Finite Group Databasebeta

← C2×#12996 #494452

Subgroups — C2×#12996 #494452

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#472C211yesmaximal, characteristic
2C2×#2913C211yesmaximal, characteristic, complemented
3C2×#2140C211yesmaximal, characteristic, complemented
4C22×#357C211yesmaximal, characteristic, complemented
5C22×#350C211yesmaximal, characteristic, complemented
6#12996C218yesmaximal, direct product factor
7C2×#1387C211yesmaximal, characteristic, complemented
8C2×#1382C211yesmaximal, characteristic, complemented
9C22×AGL(1,ℤ/8)—21no—
10C23×D8C2211yescharacteristic
11C22×#131C2211yescharacteristic
12C2×#520—21no—
13C22×M32C2211yescharacteristic
14C23×C16C2211yescharacteristic
15C2×#472—24no—
16C2×#472—22no—
17C2×#472C418yescomplemented
18C2×#472C2214yes—
19C22×#108C2211yescharacteristic
20#2913—22no—
21#2913—22no—
22#2913—22no—
23#2913C2214yescomplemented
24C2×#428—22no—
25C2×#428—21no—
26C2×#427—21no—
27C2×#427C2211yescharacteristic
28C22×#107—21no—
29C22×#107C2211yescharacteristic
30C22×#100—21no—
31#2140—22no—
32#2140C2214yescomplemented
33C2×#357—22no—
34C2×#357—22no—
35C2×#357—21no—
36C2×#357—21no—
37C2×#357C2214yescomplemented
38C2×#350—24no—
39C2×#350C2214yescomplemented
40#1387C2214yescomplemented
41#1382C2214yescomplemented
42C22×#41D411yescharacteristic
43C23×D4—21no—
44C2×AGL(1,ℤ/8)—28no—
45C2×AGL(1,ℤ/8)—24no—
46C2×AGL(1,ℤ/8)—42no—
47C2×AGL(1,ℤ/8)—44no—
48C22×SD16—41no—
49C22×D8—21no—
50C22×D8—21no—
51C22×D8—21no—
52C22×D8—21no—
53C22×D8—21no—
54C22×D8—22no—
55C22×D8—22no—
56C22×D8—22no—
57C22×D8—41no—
58C22×D8C2×C414yescomplemented
59C22×D8C2312yes—
60C2×#131—21no—
61C2×#131—21no—
62C2×#131—22no—
63C2×#131—22no—
64C2×#131C2×C414yes—
65C2×#131D414yes—
66C2×#131D412yes—
67C2×#131C2312yes—
68C22×M16—21no—
69C22×M16—21no—
70C22×M16D411yescharacteristic
71C23×C8—21no—
72C23×C8C2311yescharacteristic
73#520—24no—
74#520—24no—
75#520—42no—
76#520—42no—
77C2×C4×Q8—41no—
78C2×C4×D4—42no—
79C2×C4×D4—41no—
80C2×#117—22no—
81C2×#117—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C2×M32—21no—
85C2×M32—22no—
86C2×M32—22no—
87C2×M32C2312yes—
88C22×C16—22no—
89C22×C16—21no—
90C22×C16—21no—
91C22×C16—21no—
92C22×C16—22no—
93C22×C16C2312yes—
94#472—216no—
95#472—28no—
96#472C2×C4116yescomplemented
97C2×#108—21no—
98C2×#108—22no—
99C2×#108C2×C414yescomplemented
100C2×#108C2312yes—
101#428—24no—
102#428—28no—
103#428—24no—
104#428—28no—
105#427—24no—
106#427—21no—
107#427—21no—
108#427—24no—
109#427D412yes—
110#427C2312yes—
111C2×#107—22no—
112C2×#107—24no—
113C2×#107—21no—
114C2×#107—21no—
115C2×#107—21no—
116C2×#107—24no—
117C2×#107D411yescharacteristic
118C2×#107D412yes—
119C2×#107D412yes—
120C2×#107D411yescharacteristic
121C2×#107C2312yes—
122C2×#106—22no—
123C2×#106—21no—
124C2×#100—28no—
125C2×#100—24no—
126#357—28no—
127#357—24no—
128#357—24no—
129#350—28no—
130C25—41no—
131C2×#41—21no—
132C2×#41—22no—
133C2×#41—21no—
134C2×#41—22no—
135C2×#41C22⋊C414yes—
136C2×#41D812yes—
137C2×#41SD1614yes—
138C2×#41C2×D412yes—
139C22×Q8D811yescharacteristic
140C22×D4—21no—
141C22×D4—24no—
142C22×D4—22no—
143C22×D4—41no—
144C22×D4—41no—
145C22×D4—42no—
146C22×D4—41no—
147C22×D4—41no—
148C22×D4—42no—
149C22×D4—41no—
150C22×D4—42no—
151C22×D4D811yescharacteristic
152C23×C4—21no—
153C23×C4—42no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156AGL(1,ℤ/8)—216no—
157AGL(1,ℤ/8)—416no—
158AGL(1,ℤ/8)—48no—
