Finite Group Databasebeta

← C2×#11774 #481595

Subgroups — C2×#11774 #481595

Order 512.

1526 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic, complemented
3C2×#1465C211yesmaximal, characteristic, complemented
4#11774C218yesmaximal, direct product factor
5C2×#1344C212yesmaximal, complemented
6C2×#1343C212yesmaximal, complemented
7C22×#121—21no—
8C22×C4×D4C411yescharacteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#99—21no—
11C2×#407—21no—
12#2535—22no—
13#2535—22no—
14#2535C414yescomplemented
15#2535C2214yes—
16C2×#397C411yescharacteristic, complemented
17C2×#397C2211yescharacteristic
18C2×#392C411yescharacteristic, complemented
19C2×#392C2211yescharacteristic
20C2×#386—21no—
21C2×#386—21no—
22C2×#382C411yescharacteristic, complemented
23C2×#380—21no—
24C2×C4×C22⋊C4—21no—
25C4×#99—22no—
26C4×#99C2214yescomplemented
27C2×#342C2212yes—
28#1465—22no—
29#1465C2214yescomplemented
30C2×#324—21no—
31C2×#323C2212yes—
32#1344C2218yescomplemented
33#1343C2218yescomplemented
34C23×D4—21no—
35C23×D4—21no—
36C23×D4C2×C411yescharacteristic
37C23×D4D411yescharacteristic
38C24×C4—21no—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C22×M16—41no—
42C2×#121—24no—
43C2×#121—24no—
44C2×#121—48no—
45C2×C4×D4—22no—
46C2×C4×D4—41no—
47C2×C4×D4—41no—
48C2×C4×D4—42no—
49C2×C4×D4—42no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C2×C4×D4C2×C412yes—
53C2×C4×D4C2×C414yescomplemented
54C2×C4×D4D412yes—
55C2×C4×D4D412yes—
56C2×C4×D4C2312yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2311yescharacteristic
68C22×C42—41no—
69C22×C42C2×C411yescharacteristic
70C2×#99—28no—
71C2×#99—24no—
72#407—24no—
73#407—42no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—41no—
76#397—22no—
77#397—21no—
78#397—21no—
79#397C2×C412yes—
80#397C2×C414yescomplemented
81#397C2312yes—
82#392—22no—
83#392—21no—
84#392—21no—
85#392C2×C412yes—
86#392C2×C414yescomplemented
87#392C2312yes—
88#386—24no—
89#386—24no—
90#386—24no—
91#386—24no—
92#382—22no—
93#382C2×C414yescomplemented
94#380—24no—
95#380—24no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—42no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—41no—
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104#342—22no—
105#342C2314yes—
106#324—24no—
107#323—22no—
108#323C2314yes—
109C25—21no—
110C25C22⋊C411yescharacteristic
111C25C2×D411yescharacteristic
112C22×D4—24no—
113C22×D4—22no—
114C22×D4—24no—
115C22×D4—22no—
116C22×D4—24no—
117C22×D4—21no—
118C22×D4—41no—
119C22×D4—41no—
120C22×D4—41no—
121C22×D4—41no—
122C22×D4—44no—
123C22×D4—44no—
124C22×D4—44no—
125C22×D4—44no—
126C22×D4—44no—
127C22×D4—82no—
128C22×D4—82no—
129C22×D4C22⋊C412yes—
130C22×D4C22×C412yes—
131C22×D4C2×D412yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—44no—
147C23×C4—82no—
148C23×C4—82no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C412yes—
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C2×M16—44no—
159C2×M16—84no—
160C22×C8—42no—
161#121—416no—
162#121—416no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—44no—
172C4×D4—44no—
173C4×D4—44no—
174C4×D4—44no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22×C412yes—
209C2×C22⋊C4C22×C411yescharacteristic
210C2×C22⋊C4C2×D411yescharacteristic
211C2×C22⋊C4C2×D411yescharacteristic
212C2×C22⋊C4C2411yescharacteristic
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—44no—
217C2×C42—84no—
218C2×C42C22⋊C412yes—
219C2×C42C22×C412yes—
220#99—216no—
221C22⋊C8—24no—
222C22⋊C8—42no—
223C22⋊C8—44no—
224#95—22no—
225#95—24no—
226#95—24no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—44no—
231#95C22×C412yes—
232#95C22×C412yes—
233C24—21no—
234C24—21no—
235C24—22no—
236C24—24no—
237C24—22no—
238C24—21no—
239C24—22no—
240C24—21no—
241C24—22no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24—41no—
