Finite Group Databasebeta

← C2×#33894 #7623706

Subgroups — C2×#33894 #7623706

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3336C211yesmaximal, characteristic, complemented
2C2×#3285C211yesmaximal, characteristic
3#33894C2116yesmaximal, direct product factor
4C2×#2242C211yesmaximal, characteristic, complemented
5C2×#2236C211yesmaximal, characteristic, complemented
6C2×#2234C211yesmaximal, characteristic, complemented
7C2×#2229C211yesmaximal, characteristic, complemented
8C2×#2220C211yesmaximal, characteristic, complemented
9C2×#2219C211yesmaximal, characteristic, complemented
10C2×#2211C211yesmaximal, characteristic, complemented
11C2×#2189C211yesmaximal, characteristic, complemented
12C2×#2188C211yesmaximal, characteristic, complemented
13C2×C4×Q32C211yesmaximal, characteristic, complemented
14C2×#2169C211yesmaximal, characteristic, complemented
15C2×#2139C211yesmaximal, characteristic, complemented
16C2×#2141C211yesmaximal, characteristic, complemented
17C22×#135C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C2×#552—21no—
20C2×#548—21no—
21C22×Q32C2211yescharacteristic
22C22×Q32C2211yescharacteristic
23C22×SD32C2211yescharacteristic
24C22×M32C2211yescharacteristic
25C22×M32C2211yescharacteristic
26#3336—24no—
27#3336C2218yescomplemented
28#3285—24no—
29#3285C2218yes—
30C2×#499C2211yescharacteristic
31C2×#495C2211yescharacteristic
32C2×#485—21no—
33C2×#483—21no—
34C2×#479—21no—
35C2×#467C2211yescharacteristic
36C2×#467C2211yescharacteristic
37C2×#467C2211yescharacteristic
38C2×#466C2211yescharacteristic
39C2×#464—21no—
40C2×#452—21no—
41C2×#448—21no—
42C2×C4×Q16C2211yescharacteristic
43C2×C4×Q16C2211yescharacteristic
44C2×C8×D4C2211yescharacteristic
45C2×#427C2211yescharacteristic
46C22×#107C2211yescharacteristic
47#2242C2218yescomplemented
48#2236C2218yescomplemented
49#2234C2218yescomplemented
50#2229C2218yescomplemented
51#2220C2218yescomplemented
52#2219C2218yescomplemented
53#2211C2218yescomplemented
54#2189C2218yescomplemented
55#2188C2218yescomplemented
56C4×Q32C2218yescomplemented
57#2169C2218yescomplemented
58C2×#367C2211yescharacteristic
59C2×#366C2211yescharacteristic
60C2×#366C2211yescharacteristic
61C2×C4⋊C16C2211yescharacteristic
62#2141C2218yescomplemented
63#2139C2218yescomplemented
64C2×#358C2211yescharacteristic
65C2×#358C2211yescharacteristic
66C2×#358C2211yescharacteristic
67C2×#358C2211yescharacteristic
68C2×#358C2211yescharacteristic
69C2×#358C2211yescharacteristic
70C2×#358C2211yescharacteristic
71C2×#357C2211yescharacteristic
72C2×#357C2211yescharacteristic
73C2×#357C2211yescharacteristic
74C2×C22⋊C16C2211yescharacteristic
75C2×C22⋊C16C2211yescharacteristic
76C2×C4×C16C2211yescharacteristic
77C22×#41—21no—
78C23×Q8—21no—
79C2×#135—24no—
80C2×#135—22no—
81C2×#135—42no—
82C2×#135C2314yes—
83C22×Q16—24no—
84C22×Q16—21no—
85C22×Q16—22no—
86C22×Q16—42no—
87C22×Q16C2311yescharacteristic
88C22×Q16C2311yescharacteristic
89C22×Q16C2314yes—
90C22×Q16C2311yescharacteristic
91C22×SD16—21no—
92C22×SD16—21no—
93C22×D8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96#552—28no—
97#552—44no—
98#548—28no—
99#548—44no—
100C2×#126—41no—
101C2×#125—21no—
102C2×#125—41no—
103C2×#124—21no—
104C2×#124—41no—
105C2×#123—41no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—41no—
110C2×#121—21no—
111C2×#121—41no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×D4—41no—
115C2×C4×D4D411yescharacteristic
116C2×#117—21no—
117C2×#117—41no—
118C22×C4⋊C4—21no—
119C2×Q32—24no—
120C2×Q32—42no—
121C2×Q32C2314yes—
122C2×Q32C2314yes—
123C2×SD32—42no—
124C2×SD32C2314yes—
125C2×M32—42no—
126C2×M32—42no—
127C2×M32C2314yes—
128C2×M32C2314yes—
129C22×C16C2311yescharacteristic
130C22×C16C2311yescharacteristic
131C22×C16C2311yescharacteristic
132C22×C16C2311yescharacteristic
133#499—22no—
