Finite Group Databasebeta

← C2×#33914 #7623798

Subgroups — C2×#33914 #7623798

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3341C211yesmaximal, characteristic, complemented
2C2×#3339C211yesmaximal, characteristic, complemented
3#33914C2116yesmaximal, direct product factor
4C2×#2255C211yesmaximal, characteristic, complemented
5C2×#2234C211yesmaximal, characteristic, complemented
6C2×#2234C211yesmaximal, characteristic, complemented
7C2×#2234C211yesmaximal, characteristic, complemented
8C2×#2232C211yesmaximal, characteristic, complemented
9C2×#2228C211yesmaximal, characteristic, complemented
10C2×#2224C211yesmaximal, characteristic, complemented
11C2×#2218C211yesmaximal, characteristic, complemented
12C2×#2215C211yesmaximal, characteristic, complemented
13C2×#2177C211yesmaximal, characteristic, complemented
14C2×#2168C211yesmaximal, characteristic, complemented
15C2×#2159C211yesmaximal, characteristic, complemented
16C2×#2158C211yesmaximal, characteristic, complemented
17C2×#555—21no—
18C2×#553—21no—
19C22×D16C2211yescharacteristic
20C22×M32C2211yescharacteristic
21C23×C16C2211yescharacteristic
22#3341—24no—
23#3341C2218yescomplemented
24#3339—24no—
25#3339C2218yescomplemented
26C2×#500C2211yescharacteristic
27C2×#499C2211yescharacteristic
28C2×#481—21no—
29C2×#482—21no—
30C2×#482—21no—
31C2×#482—21no—
32C2×#478—21no—
33C2×#476—21no—
34C2×#466C2211yescharacteristic
35C2×#466C2211yescharacteristic
36C2×#466C2211yescharacteristic
37C2×#466C2211yescharacteristic
38C2×C4×D8C2211yescharacteristic
39C2×C4×D8C2211yescharacteristic
40C2×C8×D4C2211yescharacteristic
41C2×#427C2211yescharacteristic
42C2×#427C2211yescharacteristic
43C2×#427C2211yescharacteristic
44C22×#107C2211yescharacteristic
45#2255C2218yescomplemented
46#2234C2218yescomplemented
47#2234C2218yescomplemented
48#2234C2218yescomplemented
49#2232C2218yescomplemented
50#2228C2218yescomplemented
51#2224C2218yescomplemented
52#2218C2218yescomplemented
53#2215C2218yescomplemented
54#2177C2218yescomplemented
55#2168C2218yescomplemented
56#2159C2218yescomplemented
57#2158C2218yescomplemented
58C2×#367C2211yescharacteristic
59C2×#367C2211yescharacteristic
60C2×#367C2211yescharacteristic
61C2×#367C2211yescharacteristic
62C2×#366C2211yescharacteristic
63C2×#366C2211yescharacteristic
64C2×#366C2211yescharacteristic
65C2×#366C2211yescharacteristic
66C2×#366C2211yescharacteristic
67C2×C4⋊C16C2211yescharacteristic
68C2×#357C2211yescharacteristic
69C2×#357C2211yescharacteristic
70C2×#357C2211yescharacteristic
71C2×#357C2211yescharacteristic
72C2×#357C2211yescharacteristic
73C2×#357C2211yescharacteristic
74C2×C22⋊C16C2211yescharacteristic
75C2×C22⋊C16C2211yescharacteristic
76C2×#346C2211yescharacteristic
77C22×D8C2311yescharacteristic
78C22×D8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81#555—28no—
82#555—44no—
83#553—28no—
84#553—44no—
85C2×C4⋊Q8—21no—
86C2×#126—41no—
87C2×#125—21no—
88C2×#124—41no—
89C2×#124—41no—
90C2×#123—41no—
91C2×#121—21no—
92C2×#121—21no—
93C2×#121—21no—
94C2×#121—21no—
95C2×#121—41no—
96C2×#121—41no—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4—41no—
100C2×C4×D4D411yescharacteristic
101C2×#117—21no—
102C2×#117—21no—
103C2×#117—21no—
104C2×#117—41no—
105C22×C4⋊C4—21no—
106C2×D16—24no—
107C2×D16C2314yes—
108C2×M32—24no—
109C2×M32C2314yes—
110C22×C16—24no—
111C22×C16C2311yescharacteristic
112C22×C16C2311yescharacteristic
113C22×C16C2311yescharacteristic
114C22×C16C2314yes—
