Finite Group Databasebeta

← #6477784

Subgroups — #6477784

Order 512.

1648 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#2345C211yesmaximal, characteristic
3#33150C214yesmaximal, complemented
4#29973C214yesmaximal, complemented
5#26002C218yesmaximal, complemented
6#26000C218yesmaximal, complemented
7#21940C211yesmaximal, characteristic
8#21938C211yesmaximal, characteristic
9#21930C211yesmaximal, characteristic
10C2×#1824C211yesmaximal, characteristic
11C2×#1824C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2851C2214yes—
16#2646—22no—
17#2569—22no—
18#2569—22no—
19#2442C2214yes—
20#2442C2214yes—
21#2345—22no—
22#2345—22no—
23#2345C2214yes—
24#2345C2214yes—
25#2315—21no—
26#2315—21no—
27C2×#389C2212yes—
28C2×#387—22no—
29C2×#380—21no—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4C2211yescharacteristic
36#2101C2218yescomplemented
37#2088C2218yescomplemented
38#2086C2218yescomplemented
39#2059C2218yescomplemented
40#1989C2218yescomplemented
41#1949C2218yescomplemented
42#1920C2218yescomplemented
43#1918C2218yescomplemented
44#1853C2214yes—
45#1853C2214yes—
46#1852C2214yes—
47#1852C2214yes—
48#1850C2214yes—
49#1850C2214yes—
50#1824C2212yes—
51#1824C2212yes—
52#1824C2212yes—
53#1824C2212yes—
54#1824C2214yes—
55#1824C2214yes—
56#1808C2214yes—
57#1807C2214yes—
58#1797C2214yes—
59C2×#340C2214yes—
60C2×#340C2214yes—
61C24×C4—21no—
62C24×C4C2311yescharacteristic
63C23×C8C2312yes—
64C23×C8C2312yes—
65C2×#125C2314yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4Q818yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C42—22no—
85C22×C42C2312yes—
86C2×#107C2318yes—
87C2×#106C2318yes—
88C2×C4⋊C8C2318yes—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8C2312yes—
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2312yes—
94C2×C22⋊C8C2312yes—
95#400—24no—
96#400—24no—
97#398—24no—
98#398—24no—
99#389—24no—
100#389C2318yes—
101#389C2314yes—
102#389C2314yes—
103#387—24no—
104#387—24no—
105#387—28no—
106#387—28no—
107#385—24no—
108#385—24no—
109#385—24no—
110#385—24no—
111#382C2314yes—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—24no—
119#380—24no—
120#380—24no—
121#380—24no—
122#380C2312yes—
123#380C2312yes—
124#380C2312yes—
125#380C2312yes—
126#380C2312yes—
127#380C2312yes—
128C4×C4⋊C4—24no—
129C4×C4⋊C4C2318yes—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—24no—
132C4×C22⋊C4D412yes—
133C4×C22⋊C4D412yes—
134C4×C22⋊C4C2312yes—
135C4×C22⋊C4C2312yes—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—21no—
141C2×#95D412yes—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#336C2318yes—
149C25C2×D411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—28no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—44no—
159C23×C4—84no—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×Q814yes—
166C23×C4#4114yes—
167C23×C4C2411yescharacteristic
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—24no—
171C22×C8—24no—
172C22×C8—44no—
173C22×C8—44no—
174C22×C8C2414yes—
175C22×C8C2414yes—
176#125—48no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—84no—
200C2×C4⋊C4C2×Q814yes—
201C2×C4⋊C4C2×Q814yes—
202C2×C4⋊C4C2×Q818yes—
203C2×C4⋊C4C2×Q818yes—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2412yes—
208C2×C4⋊C4C2412yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4C2×D412yes—
