharacteristic
436C2×C4C2×2−1+412yes—
437C2×C4#58512yes—
438C8—48no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22#303914yes—
446C22C22×#12111yescharacteristic
447C22#348511yescharacteristic
448C22#348611yescharacteristic
449C22C22×#13711yescharacteristic
450C22C22×#13711yescharacteristic
451C22#358511yescharacteristic
452C22#358611yescharacteristic
453C4—24no—
454C4—24no—
455C4—24no—
456C4—22no—
457C4—24no—
458C4—44no—
459C4—42no—
460C4—42no—
461C4—44no—
462C4—44no—
463C4—84no—
464C4—82no—
465C4—84no—
466C4#346112yes—
467C4#349712yes—
468C2—24no—
469C2#3515311yescharacteristic
470C2C2×#303911yescharacteristic
471C2#6042511yescharacteristic
472C2#6170511yescharacteristic
473C2#6170811yescharacteristic
474C2#6182611yescharacteristic
475C2#6182911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #25467C2semidirect
9. #25290C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32362
#351732
C2×#23702
#331092
#331082
#297572
#297562
#254672
#252902
#219552
#213942