Finite Group Databasebeta

← C2×#33912 #7623796

Subgroups — C2×#33912 #7623796

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3339C211yesmaximal, characteristic, complemented
2C2×#3334C211yesmaximal, characteristic, complemented
3#33912C2116yesmaximal, direct product factor
4C2×#2255C211yesmaximal, characteristic, complemented
5C2×#2233C211yesmaximal, characteristic, complemented
6C2×#2233C211yesmaximal, characteristic, complemented
7C2×#2234C211yesmaximal, characteristic, complemented
8C2×#2232C211yesmaximal, characteristic, complemented
9C2×#2228C211yesmaximal, characteristic, complemented
10C2×#2226C211yesmaximal, characteristic, complemented
11C2×#2218C211yesmaximal, characteristic, complemented
12C2×#2215C211yesmaximal, characteristic, complemented
13C2×#2177C211yesmaximal, characteristic, complemented
14C2×#2168C211yesmaximal, characteristic, complemented
15C2×#2159C211yesmaximal, characteristic, complemented
16C22×#367C211yesmaximal, characteristic, complemented
17C2×#553—21no—
18C2×#547—21no—
19C22×D16C2211yescharacteristic
20C22×M32C2211yescharacteristic
21C23×C16C2211yescharacteristic
22#3339—24no—
23#3339C2218yescomplemented
24#3334—24no—
25#3334C2218yescomplemented
26C2×#500C2211yescharacteristic
27C2×#499C2211yescharacteristic
28C2×#481—21no—
29C2×#482—21no—
30C2×#480—21no—
31C2×#480—21no—
32C2×#478—21no—
33C2×#474—21no—
34C2×#466C2211yescharacteristic
35C2×#466C2211yescharacteristic
36C2×#466C2211yescharacteristic
37C2×#466C2211yescharacteristic
38C2×C4×D8C2211yescharacteristic
39C2×C4×D8C2211yescharacteristic
40C2×C8×D4C2211yescharacteristic
41C2×#427C2211yescharacteristic
42C22×#107C2211yescharacteristic
43C22×#107C2211yescharacteristic
44C22×#107C2211yescharacteristic
45#2255C2218yescomplemented
46#2234C2218yescomplemented
47#2233C2218yescomplemented
48#2233C2218yescomplemented
49#2232C2218yescomplemented
50#2228C2218yescomplemented
51#2226C2218yescomplemented
52#2218C2218yescomplemented
53#2215C2218yescomplemented
54#2177C2218yescomplemented
55#2168C2218yescomplemented
56#2159C2218yescomplemented
57C2×#367C2211yescharacteristic
58C2×#367C2211yescharacteristic
59C2×#367C2211yescharacteristic
60C2×#367C2211yescharacteristic
61C2×#367C2211yescharacteristic
62C2×#367C2211yescharacteristic
63C2×#367C2218yescomplemented
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×#41—41no—
78C22×D8C2311yescharacteristic
79C22×D8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82#553—28no—
83#553—44no—
84#547—28no—
85#547—44no—
86C2×C4⋊Q8—21no—
87C2×#126—41no—
88C2×#125—21no—
89C2×#123—41no—
90C2×#123—41no—
91C2×#123—41no—
92C2×C22⋊Q8—41no—
93C2×#121—21no—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121—41no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—41no—
101C2×C4×D4D411yescharacteristic
102C2×#117—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
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#481—28no—
122#480—28no—
123#480—28no—
124#478—28no—
125#474—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#427C2314yes—
142C2×#107—21no—
143C2×#107—22no—
144C2×#107—22no—
145C2×#107—22no—
146C2×#107—21no—
147C2×#107—21no—
148C2×#107—21no—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#107C2311yescharacteristic
153C2×#107C2311yescharacteristic
154C2×#107C2311yescharacteristic
155C2×#107C2311yescharacteristic
156C2×#107C2311yescharacteristic
157C2×#107C2311yescharacteristic
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#367C2314yes—
175#367C2314yes—
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—
189C2×#41—48no—
190C2×#41—88no—
191C22×Q8—41no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4—41no—
198C22×D4D811yescharacteristic
199C23×C4—41no—
200C23×C4—41no—
201C23×C4—41no—
202C23×C4—41no—
203C23×C4—41no—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C2×D8—21no—
207C2×D8—21no—
208C2×D8—22no—
209C2×D8—22no—
210C2×D8—22no—
211C2×D8—22no—
212C2×D8C2412yes—
213C2×D8C2412yes—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D412yes—
221C22×C8C2×D412yes—
222C22×C8C2×D411yescharacteristic
223C22×C8C2412yes—
224C22×C8C2412yes—
225C22×C8C2411yescharacteristic
226C4⋊Q8—24no—
227C4⋊Q8—42no—
228#126—48no—
229#125—24no—
230#125—42no—
231#123—48no—
232#123—48no—
233#123—48no—
234C22⋊Q8—48no—
235#121—24no—
236#121—24no—
237#121—24no—
238#121—24no—
239#121—48no—
240#121—42no—
241#121—42no—
242#121—42no—
243#121—42no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—48no—
247C4×D4—42no—
248C4×D4—42no—
249C4×D4D814yes—
250C4×D4C2×D414yes—
251#117—24no—