115C22×C16C2311yescharacteristic
116#500—22no—
117#500C2314yes—
118#499—22no—
119#499C2314yes—
120#482—28no—
121#482—28no—
122#482—28no—
123#481—28no—
124#478—28no—
125#476—28no—
126#466—22no—
127#466—22no—
128#466—22no—
129#466—22no—
130#466C2314yes—
131#466C2314yes—
132#466C2314yes—
133#466C2314yes—
134C4×D8—22no—
135C4×D8—22no—
136C4×D8C2314yes—
137C4×D8C2314yes—
138C8×D4D414yes—
139C8×D4C2314yes—
140#427—22no—
141#427—22no—
142#427—22no—
143#427C2314yes—
144#427C2314yes—
145#427C2314yes—
146C2×#107—21no—
147C2×#107—22no—
148C2×#107—21no—
149C2×#107C2314yes—
150C2×#107C2311yescharacteristic
151C2×#107C2311yescharacteristic
152C2×#107C2311yescharacteristic
153C2×#107C2311yescharacteristic
154C2×#107C2311yescharacteristic
155C2×#107C2311yescharacteristic
156C2×#106—21no—
157C2×#106—21no—
158C2×#106—21no—
159C2×#106—21no—
160C2×C4⋊C8D411yescharacteristic
161C2×#102—21no—
162C2×#102—21no—
163C2×#102—21no—
164C2×#102—21no—
165C2×#102—21no—
166C2×#102—21no—
167C2×C22⋊C8D411yescharacteristic
168C2×C22⋊C8D411yescharacteristic
169C2×C4×C8C2311yescharacteristic
170#367C2314yes—
171#367C2314yes—
172#367C2314yes—
173#367C2314yes—
174#366C2314yes—
175#366C2314yes—
176#366C2314yes—
177#366C2314yes—
178#366C2314yes—
179C4⋊C16C2314yes—
180#357C2314yes—
181#357C2314yes—
182#357C2314yes—
183#357C2314yes—
184#357C2314yes—
185#357C2314yes—
186C22⋊C16C2314yes—
187C22⋊C16C2314yes—
188#346C2314yes—
189C22×Q8—41no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4D811yescharacteristic
197C23×C4—41no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C2×D8—21no—
203C2×D8—21no—
204C2×D8—22no—
205C2×D8—22no—
206C2×D8—22no—
207C2×D8—22no—
208C2×D8C2412yes—
209C2×D8C2412yes—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8C2×D411yescharacteristic
219C22×C8C2412yes—
220C22×C8C2412yes—
221C22×C8C2411yescharacteristic
222C4⋊Q8—24no—
223C4⋊Q8—42no—
224#126—48no—
225#125—24no—
226#125—42no—
227#124—48no—
228#124—48no—
229#123—48no—
230#121—24no—
231#121—24no—
232#121—24no—
233#121—24no—
234#121—48no—
235#121—48no—
236#121—42no—
237#121—42no—
238#121—42no—
239#121—42no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—48no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4D814yes—
246C4×D4C2×D414yes—
247#117—24no—
248#117—24no—
249#117—24no—
250#117—42no—
251#117—42no—
252#117—42no—
253#117—48no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4D811yescharacteristic
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4D811yescharacteristic
277C2×C22⋊C4D811yescharacteristic
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42C2×D411yescharacteristic
282D16—28no—
283M32—28no—
284C2×C16—22no—
285C2×C16—22no—
286C2×C16—22no—
287C2×C16—28no—
288C2×C16#4114yes—
289C2×C16C2412yes—
290C2×C16C2412yes—
291C2×C16C2412yes—
292C2×C16C2412yes—
293#107—21no—
294#107—21no—
295#107—24no—
296#107—24no—
297#107—21no—
298#107—21no—
299#107—21no—
300#107—21no—
301#107C2412yes—
302#107C2412yes—
303#107C2412yes—
304#107C2412yes—
305#107C2412yes—
306#107C2412yes—
307#106—24no—
308#106—24no—
309#106—24no—
310#106—24no—
311C4⋊C8C2×D414yes—
312#102—24no—
313#102—24no—
314#102—24no—
315#102—24no—
316#102—24no—
317#102—24no—
318C22⋊C8C2×D414yes—
319C22⋊C8C2×D414yes—
320C4×C8C2×D412yes—
321C4×C8C2412yes—
322C24—41no—
323C24—41no—
324C24C2×D811yescharacteristic
325C24C2×D811yescharacteristic
326C2×Q8—44no—
327C2×Q8—162no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—84no—
342C2×D4—162no—
343C2×D4—162no—
