Finite Group Databasebeta

← #6433103

Subgroups — #6433103

Order 512.

2586 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic
3#29957C218yesmaximal, complemented
4#29953C214yesmaximal, complemented
5#29951C214yesmaximal, complemented
6#23732C212yesmaximal, complemented
7#23619C214yesmaximal, complemented
8#23455C212yesmaximal, complemented
9#20844C212yesmaximal
10#20626C211yesmaximal, characteristic
11#20599C212yesmaximal
12C24×Q8C2211yescharacteristic
13C2×#533C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#428C2214yes—
17C2×#423C2212yes—
18C2×#419C2214yes—
19C2×#419C2214yes—
20C2×#407C2212yes—
21#2857C2212yes—
22#2857C2212yes—
23#2855—22no—
24#2851C2214yes—
25#2848—22no—
26C4×C22⋊Q8—22no—
27C4×C22⋊Q8—22no—
28#2296C2212yes—
29C2×#391—22no—
30C2×#391—22no—
31C2×#391—22no—
32C2×#391C2214yes—
33C2×#391C2214yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36#2285C2211yescharacteristic
37#2280C2212yes—
38C22×#95—22no—
39#2097C2218yescomplemented
40#2096C2214yescomplemented
41#2095C2214yescomplemented
42#2068C22116yescomplemented
43#2066C22116yescomplemented
44#2026C2218yescomplemented
45#2024C2218yescomplemented
46#1853C2218yes—
47#1852C2218yes—
48#1797C2214yes—
49#1797C2214yes—
50#1787C2218yes—
51#1787C2218yes—
52#1736C2214yes—
53#1728C2212yes—
54C23×Q8—22no—
55C23×Q8—21no—
56C23×Q8—24no—
57C23×Q8—44no—
58C23×Q8—44no—
59C23×Q8Q818yescomplemented
60C23×Q8C2312yes—
61C23×Q8C2314yes—
62C24×C4—21no—
63C24×C4—21no—
64C24×C4D412yes—
65C24×C4C2311yescharacteristic
66C22×M16C2314yes—
67#533—44no—
68C2×C4⋊Q8C2312yes—
69C2×#125C2312yes—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C4×Q8—22no—
74C2×#117C2312yes—
75C2×#117C2312yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C42—44no—
82C22×C42D412yes—
83C22×C42D412yes—
84C22×C42D412yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2312yes—
88C22×C42C2312yes—
89#428—48no—
90C2×#107C2318yes—
91C2×#106C2318yes—
92#423—44no—
93C2×C4⋊C8C2318yes—
94#419—48no—
95#419—48no—
96C2×#103C2318yes—
97C2×#103C23116yes—
98C2×#103C2318yes—
99#407—28no—
100C2×C22⋊C8C2318yes—
101#391—22no—
102#391—22no—
103#391—24no—
104#391—22no—
105#391—22no—
106#391—24no—
107#391—24no—
108#391—24no—
109#391—24no—
110#391C2318yes—
111#391C2314yes—
112#391C2314yes—
113#389—22no—
114#389—22no—
115#389C2314yes—
116#389C2314yes—
117#387—24no—
118#387—24no—
119#385—24no—
120#385—24no—
121#385—24no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382C2312yes—
125#381—22no—
126#380—22no—
127#380—22no—
128#380—22no—
129#380C2314yes—
130#380C2314yes—
131#379—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C22⋊C4—24no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95—24no—
139C2×#95—24no—
140C2×#95—24no—
141C2×#95—24no—
142#340C23116yes—
143#336C2314yes—
144#336C2314yes—
145C25C2×D411yescharacteristic
146C22×Q8—216no—
147C22×Q8—28no—
148C22×Q8—24no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—28no—
152C22×Q8—24no—
153C22×Q8—28no—
154C22×Q8—28no—
155C22×Q8—432no—
156C22×Q8—48no—
157C22×Q8—48no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8—48no—
161C22×Q8—48no—
162C22×Q8—432no—
163C22×Q8C2×Q818yes—
164C22×Q8C2×Q8116yes—
165C22×Q8C2414yes—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—28no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—28no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—48no—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×Q814yes—
189C23×C4#4114yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C2×M16—28no—
193C2×M16—28no—
194C2×M16—28no—
195C22×C8C2418yes—
196C4⋊Q8—44no—
197#125—44no—
198C22⋊Q8—48no—
199C22⋊Q8—48no—
200C22⋊Q8—48no—
201C4×Q8—48no—
202#117—28no—
203#117—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2414yes—
218C2×C4⋊C4C2412yes—
219C2×C4⋊C4C2412yes—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—44no—
