Finite Group Databasebeta

← #6487851

Subgroups — #6487851

Order 512.

1678 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#32960C214yesmaximal
4#29973C214yesmaximal
5#26113C214yesmaximal, complemented
6#26109C214yesmaximal, complemented
7#26047C214yesmaximal, complemented
8#26043C214yesmaximal, complemented
9#21982C211yesmaximal, characteristic
10#21981C211yesmaximal, characteristic
11#21951C211yesmaximal, characteristic
12#20734C211yesmaximal, characteristic
13#20734C211yesmaximal, characteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2212yes—
16C2×#479C2218yes—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2851C2214yes—
20#2646—22no—
21#2571C2212yes—
22#2571C2212yes—
23#2570—22no—
24#2570—22no—
25#2552C2212yes—
26#2552C2212yes—
27Q8×C4⋊C4—22no—
28Q8×C4⋊C4—22no—
29Q8×C4⋊C4C2214yes—
30Q8×C4⋊C4C2214yes—
31#2317—21no—
32#2317—21no—
33C2×#389—22no—
34C2×#389C2212yes—
35#2285C2211yescharacteristic
36#2285C2211yescharacteristic
37C2×#384—21no—
38C2×#384C2211yescharacteristic
39C2×C4×C4⋊C4—21no—
40C2×C4×C4⋊C4C2211yescharacteristic
41#2101C2218yes—
42#2088C2218yes—
43#2086C2218yes—
44#2059C2218yes—
45#1948C2214yes—
46#1947C2214yes—
47#1939C2214yes—
48#1939C2214yes—
49#1865C2218yes—
50#1863C2218yes—
51#1853C2214yes—
52#1852C2214yes—
53#1845C2212yes—
54#1844C2212yes—
55#1834C2212yes—
56#1834C2212yes—
57#1824C2212yes—
58#1824C2212yes—
59#1805C2218yes—
60#1803C2212yes—
61#1803C2212yes—
62#1803C2212yes—
63#1803C2212yes—
64#1797C2214yes—
65#1787C2212yes—
66#1787C2212yes—
67C4×#103C2214yes—
68C4×#103C2214yes—
69C23×Q8C2311yescharacteristic
70C2×#125C2314yes—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—24no—
74C2×C4×Q8—24no—
75C2×C4×Q8—48no—
76C2×C4×Q8Q818yes—
77C2×C4×Q8C2318yes—
78C2×C4×Q8C2312yes—
79C2×C4×Q8C2312yes—
80C2×#117—21no—
81C2×#117—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C42—21no—
92C22×C42—22no—
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#479—416no—
96#431—24no—
97#431—24no—
98C2×#107C2318yes—
99C2×#106C2318yes—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102C2×C4⋊C8C2318yes—
103C2×#103C2314yes—
104C2×#103C2314yes—
105C2×#103C2314yes—
106C2×#103C2314yes—
107C2×C22⋊C8C2312yes—
108C2×C22⋊C8C2312yes—
109C2×C4×C8C2312yes—
110C2×C4×C8C2312yes—
111#400—24no—
112#400—24no—
113#398—24no—
114#398—24no—
115#389—22no—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—24no—
120#389—28no—
121#389—28no—
122#389C2318yes—
123#389C2314yes—
124#389C2314yes—
125#387—24no—
126#387—24no—
127#385—24no—
128#385—24no—
129#384—22no—
130#384—22no—
131#384—24no—
132#384—24no—
133#384—21no—
134#384—21no—
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2312yes—
140#384C2312yes—
141#382—22no—
142#382—22no—
143#382—21no—
144#382—21no—
145#382C2314yes—
146#380—21no—
147#380—21no—
148#380C2312yes—
149#380C2312yes—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4D412yes—
155C4×C4⋊C4D412yes—
156C4×C4⋊C4C2312yes—
157C4×C4⋊C4C2312yes—
158C4×C4⋊C4C2312yes—
159C4×C4⋊C4C2312yes—
160C2×#95—22no—
161C2×#95D412yes—
162#340C2314yes—
163#340C2314yes—
164#340C2314yes—
165#340C2314yes—
166#340C2318yes—
167#340C2318yes—
168#336C2318yes—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—21no—
172C22×Q8—22no—
173C22×Q8C2×Q814yes—
174C22×Q8C2412yes—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C22×C8C2414yes—
184C22×C8C2414yes—
185#125—48no—
186C4×Q8—28no—
187C4×Q8—28no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—216no—
191C4×Q8—48no—
192C4×Q8—416no—
193C4×Q8C2×Q818yes—
194C4×Q8C2×Q818yes—
195#117—44no—