250C24—41no—
251C24—42no—
252C24—42no—
253C24C2×C22⋊C411yescharacteristic
254C24C2×C22⋊C412yes—
255C24C2×C22⋊C411yescharacteristic
256C24AGL(1,ℤ/8)14yes—
257C24C22×D411yescharacteristic
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—48no—
265C2×D4—48no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—48no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—48no—
274C2×D4—44no—
275C2×D4—48no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—82no—
280C2×D4—84no—
281C2×D4—82no—
282C2×D4—816no—
283C2×D4—816no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—48no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—48no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4#9911yescharacteristic
338C22×C4#9911yescharacteristic
339C22×C4#9912yes—
340C22×C4#10414yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C412yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D411yescharacteristic
348M16—88no—
349C2×C8—44no—
350C2×C8—84no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—48no—
356C22⋊C4—28no—
357C22⋊C4—42no—
358C22⋊C4—41no—
359C22⋊C4—42no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—82no—
371C22⋊C4—84no—
372C22⋊C4—82no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—88no—
378C23—22no—
379C23—22no—
380C23—21no—
381C23—21no—
382C23—24no—
383C23—21no—
384C23—21no—
385C23—24no—
386C23—24no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—41no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—41no—
396C23—48no—
397C23—42no—
398C23—42no—
399C23—44no—
400C23—42no—
401C23—48no—
402C23—48no—
403C23—48no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—84no—
419C23—84no—
420C23—82no—
421C23—84no—
422C23—84no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23C2×#9911yescharacteristic
427C23C2×#9911yescharacteristic
428C23#41812yes—
429C23#41812yes—
430C23C2×#10412yes—
431C23C22×C22⋊C411yescharacteristic
432C23C2×AGL(1,ℤ/8)12yes—
433D4—48no—
434D4—48no—
435D4—44no—
436D4—44no—
437D4—48no—
438D4—88no—
439D4—88no—
440D4—816no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—48no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—816no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—816no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4C2×#9912yes—
492C2×C4C2×#9912yes—
493C2×C4C2×#10414yes—
494C8—84no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—24no—
500C22—24no—
501C22—22no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—48no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—81no—
518C22—81no—
519C22—84no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—84no—
524C22—88no—
525C22—82no—
526C22—82no—
527C22—84no—
528C22—88no—
529C22#150712yes—
530C22#150712yes—
531C22#212012yes—
532C22#212712yes—
533C22C22×#9911yescharacteristic
534C22C2×#41811yescharacteristic
535C22C22×#10411yescharacteristic
536C4—22no—
537C4—22no—
538C4—24no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—42no—
543C4—41no—
544C4—82no—
545C4—82no—
546C4—88no—
547C4—82no—
548C2—21no—
549C2—21no—
550C2—22no—
551C2—42no—
552C2—42no—
553C2—82no—
554C2—84no—
555C2#1177414yesdirect product factor
556C2C2×#150711yescharacteristic
557C2C2×#212011yescharacteristic
558C2C2×#212711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#99C2semidirect
3. C2×#1465C2semidirect
4. #11774C2direct (N × H)
5. C2×#1344C2semidirect
6. C2×#1343C2semidirect
8. C22×C4×D4C4semidirect
14. #2535C4semidirect
16. C2×#397C4semidirect
18. C2×#392C4semidirect
22. C2×#382C4semidirect
26. C4×#99C22semidirect
29. #1465C22semidirect
32. #1344C22semidirect
33. #1343C22semidirect
53. C2×C4×D4C2×C4semidirect
80. #397C2×C4semidirect
86. #392C2×C4semidirect
93. #382C2×C4semidirect
555. C2#11774direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25352
C2×C4×#992
C2×#14652
#117742
C2×#13442
C2×#13432