Finite Group Databasebeta

← C2×#18325 #6346872

Subgroups — C2×#18325 #6346872

Order 512.

4234 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#537C211yesmaximal, characteristic
2C2×#3193C212yesmaximal, complemented
3C2×#2583C211yesmaximal, characteristic, complemented
4#18325C2116yesmaximal, direct product factor
5C2×#1610C214yesmaximal, complemented
6C2×#1604C214yesmaximal, complemented
7C2×#1507C212yesmaximal, complemented
8C2×#1463C211yesmaximal, characteristic, complemented
9C23×#41—21no—
10C2×#537—28no—
11C2×#537—24no—
12C2×#537—24no—
13C2×#537—22no—
14C2×#537C2218yes—
15C2×#537C2214yes—
16C22×C22⋊Q8—22no—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121—22no—
19C22×#121—21no—
20C22×#121C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22#3193—28no—
23#3193C22116yescomplemented
24C2×#484—22no—
25C2×#480C2214yescomplemented
26C2×#468—24no—
27C2×#448C2214yescomplemented
28C2×#450—22no—
29C2×#407C2212yes—
30#2583—24no—
31#2583C2218yescomplemented
32C2×#398C2212yescomplemented
33C2×#395C2212yescomplemented
34C2×#389—22no—
35C2×#385—21no—
36C2×#385—21no—
37C2×C4×C22⋊C4C2211yescharacteristic
38#1610C22132yescomplemented
39#1604C22132yescomplemented
40#1507C22116yescomplemented
41C2×#327C2218yescomplemented
42#1463C2218yescomplemented
43C2×#324C2214yescomplemented
44C22×#41—28no—
45C22×#41—24no—
46C22×#41—28no—
47C22×#41—24no—
48C22×#41—416no—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C23×D4—21no—
52C23×D4—22no—
53C23×D4—21no—
54C23×D4D412yes—
55C24×C4—22no—
56C24×C4D411yescharacteristic
57C24×C4D411yescharacteristic
58C24×C4C2311yescharacteristic
59C22×SD16—24no—
60C22×M16—22no—
61#537—216no—
62#537—232no—
63#537—216no—
64#537—232no—
65#537—216no—
66#537—28no—
67#537C23116yes—
68C2×C22⋊Q8—216no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—216no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8C2314yes—
77C2×#121—28no—
78C2×#121—24no—
79C2×#121—216no—
80C2×#121—28no—
81C2×#121—28no—
82C2×#121—216no—
83C2×#121—24no—
84C2×#121—28no—
85C2×#121—22no—
86C2×#121—24no—
87C2×#121—22no—
88C2×#121—24no—
89C2×#121D418yes—
90C2×#121D414yes—
91C2×#121C2314yes—
92C2×#121C2318yes—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4D414yes—
101C22×C4⋊C4D412yes—
102C22×C4⋊C4D412yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—22no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—22no—
109C22×C22⋊C4—21no—
110C22×C42—21no—
111#484—216no—
112#480—28no—
113#480C23116yescomplemented
114#468—232no—
115#450—216no—
116#448—28no—
117#448C23116yescomplemented
118C2×#107—24no—
119C2×#103—24no—
120C2×#102—24no—
121#407—24no—
122#407C2318yes—
123C2×C22⋊C8—22no—
124C2×C22⋊C8C2314yes—
125#398—24no—
126#398C2318yescomplemented
127#395—24no—
128#395C2318yescomplemented
129#389—216no—
130#385—28no—
131#385—28no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4C2314yes—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95C2312yes—
137#327C23132yescomplemented
138#324C23116yescomplemented
139C25—22no—
140C25C2×D411yescharacteristic
141C2×#41—232no—
142C2×#41—216no—
143C2×#41—216no—
144C2×#41—464no—
145C2×#41—432no—
146C2×#41—416no—
147C2×#41—464no—
148C2×#41—416no—
149C22×Q8—28no—
150C22×Q8—24no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—48no—
155C22×Q8—82no—
156C22×Q8C2412yes—
157C22×D4—28no—
158C22×D4—24no—
159C22×D4—28no—
160C22×D4—24no—
161C22×D4—28no—
162C22×D4—24no—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—22no—
166C22×D4—24no—
167C22×D4—48no—
168C22×D4—416no—
169C22×D4—84no—
170C22×D4—84no—
171C22×D4C2×D414yes—
172C22×D4C2×D418yes—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—28no—
177C23×C4—216no—
178C23×C4—28no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—416no—
187C23×C4—82no—
188C23×C4—82no—
189C23×C4C2×D414yes—
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D414yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C2×SD16—216no—
201C2×SD16—88no—
202C2×M16—28no—
203C2×M16—84no—
204C22×C8—24no—
205C22⋊Q8—264no—
206C22⋊Q8—232no—
207C22⋊Q8—232no—
208C22⋊Q8—28no—
209C22⋊Q8—28no—
210C22⋊Q8—44no—
211C22⋊Q8—44no—
212#121—216no—
213#121—232no—
214#121—264no—
215#121—232no—
216#121—232no—
217#121—28no—
218#121—28no—
219#121—28no—
220#121—216no—
221#121—24no—
222#121—28no—
223#121—24no—
224#121—28no—
225#121C2×D418yes—
226#121C2×D4116yes—
227#121C2418yes—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—84no—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D418yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2414yes—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—216no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C42—24no—
