Finite Group Databasebeta

← C2×#33875 #7623688

Subgroups — C2×#33875 #7623688

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3334C211yesmaximal, characteristic, complemented
2C2×#3285C211yesmaximal, characteristic, complemented
3#33875C2116yesmaximal, direct product factor
4C2×#2243C211yesmaximal, characteristic, complemented
5C2×#2236C211yesmaximal, characteristic, complemented
6C2×#2232C211yesmaximal, characteristic, complemented
7C2×#2227C211yesmaximal, characteristic, complemented
8C2×#2220C211yesmaximal, characteristic, complemented
9C2×#2215C211yesmaximal, characteristic, complemented
10C2×#2210C211yesmaximal, characteristic, complemented
11C2×#2186C211yesmaximal, characteristic, complemented
12C2×#2185C211yesmaximal, characteristic, complemented
13C2×#2175C211yesmaximal, characteristic, complemented
14C2×#2168C211yesmaximal, characteristic, complemented
15C2×#2139C211yesmaximal, characteristic, complemented
16C22×#358C211yesmaximal, characteristic, complemented
17C22×#135C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C2×#548—21no—
20C2×#547—21no—
21C22×SD32C2211yescharacteristic
22C22×SD32C2211yescharacteristic
23C22×D16C2211yescharacteristic
24C22×M32C2211yescharacteristic
25C23×C16C2211yescharacteristic
26#3334—24no—
27#3334C2218yescomplemented
28#3285—24no—
29#3285C2218yescomplemented
30C2×#500C2211yescharacteristic
31C2×#495C2211yescharacteristic
32C2×#480—21no—
33C2×#480—21no—
34C2×#474—21no—
35C2×#467C2211yescharacteristic
36C2×#466C2211yescharacteristic
37C2×#466C2211yescharacteristic
38C2×#466C2211yescharacteristic
39C2×#464—21no—
40C2×#452—21no—
41C2×#448—21no—
42C2×C4×Q16C2211yescharacteristic
43C2×C4×D8C2211yescharacteristic
44C2×C8×D4C2211yescharacteristic
45C22×#107C2211yescharacteristic
46C22×#107C2211yescharacteristic
47#2243C2218yescomplemented
48#2236C2218yescomplemented
49#2232C2218yescomplemented
50#2227C2218yescomplemented
51#2220C2218yescomplemented
52#2215C2218yescomplemented
53#2210C2218yescomplemented
54#2186C2218yescomplemented
55#2185C2218yescomplemented
56#2175C2218yescomplemented
57#2168C2218yescomplemented
58C2×#366C2211yescharacteristic
59C2×#366C2211yescharacteristic
60C2×#366C2211yescharacteristic
61C2×C4⋊C16C2211yescharacteristic
62#2139C2218yescomplemented
63C2×#358C2211yescharacteristic
64C2×#358C2211yescharacteristic
65C2×#358C2211yescharacteristic
66C2×#358C2211yescharacteristic
67C2×#358C2211yescharacteristic
68C2×#358C2211yescharacteristic
69C2×#358C2218yescomplemented
70C2×#357C2211yescharacteristic
71C2×#357C2211yescharacteristic
72C2×#357C2211yescharacteristic
73C2×#357C2211yescharacteristic
74C2×C22⋊C16C2211yescharacteristic
75C2×C22⋊C16C2211yescharacteristic
76C2×#346C2211yescharacteristic
77C22×#41—21no—
78C22×#41—41no—
79C23×Q8—21no—
80C2×#135—24no—
81C2×#135—22no—
82C2×#135—42no—
83C2×#135C2314yes—
84C22×Q16—24no—
85C22×Q16—21no—
86C22×Q16—22no—
87C22×Q16—42no—
88C22×Q16C2311yescharacteristic
89C22×Q16C2314yes—
90C22×Q16C2311yescharacteristic
91C22×SD16—21no—
92C22×SD16—21no—
93C22×D8C2311yescharacteristic
94C22×D8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97#548—28no—
98#548—44no—
99#547—28no—
100#547—44no—
101C2×C4⋊Q8—21no—
102C2×#124—21no—
103C2×#124—41no—
104C2×#123—41no—
105C2×#123—41no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—41no—
108C2×C22⋊Q8—41no—
109C2×#121—21no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121—41no—
113C2×C4×Q8—21no—
114C2×C4×D4—21no—
115C2×C4×D4—41no—
116C2×C4×D4D411yescharacteristic
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C2×SD32—24no—
120C2×SD32—42no—
121C2×SD32C2314yes—
122C2×SD32C2314yes—
123C2×D16—42no—
124C2×D16C2314yes—
125C2×M32—42no—
126C2×M32C2314yes—
127C22×C16—42no—
128C22×C16C2311yescharacteristic
129C22×C16C2311yescharacteristic
130C22×C16C2311yescharacteristic
131C22×C16C2311yescharacteristic
132C22×C16C2314yes—
133#500—22no—
134#500C2314yes—
135#495—22no—
136#495C2314yes—
137#480—28no—
138#480—28no—
