Finite Group Databasebeta

← C2×#33684 #7623091

Subgroups — C2×#33684 #7623091

Order 512.

1552 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×#3175C211yesmaximal, characteristic, complemented
2C2×#2951C211yesmaximal, characteristic, complemented
3#33684C2116yesmaximal, direct product factor
4C2×#2236C211yesmaximal, characteristic, complemented
5C2×#2232C211yesmaximal, characteristic, complemented
6C2×#2227C211yesmaximal, characteristic, complemented
7C2×#2224C211yesmaximal, characteristic, complemented
8C2×#2211C211yesmaximal, characteristic, complemented
9C2×#2210C211yesmaximal, characteristic, complemented
10C2×#2189C211yesmaximal, characteristic, complemented
11C2×#2186C211yesmaximal, characteristic
12C2×#2175C211yesmaximal, characteristic, complemented
13C2×#2175C211yesmaximal, characteristic, complemented
14C2×C4×Q32C211yesmaximal, characteristic, complemented
15C2×C4×D16C211yesmaximal, characteristic, complemented
16C2×#2164C211yesmaximal, characteristic, complemented
17C22×#135C2211yescharacteristic
18C2×#542—21no—
19C2×#520—21no—
20C22×Q32C2211yescharacteristic
21C22×SD32C2211yescharacteristic
22C22×SD32C2211yescharacteristic
23C22×D16C2211yescharacteristic
24#3175—24no—
25#3175C2218yescomplemented
26C2×#500C2211yescharacteristic
27C2×#495C2211yescharacteristic
28C2×#489—21no—
29C2×#468—21no—
30C2×#467C2211yescharacteristic
31C2×#466C2211yescharacteristic
32#2951—24no—
33#2951C2218yescomplemented
34C2×#440—21no—
35C2×#440—21no—
36C2×#440—21no—
37C2×C4×Q16C2211yescharacteristic
38C2×C4×Q16C2211yescharacteristic
39C2×C4×Q16C2211yescharacteristic
40C2×C4×D8C2211yescharacteristic
41C2×C4×D8C2211yescharacteristic
42C2×C4×D8C2211yescharacteristic
43C22×#107C2211yescharacteristic
44C2×#417—21no—
45C2×#405C2211yescharacteristic
46#2236C2218yescomplemented
47#2232C2218yescomplemented
48#2227C2218yescomplemented
49#2224C2218yescomplemented
50#2211C2218yescomplemented
51#2210C2218yescomplemented
52#2189C2218yescomplemented
53#2186C2218yes—
54#2175C2218yescomplemented
55#2175C2218yescomplemented
56C4×Q32C2218yescomplemented
57C4×D16C2218yescomplemented
58#2164C2218yescomplemented
59C2×#366C2211yescharacteristic
60C2×#366C2211yescharacteristic
61C2×#366C2211yescharacteristic
62C2×#366C2211yescharacteristic
63C2×C4⋊C16C2211yescharacteristic
64C2×C4⋊C16C2211yescharacteristic
65C2×#358C2211yescharacteristic
66C2×#358C2211yescharacteristic
67C2×#358C2211yescharacteristic
68C2×#358C2211yescharacteristic
69C2×#357C2211yescharacteristic
70C2×#357C2211yescharacteristic
71C2×#357C2211yescharacteristic
72C2×#357C2211yescharacteristic
73C2×C22⋊C16C2211yescharacteristic
74C2×C22⋊C16C2211yescharacteristic
75C2×#346C2211yescharacteristic
76C2×C4×C16C2211yescharacteristic
77C22×#41—21no—
78C2×#135—24no—
79C2×#135—24no—
80C2×#135—22no—
81C2×#135C2314yes—
82C22×Q16C2311yescharacteristic
83C22×Q16C2311yescharacteristic
84C22×SD16—21no—
85C22×SD16—21no—
86C22×D8C2311yescharacteristic
87C22×D8C2311yescharacteristic
88C22×M16D411yescharacteristic
89C23×C8C2311yescharacteristic
90#542—28no—
91#542—44no—
92C2×C4⋊Q8—21no—
93C2×#126—41no—
94C2×#124—21no—
95C2×#124—41no—
96C2×#123—41no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—41no—
99C2×#121—21no—
100#520—28no—
101#520—44no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—41no—
109C2×C4×D4—41no—
110C2×#117—41no—
111C2×#117D411yescharacteristic
112C22×C4⋊C4—21no—
113C22×C4⋊C4—41no—
114C2×Q32—24no—
115C2×Q32C2314yes—
116C2×SD32—24no—
117C2×SD32—24no—
118C2×SD32C2314yes—
119C2×SD32C2314yes—
120C2×D16—24no—
121C2×D16C2314yes—
122C22×C16C2311yescharacteristic
123C22×C16C2311yescharacteristic
124C22×C16C2311yescharacteristic
125C22×C16C2311yescharacteristic
126#500—22no—
127#500C2314yes—
128#495—22no—
