Finite Group Databasebeta

← #6433152

Subgroups — #6433152

Order 512.

2500 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2616C211yesmaximal, characteristic
2#34044C211yesmaximal, characteristic
3#29901C218yesmaximal, complemented
4#29900C218yesmaximal, complemented
5#23642C214yesmaximal, complemented
6#23481C214yesmaximal, complemented
7#20911C212yesmaximal, complemented
8#20846C212yesmaximal, complemented
9#20580C211yesmaximal, characteristic
10C23×#41C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×#117C2212yes—
14C2×#428C2214yes—
15C2×#419C2214yes—
16C2×#407C2212yes—
17C2×#407C2214yes—
18#2616—24no—
19#2616C2218yes—
20C2×#395—21no—
21C2×#394—22no—
22C2×#394—21no—
23#2311C2214yes—
24#2311C2214yes—
25#2310—21no—
26#2309—21no—
27#2308—22no—
28#2297—24no—
29#2280C2214yes—
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32#2077C2218yescomplemented
33#2077C2218yescomplemented
34#2076C2218yescomplemented
35#2075C2218yescomplemented
36#2036C2218yescomplemented
37#2036C2218yescomplemented
38#2035C2218yescomplemented
39#2035C2218yescomplemented
40#1873C2218yescomplemented
41#1872C2214yescomplemented
42#1872C2214yescomplemented
43#1854C2218yescomplemented
44#1853C2214yescomplemented
45#1852C2214yescomplemented
46#1780C2218yes—
47#1736C2214yes—
48#1736C2214yes—
49C22×#41—28no—
50C22×#41—48no—
51C22×#41D414yes—
52C22×#41C2314yes—
53C23×Q8D411yescharacteristic
54C23×Q8C2311yescharacteristic
55C23×D4D412yes—
56C23×D4D411yescharacteristic
57C24×C4C2312yes—
58C24×C4C2311yescharacteristic
59C22×M16C2314yes—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—48no—
64C2×C22⋊Q8C2314yes—
65C2×C22⋊Q8C2318yes—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—48no—
69C2×#117C2314yes—
70C2×#117C2314yes—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—21no—
82C22×C42—22no—
83#428—48no—
84C2×#107C2318yes—
85C2×#106C2318yes—
86#419—48no—
87C2×#103C2318yes—
88C2×#103C2318yes—
89#407—28no—
90#407—48no—
91C2×C22⋊C8C2314yes—
92C2×C22⋊C8C2314yes—
93C2×C22⋊C8C2318yes—
94C2×C22⋊C8C2318yes—
95#395—28no—
96#394—28no—
97#394—28no—
98#394—28no—
99#391—24no—
100#391—24no—
101#390—24no—
102#390—24no—
103#386—24no—
104#386—24no—
105#386—24no—
106#386—24no—
107#381—24no—
108#380—24no—
109#380—22no—
110#380C2314yes—
111#380C2314yes—
112#380C2314yes—
113#380C2314yes—
114#380C2318yes—
115#380C2314yes—
116#379—24no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4—24no—
119C2×#95—22no—
120C2×#95—21no—
121#340C2318yes—
122#340C2318yes—
123#340C2318yes—
124#340C2318yes—
125C25C2×D412yes—
126C25C2×D411yescharacteristic
127C2×#41—216no—
128C2×#41—28no—
129C2×#41—28no—
130C2×#41—48no—
131C2×#41—48no—
132C2×#41—44no—
133C2×#41—44no—
134C2×#41—44no—
135C2×#41—44no—
136C2×#41—464no—
137C22×Q8—24no—
138C22×Q8—21no—
139C22×Q8—24no—
140C22×Q8—28no—
141C22×Q8—44no—
142C22×Q8C2×D414yes—
143C22×Q8C2412yes—
144C22×D4—28no—
145C22×D4—24no—
146C22×D4—44no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4C2×D414yes—
150C22×D4C2×D414yes—
151C22×D4C2×D414yes—
152C23×C4—28no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—28no—
156C23×C4—28no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4C2×D412yes—
163C23×C4C2×D414yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4#4118yes—
167C23×C4C2414yes—
168C23×C4C2412yes—
169C23×C4C2411yescharacteristic
170C2×M16—28no—
171C2×M16—28no—
172C2×M16—28no—
173C22×C8C2418yes—
174C22⋊Q8—28no—
175C22⋊Q8—28no—
176C22⋊Q8—48no—
177C22⋊Q8—48no—
178C22⋊Q8—416no—
179C22⋊Q8—48no—
180#117—28no—
181#117—28no—
182#117—44no—
183#117—44no—
184#117—48no—
185#117—48no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4C2414yes—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2412yes—
201C2×C4⋊C4C2414yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—48no—
213C2×C22⋊C4—48no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—82no—