139#474—28no—
140#467—22no—
141#467C2314yes—
142#466—22no—
143#466—22no—
144#466—22no—
145#466C2314yes—
146#466C2314yes—
147#466C2314yes—
148#464—28no—
149#452—28no—
150#448—28no—
151C4×Q16—22no—
152C4×Q16C2314yes—
153C4×D8—22no—
154C4×D8C2314yes—
155C8×D4D414yes—
156C8×D4C2314yes—
157C2×#107—22no—
158C2×#107—22no—
159C2×#107—21no—
160C2×#107—21no—
161C2×#107—21no—
162C2×#107C2314yes—
163C2×#107C2314yes—
164C2×#107C2311yescharacteristic
165C2×#107C2311yescharacteristic
166C2×#107C2311yescharacteristic
167C2×#107C2311yescharacteristic
168C2×C4⋊C8D411yescharacteristic
169C2×#103—21no—
170C2×#103—21no—
171C2×#102—21no—
172C2×#102—21no—
173C2×#102—21no—
174C2×#102—21no—
175C2×C22⋊C8D411yescharacteristic
176C2×C22⋊C8D411yescharacteristic
177C2×C4×C8C2311yescharacteristic
178#366C2314yes—
179#366C2314yes—
180#366C2314yes—
181C4⋊C16C2314yes—
182#358C2314yes—
183#358C2314yes—
184#358C2314yes—
185#358C2314yes—
186#358C2314yes—
187#358C2314yes—
188#357C2314yes—
189#357C2314yes—
190#357C2314yes—
191#357C2314yes—
192C22⋊C16C2314yes—
193C22⋊C16C2314yes—
194#346C2314yes—
195C2×#41—24no—
196C2×#41—44no—
197C2×#41—48no—
198C2×#41—42no—
199C2×#41—82no—
200C2×#41—88no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—24no—
204C22×Q8—44no—
205C22×Q8—41no—
206C22×Q8—41no—
207C22×Q8—42no—
208C22×Q8—82no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—41no—
212C22×D4—41no—
213C22×D4—41no—
214C22×D4—41no—
215C22×D4D811yescharacteristic
216C23×C4—41no—
217C23×C4—41no—
218C23×C4—41no—
219C23×C4—41no—
220C23×C4—41no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223#135—28no—
224#135—44no—
225#135—48no—
226C2×Q16—28no—
227C2×Q16—21no—
228C2×Q16—24no—
229C2×Q16—21no—
230C2×Q16—22no—
231C2×Q16—22no—
232C2×Q16—22no—
233C2×Q16—44no—
234C2×Q16—48no—
235C2×Q16—44no—
236C2×Q16#4114yescomplemented
237C2×Q16C2412yes—
238C2×Q16C2412yes—
239C2×SD16—24no—
240C2×SD16—24no—
241C2×SD16—44no—
242C2×SD16—82no—
243C2×D8—21no—
244C2×D8—21no—
245C2×D8—22no—
246C2×D8—22no—
247C2×D8—22no—
248C2×D8—22no—
249C2×D8C2412yes—
250C2×D8C2412yes—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8C2×D411yescharacteristic
256C22×C8C2×D411yescharacteristic
257C22×C8C2×D412yes—
258C22×C8C2×D412yes—
259C22×C8C2×D411yescharacteristic
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2411yescharacteristic
263C4⋊Q8—24no—
264C4⋊Q8—42no—
265#124—24no—
266#124—48no—
267#124—42no—
268#123—48no—
269#123—48no—
270C22⋊Q8—24no—
271C22⋊Q8—42no—
272C22⋊Q8—48no—
273C22⋊Q8—48no—
274#121—24no—
275#121—24no—
276#121—24no—
277#121—48no—
278#121—42no—
279#121—42no—
280#121—42no—
281C4×Q8—24no—
282C4×Q8—42no—
283C4×D4—24no—
284C4×D4—48no—
285C4×D4—42no—
286C4×D4D814yes—
287C4×D4C2×D414yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—41no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—41no—
299C2×C4⋊C4—41no—
300C2×C4⋊C4—41no—
301C2×C4⋊C4D811yescharacteristic
302C2×C22⋊C4—41no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4—41no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4—41no—
308C2×C22⋊C4—41no—
309C2×C22⋊C4D811yescharacteristic
310C2×C22⋊C4D811yescharacteristic
311C2×C42—41no—
312C2×C42C2×D411yescharacteristic
313SD32—28no—
314SD32—44no—
315D16—44no—
316M32—44no—
317C2×C16—22no—
318C2×C16—22no—
319C2×C16—22no—
320C2×C16—22no—
321C2×C16—44no—
322C2×C16C2412yes—
323C2×C16C2412yes—
324C2×C16C2412yes—
325C2×C16C2412yes—
326#107—21no—
327#107—21no—
328#107—21no—
329#107—24no—
330#107—24no—
331#107—24no—
332#107—21no—
333#107C2412yes—
334#107C2412yes—
335#107C2412yes—
336#107C2412yes—
337C4⋊C8C2×D414yes—
338#103—24no—
339#103—24no—
340#102—24no—
341#102—24no—
342#102—24no—
343#102—24no—
344C22⋊C8C2×D414yes—
345C22⋊C8C2×D414yes—
346C4×C8C2×D412yes—
347C4×C8C2412yes—
348C24—41no—
349C24—41no—
350C24C2×D811yescharacteristic
351C24C2×D811yescharacteristic
352#41—48no—
353#41—84no—
354#41—88no—
355#41—832no—
356C2×Q8—24no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—48no—
360C2×Q8—41no—