129#495C2314yes—
130#489—28no—
131#468—28no—
132#467—22no—
133#467C2314yes—
134#466—22no—
135#466C2314yes—
136#440—28no—
137#440—28no—
138#440—28no—
139C4×Q16—22no—
140C4×Q16—22no—
141C4×Q16—22no—
142C4×Q16C2314yes—
143C4×Q16C2314yes—
144C4×Q16C2314yes—
145C4×D8—22no—
146C4×D8—22no—
147C4×D8—22no—
148C4×D8C2314yes—
149C4×D8C2314yes—
150C4×D8C2314yes—
151C2×#107—21no—
152C2×#107—21no—
153C2×#107—22no—
154C2×#107C2311yescharacteristic
155C2×#107C2311yescharacteristic
156C2×#107C2311yescharacteristic
157C2×#107C2311yescharacteristic
158C2×#107C2314yes—
159#417—28no—
160C2×#103—21no—
161C2×#103—21no—
162C2×#103—21no—
163C2×#103—21no—
164C2×#102—21no—
165C2×#102—21no—
166C2×#102—21no—
167C2×#102—21no—
168#405D414yes—
169#405C2314yes—
170C2×#97D411yescharacteristic
171C2×#97D411yescharacteristic
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2311yescharacteristic
174#366C2314yes—
175#366C2314yes—
176#366C2314yes—
177#366C2314yes—
178C4⋊C16C2314yes—
179C4⋊C16C2314yes—
180#358C2314yes—
181#358C2314yes—
182#358C2314yes—
183#358C2314yes—
184#357C2314yes—
185#357C2314yes—
186#357C2314yes—
187#357C2314yes—
188C22⋊C16C2314yes—
189C22⋊C16C2314yes—
190#346C2314yes—
191C4×C16C2314yes—
192C2×#41—28no—
193C2×#41—24no—
194C2×#41—44no—
195C2×#41—42no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—41no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4—41no—
202C22×D4—41no—
203C23×C4—41no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—41no—
207C23×C4—41no—
208C23×C4C2×D411yescharacteristic
209#135—216no—
210#135—28no—
211#135—28no—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—22no—
215C2×Q16—22no—
216C2×Q16—22no—
217C2×Q16#4114yes—
218C2×Q16C2412yes—
219C2×Q16C2412yes—
220C2×SD16—28no—
221C2×SD16—24no—
222C2×SD16—24no—
223C2×SD16—44no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—22no—
227C2×D8—22no—
228C2×D8—22no—
229C2×D8#4114yes—
230C2×D8C2412yes—
231C2×D8C2412yes—
232C2×M16—42no—
233C2×M16C2×D414yes—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D412yes—
238C22×C8C2×D411yescharacteristic
239C22×C8C2×D411yescharacteristic
240C22×C8C2412yes—
241C22×C8C2411yescharacteristic
242C4⋊Q8—24no—
243C4⋊Q8—42no—
244#126—48no—
245#124—24no—
246#124—48no—
247#124—42no—
248#123—48no—
249C22⋊Q8—24no—
250C22⋊Q8—42no—
251C22⋊Q8—48no—
252#121—24no—
253#121—42no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—42no—
258C4×Q8—42no—
259C4×Q8—42no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—48no—
264C4×D4—48no—
265C4×D4—42no—
266C4×D4—42no—
267C4×D4—42no—
268#117—48no—
269#117D814yes—
270#117C2×D414yes—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—48no—
284C2×C4⋊C4D811yescharacteristic
285C2×C4⋊C4D811yescharacteristic
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4D811yescharacteristic
294C2×C22⋊C4D811yescharacteristic
295C2×C42—41no—
296C2×C42—41no—
297C2×C42—41no—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300Q32—28no—
301SD32—28no—
302SD32—28no—
303D16—28no—
304C2×C16—22no—
305C2×C16—22no—
306C2×C16—22no—
307C2×C16—22no—
308C2×C16C2412yes—
309C2×C16C2412yes—
310C2×C16C2412yes—
311C2×C16C2412yes—
312#107—21no—
313#107—21no—
314#107—24no—
315#107—24no—
316#107—21no—
317#107—21no—
318#107C2412yes—
319#107C2412yes—
320#107C2412yes—
321#107C2412yes—
322#103—24no—
323#103—24no—
324#103—24no—
325#103—24no—
326#102—24no—
327#102—24no—
328#102—24no—
329#102—24no—
330#97C2×D414yes—
331#97C2×D414yes—
332C4×C8C2×D412yes—
333C4×C8C2×D412yes—
334C4×C8C2412yes—
335C4×C8C2412yes—
336C24—41no—
337C24—41no—
338C24C2×D811yescharacteristic
339#41—216no—
340#41—416no—