196#117—44no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—84no—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×Q814yes—
233C2×C4⋊C4C2414yes—
234C2×C4⋊C4C2412yes—
235C2×C4⋊C4C2412yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2412yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C42—28no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—28no—
247C2×C42—44no—
248C2×C42—48no—
249C2×C42C2×D414yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×Q814yes—
253C2×C42#4114yes—
254C2×C42C2414yes—
255C2×C42C2411yescharacteristic
256C2×C42C2411yescharacteristic
257#107—48no—
258#106—48no—
259C4⋊C8—24no—
260C4⋊C8—24no—
261C4⋊C8—48no—
262#103—28no—
263#103—28no—
264#103—28no—
265#103—28no—
266C22⋊C8—24no—
267C22⋊C8—24no—
268C4×C8—24no—
269C4×C8—24no—
270#95—28no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95C2×D414yes—
280#95C2×D414yes—
281C24C22×D411yescharacteristic
282C2×Q8—24no—
283C2×Q8—216no—
284C2×Q8—24no—
285C2×Q8—216no—
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8C22×Q814yes—
291C2×Q8H(5,𝔽2)14yes—
292C2×Q82−1+414yes—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4C22⋊Q814yes—
320C22×C4C22⋊Q814yes—
321C22×C4#13514yes—
322C22×C4#13714yes—
323C22×C4C22×D414yes—
324C22×C4C22×D414yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×Q812yes—
331C22×C4C2×#4112yes—
332C22×C4C2511yescharacteristic
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—416no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—88no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4C22×Q814yes—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C42—28no—
375C42—28no—
376C42—28no—
377C42—44no—
378C42—416no—
379C42—44no—
380C42—44no—
381C42C22⋊Q814yes—
382C42C22⋊Q814yes—
383C42C22×Q812yes—
384C42C22×Q812yes—
385C42C2×#4112yes—
386C42C2×#4112yes—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23C2×C22⋊Q812yes—
392C23C2×#13512yes—
393C23C2×#13712yes—
394C23C23×D411yescharacteristic
395Q8—216no—
396Q8—24no—
397Q8—24no—
398Q8—24no—
399Q8#54418yes—
400C2×C4—216no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—84no—
440C2×C4—88no—
441C2×C4#479116yescomplemented
442C2×C4C2×C22⋊Q812yes—
443C2×C4C2×C22⋊Q812yes—
444C2×C4C2×C22⋊Q812yes—
445C2×C4C2×C22⋊Q812yes—
446C2×C4C2×C22⋊Q812yes—
447C2×C4C2×#13512yes—
448C2×C4C2×#13512yes—
449C2×C4C2×#13712yes—
450C2×C4C2×#13712yes—
451C2×C4C23×Q811yescharacteristic
452C2×C4C22×#4111yescharacteristic
453C2×C4C2×H(5,𝔽2)12yes—
454C2×C4C2×2−1+412yes—
455C8—44no—
456C8—44no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22C2×#47914yes—
464C22C22×C22⋊Q811yescharacteristic
465C22#348411yescharacteristic
466C22#348611yescharacteristic
467C22C22×#13511yescharacteristic
468C22C22×#13711yescharacteristic
469C22#358511yescharacteristic
470C22#358611yescharacteristic
471C4—22no—
472C4—22no—
473C4—28no—
474C4—22no—
475C4—44no—
476C4—44no—
477C4—44no—
478C4—82no—
479C4—84no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4C2×#47914yes—
484C4C2×#47914yes—
485C4C2×#54412yes—
486C4#349212yes—
487C2—24no—
488C2#3522411yescharacteristic
489C2C22×#47911yescharacteristic
490C2#6049211yescharacteristic
491C2#6156611yescharacteristic
492C2#6158411yescharacteristic
493C2#6182211yescharacteristic
494C2#6183011yescharacteristic
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
5. #26113C2semidirect
6. #26109C2semidirect
7. #26047C2semidirect
8. #26043C2semidirect
441. C2×C4#479semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×Q8×C4⋊C42
#329602
#299732
#261132
#261092
#260472
#260432
#219822
#219812
#219512
#207342