159C2×SD16—48no—
160C2×SD16—44no—
161C2×D8—24no—
162C2×D8—24no—
163C2×D8—24no—
164C2×D8—24no—
165C2×D8—24no—
166C2×D8—22no—
167C2×D8—22no—
168C2×D8—22no—
169C2×D8—22no—
170C2×D8—22no—
171C2×D8—48no—
172C2×D8—44no—
173C2×D8—41no—
174C2×D8—41no—
175C2×D8—42no—
176C2×D8—42no—
177C2×D8—42no—
178C2×D8—42no—
179C2×D8—42no—
180C2×D8C22×C414yescomplemented
181#131—24no—
182#131—24no—
183#131—24no—
184#131—24no—
185#131—24no—
186#131—22no—
187#131—22no—
188#131C22⋊C418yes—
189#131C22×C414yes—
190#131C2×D414yes—
191C2×M16—21no—
192C2×M16—22no—
193C2×M16—24no—
194C2×M16—22no—
195C2×M16—24no—
196C2×M16—22no—
197C2×M16—24no—
198C2×M16—22no—
199C2×M16—41no—
200C2×M16—42no—
201C2×M16C22⋊C414yes—
202C2×M16C2×D412yes—
203C22×C8—21no—
204C22×C8—24no—
205C22×C8—22no—
206C22×C8—24no—
207C22×C8—22no—
208C22×C8—41no—
209C22×C8C22×C412yes—
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D412yes—
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8#4112yes—
217C22×C8#4112yes—
218C22×C8C2411yescharacteristic
219C4×Q8—44no—
220C4×Q8—44no—
221C4×D4—48no—
222C4×D4—48no—
223C4×D4—44no—
224C4×D4—44no—
225#117—28no—
226#117—22no—
227#117—22no—
228#117—44no—
229#117—41no—
230#117—41no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—41no—
244C2×C42—42no—
245C2×C42—41no—
246M32—22no—
247M32—22no—
248M32—42no—
249C2×C16—22no—
250C2×C16—22no—
251C2×C16—22no—
252C2×C16—22no—
253C2×C16—42no—
254C2×C16—42no—
255C2×C16—42no—
256#108—24no—
257#108—22no—
258#108C22×C414yescomplemented
259#107—28no—
260#107—21no—
261#107—22no—
262#107—22no—
263#107—21no—
264#107—22no—
265#107C2×D412yes—
266#107C2×D414yes—
267#107C2×D412yes—
268#106—28no—
269#106—22no—
270#106—22no—
271#100—216no—
272C24—21no—
273C24—42no—
274C24—44no—
275C24—81no—
276C24—81no—
277C24—82no—
278C24—162no—
279C24—162no—
280C24AGL(1,ℤ/8)11yescharacteristic
281#41—22no—
282#41—22no—
283#41—24no—
284#41—24no—
285#41—24no—
286#41—24no—
287#41—24no—
288#41#10218yes—
289#41C2×C22⋊C414yes—
290#41C2×SD1614yes—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8#10214yes—
294C2×Q8C2×D812yes—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—21no—
301C2×D4—22no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—81no—
313C2×D4—81no—
314C2×D4—82no—
315C2×D4—82no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4#10214yes—
320C2×D4C2×D812yes—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—48no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—162no—
333C22×C4—164no—
334C22×C4C2×C22⋊C412yes—
335C22×C4#12011yescharacteristic
336C22×C4#12011yescharacteristic
337C22×C4#12312yes—
338C22×C4C2×D811yescharacteristic
339C22×C4C2×D811yescharacteristic
340C22×C4C2×SD1612yes—
341C22×C4AGL(1,ℤ/8)11yescharacteristic
342C22×C4#13712yes—
343C22×C4C22×D411yescharacteristic
344SD16—416no—
345D8—24no—
346D8—24no—
347D8—416no—
348D8—42no—
349D8—42no—
350D8—44no—
351D8—44no—
352M16—22no—
353M16—24no—
354M16—24no—
355M16—24no—
356M16—28no—
357M16—24no—
358M16—42no—
359M16—44no—
360M16C2×C22⋊C414yes—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—28no—
366C2×C8—42no—
367C2×C8—44no—
368C2×C8C2×C22⋊C412yes—
369C2×C8C4×D412yes—
370C2×C8C4×D412yes—
371C2×C8C4×D412yes—
372C2×C8C4×D412yes—
373C2×C8#12011yescharacteristic
374C2×C8#12011yescharacteristic
375C2×C8#12112yes—
376C2×C8#12112yes—
377C2×C8#12112yes—
378C2×C8#12112yes—
379C2×C8#12312yes—
380C2×C8C23×C411yescharacteristic
