Finite Group Databasebeta

← #6434674

Subgroups — #6434674

Order 512.

4209 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#3092C212yesmaximal
3#36626C211yesmaximal, characteristic
4#29642C214yesmaximal, complemented
5#29641C214yesmaximal, complemented
6#29639C214yesmaximal, complemented
7#29638C214yesmaximal, complemented
8#23552C214yesmaximal, complemented
9#23452C214yesmaximal, complemented
10#20700C212yesmaximal, complemented
11#20682C211yesmaximal, characteristic, complemented
12C24×Q8—21no—
13C22×C4⋊Q8—22no—
14C22×C4⋊Q8—22no—
15C22×C4⋊Q8—22no—
16C22×C4⋊Q8—216no—
17C22×C4⋊Q8C2214yes—
18C22×C4⋊Q8C2214yes—
19C22×C4⋊Q8C2214yes—
20C2×#525C2211yescharacteristic
21C23×C4⋊C4—21no—
22C23×C4⋊C4C2212yes—
23C23×C42C2211yescharacteristic
24#3092—216no—
25C2×#484C2214yes—
26C2×#481C2214yes—
27C2×#483C2214yes—
28C2×#480C2214yes—
29C2×#428C2214yes—
30C2×#417C2214yes—
31C2×#407C2212yes—
32#2855C2212yes—
33#2855C2212yes—
34#2854C2212yes—
35#2854C2212yes—
36#2853—22no—
37#2849—21no—
38#2849—21no—
39C4×#123—22no—
40C4×#123—22no—
41#2297C2212yes—
42#2296C2212yes—
43#2280C2211yescharacteristic
44#2058C22116yescomplemented
45#2057C22116yescomplemented
46#2024C22116yescomplemented
47#2023C22116yescomplemented
48#1824C2218yes—
49#1823C2218yes—
50#1817C2214yes—
51#1817C2214yes—
52#1736C2218yes—
53#1728C2214yes—
54C22×#41C2311yescharacteristic
55C23×Q8—216no—
56C23×Q8—24no—
57C23×Q8—24no—
58C23×Q8—24no—
59C23×Q8—416no—
60C24×C4—22no—
61C24×C4D411yescharacteristic
62C24×C4D411yescharacteristic
63C24×C4C2311yescharacteristic
64C22×M16C2314yes—
65C2×C4⋊Q8—232no—
66C2×C4⋊Q8—232no—
67C2×C4⋊Q8—232no—
68C2×C4⋊Q8—232no—
69C2×C4⋊Q8—232no—
70C2×C4⋊Q8—232no—
71C2×C4⋊Q8—28no—
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8—28no—
74C2×C4⋊Q8C23116yes—
75#525—28no—
76C2×#123—22no—
77C2×#123—22no—
78C2×#123—21no—
79C2×C22⋊Q8C2311yescharacteristic
80C2×C22⋊Q8C2311yescharacteristic
81C2×#121C2311yescharacteristic
82C2×#121C2311yescharacteristic
83C2×#120—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×#117C2311yescharacteristic
87C2×#117C2312yes—
88C22×C4⋊C4—216no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4D4116yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C42—28no—
101C22×C42D412yes—
102C22×C42D412yes—
103C22×C42D412yes—
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2312yes—
107#484—216no—
108#483—216no—
109#481—216no—
110#480—216no—
111#428—216no—
112C2×#107C2318yes—
113C2×#106C2318yes—
114#417—216no—
115C2×#103C2318yes—
116C2×#102C2318yes—
117#407—28no—
118C2×C22⋊C8C2314yes—
119C2×C22⋊C8C2314yes—
120#391C2314yes—
121#390C2314yes—
122#389C2314yes—
123#389C2314yes—
124#388—24no—
125#388—24no—
126#387—24no—
127#387—24no—
128#386—22no—
129#386—22no—
130#386—22no—
131#386—22no—
132#385—22no—
133#385—22no—
134#385—22no—
135#385—22no—
136#381—22no—
137#381—22no—
138#381—21no—
139#380C2312yes—
140#380C2312yes—
141#379—21no—
142C4×C4⋊C4—24no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—24no—
146#340C23132yes—
147#336C23116yes—
148C25C2×D411yescharacteristic
149C2×#41—24no—
150C2×#41—28no—
151C22×Q8—232no—
152C22×Q8—232no—
153C22×Q8—232no—
154C22×Q8—216no—
155C22×Q8—4128no—
156C22×Q8—416no—
157C22×Q8—432no—
158C22×Q8—432no—
159C22×Q8—416no—
160C22×Q8C2411yescharacteristic
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C2411yescharacteristic
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—28no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—216no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—21no—
175C23×C4—416no—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D418yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C2×M16—28no—
190C2×M16—216no—
191C22×C8C2418yes—
192C4⋊Q8—264no—
193C4⋊Q8—264no—
194C4⋊Q8—264no—
195C4⋊Q8—264no—
196#123—44no—
197#123—48no—
198#123—48no—
199C22⋊Q8—24no—
200C22⋊Q8—24no—
201#121—24no—
202#121—24no—
203#120—44no—
204C4×D4—44no—
205C4×D4—44no—
206#117—24no—
207#117—28no—
208C2×C4⋊C4—232no—
209C2×C4⋊C4—232no—
210C2×C4⋊C4—232no—
211C2×C4⋊C4—216no—
212C2×C4⋊C4—216no—
213C2×C4⋊C4—216no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4C2×D4116yes—
226C2×C4⋊C4C2×D4116yes—
