Finite Group Databasebeta

← #6433201

Subgroups — #6433201

Order 512.

2220 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3092C212yesmaximal, complemented
2#34633C211yesmaximal, characteristic
3C2×#2316C211yesmaximal, characteristic, complemented
4#29768C214yesmaximal, complemented
5#29767C214yesmaximal, complemented
6#29764C214yesmaximal, complemented
7#29763C214yesmaximal, complemented
8#23657C212yesmaximal, complemented
9#23617C214yesmaximal, complemented
10#23508C212yesmaximal, complemented
11#20910C212yesmaximal, complemented
12#20627C211yesmaximal, characteristic, complemented
13C2×#532C2211yescharacteristic
14C2×#525C2212yes—
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#3092—216no—
19C2×#484C2214yes—
20C2×#481C2214yes—
21C2×#483C2214yes—
22C2×#480C2214yes—
23C2×#428C2214yes—
24C2×#423C2212yes—
25C2×#417C2214yes—
26C2×#417C2214yes—
27C2×#407C2212yes—
28#2621C2212yes—
29#2620C2212yes—
30#2620C2212yes—
31#2619C2212yes—
32#2618—22no—
33#2617—22no—
34#2392—24no—
35#2316—24no—
36#2316C2214yes—
37#2316C2214yes—
38#2297C2212yes—
39C2×#389—22no—
40C2×#389C2212yes—
41C2×#389C2212yes—
42#2285C2211yescharacteristic
43C2×#380—22no—
44#2085C2218yescomplemented
45#2084C2214yescomplemented
46#2083C2214yescomplemented
47#2068C2218yescomplemented
48#2067C2218yescomplemented
49#2066C2218yescomplemented
50#2065C2218yescomplemented
51#2043C2214yescomplemented
52#2042C2218yescomplemented
53#2041C2214yescomplemented
54#1875C2214yes—
55#1875C2214yes—
56#1873C2214yes—
57#1873C2214yes—
58#1798C2214yes—
59#1798C2214yes—
60#1736C2214yes—
61#1728C2212yes—
62C22×#41C2312yes—
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C22×M16C2314yes—
66#532—44no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#124C2312yes—
70#525—28no—
71#525—28no—
72C2×#123—22no—
73C2×C22⋊Q8C2312yes—
74C2×C22⋊Q8C2312yes—
75C2×#121—22no—
76C2×#121C2312yes—
77C2×#121C2312yes—
78C2×#120—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×#117—22no—
83C2×#117—44no—
84C2×#117—44no—
85C2×#117D414yes—
86C2×#117C2312yes—
87C2×#117C2314yes—
88C2×#117C2312yes—
89C2×#117C2312yes—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4D412yes—
96C22×C4⋊C4D418yescomplemented
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4—21no—
101C22×C22⋊C4D412yes—
102C22×C42—21no—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105#484—216no—
106#483—216no—
107#481—216no—
108#480—216no—
109#428—216no—
110C2×#107C2318yes—
111C2×#106C2318yes—
112#423—44no—
113C2×C4⋊C8C2318yes—
114#417—216no—
115#417—48no—
116C2×#103C2318yes—
117C2×#103C2318yes—
118C2×#102C2318yes—
119C2×#102C2318yes—
120#407—28no—
121C2×C22⋊C8C2314yes—
122C2×C22⋊C8C2314yes—
123#398—24no—
124#397—24no—
125#396—24no—
126#396—24no—
127#394—24no—
128#394—24no—
129#394—24no—
130#392—24no—
131#391C2312yes—
132#391C2312yes—
133#390C2312yes—
134#390C2312yes—
135#389—28no—
136#389—28no—
137#389—22no—
138#389—22no—
139#389—24no—
140#389C2318yes—
141#389C2314yes—
142#389C2314yes—
143#386—24no—
144#384C2311yescharacteristic
145#384C2311yescharacteristic
146#382C2312yes—
147#380—22no—
148#380—28no—
149#380—28no—
150C4×C4⋊C4—22no—
151C4×C22⋊C4—24no—
152C2×#95—22no—
153C2×#95—22no—
154#340C2318yes—
155#340C2318yes—
156#336C2314yes—
157#336C2314yes—
158C25C2×D411yescharacteristic
159C2×#41—24no—
160C2×#41—24no—
161C2×#41—28no—
162C2×#41—28no—
163C22×Q8C2412yes—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4C2412yes—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—28no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—84no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D414yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4#4114yes—
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C2×M16—28no—
186C2×M16—28no—
187C2×M16—28no—
188C22×C8C2418yes—
189C4⋊Q8—42no—
190#127—42no—
191#124—44no—
192#123—48no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197#121—24no—
198#121—24no—
199#121—24no—
200#121—24no—
201#121—48no—
202#120—48no—
203C4×D4—48no—
204C4×D4—48no—
205C4×D4—48no—
206#117—24no—
207#117—24no—
208#117—48no—
209#117—44no—
210#117—44no—
211#117—48no—
212#117—48no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—84no—
231C2×C4⋊C4—84no—
232C2×C4⋊C4C2×D418yes—
