Finite Group Databasebeta

← #6343642

Subgroups — #6343642

Order 512.

3329 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C2×#2972C211yesmaximal, characteristic
3#34181C211yesmaximal, characteristic
4#17614C2116yesmaximal, complemented
5#17575C214yesmaximal, complemented
6C2×#1520C218yesmaximal, complemented
7#3501—24no—
8#3501—24no—
9#3501C418yescomplemented
10#3501C2218yes—
11C2×#554—24no—
12C2×#554C414yescomplemented
13C2×D4×Q8C414yescomplemented
14C2×#544—21no—
15C2×#543—22no—
16C2×#533C2212yes—
17C22×C22⋊Q8C2214yes—
18C2×#518—21no—
19C2×#518—21no—
20C22×C4×D4C2211yescharacteristic
21#2972—24no—
22#2972C2218yes—
23C2×C8×D4—24no—
24C2×#431C2212yes—
25C22×C22⋊C8C2214yes—
26#2349C2218yes—
27D4×C4⋊C4—24no—
28#2316—21no—
29#2315—22no—
30#2285C2212yes—
31C2×#380C2214yes—
32#1520C22116yes—
33#1520C22132yescomplemented
34#1518C22116yescomplemented
35#1503C22116yescomplemented
36C2×#328C22132yescomplemented
37C23×Q8C2×C414yescomplemented
38C23×D4D411yescharacteristic
39C24×C4C2312yes—
40C23×C8—24no—
41#554—232no—
42#554—216no—
43#554—232no—
44#554C2×C4132yescomplemented
45D4×Q8—216no—
46D4×Q8C2×C4132yescomplemented
47#544—28no—
48#544—28no—
49#543—216no—
50#543—216no—
51#533—24no—
52#533—24no—
53#533C2×C418yescomplemented
54#533C2318yes—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8C2×C412yescomplemented
57C2×C4⋊Q8C2×C412yescomplemented
58C2×#125—24no—
59C2×#125—22no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—28no—
66C2×C22⋊Q8—28no—
67C2×C22⋊Q8C2×C418yescomplemented
68C2×C22⋊Q8C2×C4116yescomplemented
69C2×C22⋊Q8C2314yes—
70C2×C22⋊Q8C23116yes—
71#518—28no—
72#518—28no—
73#518—48no—
74C2×C4×Q8—22no—
75C2×C4×Q8C2×C414yescomplemented
76C2×C4×D4—24no—
77C2×C4×D4—24no—
78C2×C4×D4—28no—
79C2×C4×D4C2×C414yes—
80C2×C4×D4C2×C414yes—
81C2×C4×D4D414yes—
82C2×C4×D4D414yes—
83C2×C4×D4D414yes—
84C2×C4×D4C2314yes—
85C2×#117—22no—
86C2×#117—22no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4C2×C414yescomplemented
90C22×C4⋊C4D411yescharacteristic
91C22×C22⋊C4—24no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4D412yes—
94C22×C42C2311yescharacteristic
95C8×D4—232no—
96#431—24no—
97#431C2318yes—
98C2×C4⋊C8—22no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—28no—
101C2×C22⋊C8C2314yes—
102C2×C22⋊C8C23116yes—
103C2×C4×C8—22no—
104#389—28no—
105#385—216no—
106#384—22no—
107#382C2314yes—
108#380—22no—
109#380—24no—
110#380—24no—
111#380C2318yes—
112#380C2314yes—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2314yes—
115C2×#95C2318yes—
116#328C23164yes—
117C25C2×D412yes—
118C22×Q8—22no—
119C22×Q8—24no—
120C22×Q8—28no—
121C22×Q8—88no—
122C22×Q8C22×C414yes—
123C22×Q8C22×C4116yes—
124C22×D4C22⋊C414yes—
125C22×D4C22⋊C414yes—
126C22×D4SD1618yes—
127C22×D4Q1618yes—
128C22×D4C2×D414yes—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—28no—
132C23×C4—84no—
133C23×C4C22×C414yes—
134C23×C4C22×C414yes—
135C23×C4C2×D414yes—
136C23×C4C2×D414yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D414yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C23×C4C2414yes—
143C22×C8—24no—
144C22×C8—216no—
145C22×C8—88no—
146C4⋊Q8—232no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8C22×C418yes—
150C4⋊Q8C22×C418yes—
151#125—232no—
152#125—28no—
153#125—28no—
154C22⋊Q8—216no—
155C22⋊Q8—216no—
156C22⋊Q8—216no—
157C22⋊Q8—28no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8—216no—
161C22⋊Q8—216no—
162C22⋊Q8—28no—
163C22⋊Q8—264no—
164C22⋊Q8—264no—
165C22⋊Q8C22×C418yes—
166C22⋊Q8C22×C4132yes—
167C22⋊Q8C2418yes—
168C4×Q8—28no—
169C4×Q8—28no—
170C4×Q8—28no—
171C4×Q8C22×C4116yes—
172C4×D4—232no—
173C4×D4—232no—
174C4×D4—432no—
175C4×D4C22⋊C4116yes—
176C4×D4C22⋊C4116yes—
177C4×D4C22×C4116yes—
178C4×D4C2×D4116yes—
179#117—28no—
180#117—28no—
181#117—48no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—216no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—216no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—88no—
197C2×C4⋊C4C22⋊C414yes—
198C2×C4⋊C4C22⋊C414yes—
199C2×C4⋊C4C22×C414yes—
200C2×C4⋊C4C22×C4116yes—
201C2×C4⋊C4C2×D414yes—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—216no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—216no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—416no—
209C2×C22⋊C4C22⋊C418yes—
210C2×C22⋊C4C22⋊C418yes—