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2411yescharacteristic
241C2×C42C2411yescharacteristic
242C2×C42C2412yes—
243#107—48no—
244#106—48no—
245C4⋊C8—48no—
246#103—48no—
247#103—48no—
248#103—416no—
249C22⋊C8—216no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259C24—22no—
260C24—22no—
261C24—24no—
262C24—24no—
263C24C22⋊Q814yes—
264C24C22×D411yescharacteristic
265C24C22×D412yes—
266C2×Q8—216no—
267C2×Q8—28no—
268C2×Q8—24no—
269C2×Q8—232no—
270C2×Q8—28no—
271C2×Q8—28no—
272C2×Q8—432no—
273C2×Q8—432no—
274C2×Q8—48no—
275C2×Q8—48no—
276C2×Q8—464no—
277C2×Q8—432no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8—416no—
283C2×Q8—44no—
284C2×Q8C22×Q818yes—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—216no—
290C22×C4—216no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—216no—
294C22×C4—28no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—432no—
318C22×C4—432no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—44no—
327C22×C4—48no—
328C22×C4—816no—
329C22×C4C22⋊Q814yes—
330C22×C4C22⋊Q818yes—
331C22×C4#13718yes—
332C22×C4C22×D412yes—
333C22×C4C22×D412yes—
334C22×C4C22×D412yes—
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×Q814yes—
339C22×C4C22×Q812yes—
340C22×C4C2×#4114yes—
341C22×C4C2×#4112yes—
342C22×C4H(5,𝔽2)14yes—
343C22×C4H(5,𝔽2)14yes—
344C22×C4C2511yescharacteristic
345M16—416no—
346C2×C8—28no—
347C2×C8—28no—
348C2×C8—28no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—28no—
352C4⋊C4—28no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—48no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—48no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C42—22no—
366C42—22no—
367C42—24no—
368C42—24no—
369C42—22no—
370C42—22no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—44no—
376C42—44no—
377C23—216no—
378C23—28no—
379C23—28no—
380C23—216no—
381C23—28no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23C2×C22⋊Q814yes—
389C23C2×C22⋊Q812yes—
390C23#53412yes—
391C23#53512yes—
392C23C2×#13714yes—
393C23#57514yes—
394C23C23×D411yescharacteristic
395Q8—216no—
396Q8—432no—
397Q8—416no—
398Q8—48no—
399Q8—88no—
400C2×C4—28no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—216no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—432no—
424C2×C4—416no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—48no—
436C2×C4—88no—
437C2×C4—816no—
438C2×C4C2×C22⋊Q814yes—
439C2×C4C2×C22⋊Q812yes—
440C2×C4#53412yes—
441C2×C4#53512yes—
442C2×C4#53818yes—
443C2×C4#54318yes—
444C2×C4C2×#13714yes—
445C2×C4C2×#13714yes—
446C2×C4C23×Q811yescharacteristic
447C2×C4C22×#4111yescharacteristic
448C2×C4C2×H(5,𝔽2)12yes—
449C2×C4C2×H(5,𝔽2)12yes—
450C8—48no—
451C22—232no—
452C22—24no—
453C22—24no—
454C22—24no—
455C22—24no—
456C22—24no—
457C22—22no—
458C22—24no—
459C22—22no—
460C22—24no—
461C22—24no—
462C22#262614yes—
463C22#307818yes—
464C22C22×C22⋊Q811yescharacteristic
465C22C2×#53411yescharacteristic
466C22C2×#53511yescharacteristic
467C22C22×#13712yes—
468C22C2×#57512yes—
469C4—24no—
470C4—24no—
471C4—24no—
472C4—42no—
473C4—42no—
474C4—48no—
475C4—84no—
476C4—82no—
477C4—84no—
478C4—82no—
479C4#262614yes—
480C4C2×#53812yes—
481C4C2×#54312yes—
482C2—28no—
483C2—24no—
484C2C2×#262611yescharacteristic
485C2C2×#307812yes—
486C2#6075112yes—
487C2#6079911yescharacteristic
488C2#6080011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29957C2semidirect
4. #29953C2semidirect
5. #29951C2semidirect
6. #23732C2semidirect
7. #23619C2semidirect
8. #23455C2semidirect
39. #2097C22semidirect
40. #2096C22semidirect
41. #2095C22semidirect
42. #2068C22semidirect
43. #2066C22semidirect
44. #2026C22semidirect
45. #2024C22semidirect
59. C23×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#366252
C22×#3912
#299572
#299532
#299512
#237322
#236192
#234552
#208442
#206262
#205992