216C2×C22⋊C4—84no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—48no—
220#107—48no—
221#106—48no—
222#103—48no—
223#103—48no—
224C22⋊C8—28no—
225C22⋊C8—28no—
226C22⋊C8—48no—
227C22⋊C8—48no—
228#95—28no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234C24—24no—
235C24—24no—
236C24—24no—
237C24—24no—
238C24—41no—
239C24—42no—
240C24—41no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24#12314yes—
246C24C22×D414yes—
247C24C22×D412yes—
248C24C22×D411yescharacteristic
249#41—464no—
250#41—416no—
251#41—416no—
252#41—416no—
253#41—416no—
254C2×Q8—28no—
255C2×Q8—24no—
256C2×Q8—28no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—416no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—416no—
263C2×D4—28no—
264C2×D4—28no—
265C2×D4—28no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—416no—
273C2×D4—432no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—232no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—28no—
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—416no—
298C22×C4—416no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4#12314yes—
314C22×C4#12318yes—
315C22×C4C22×D414yes—
316C22×C4C22×D412yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4114yes—
319C22×C4C2×#4114yes—
320C22×C4C2×#4114yes—
321C22×C4H(5,𝔽2)18yes—
322C22×C4C2511yescharacteristic
323M16—416no—
324C2×C8—28no—
325C2×C8—28no—
326C2×C8—28no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—28no—
332C4⋊C4—24no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—416no—
336C4⋊C4—42no—
337C4⋊C4—48no—
338C4⋊C4—44no—
339C22⋊C4—48no—
340C22⋊C4—416no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4—84no—
344C22⋊C4—84no—
345C22⋊C4—84no—
346C22⋊C4—84no—
347C22⋊C4—84no—
348C22⋊C4—88no—
349C42—48no—
350C42—48no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—28no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—44no—
369C23—44no—
370C23—48no—
371C23—48no—
372C23—48no—
373C23—84no—
374C23—84no—
375C23—84no—
376C23—84no—
377C23—84no—
378C23—84no—
379C23C2×#12314yes—
380C23C2×#12312yes—
381C23#53414yes—
382C23C23×D411yescharacteristic
383Q8—416no—
384Q8—48no—
385Q8—84no—
386D4—416no—
387D4—416no—
388D4—416no—
389C2×C4—28no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—216no—
393C2×C4—28no—
394C2×C4—28no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—48no—
399C2×C4—416no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—88no—
419C2×C4—84no—
420C2×C4—88no—
421C2×C4—88no—
422C2×C4—84no—
423C2×C4C2×#12314yes—
424C2×C4C2×#12312yes—
425C2×C4#53414yes—
426C2×C4#538116yes—
427C2×C4C22×#4112yes—
428C2×C4C2×H(5,𝔽2)14yes—
429C8—48no—
430C22—24no—
431C22—24no—
432C22—22no—
433C22—22no—
434C22—24no—
435C22—24no—
436C22—24no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—44no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—88no—
449C22—816no—
450C22#262214yes—
451C22C22×#12311yescharacteristic
452C22C2×#53412yes—
453C22#358511yescharacteristic
454C22#358611yescharacteristic
455C22#358612yes—
456C4—24no—
457C4—44no—
458C4—84no—
459C4—84no—
460C4—84no—
461C4—82no—
462C4—82no—
463C4#262214yes—
464C4C2×#53814yes—
465C2—24no—
466C2—44no—
467C2—84no—
468C2C2×#262211yescharacteristic
469C2#6051911yescharacteristic
470C2#6052011yescharacteristic
471C2#6075412yes—
472C2#6075512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29901C2semidirect
4. #29900C2semidirect
5. #23642C2semidirect
6. #23481C2semidirect
7. #20911C2semidirect
8. #20846C2semidirect
32. #2077C22semidirect
33. #2077C22semidirect
34. #2076C22semidirect
35. #2075C22semidirect
36. #2036C22semidirect
37. #2036C22semidirect
38. #2035C22semidirect
39. #2035C22semidirect
40. #1873C22semidirect
41. #1872C22semidirect
42. #1872C22semidirect
43. #1854C22semidirect
44. #1853C22semidirect
45. #1852C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26162
#340442
#299012
#299002
#236422
#234812
#209112
#208462
#205802