269C2×C42—82no—
270#107—216no—
271#103—216no—
272#102—216no—
273C22⋊C8—28no—
274C22⋊C8—24no—
275C22⋊C8C2418yes—
276#95—28no—
277#95—28no—
278#95—22no—
279#95C2414yes—
280C24—24no—
281C24—24no—
282C24—28no—
283C24—24no—
284C24—42no—
285C24—48no—
286C24—48no—
287C24—48no—
288C24#12011yescharacteristic
289C24#12012yes—
290C24#12011yescharacteristic
291C24C22×D412yes—
292C24C22×D411yescharacteristic
293#41—4128no—
294#41—464no—
295#41—432no—
296#41—432no—
297C2×Q8—216no—
298C2×Q8—24no—
299C2×Q8—432no—
300C2×Q8—416no—
301C2×Q8—48no—
302C2×Q8—42no—
303C2×Q8—82no—
304C2×Q8—82no—
305C2×Q8—84no—
306C2×D4—216no—
307C2×D4—216no—
308C2×D4—28no—
309C2×D4—28no—
310C2×D4—216no—
311C2×D4—28no—
312C2×D4—24no—
313C2×D4—432no—
314C2×D4—416no—
315C2×D4—48no—
316C2×D4—432no—
317C2×D4—432no—
318C2×D4—416no—
319C2×D4—416no—
320C2×D4—84no—
321C2×D4—88no—
322C2×D4—88no—
323C2×D4—816no—
324C2×D4—88no—
325C2×D4C22×D418yes—
326C22×C4—28no—
327C22×C4—216no—
328C22×C4—28no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—216no—
335C22×C4—28no—
336C22×C4—232no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—464no—
351C22×C4—432no—
352C22×C4—44no—
353C22×C4—48no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—44no—
363C22×C4—82no—
364C22×C4—84no—
365C22×C4—82no—
366C22×C4—84no—
367C22×C4—82no—
368C22×C4—84no—
369C22×C4#12012yes—
370C22×C4#12014yes—
371C22×C4#12012yes—
372C22×C4#12014yes—
373C22×C4AGL(1,ℤ/8)18yes—
374C22×C4C22×D412yes—
375C22×C4C22×D414yes—
376C22×C4C22×D414yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D412yes—
379C22×C4C22×D414yes—
380C22×C4C22×D412yes—
381C22×C4C22×D414yes—
382C22×C4C2511yescharacteristic
383SD16—816no—
384M16—88no—
385C2×C8—28no—
386C2×C8—48no—
387C4⋊C4—216no—
388C4⋊C4—216no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—24no—
392C4⋊C4—28no—
393C4⋊C4—24no—
394C4⋊C4—42no—
395C4⋊C4—48no—
396C4⋊C4—416no—
397C4⋊C4—88no—
398C4⋊C4C22×D418yes—
399C22⋊C4—216no—
400C22⋊C4—232no—
401C22⋊C4—216no—
402C22⋊C4—28no—
403C22⋊C4—28no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C42—84no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—24no—
414C23—24no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—28no—
419C23—42no—
420C23—42no—
421C23—416no—
422C23—416no—
423C23—416no—
424C23—48no—
425C23—416no—
426C23—48no—
427C23—48no—
428C23—832no—
429C23#45614yes—
430C23C2×#12012yes—
431C23C2×#12011yescharacteristic
432C23C2×#12012yes—
433C23C2×#12011yescharacteristic
434C23C2×AGL(1,ℤ/8)14yes—
435C23C23×D411yescharacteristic
436Q8—432no—
437Q8—48no—
438Q8—84no—
439D4—48no—
440D4—432no—
441D4—432no—
442D4—416no—
443D4—416no—
444D4—84no—
445D4—88no—
446D4—816no—
447C2×C4—216no—
448C2×C4—216no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—464no—
468C2×C4—44no—
469C2×C4—416no—
470C2×C4—416no—
471C2×C4—48no—
472C2×C4—48no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4C2×#12014yes—
487C2×C4C2×#12012yes—
488C2×C4C2×AGL(1,ℤ/8)18yes—
489C2×C4C23×D412yes—
490C8—48no—
491C22—28no—
492C22—28no—
493C22—22no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—48no—
503C22—48no—
504C22—44no—
505C22—48no—
506C22—48no—
507C22—48no—
508C22—832no—
509C22—816no—
510C22#300312yes—
511C22#300312yes—
512C22C2×#45614yes—
513C22C2×#45612yes—
514C22#302612yes—
515C22C22×#12011yescharacteristic
516C22C22×AGL(1,ℤ/8)12yes—
517C4—28no—
518C4—22no—
519C4—24no—
520C4—22no—
521C4—48no—
522C4—48no—
523C4—44no—
524C4—82no—
525C4—82no—
526C2—22no—
527C2—24no—
528C2—22no—
529C2—42no—
530C2—88no—
531C2#1832514yesdirect product factor
532C2C2×#300311yescharacteristic
533C2C22×#45611yescharacteristic
534C2C2×#302611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3193C2semidirect
3. C2×#2583C2semidirect
4. #18325C2direct (N × H)
5. C2×#1610C2semidirect
6. C2×#1604C2semidirect
7. C2×#1507C2semidirect
8. C2×#1463C2semidirect
23. #3193C22semidirect
25. C2×#480C22semidirect
27. C2×#448C22semidirect
31. #2583C22semidirect
32. C2×#398C22semidirect
33. C2×#395C22semidirect
38. #1610C22semidirect
39. #1604C22semidirect
40. #1507C22semidirect
41. C2×#327C22semidirect
42. #1463C22semidirect
43. C2×#324C22semidirect
113. #480C23semidirect
117. #448C23semidirect
126. #398C23semidirect
128. #395C23semidirect
137. #327C23semidirect
138. #324C23semidirect
531. C2#18325direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5372
C2×#31932
C2×#25832
#183252
C2×#16102
C2×#16042
C2×#15072
C2×#14632