Finite Group Databasebeta

← #6434676

Subgroups — #6434676

Order 512.

3221 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic, complemented
2C2×#3093C212yesmaximal
3#36626C211yesmaximal, characteristic
4#29645C218yesmaximal, complemented
5#29644C218yesmaximal, complemented
6#23553C212yesmaximal, complemented
7#23552C212yesmaximal, complemented
8#23453C212yesmaximal, complemented
9#23452C212yesmaximal, complemented
10#20701C212yesmaximal, complemented
11#20682C211yesmaximal, characteristic, complemented
12C2×#533—24no—
13C2×#533—216no—
14C2×#533C2218yes—
15C22×C4⋊Q8C2212yes—
16C22×#125C2212yes—
17C2×#525C2211yescharacteristic
18C22×C22⋊Q8—22no—
19C22×C4×Q8—22no—
20C22×#117C2212yes—
21C23×C42C2211yescharacteristic
22#3093—216no—
23C2×#485C2218yes—
24C2×#482C2218yes—
25C2×#428C2214yes—
26C2×#417C2214yes—
27C2×#407C2212yes—
28#2855C2214yes—
29#2854C2214yes—
30#2853—22no—
31#2849—22no—
32C4×#123—24no—
33#2297C2211yescharacteristic
34#2297C2211yescharacteristic
35#2296C2211yescharacteristic
36#2296C2211yescharacteristic
37#2280C2211yescharacteristic
38#2059C22116yescomplemented
39#2058C2218yescomplemented
40#2057C2218yescomplemented
41#2026C2218yescomplemented
42#2025C2218yescomplemented
43#2024C2218yescomplemented
44#2023C2218yescomplemented
45#1825C22116yes—
46#1817C2218yes—
47#1736C2214yes—
48#1736C2214yes—
49#1728C2212yes—
50#1728C2212yes—
51C22×#41C2311yescharacteristic
52C23×Q8—22no—
53C24×C4D412yes—
54C24×C4C2311yescharacteristic
55C22×M16C2314yes—
56#533—264no—
57#533—264no—
58C2×C4⋊Q8—216no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8C2318yes—
61C2×#125—216no—
62C2×#125—24no—
63C2×#125C2318yes—
64#525—28no—
65C2×#123—24no—
66C2×#123—21no—
67C2×C22⋊Q8—232no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8C2312yes—
71C2×#121C2312yes—
72C2×#120—21no—
73C2×C4×Q8—232no—
74C2×C4×Q8—28no—
75C2×C4×Q8—28no—
76C2×C4×D4—22no—
77C2×#117—28no—
78C2×#117—24no—
79C2×#117D4116yes—
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C2×#117C2311yescharacteristic
83C2×#117C2318yes—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4C2314yes—
87C22×C22⋊C4—22no—
88C22×C42—28no—
89C22×C42—22no—
90C22×C42D412yes—
91C22×C42D414yes—
92C22×C42C2314yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#485—232no—
96#482—232no—
97#428—216no—
98C2×#107C2318yes—
99C2×#106C2318yes—
100#417—216no—
101C2×#103C2318yes—
102C2×#102C2318yes—
103#407—28no—
104C2×C22⋊C8C2318yes—
105#391C2312yes—
106#391C2312yes—
107#390C2312yes—
108#390C2312yes—
109#389C2314yes—
110#389C2314yes—
111#388—24no—
112#388—24no—
113#387—24no—
114#387—24no—
115#386—22no—
116#386—22no—
117#386—24no—
118#385—22no—
119#385—22no—
120#385—24no—
121#381—24no—
122#381—21no—
123#380C2314yes—
124#379—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130#340C23116yes—
131#340C23116yes—
132#336C2318yes—
133#336C2318yes—
134C25C2×D411yescharacteristic
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—24no—
138C22×Q8—216no—
139C22×Q8—28no—
140C22×Q8—416no—
141C22×Q8C2411yescharacteristic
142C22×D4—22no—
143C22×D4—21no—
144C22×D4C2411yescharacteristic
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—21no—
150C23×C4—48no—
151C23×C4C2×D412yes—
152C23×C4C2×D414yes—
153C23×C4C2×D414yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D418yes—
157C23×C4C2×D412yes—
158C23×C4C2412yes—
159C23×C4C2411yescharacteristic
160C2×M16—28no—
161C2×M16—28no—
162C2×M16—28no—
163C22×C8C2418yes—
164C4⋊Q8—232no—
165C4⋊Q8—232no—
166#125—232no—
167#125—232no—
168#123—416no—
169#123—44no—
170C22⋊Q8—264no—
171C22⋊Q8—264no—
172C22⋊Q8—28no—
173#121—28no—
174#120—44no—
175C4×Q8—264no—
176C4×Q8—264no—
177C4×D4—48no—
178#117—232no—
179#117—216no—
180#117—24no—
181#117—24no—
182#117—24no—
183#117C2×D4132yes—
184C2×C4⋊C4—232no—
185C2×C4⋊C4—216no—
186C2×C4⋊C4—216no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4C2×D4116yes—
194C2×C4⋊C4C2412yes—
195C2×C4⋊C4C2412yes—