211C2×C22⋊C4C2×D418yes—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2412yes—
220C4⋊C8—28no—
221C22⋊C8—24no—
222C22⋊C8—216no—
223C22⋊C8C2418yes—
224C4×C8—28no—
225#95—28no—
226#95—24no—
227#95C2418yes—
228C24—82no—
229C24—82no—
230C24C2×C22⋊C414yes—
231C24C2×C22⋊C414yes—
232C24C2×SD1618yes—
233C24C2×Q1618yes—
234C24C22×D411yescharacteristic
235C24C22×D414yes—
236C2×Q8—24no—
237C2×Q8—28no—
238C2×Q8—216no—
239C2×Q8—48no—
240C2×Q8—816no—
241C2×Q8—84no—
242C2×Q8—88no—
243C2×Q8—816no—
244C2×Q8C23×C418yes—
245C2×D4#103132yes—
246C2×D4#103132yes—
247C2×D4C2×C22⋊C4116yes—
248C2×D4C2×SD16116yes—
249C2×D4C2×Q16116yes—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—216no—
253C22×C4—216no—
254C22×C4—28no—
255C22×C4—48no—
256C22×C4—48no—
257C22×C4—42no—
258C22×C4—48no—
259C22×C4—416no—
260C22×C4—84no—
261C22×C4—82no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4—81no—
265C22×C4—88no—
266C22×C4—88no—
267C22×C4—88no—
268C22×C4—88no—
269C22×C4C2×C22⋊C418yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C414yes—
273C22×C4C2×C22⋊C414yes—
274C22×C4C2×C22⋊C418yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×SD1614yes—
278C22×C4C2×Q1614yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C23×C411yescharacteristic
281C22×C4C23×C418yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D412yes—
285C22×C4C22×D414yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D418yes—
288C22×C4C22×D412yes—
289C22×C4C2511yescharacteristic
290C2×C8—28no—
291C2×C8—48no—
292C2×C8—816no—
293C4⋊C4—28no—
294C4⋊C4—24no—
295C4⋊C4—232no—
296C4⋊C4—216no—
297C4⋊C4—28no—
298C4⋊C4—28no—
299C4⋊C4—28no—
300C4⋊C4—216no—
301C4⋊C4—28no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—84no—
305C4⋊C4—88no—
306C4⋊C4C2×C22⋊C4116yes—
307C4⋊C4C23×C418yes—
308C22⋊C4—28no—
309C22⋊C4—232no—
310C22⋊C4—28no—
311C22⋊C4—48no—
312C22⋊C4—416no—
313C22⋊C4C2×C22⋊C4132yes—
314C42—216no—
315C42—48no—
316C42C2×C22⋊C414yes—
317C42C2×C22⋊C414yes—
318C42C23×C414yes—
319C42C22×D414yes—
320C23—44no—
321C23—44no—
322C23—41no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—41no—
327C23—44no—
328C23—88no—
329C23—88no—
330C23C2×#103116yes—
331C23C2×#103116yes—
332C23C22×C22⋊C411yescharacteristic
333C23C22×C22⋊C418yes—
334C23C22×C22⋊C411yescharacteristic
335C23C22×SD1612yes—
336C23C22×SD1618yes—
337C23C22×Q1612yes—
338C23C22×Q1618yes—
339C23C23×D411yescharacteristic
340Q8—48no—
341Q8—48no—
342Q8—88no—
343Q8—816no—
344D4C2×#103164yes—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—216no—
349C2×C4—28no—
350C2×C4—28no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—48no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—816no—
369C2×C4—816no—
370C2×C4—1616no—
371C2×C4C2×#10318yes—
372C2×C4C2×#10318yes—
373C2×C4C22×C22⋊C411yescharacteristic
374C2×C4C22×C22⋊C414yes—
375C2×C4C22×C22⋊C418yes—
376C2×C4C22×C22⋊C411yescharacteristic
377C2×C4C22×C22⋊C414yes—
378C2×C4C22×SD1614yes—
379C2×C4C22×Q1614yes—
380C2×C4C24×C411yescharacteristic
381C2×C4C23×D411yescharacteristic
382C8—48no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—28no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—28no—
391C22—48no—
392C22—42no—
393C22—42no—
394C22—48no—
395C22—416no—
396C22—416no—
397C22C22×#10312yes—
398C22C22×#10312yes—
399C22C22×#103116yes—
400C22C23×C22⋊C411yescharacteristic
401C22C23×SD1611yescharacteristic
402C22C23×Q1611yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—28no—
407C4—24no—
408C4—48no—
409C4—48no—
410C4—88no—
411C4#288312yes—
412C4C22×#10318yes—
413C2—21no—
414C2—22no—
415C2—22no—
416C2—21no—
417C2—216no—
418C2C2×#288311yescharacteristic
419C2C23×#10311yescharacteristic
420C2#6009311yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17614C2semidirect
5. #17575C2semidirect
6. C2×#1520C2semidirect
9. #3501C4semidirect
12. C2×#554C4semidirect
13. C2×D4×Q8C4semidirect
33. #1520C22semidirect
34. #1518C22semidirect
35. #1503C22semidirect
36. C2×#328C22semidirect
37. C23×Q8C2×C4semidirect
44. #554C2×C4semidirect
46. D4×Q8C2×C4semidirect
53. #533C2×C4semidirect
56. C2×C4⋊Q8C2×C4semidirect
57. C2×C4⋊Q8C2×C4semidirect
67. C2×C22⋊Q8C2×C4semidirect
68. C2×C22⋊Q8C2×C4semidirect
75. C2×C4×Q8C2×C4semidirect
89. C22×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#29722
#341812
#176142
#175752
C2×#15202