Finite Group Databasebeta

← #6643192

Subgroups — #6643192

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2496C211yesmaximal, characteristic
3#33146C212yesmaximal, complemented
4#31686C218yesmaximal, complemented
5#31422C218yesmaximal, complemented
6#29966C212yesmaximal, complemented
7#29296C212yesmaximal, complemented
8#27960C212yesmaximal, complemented
9#27600C212yesmaximal
10#27600C212yesmaximal
11#21902C211yesmaximal, characteristic
12C2×#543—21no—
13C22×#126—21no—
14#2857C2212yes—
15#2857C2212yes—
16#2659—21no—
17#2659—21no—
18#2549—24no—
19#2496—24no—
20#2496C2218yes—
21#2440C2214yes—
22#2440C2214yes—
23C2×#400—21no—
24C2×#397—21no—
25#2315—21no—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#387—21no—
29C2×#384C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35#2099C2214yescomplemented
36#2099C2218yescomplemented
37#2086C2218yescomplemented
38#2083C2214yescomplemented
39#2075C2218yes—
40#2075C2218yes—
41#2057C2214yescomplemented
42#2057C2214yescomplemented
43#1990C2214yescomplemented
44#1988C2214yescomplemented
45#1947C2214yescomplemented
46#1942C2218yescomplemented
47#1923C2218yes—
48#1918C2218yescomplemented
49#1852C2214yes—
50#1852C2214yes—
51#1852C2214yes—
52#1852C2214yes—
53#1834C2214yescomplemented
54#1823C2214yes—
55#1823C2214yes—
56#1823C2214yes—
57#1823C2214yes—
58#1797C2212yes—
59#1779C2212yes—
60C4×C22⋊C8C2212yes—
61C24×C4C2311yescharacteristic
62#543—48no—
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×#126—28no—
66C2×#126—48no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#107C2318yes—
90C2×#107C2318yes—
91C2×C4⋊C8C2314yes—
92C2×C22⋊C8C2314yes—
93C2×C22⋊C8C2314yes—
94C2×C22⋊C8C2314yes—
95C2×C4×C8C2314yes—
96#400—28no—
97#398—24no—
98#398—24no—
99#397—28no—
100#395—24no—
101#395—24no—
102#391—22no—
103#391—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389C2314yes—
109#389C2314yes—
110#389C2314yes—
111#389C2314yes—
112#387—28no—
113#385—24no—
114#385—24no—
115#384—22no—
116#384C2312yes—
117#384C2312yes—
118#384C2314yes—
119#380—22no—
120#380—22no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380—22no—
125#380C2314yes—
126#380C2314yes—
127#380C2314yes—
128#380C2314yes—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2314yes—
131C4×C22⋊C4D414yes—
132C4×C22⋊C4C2314yes—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95D411yescharacteristic
140C2×#95D411yescharacteristic
141#340C2318yes—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#336C2314yes—
148#336C2314yes—
149C25C2×D411yescharacteristic
150C22×Q8—22no—
151C22×Q8—22no—
152C23×C4—24no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2412yes—
168C23×C4C2411yescharacteristic
169C22×C8C2418yes—
170C4⋊Q8—44no—
171C4⋊Q8—44no—
172#126—432no—
173C22⋊Q8—48no—
174C22⋊Q8—48no—
175C22⋊Q8—416no—
176C22⋊Q8—416no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2412yes—
208C2×C4⋊C4C2412yes—
209C2×C4⋊C4C2412yes—
210C2×C4⋊C4C2412yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—48no—
219C2×C22⋊C4D818yes—
220C2×C22⋊C4C2×D414yes—
221C2×C22⋊C4C2×D414yes—
222C2×C42—24no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—44no—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2412yes—
229C2×C42C2412yes—
230#107—48no—
231#107—48no—
232C4⋊C8—44no—
233C22⋊C8—28no—
234C22⋊C8—44no—
235C22⋊C8—44no—
236C4×C8—44no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95C2×D414yes—
247#95C2×D414yes—
248C24—22no—
249C24—22no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24C2×D814yes—
255C24C22×D412yes—
256C24C22×D411yescharacteristic
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—82no—
260C2×Q8—82no—
261C2×Q8—82no—
262C2×Q8—82no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—88no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4C2×D814yes—
309C22×C4C2×D814yes—
310C22×C4#13714yes—
311C22×C4C22×D412yes—
312C22×C4C22×D412yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D412yes—
315C22×C4C22×D412yes—
316C22×C4C22×D412yes—
317C22×C42−1+414yes—
318C22×C4C2511yescharacteristic
319C2×C8—28no—
320C2×C8—28no—
321C2×C8—28no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—44no—
336C4⋊C4—48no—
337C4⋊C4—44no—
338C4⋊C4—48no—
339C4⋊C4—48no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C22⋊C4—28no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4C2×D8116yes—
352C42—24no—
353C42—24no—
354C42—42no—
355C42—42no—
356C42—48no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—28no—
368C23—44no—
369C23—44no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23#53612yes—
375C23C22×D814yes—
376C23C22×D812yes—
377C23C2×#13712yes—
378C23C23×D411yescharacteristic
379Q8—164no—
380Q8—164no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—28no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—48no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—84no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—164no—
426C2×C4—164no—
427C2×C4#53612yes—
428C2×C4C22×D814yes—
429C2×C4C22×D814yes—
430C2×C4C2×#13714yes—
431C2×C4C2×2−1+412yes—
432C2×C4#58512yes—
433C2×C4#58512yes—
434C8—48no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—88no—
449C22#322614yes—
450C22C2×#53611yescharacteristic
451C22#348511yescharacteristic
452C22#348511yescharacteristic
453C22C23×D811yescharacteristic
454C22C22×#13711yescharacteristic
455C22#358611yescharacteristic
456C22#358611yescharacteristic
457C4—24no—
458C4—22no—
459C4—24no—
460C4—24no—
461C4—42no—
462C4—42no—
463C4—82no—
464C4—82no—
465C4—82no—
466C4—82no—
467C4—84no—
468C4—84no—
469C4#356414yes—
470C2—24no—
471C2—44no—
472C2#3697011yescharacteristic
473C2C2×#322611yescharacteristic
474C2#6086911yescharacteristic
475C2#6169411yescharacteristic
476C2#6169411yescharacteristic
477C2#6170511yescharacteristic
478C2#6172311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33146C2semidirect
4. #31686C2semidirect
5. #31422C2semidirect
6. #29966C2semidirect
7. #29296C2semidirect
8. #27960C2semidirect
35. #2099C22semidirect
36. #2099C22semidirect
37. #2086C22semidirect
38. #2083C22semidirect
41. #2057C22semidirect
42. #2057C22semidirect
43. #1990C22semidirect
44. #1988C22semidirect
45. #1947C22semidirect
46. #1942C22semidirect
48. #1918C22semidirect
53. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#24962
#331462
#316862
#314222
#299662
#292962
#279602
#276002
#219022