134#499C2314yes—
135#495—22no—
136#495C2314yes—
137#485—28no—
138#483—28no—
139#479—28no—
140#467—22no—
141#467—22no—
142#467—22no—
143#467C2314yes—
144#467C2314yes—
145#467C2314yes—
146#466—22no—
147#466C2314yes—
148#464—28no—
149#452—28no—
150#448—28no—
151C4×Q16—22no—
152C4×Q16—22no—
153C4×Q16C2314yes—
154C4×Q16C2314yes—
155C8×D4D414yes—
156C8×D4C2314yes—
157#427—22no—
158#427C2314yes—
159C2×#107—21no—
160C2×#107—22no—
161C2×#107C2314yes—
162C2×#107C2311yescharacteristic
163C2×#107C2311yescharacteristic
164C2×#107C2311yescharacteristic
165C2×#107C2311yescharacteristic
166C2×#106—21no—
167C2×#106—21no—
168C2×C4⋊C8D411yescharacteristic
169C2×#103—21no—
170C2×#103—21no—
171C2×#103—21no—
172C2×#103—21no—
173C2×#103—21no—
174C2×#102—21no—
175C2×C22⋊C8D411yescharacteristic
176C2×C22⋊C8D411yescharacteristic
177C2×C4×C8C2311yescharacteristic
178#367C2314yes—
179#366C2314yes—
180#366C2314yes—
181C4⋊C16C2314yes—
182#358C2314yes—
183#358C2314yes—
184#358C2314yes—
185#358C2314yes—
186#358C2314yes—
187#358C2314yes—
188#358C2314yes—
189#357C2314yes—
190#357C2314yes—
191#357C2314yes—
192C22⋊C16C2314yes—
193C22⋊C16C2314yes—
194C4×C16C2314yes—
195C2×#41—24no—
196C2×#41—44no—
197C2×#41—42no—
198C2×#41—82no—
199C22×Q8—21no—
200C22×Q8—21no—
201C22×Q8—21no—
202C22×Q8—24no—
203C22×Q8—44no—
204C22×Q8—41no—
205C22×Q8—42no—
206C22×Q8—82no—
207C22×D4—21no—
208C22×D4—41no—
209C22×D4—41no—
210C22×D4D811yescharacteristic
211C23×C4—41no—
212C23×C4—41no—
213C23×C4—41no—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216#135—28no—
217#135—44no—
218#135—48no—
219C2×Q16—28no—
220C2×Q16—21no—
221C2×Q16—24no—
222C2×Q16—21no—
223C2×Q16—21no—
224C2×Q16—22no—
225C2×Q16—22no—
226C2×Q16—22no—
227C2×Q16—22no—
228C2×Q16—22no—
229C2×Q16—44no—
230C2×Q16—48no—
231C2×Q16—44no—
232C2×Q16#4114yes—
233C2×Q16C2412yes—
234C2×Q16C2412yes—
235C2×Q16C2412yes—
236C2×SD16—24no—
237C2×SD16—24no—
238C2×SD16—44no—
239C2×SD16—82no—
240C2×D8—21no—
241C2×D8—22no—
242C2×D8—22no—
243C2×D8C2412yes—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8C2×D411yescharacteristic
249C22×C8C2×D411yescharacteristic
250C22×C8C2×D412yes—
251C22×C8C2×D412yes—
252C22×C8C2×D411yescharacteristic
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2411yescharacteristic
256#126—48no—
257#125—24no—
258#125—48no—
259#125—42no—
260#124—24no—
261#124—48no—
262#124—42no—
263#123—48no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—42no—
268C22⋊Q8—42no—
269C22⋊Q8—42no—
270C22⋊Q8—48no—
271#121—24no—
272#121—48no—
273#121—42no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—42no—
277C4×Q8—42no—
278C4×D4—48no—
279C4×D4D814yes—
280C4×D4C2×D414yes—
281#117—24no—
282#117—42no—
283#117—48no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—41no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—41no—
296C2×C4⋊C4—41no—
297C2×C4⋊C4—41no—
298C2×C4⋊C4D811yescharacteristic
299C2×C22⋊C4—41no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4—41no—
306C2×C22⋊C4D811yescharacteristic
307C2×C22⋊C4D811yescharacteristic
308C2×C42—41no—
309C2×C42—41no—
310C2×C42—41no—
311C2×C42C2×D411yescharacteristic
312Q32—28no—
313Q32—44no—
314SD32—44no—
315M32—44no—
316M32—44no—
317C2×C16—22no—
318C2×C16—22no—
319C2×C16—22no—
320C2×C16—22no—
321C2×C16C2412yes—
322C2×C16C2412yes—
323C2×C16C2412yes—
324C2×C16C2412yes—
325#107—21no—
326#107—21no—
327#107—24no—
328#107—21no—
329#107—21no—
330#107C2412yes—
331#107C2412yes—
332#107C2412yes—
333#107C2412yes—
334#106—24no—
335#106—24no—
336C4⋊C8C2×D414yes—
337#103—24no—
338#103—24no—
339#103—24no—
340#103—24no—
341#103—24no—
342#102—24no—
343C22⋊C8C2×D414yes—
344C22⋊C8C2×D414yes—
345C4×C8C2×D412yes—
346C4×C8C2412yes—
347C24—41no—
348C24C2×D811yescharacteristic
349C24C2×D811yescharacteristic
350#41—48no—
351#41—84no—