196C2×C4⋊C4C2412yes—
197C2×C4⋊C4C2418yes—
198C2×C22⋊C4—216no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C42—216no—
209C2×C42—28no—
210C2×C42—216no—
211C2×C42—28no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—216no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C2×D414yes—
219C2×C42C2×D414yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2412yes—
223C2×C42C2412yes—
224#107—216no—
225#106—216no—
226#103—216no—
227#102—216no—
228C22⋊C8—216no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234C24—22no—
235C24—22no—
236C24—21no—
237C24—44no—
238C24#12014yes—
239C24C22×D412yes—
240C24C22×D411yescharacteristic
241#41—416no—
242C2×Q8—232no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—464no—
247C2×Q8—416no—
248C2×Q8—416no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—48no—
253C2×D4—44no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—232no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—216no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—432no—
281C22×C4—416no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4#12014yes—
286C22×C4#12018yes—
287C22×C4C22×D414yes—
288C22×C4C22×D412yes—
289C22×C4C22×D418yes—
290C22×C4C22×D414yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D414yes—
293C22×C42−1+414yes—
294C22×C42−1+414yes—
295C22×C4C2511yescharacteristic
296M16—416no—
297C2×C8—28no—
298C2×C8—28no—
299C2×C8—28no—
300C4⋊C4—264no—
301C4⋊C4—232no—
302C4⋊C4—216no—
303C4⋊C4—28no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—48no—
308C4⋊C4—44no—
309C4⋊C4—48no—
310C4⋊C4C22×D4116yes—
311C22⋊C4—232no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C22⋊C4—42no—
315C22⋊C4—44no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—48no—
320C42—28no—
321C42—28no—
322C42—24no—
323C42—24no—
324C42—232no—
325C42—24no—
326C42—24no—
327C42—42no—
328C42—42no—
329C42—42no—
330C23—24no—
331C23—28no—
332C23—22no—
333C23—22no—
334C23—24no—
335C23—24no—
336C23—48no—
337C23—48no—
338C23—48no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—44no—
343C23C2×#12014yes—
344C23C2×#12012yes—
345C23#53612yes—
346C23#53612yes—
347C23C23×D411yescharacteristic
348Q8—464no—
349Q8—44no—
350D4—44no—
351D4—88no—
352D4—84no—
353C2×C4—216no—
354C2×C4—28no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—216no—
360C2×C4—216no—
361C2×C4—24no—
362C2×C4—216no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—416no—
366C2×C4—416no—
367C2×C4—48no—
368C2×C4—432no—
369C2×C4—416no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—84no—
380C2×C4C2×#12014yes—
381C2×C4C2×#12012yes—
382C2×C4#53612yes—
383C2×C4#53612yes—
384C2×C4#536116yes—
385C2×C4#57518yes—
386C2×C4C23×D412yes—
387C2×C4C2×2−1+412yes—
388C2×C4C2×2−1+412yes—
389C8—48no—
390C22—24no—
391C22—24no—
392C22—24no—
393C22—24no—
394C22—24no—
395C22—24no—
396C22—416no—
397C22—44no—
398C22—44no—
399C22—81no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22#261414yes—
404C22C22×#12011yescharacteristic
405C22C2×#53611yescharacteristic
406C22C2×#53611yescharacteristic
407C22C2×#57514yes—
408C4—24no—
409C4—28no—
410C4—44no—
411C4—416no—
412C4—81no—
413C4—82no—
414C4—82no—
415C4—82no—
416C4#261414yes—
417C4C2×#53614yes—
418C2—24no—
419C2—44no—
420C2—161no—
421C2C2×#261411yescharacteristic
422C2#6035912yes—
423C2#6086212yes—
424C2#6086312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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
1. C22×#533C2semidirect
4. #29645C2semidirect
5. #29644C2semidirect
6. #23553C2semidirect
7. #23552C2semidirect
8. #23453C2semidirect
9. #23452C2semidirect
10. #20701C2semidirect
11. #20682C2semidirect
38. #2059C22semidirect
39. #2058C22semidirect
40. #2057C22semidirect
41. #2026C22semidirect
42. #2025C22semidirect
43. #2024C22semidirect
44. #2023C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#30932
#366262
#296452
#296442
#235532
#235522
#234532
#234522
#207012
#206822