344C2×D4—162no—
345C2×D4C2×D814yes—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—84no—
360C22×C4—162no—
361C22×C4—162no—
362C22×C4C2×D812yes—
363C22×C4C2×D812yes—
364C22×C4C2×D811yescharacteristic
365C22×C4C2×D811yescharacteristic
366C22×C4C2×D811yescharacteristic
367C22×C4C2×D811yescharacteristic
368C22×C4C22×D412yes—
369C22×C4C22×D412yes—
370C22×C4C22×D411yescharacteristic
371D8—22no—
372D8—22no—
373D8—22no—
374D8—22no—
375D8—42no—
376D8—42no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8C22×D412yes—
387C2×C8C22×D412yes—
388C2×C8C22×D412yes—
389C2×C8C22×D411yescharacteristic
390C2×C8C2×#4112yes—
391C2×C82−1+412yes—
392C2×C8C2511yescharacteristic
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—44no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4C2×D814yes—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4C2×D814yes—
420C22⋊C4C2×D814yes—
421C42—44no—
422C42—44no—
423C42—44no—
424C42C2×D812yes—
425C42C22×D412yes—
426C16—22no—
427C16—22no—
428C16—22no—
429C16C2×#4114yes—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—42no—
437C23—42no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23C22×D812yes—
445C23C22×D812yes—
446C23C22×D811yescharacteristic
447Q8—164no—
448D4—44no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—88no—
454D4—82no—
455D4—82no—
456D4—164no—
457D4—164no—
458D4—164no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—88no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—164no—
483C2×C4—164no—
484C2×C4#54712yes—
485C2×C4C22×D812yes—
486C2×C4C22×D812yes—
487C2×C4C22×D812yes—
488C2×C4C22×D812yes—
489C2×C4C22×D811yescharacteristic
490C2×C4#57812yes—
491C2×C4C23×D411yescharacteristic
492C8—22no—
493C8—22no—
494C8—22no—
495C8#54712yes—
496C8C22×#4111yescharacteristic
497C8C2×2−1+411yescharacteristic
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—44no—
505C22—44no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—164no—
513C22—164no—
514C22#333412yes—
515C22#341412yes—
516C22#341512yes—
517C22C23×D811yescharacteristic
518C4—22no—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—82no—
523C4—82no—
524C4—82no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4#333412yes—
529C4C2×#54711yescharacteristic
530C4C2×#57811yescharacteristic
531C2—42no—
532C2—42no—
533C2—162no—
534C2—162no—
535C2#3391414yesdirect product factor
536C2C2×#333411yescharacteristic
537C2C2×#341411yescharacteristic
538C2C2×#341511yescharacteristic
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×#3341C2semidirect
2. C2×#3339C2semidirect
3. #33914C2direct (N × H)
4. C2×#2255C2semidirect
5. C2×#2234C2semidirect
6. C2×#2234C2semidirect
7. C2×#2234C2semidirect
8. C2×#2232C2semidirect
9. C2×#2228C2semidirect
10. C2×#2224C2semidirect
11. C2×#2218C2semidirect
12. C2×#2215C2semidirect
13. C2×#2177C2semidirect
14. C2×#2168C2semidirect
15. C2×#2159C2semidirect
16. C2×#2158C2semidirect
23. #3341C22semidirect
25. #3339C22semidirect
45. #2255C22semidirect
46. #2234C22semidirect
47. #2234C22semidirect
48. #2234C22semidirect
49. #2232C22semidirect
50. #2228C22semidirect
51. #2224C22semidirect
52. #2218C22semidirect
53. #2215C22semidirect
54. #2177C22semidirect
55. #2168C22semidirect
56. #2159C22semidirect
57. #2158C22semidirect
535. C2#33914direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33412
C2×#33392
#339142
C2×#22552
C2×#22342
C2×#22322
C2×#22282
C2×#22242
C2×#22182
C2×#22152
C2×#21772
C2×#21682
C2×#21592
C2×#21582