352#41—88no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—48no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—44no—
361C2×Q8—41no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—84no—
368C2×Q8—84no—
369C2×Q8—88no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4C2×D814yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—41no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—44no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4C2×D812yes—
398C22×C4C2×D812yes—
399C22×C4C2×D811yescharacteristic
400C22×C4C2×D811yescharacteristic
401C22×C4C2×D811yescharacteristic
402C22×C4C2×D811yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406Q16—24no—
407Q16—22no—
408Q16—22no—
409Q16—22no—
410Q16—22no—
411Q16—22no—
412Q16—48no—
413Q16—42no—
414Q16—42no—
415Q16C2×#4114yes—
416SD16—48no—
417SD16—84no—
418D8—22no—
419D8—22no—
420D8—42no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8C22×D412yes—
431C2×C8C22×D412yes—
432C2×C8C22×D412yes—
433C2×C8C22×D411yescharacteristic
434C2×C8C2×#4112yes—
435C2×C8H(5,𝔽2)12yes—
436C2×C8C2511yescharacteristic
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—44no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4C2×D814yes—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4C2×D814yes—
462C22⋊C4C2×D814yes—
463C42—44no—
464C42—44no—
465C42—44no—
466C42C2×D812yes—
467C42C22×D412yes—
468C16—22no—
469C16—22no—
470C16—22no—
471C16—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—42no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23C22×D812yes—
483C23C22×D812yes—
484C23C22×D811yescharacteristic
485Q8—24no—
486Q8—44no—
487Q8—42no—
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—88no—
493Q8—82no—
494Q8—82no—
495D4—44no—
496D4—42no—
497D4—42no—
498D4—88no—
499D4—88no—
500D4—82no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—88no—
520C2×C4—88no—
521C2×C4—88no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#54612yes—
529C2×C4C22×D812yes—
530C2×C4C22×D812yes—
531C2×C4C22×D812yes—
532C2×C4C22×D812yes—
533C2×C4C22×D811yescharacteristic
534C2×C4#57712yes—
535C2×C4C23×D411yescharacteristic
536C8—22no—
537C8—22no—
538C8—22no—
539C8#54612yes—
540C8C22×#4111yescharacteristic
541C8C2×H(5,𝔽2)11yescharacteristic
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—44no—
549C22—44no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—164no—
554C22#329412yes—
555C22#341412yes—
556C22#341712yes—
557C22C23×D811yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—44no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4#329412yes—
570C4C2×#54611yescharacteristic
571C4C2×#57711yescharacteristic
572C2—42no—
573C2—42no—
574C2—162no—
575C2#3389414yesdirect product factor
576C2C2×#329411yescharacteristic
577C2C2×#341411yescharacteristic
578C2C2×#341711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC2×#3285C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3336C2semidirect
3. #33894C2direct (N × H)
4. C2×#2242C2semidirect
5. C2×#2236C2semidirect
6. C2×#2234C2semidirect
7. C2×#2229C2semidirect
8. C2×#2220C2semidirect
9. C2×#2219C2semidirect
10. C2×#2211C2semidirect
11. C2×#2189C2semidirect
12. C2×#2188C2semidirect
13. C2×C4×Q32C2semidirect
14. C2×#2169C2semidirect
15. C2×#2139C2semidirect
16. C2×#2141C2semidirect
27. #3336C22semidirect
47. #2242C22semidirect
48. #2236C22semidirect
49. #2234C22semidirect
50. #2229C22semidirect
51. #2220C22semidirect
52. #2219C22semidirect
53. #2211C22semidirect
54. #2189C22semidirect
55. #2188C22semidirect
56. C4×Q32C22semidirect
57. #2169C22semidirect
62. #2141C22semidirect
63. #2139C22semidirect
575. C2#33894direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33362
C2×#32852
#338942
C2×#22422
C2×#22362
C2×#22342
C2×#22292
C2×#22202
C2×#22192
C2×#22112
C2×#21892
C2×#21882
C2×C4×Q322
C2×#21692
C2×#21392
C2×#21412