233C2×C4⋊C4C2×D4116yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2414yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2414yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C2×D414yes—
252C2×C22⋊C4C2×D414yes—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—44no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2411yescharacteristic
261C2×C42C2411yescharacteristic
262C2×C42C2412yes—
263#107—216no—
264#106—216no—
265C4⋊C8—48no—
266#103—216no—
267#103—48no—
268#102—216no—
269#102—48no—
270C22⋊C8—28no—
271C22⋊C8—28no—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—42no—
282C24—42no—
283C24#12114yes—
284C24C22×D412yes—
285C24C22×D411yescharacteristic
286#41—416no—
287#41—416no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4—84no—
305C2×D4—84no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—216no—
311C22×C4—216no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—24no—
316C22×C4—216no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—88no—
336C22×C4—816no—
337C22×C4#12118yes—
338C22×C4#12114yes—
339C22×C4C22×D414yes—
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×D414yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D412yes—
346C22×C4C2×#4114yes—
347C22×C4C2×#4112yes—
348C22×C4H(5,𝔽2)14yes—
349C22×C42−1+414yes—
350C22×C4C2511yescharacteristic
351M16—416no—
352C2×C8—28no—
353C2×C8—28no—
354C2×C8—28no—
355C4⋊C4—216no—
356C4⋊C4—28no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—28no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—88no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4C22×D418yes—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—48no—
378C22⋊C4—48no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—84no—
385C22⋊C4—84no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—22no—
390C42—22no—
391C42—48no—
392C42—44no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—24no—
397C23—28no—
398C23—22no—
399C23—22no—
400C23—44no—
401C23—44no—
402C23—48no—
403C23—48no—
404C23—44no—
405C23—44no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23C2×#12114yes—
410C23C2×#12112yes—
411C23#53412yes—
412C23#53612yes—
413C23C23×D411yescharacteristic
414Q8—44no—
415Q8—44no—
416D4—44no—
417D4—44no—
418D4—88no—
419D4—88no—
420D4—88no—
421D4—88no—
422C2×C4—28no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—216no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—28no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—416no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—816no—
454C2×C4—88no—
455C2×C4—88no—
456C2×C4C2×#12114yes—
457C2×C4C2×#12112yes—
458C2×C4#53412yes—
459C2×C4#53612yes—
460C2×C4#53718yes—
461C2×C4#53918yes—
462C2×C4C23×D411yescharacteristic
463C2×C4C22×#4111yescharacteristic
464C2×C4C2×H(5,𝔽2)12yes—
465C2×C4C2×2−1+412yes—
466C8—48no—
467C22—22no—
468C22—22no—
469C22—24no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22—24no—
474C22—416no—
475C22—44no—
476C22—44no—
477C22—82no—
478C22—82no—
479C22—84no—
480C22—84no—
481C22—82no—
482C22#261814yes—
483C22C22×#12111yescharacteristic
484C22C2×#53411yescharacteristic
485C22C2×#53611yescharacteristic
486C22#358512yes—
487C22#358612yes—
488C4—24no—
489C4—48no—
490C4—44no—
491C4—42no—
492C4—42no—
493C4—82no—
494C4—82no—
495C4—84no—
496C4—82no—
497C4#261814yes—
498C4C2×#53712yes—
499C4C2×#53912yes—
500C2—24no—
501C2—44no—
502C2—162no—
503C2C2×#261811yescharacteristic
504C2#6042311yescharacteristic
505C2#6042411yescharacteristic
506C2#6075512yes—
507C2#6086712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34633C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3092C2semidirect
3. C2×#2316C2semidirect
4. #29768C2semidirect
5. #29767C2semidirect
6. #29764C2semidirect
7. #29763C2semidirect
8. #23657C2semidirect
9. #23617C2semidirect
10. #23508C2semidirect
11. #20910C2semidirect
12. #20627C2semidirect
44. #2085C22semidirect
45. #2084C22semidirect
46. #2083C22semidirect
47. #2068C22semidirect
48. #2067C22semidirect
49. #2066C22semidirect
50. #2065C22semidirect
51. #2043C22semidirect
52. #2042C22semidirect
53. #2041C22semidirect
96. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30922
#346332
C2×#23162
#297682
#297672
#297642
#297632
#236572
#236172
#235082
#209102
#206272