381C2×C8C22×D411yescharacteristic
382C2×C8C22×D411yescharacteristic
383C2×C8C22×D411yescharacteristic
384C2×C8C2×#4111yescharacteristic
385C2×C8C2×#4111yescharacteristic
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—48no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C22⋊C4—48no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C42—48no—
399C42—42no—
400C42—42no—
401C16—42no—
402C16—42no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—22no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—82no—
411C23—84no—
412C23—84no—
413C23—82no—
414C23—162no—
415C23—168no—
416C23—162no—
417C23—162no—
418C23—168no—
419C23—162no—
420C23—162no—
421C23—162no—
422C23#41712yes—
423C23#44711yescharacteristic
424C23#44711yescharacteristic
425C23#48112yes—
426C23C2×AGL(1,ℤ/8)11yescharacteristic
427Q8—22no—
428Q8—24no—
429Q8C2×#10214yes—
430D4—24no—
431D4—24no—
432D4—28no—
433D4—22no—
434D4—24no—
435D4—44no—
436D4—44no—
437D4—82no—
438D4—84no—
439D4—84no—
440D4—82no—
441D4C2×#10214yes—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—84no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—162no—
454C2×C4—162no—
455C2×C4—168no—
456C2×C4#38612yes—
457C2×C4#38612yes—
458C2×C4C2×#10212yes—
459C2×C4C2×#10212yes—
460C2×C4#41712yes—
461C2×C4#44711yescharacteristic
462C2×C4#44711yescharacteristic
463C2×C4#44812yes—
464C2×C4#44812yes—
465C2×C4#48012yes—
466C2×C4#48012yes—
467C2×C4#48112yes—
468C2×C4C22×C22⋊C411yescharacteristic
469C2×C4C2×#12011yescharacteristic
470C2×C4C2×#12311yescharacteristic
471C2×C4C22×D811yescharacteristic
472C2×C4C22×SD1611yescharacteristic
473C2×C4C2×#13711yescharacteristic
474C8—24no—
475C8—44no—
476C8#38612yes—
477C8#38612yes—
478C8C2×C4×D411yescharacteristic
479C8C2×C4×D411yescharacteristic
480C8C2×#12111yescharacteristic
481C8C2×#12111yescharacteristic
482C22—21no—
483C22—21no—
484C22—24no—
485C22—24no—
486C22—21no—
487C22—21no—
488C22—24no—
489C22—42no—
490C22—48no—
491C22—42no—
492C22—84no—
493C22—164no—
494C22—164no—
495C22—168no—
496C22—162no—
497C22—168no—
498C22—162no—
499C22#187412yes—
500C22#187412yes—
501C22#219212yes—
502C22#219312yes—
503C22C2×#41711yescharacteristic
504C22C2×#44711yescharacteristic
505C22C2×#48111yescharacteristic
506C4—24no—
507C4—82no—
508C4—84no—
509C4—82no—
510C4#187412yes—
511C4#187412yes—
512C4C2×#38611yescharacteristic
513C4C22×#10211yescharacteristic
514C4C2×#44811yescharacteristic
515C4C2×#48011yescharacteristic
516C2—21no—
517C2—21no—
518C2—22no—
519C2—44no—
520C2—164no—
521C2#1299614yesdirect product factor
522C2C2×#187411yescharacteristic
523C2C2×#219211yescharacteristic
524C2C2×#219311yescharacteristic
subgroupG/H
center, socleC23#447
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×#472C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2913C2semidirect
3. C2×#2140C2semidirect
4. C22×#357C2semidirect
5. C22×#350C2semidirect
6. #12996C2direct (N × H)
7. C2×#1387C2semidirect
8. C2×#1382C2semidirect
17. C2×#472C4semidirect
23. #2913C22semidirect
32. #2140C22semidirect
37. C2×#357C22semidirect
39. C2×#350C22semidirect
40. #1387C22semidirect
41. #1382C22semidirect
58. C22×D8C2×C4semidirect
96. #472C2×C4semidirect
99. C2×#108C2×C4semidirect
180. C2×D8C22×C4semidirect
258. #108C22×C4semidirect
521. C2#12996direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4722
C2×#29132
C2×#21402
C22×#3572
C22×#3502
#129962
C2×#13872
C2×#13822