361C2×Q8—44no—
362C2×Q8—41no—
363C2×Q8—44no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—84no—
368C2×Q8—84no—
369C2×Q8—88no—
370C2×Q8—84no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4C2×D814yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—41no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4C2×D812yes—
410C22×C4C2×D812yes—
411C22×C4C2×D811yescharacteristic
412C22×C4C2×D811yescharacteristic
413C22×C4C2×D811yescharacteristic
414C22×C4C2×D811yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418Q16—24no—
419Q16—22no—
420Q16—22no—
421Q16—22no—
422Q16—48no—
423Q16—42no—
424Q16C2×#4114yescomplemented
425SD16—48no—
426SD16—84no—
427D8—22no—
428D8—22no—
429D8—22no—
430D8—22no—
431D8—42no—
432D8—42no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8C22×D412yes—
443C2×C8C22×D412yes—
444C2×C8C22×D412yes—
445C2×C8C22×D411yescharacteristic
446C2×C8C2×#4112yes—
447C2×C8H(5,𝔽2)12yes—
448C2×C8C2511yescharacteristic
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—41no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4C2×D814yes—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4C2×D814yes—
471C22⋊C4C2×D814yes—
472C42—44no—
473C42C2×D812yes—
474C42C22×D412yes—
475C16—22no—
476C16—22no—
477C16—22no—
478C16—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—42no—
485C23—42no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23#50914yes—
493C23C22×D812yes—
494C23C22×D812yes—
495C23C22×D811yescharacteristic
496Q8—24no—
497Q8—44no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501Q8—88no—
502Q8—88no—
503Q8—82no—
504D4—28no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—88no—
510D4—88no—
511D4—88no—
512D4—88no—
513D4—82no—
514D4—82no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—88no—
533C2×C4—88no—
534C2×C4—88no—
535C2×C4—88no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—88no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4#54612yes—
543C2×C4C22×D812yes—
544C2×C4C22×D812yes—
545C2×C4C22×D812yes—
546C2×C4C22×D812yes—
547C2×C4C22×D811yescharacteristic
548C2×C4#57712yes—
549C2×C4C23×D411yescharacteristic
550C8—22no—
551C8—22no—
552C8—22no—
553C8#54612yes—
554C8C22×#4111yescharacteristic
555C8C2×H(5,𝔽2)11yescharacteristic
556C22—28no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—44no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—164no—
570C22—164no—
571C22#329412yes—
572C22C2×#50914yes—
573C22C2×#50912yes—
574C22#341712yes—
575C22C23×D811yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—44no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4#329412yes—
587C4C2×#54611yescharacteristic
588C4C2×#57711yescharacteristic
589C2—24no—
590C2—42no—
591C2—162no—
592C2—162no—
593C2#3387514yesdirect product factor
594C2C2×#329411yescharacteristic
595C2C22×#50911yescharacteristic
596C2C2×#341711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC2×C8C25
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×#3334C2semidirect
2. C2×#3285C2semidirect
3. #33875C2direct (N × H)
4. C2×#2243C2semidirect
5. C2×#2236C2semidirect
6. C2×#2232C2semidirect
7. C2×#2227C2semidirect
8. C2×#2220C2semidirect
9. C2×#2215C2semidirect
10. C2×#2210C2semidirect
11. C2×#2186C2semidirect
12. C2×#2185C2semidirect
13. C2×#2175C2semidirect
14. C2×#2168C2semidirect
15. C2×#2139C2semidirect
16. C22×#358C2semidirect
27. #3334C22semidirect
29. #3285C22semidirect
47. #2243C22semidirect
48. #2236C22semidirect
49. #2232C22semidirect
50. #2227C22semidirect
51. #2220C22semidirect
52. #2215C22semidirect
53. #2210C22semidirect
54. #2186C22semidirect
55. #2185C22semidirect
56. #2175C22semidirect
57. #2168C22semidirect
62. #2139C22semidirect
69. C2×#358C22semidirect
236. C2×Q16#41semidirect
424. Q16C2×#41semidirect
593. C2#33875direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33342
C2×#32852
#338752
C2×#22432
C2×#22362
C2×#22322
C2×#22272
C2×#22202
C2×#22152
C2×#22102
C2×#21862
C2×#21852
C2×#21752
C2×#21682
C2×#21392
C22×#3582