252#117—42no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4D811yescharacteristic
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4D811yescharacteristic
280C2×C22⋊C4D811yescharacteristic
281C2×C42—41no—
282C2×C42C2×D411yescharacteristic
283D16—28no—
284M32—28no—
285C2×C16—22no—
286C2×C16—22no—
287C2×C16—22no—
288C2×C16—28no—
289C2×C16#4114yescomplemented
290C2×C16C2412yes—
291C2×C16C2412yes—
292C2×C16C2412yes—
293C2×C16C2412yes—
294#107—21no—
295#107—21no—
296#107—21no—
297#107—21no—
298#107—21no—
299#107—24no—
300#107—24no—
301#107—24no—
302#107—24no—
303#107—21no—
304#107C2412yes—
305#107C2412yes—
306#107C2412yes—
307#107C2412yes—
308#107C2412yes—
309#107C2412yes—
310#106—24no—
311#106—24no—
312C4⋊C8C2×D414yes—
313#102—24no—
314#102—24no—
315#102—24no—
316#102—24no—
317#102—24no—
318#102—24no—
319C22⋊C8C2×D414yes—
320C22⋊C8C2×D414yes—
321C4×C8C2×D412yes—
322C4×C8C2412yes—
323C24—41no—
324C24—41no—
325C24C2×D811yescharacteristic
326C24C2×D811yescharacteristic
327#41—832no—
328C2×Q8—44no—
329C2×Q8—84no—
330C2×D4—24no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—162no—
347C2×D4C2×D814yes—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4—162no—
367C22×C4—162no—
368C22×C4C2×D812yes—
369C22×C4C2×D812yes—
370C22×C4C2×D811yescharacteristic
371C22×C4C2×D811yescharacteristic
372C22×C4C2×D811yescharacteristic
373C22×C4C2×D811yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377D8—22no—
378D8—22no—
379D8—22no—
380D8—22no—
381D8—42no—
382D8—42no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8C22×D412yes—
393C2×C8C22×D412yes—
394C2×C8C22×D412yes—
395C2×C8C22×D411yescharacteristic
396C2×C8C2×#4112yes—
397C2×C82−1+412yes—
398C2×C8C2511yescharacteristic
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—41no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4C2×D814yes—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4C2×D814yes—
426C22⋊C4C2×D814yes—
427C42—44no—
428C42C2×D812yes—
429C42C22×D412yes—
430C16—22no—
431C16—22no—
432C16—22no—
433C16C2×#4114yescomplemented
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—42no—
440C23—42no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23#50914yes—
448C23C22×D812yes—
449C23C22×D812yes—
450C23C22×D811yescharacteristic
451Q8—88no—
452D4—28no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—88no—
458D4—88no—
459D4—88no—
460D4—82no—
461D4—82no—
462D4—164no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—88no—
480C2×C4—88no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—88no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—164no—
489C2×C4—164no—
490C2×C4#54712yes—
491C2×C4C22×D812yes—
492C2×C4C22×D812yes—
493C2×C4C22×D812yes—
494C2×C4C22×D812yes—
495C2×C4C22×D811yescharacteristic
496C2×C4#57812yes—
497C2×C4C23×D411yescharacteristic
498C8—22no—
499C8—22no—
500C8—22no—
501C8#54712yes—
502C8C22×#4111yescharacteristic
503C8C2×2−1+411yescharacteristic
504C22—22no—
505C22—28no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—44no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—164no—
518C22—164no—
519C22#333412yes—
520C22C2×#50914yes—
521C22C2×#50912yes—
522C22#341612yes—
523C22C23×D811yescharacteristic
524C4—22no—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4#333412yes—
535C4C2×#54711yescharacteristic
536C4C2×#57811yescharacteristic
537C2—24no—
538C2—42no—
539C2—162no—
540C2—162no—
541C2#3391214yesdirect product factor
542C2C2×#333411yescharacteristic
543C2C22×#50911yescharacteristic
544C2C2×#341611yescharacteristic
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×#3339C2semidirect
2. C2×#3334C2semidirect
3. #33912C2direct (N × H)
4. C2×#2255C2semidirect
5. C2×#2233C2semidirect
6. C2×#2233C2semidirect
7. C2×#2234C2semidirect
8. C2×#2232C2semidirect
9. C2×#2228C2semidirect
10. C2×#2226C2semidirect
11. C2×#2218C2semidirect
12. C2×#2215C2semidirect
13. C2×#2177C2semidirect
14. C2×#2168C2semidirect
15. C2×#2159C2semidirect
16. C22×#367C2semidirect
23. #3339C22semidirect
25. #3334C22semidirect
45. #2255C22semidirect
46. #2234C22semidirect
47. #2233C22semidirect
48. #2233C22semidirect
49. #2232C22semidirect
50. #2228C22semidirect
51. #2226C22semidirect
52. #2218C22semidirect
53. #2215C22semidirect
54. #2177C22semidirect
55. #2168C22semidirect
56. #2159C22semidirect
63. C2×#367C22semidirect
289. C2×C16#41semidirect
433. C16C2×#41semidirect
541. C2#33912direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33392
C2×#33342
#339122
C2×#22552
C2×#22332
C2×#22342
C2×#22322
C2×#22282
C2×#22262
C2×#22182
C2×#22152
C2×#21772
C2×#21682
C2×#21592
C22×#3672