227C2×C4⋊C4C2×D4116yes—
228C2×C4⋊C4C2411yescharacteristic
229C2×C4⋊C4C2411yescharacteristic
230C2×C4⋊C4C2414yes—
231C2×C4⋊C4C2418yes—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C42—28no—
243C2×C42—28no—
244C2×C42—28no—
245C2×C42—28no—
246C2×C42—28no—
247C2×C42—28no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42C2×D414yes—
251C2×C42C2×D414yes—
252C2×C42C2×D414yes—
253C2×C42C2414yes—
254#107—216no—
255#106—216no—
256#103—216no—
257#102—216no—
258C22⋊C8—28no—
259C22⋊C8—28no—
260#95—22no—
261#95—22no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266C24—24no—
267C24—24no—
268C24—24no—
269C24—21no—
270C24#12014yes—
271C24C22×D411yescharacteristic
272C24C22×D411yescharacteristic
273C24C22×D411yescharacteristic
274#41—416no—
275C2×Q8—264no—
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—4128no—
279C2×Q8—4128no—
280C2×Q8—4128no—
281C2×Q8—432no—
282C2×Q8—432no—
283C2×D4—22no—
284C2×D4—24no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C22×C4—232no—
292C22×C4—232no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—232no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—216no—
312C22×C4—216no—
313C22×C4—216no—
314C22×C4—28no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—464no—
320C22×C4—416no—
321C22×C4—416no—
322C22×C4—416no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4#12014yes—
326C22×C4#12014yes—
327C22×C4#12014yes—
328C22×C4AGL(1,ℤ/8)18yes—
329C22×C4#13718yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D414yes—
333C22×C4C22×D414yes—
334C22×C4C22×D414yes—
335C22×C4C22×D414yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D414yes—
338C22×C42−1+418yes—
339C22×C4C2511yescharacteristic
340M16—416no—
341C2×C8—28no—
342C2×C8—216no—
343C4⋊C4—264no—
344C4⋊C4—216no—
345C4⋊C4—216no—
346C4⋊C4—216no—
347C4⋊C4—28no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—28no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C4⋊C4C22×D4116yes—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—48no—
365C22⋊C4—48no—
366C42—28no—
367C42—28no—
368C42—28no—
369C42—28no—
370C42—42no—
371C42—44no—
372C23—232no—
373C23—28no—
374C23—22no—
375C23—22no—
376C23—28no—
377C23—28no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23C2×#12012yes—
387C23C2×#12012yes—
388C23C2×#12012yes—
389C23#53614yes—
390C23C2×AGL(1,ℤ/8)14yes—
391C23C2×#13714yes—
392C23C23×D411yescharacteristic
393Q8—4128no—
394Q8—44no—
395D4—44no—
396D4—84no—
397D4—84no—
398D4—84no—
399C2×C4—216no—
400C2×C4—216no—
401C2×C4—216no—
402C2×C4—216no—
403C2×C4—28no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—216no—
409C2×C4—216no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—216no—
417C2×C4—432no—
418C2×C4—432no—
419C2×C4—432no—
420C2×C4—416no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—84no—
431C2×C4C2×#12012yes—
432C2×C4C2×#12012yes—
433C2×C4C2×#12012yes—
434C2×C4#53614yes—
435C2×C4#536116yes—
436C2×C4C2×AGL(1,ℤ/8)18yes—
437C2×C4C2×#13718yes—
438C2×C4C23×D412yes—
439C2×C4C2×2−1+414yes—
440C8—48no—
441C22—232no—
442C22—22no—
443C22—22no—
444C22—28no—
445C22—28no—
446C22—28no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—81no—
452C22—84no—
453C22—81no—
454C22—81no—
455C22#261414yes—
456C22#300314yes—
457C22#300414yes—
458C22C22×#12011yescharacteristic
459C22C2×#53612yes—
460C22C22×AGL(1,ℤ/8)12yes—
461C22C22×#13712yes—
462C4—22no—
463C4—22no—
464C4—28no—
465C4—28no—
466C4—416no—
467C4—81no—
468C4—84no—
469C4—81no—
470C4—81no—
471C4#261414yes—
472C4C2×#53614yes—
473C2—24no—
474C2—24no—
475C2—24no—
476C2—161no—
477C2C2×#261411yescharacteristic
478C2C2×#300311yescharacteristic
479C2C2×#300411yescharacteristic
480C2#6086014yes—
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. C23×C4⋊Q8C2semidirect
4. #29642C2semidirect
5. #29641C2semidirect
6. #29639C2semidirect
7. #29638C2semidirect
8. #23552C2semidirect
9. #23452C2semidirect
10. #20700C2semidirect
11. #20682C2semidirect
44. #2058C22semidirect
45. #2057C22semidirect
46. #2024C22semidirect
47. #2023C22semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C2×#30922
#366262
#296422
#296412
#296392
#296382
#235522
#234522
#207002
#206822