341#41—48no—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—44no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—162no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—48no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—42no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—162no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—48no—
367C22×C4—41no—
368C22×C4—44no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—162no—
379C22×C4—162no—
380C22×C4—162no—
381C22×C4—162no—
382C22×C4C2×D812yes—
383C22×C4C2×D811yescharacteristic
384C22×C4C2×D811yescharacteristic
385C22×C4C2×D811yescharacteristic
386C22×C4C2×D811yescharacteristic
387C22×C4C2×D811yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390Q16—24no—
391Q16—22no—
392Q16—22no—
393Q16—22no—
394Q16—42no—
395Q16C2×#4114yes—
396SD16—216no—
397SD16—48no—
398D8—24no—
399D8—22no—
400D8—22no—
401D8—22no—
402D8—42no—
403D8C2×#4114yes—
404M16—44no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8C22×D412yes—
412C2×C8C22×D412yes—
413C2×C8C22×D412yes—
414C2×C8C22×D411yescharacteristic
415C2×C8C2×#4112yes—
416C2×C8C2×#4112yes—
417C2×C8C2511yescharacteristic
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—44no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4C2×D814yes—
433C4⋊C4C2×D814yes—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4C2×D814yes—
442C22⋊C4C2×D814yes—
443C42—44no—
444C42—44no—
445C42—44no—
446C42C2×D812yes—
447C42C2×D812yes—
448C42C22×D412yes—
449C42C22×D412yes—
450C16—22no—
451C16—22no—
452C16—22no—
453C16—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23C22×D812yes—
466C23C22×D811yescharacteristic
467Q8—24no—
468Q8—44no—
469Q8—42no—
470Q8—42no—
471Q8—42no—
472Q8—82no—
473Q8—164no—
474D4—24no—
475D4—416no—
476D4—44no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—82no—
481D4—164no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—416no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—164no—
504C2×C4—164no—
505C2×C4—164no—
506C2×C4—164no—
507C2×C4#52512yes—
508C2×C4C22×D812yes—
509C2×C4C22×D812yes—
510C2×C4C22×D812yes—
511C2×C4C22×D812yes—
512C2×C4C22×D812yes—
513C2×C4C22×D811yescharacteristic
514C2×C4#57512yes—
515C2×C4C23×D411yescharacteristic
516C8—22no—
517C8—22no—
518C8—22no—
519C8#52512yes—
520C8C22×#4111yescharacteristic
521C8C22×#4111yescharacteristic
522C22—22no—
523C22—22no—
524C22—22no—
525C22—44no—
526C22—44no—
527C22—88no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—164no—
534C22#310012yes—
535C22#341512yes—
536C22#341712yes—
537C22C23×D811yescharacteristic
538C4—22no—
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4#310012yes—
550C4C2×#52511yescharacteristic
551C4C2×#57511yescharacteristic
552C2—42no—
553C2—84no—
554C2—162no—
555C2#3368414yesdirect product factor
556C2C2×#310011yescharacteristic
557C2C2×#341511yescharacteristic
558C2C2×#341711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC2×#2186C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3175C2semidirect
2. C2×#2951C2semidirect
3. #33684C2direct (N × H)
4. C2×#2236C2semidirect
5. C2×#2232C2semidirect
6. C2×#2227C2semidirect
7. C2×#2224C2semidirect
8. C2×#2211C2semidirect
9. C2×#2210C2semidirect
10. C2×#2189C2semidirect
12. C2×#2175C2semidirect
13. C2×#2175C2semidirect
14. C2×C4×Q32C2semidirect
15. C2×C4×D16C2semidirect
16. C2×#2164C2semidirect
25. #3175C22semidirect
33. #2951C22semidirect
46. #2236C22semidirect
47. #2232C22semidirect
48. #2227C22semidirect
49. #2224C22semidirect
50. #2211C22semidirect
51. #2210C22semidirect
52. #2189C22semidirect
54. #2175C22semidirect
55. #2175C22semidirect
56. C4×Q32C22semidirect
57. C4×D16C22semidirect
58. #2164C22semidirect
555. C2#33684direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31752
C2×#29512
#336842
C2×#22362
C2×#22322
C2×#22272
C2×#22242
C2×#22112
C2×#22102
C2×#21892
C2×#21862
C2×#21752
C2×C4×Q322
C2×C4×D162
C2×#21642