221C2×C22⋊C4C2×D412yes—
222C2×C22⋊C4C2×D412yes—
223C2×C22⋊C4C2×D412yes—
224C2×C42—28no—
225C2×C42—44no—
226C2×C42—84no—
227C2×C42C2×D414yes—
228C2×C42C2414yes—
229#107—48no—
230#106—48no—
231C4⋊C8—48no—
232C22⋊C8—24no—
233C22⋊C8—24no—
234C22⋊C8—24no—
235C22⋊C8—24no—
236#95—28no—
237#95—28no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95C2×D414yes—
249#95C2×D414yes—
250C24—42no—
251C24—42no—
252C24C22⋊Q814yes—
253C24#13514yes—
254C24#13714yes—
255C24C22×D411yescharacteristic
256C24C22×D411yescharacteristic
257C24C22×D411yescharacteristic
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4—816no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4C22⋊Q814yes—
296C22×C4C22⋊Q814yes—
297C22×C4C22⋊Q814yes—
298C22×C4C22×D414yes—
299C22×C4C22×D414yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×Q812yes—
304C22×C4C22×Q812yes—
305C22×C4C22×Q812yes—
306C22×C4C2×#4112yes—
307C22×C4C2×#4112yes—
308C22×C4C2×#4112yes—
309C22×C4C2511yescharacteristic
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—28no—
313C2×C8—28no—
314C2×C8—84no—
315C2×C8—84no—
316C4⋊C4—24no—
317C4⋊C4—28no—
318C4⋊C4—28no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—28no—
323C4⋊C4—28no—
324C4⋊C4—28no—
325C4⋊C4—44no—
326C4⋊C4—84no—
327C4⋊C4—84no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—88no—
331C4⋊C4—84no—
332C4⋊C4—84no—
333C4⋊C4C22×Q814yes—
334C4⋊C4C22×Q814yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C42—28no—
346C42—44no—
347C42—88no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—44no—
355C23—44no—
356C23—48no—
357C23—48no—
358C23#479116yes—
359C23C2×C22⋊Q812yes—
360C23C2×C22⋊Q812yes—
361C23C2×C22⋊Q812yes—
362C23C2×#13512yes—
363C23C2×#13512yes—
364C23C2×#13512yes—
365C23C2×#13712yes—
366C23C2×#13712yes—
367C23C2×#13712yes—
368C23C23×D411yescharacteristic
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—84no—
404C2×C4—88no—
405C2×C4—88no—
406C2×C4—88no—
407C2×C4—164no—
408C2×C4—164no—
409C2×C4C2×C22⋊Q812yes—
410C2×C4C2×C22⋊Q812yes—
411C2×C4C2×C22⋊Q812yes—
412C2×C4C23×Q811yescharacteristic
413C2×C4C22×#4111yescharacteristic
414C2×C4#58514yes—
415C8—44no—
416C8—44no—
417C22—24no—
418C22—24no—
419C22—24no—
420C22—24no—
421C22—24no—
422C22—24no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—416no—
430C22C2×#47914yes—
431C22C2×#47914yes—
432C22C2×#47914yes—
433C22C22×C22⋊Q811yescharacteristic
434C22#348512yes—
435C22C22×#13511yescharacteristic
436C22C22×#13711yescharacteristic
437C22#358512yes—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—44no—
442C4—44no—
443C4—44no—
444C4—44no—
445C4—84no—
446C4—84no—
447C4—84no—
448C4#345812yes—
449C4#349112yes—
450C2—24no—
451C2—24no—
452C2—24no—
453C2#3509011yescharacteristic
454C2C22×#47911yescharacteristic
455C2#6049011yescharacteristic
456C2#6169912yes—
457C2#6170912yes—
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. #33150C2semidirect
4. #29973C2semidirect
5. #26002C2semidirect
6. #26000C2semidirect
36. #2101C22semidirect
37. #2088C22semidirect
38. #2086C22semidirect
39. #2059C22semidirect
40. #1989C22semidirect
41. #1949C22semidirect
42. #1920C22semidirect
43. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#23452
#331502
#299732
#260022
#260002
#219402
#219382
#219302
C2×#18242