Finite Group Databasebeta

← C2×#33745 #7623239

Subgroups — C2×#33745 #7623239

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3177C211yesmaximal, characteristic, complemented
2C2×#3175C211yesmaximal, characteristic
3#33745C2116yesmaximal, direct product factor
4C2×#2255C211yesmaximal, characteristic, complemented
5C2×#2251C211yesmaximal, characteristic, complemented
6C2×#2242C211yesmaximal, characteristic, complemented
7C2×#2241C211yesmaximal, characteristic, complemented
8C2×#2220C211yesmaximal, characteristic
9C2×#2219C211yesmaximal, characteristic, complemented
10C2×#2217C211yesmaximal, characteristic
11C2×#2216C211yesmaximal, characteristic, complemented
12C2×#2176C211yesmaximal, characteristic, complemented
13C2×#2175C211yesmaximal, characteristic, complemented
14C2×C4×Q32C211yesmaximal, characteristic, complemented
15C2×C4×SD32C211yesmaximal, characteristic, complemented
16C2×#2108C211yesmaximal, characteristic, complemented
17C2×#542—21no—
18C2×#541—21no—
19C22×Q32C2211yescharacteristic
20C22×Q32C2211yescharacteristic
21C22×SD32C2211yescharacteristic
22C22×SD32C2211yescharacteristic
23C22×M32C2211yescharacteristic
24C23×C16C2211yescharacteristic
25#3177—24no—
26#3177C2218yescomplemented
27#3175—24no—
28#3175C2218yes—
29C2×#500C2211yescharacteristic
30C2×#499C2211yescharacteristic
31C2×#495C2211yescharacteristic
32C2×#494C2211yescharacteristic
33C2×#491—21no—
34C2×#489—21no—
35C2×#470—21no—
36C2×#468—21no—
37C2×#467C2211yescharacteristic
38C2×#467C2211yescharacteristic
39C2×#467C2211yescharacteristic
40C2×#466C2211yescharacteristic
41C2×#441—21no—
42C2×#440—21no—
43C2×C4×Q16C2211yescharacteristic
44C2×C4×Q16C2211yescharacteristic
45C2×C4×Q16C2211yescharacteristic
46C2×C4×D8C2211yescharacteristic
47C2×#405C2211yescharacteristic
48#2255C2218yescomplemented
49#2251C2218yescomplemented
50#2242C2218yescomplemented
51#2241C2218yescomplemented
52#2220C2218yes—
53#2219C2218yescomplemented
54#2217C2218yes—
55#2216C2218yescomplemented
56#2176C2218yescomplemented
57#2175C2218yescomplemented
58C4×Q32C2218yescomplemented
59C4×SD32C2218yescomplemented
60C2×#367C2211yescharacteristic
61C2×#367C2211yescharacteristic
62C2×#366C2211yescharacteristic
63C2×#366C2211yescharacteristic
64C2×#358C2211yescharacteristic
65C2×#358C2211yescharacteristic
66C2×#358C2211yescharacteristic
67C2×#358C2211yescharacteristic
68C2×#358C2211yescharacteristic
69C2×#358C2211yescharacteristic
70C2×#357C2211yescharacteristic
71C2×#357C2211yescharacteristic
72#2108C2218yescomplemented
73C2×#346C2211yescharacteristic
74C2×#346C2211yescharacteristic
75C2×C4×C16C2211yescharacteristic
76C2×C4×C16C2211yescharacteristic
77C22×Q16—21no—
78C22×Q16C2311yescharacteristic
79C22×Q16C2311yescharacteristic
80C22×Q16C2311yescharacteristic
81C22×SD16—21no—
82C22×D8C2311yescharacteristic
83C22×M16D411yescharacteristic
84C23×C8C2311yescharacteristic
85#542—28no—
86#542—44no—
87#541—28no—
88#541—44no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—21no—
91C2×#126—41no—
92C2×#126—41no—
93C2×#126—41no—
94C2×#125—21no—
95C2×#125—41no—
96C2×#125—41no—
97C2×#124—21no—
98C2×#124—41no—
99C2×#123—41no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—41no—
104C2×#121—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×D4—21no—
109C2×#117D411yescharacteristic
110C2×Q32—24no—
111C2×Q32—24no—
112C2×Q32C2314yes—
113C2×Q32C2314yes—
114C2×SD32—24no—
115C2×SD32—24no—
116C2×SD32C2314yes—
117C2×SD32C2314yes—
118C2×M32—24no—
119C2×M32C2314yes—
120C22×C16—24no—
121C22×C16C2311yescharacteristic
122C22×C16C2311yescharacteristic
123C22×C16C2314yes—
124C22×C16C2311yescharacteristic
125C22×C16C2311yescharacteristic
126#500—22no—
127#500C2314yes—
128#499—22no—
129#499C2314yes—
130#495—22no—
131#495C2314yes—
132#494—22no—
133#494C2314yes—
134#491—28no—
135#489—28no—
136#470—28no—
137#468—28no—
138#467—22no—
139#467—22no—
140#467—22no—
141#467C2314yes—
142#467C2314yes—
143#467C2314yes—
144#466—22no—
145#466C2314yes—
146#441—28no—
147#440—28no—
148C4×Q16—22no—
149C4×Q16—22no—
150C4×Q16—22no—
151C4×Q16C2314yes—
152C4×Q16C2314yes—
153C4×Q16C2314yes—
154C4×D8—22no—
155C4×D8C2314yes—
156C2×#107—21no—
157C2×#107C2311yescharacteristic
158C2×#107C2311yescharacteristic
159C2×#107C2311yescharacteristic
160C2×#107C2311yescharacteristic
161C2×#106—21no—
162C2×#103—21no—
163C2×#103—21no—
164C2×#103—21no—
165C2×#103—21no—
166C2×#103—21no—
167C2×#103—21no—
168C2×#102—21no—
169C2×#102—21no—
170#405D414yes—
171#405C2314yes—
172C2×#97D411yescharacteristic
173C2×#97D411yescharacteristic
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176#367C2314yes—
177#367C2314yes—
178#366C2314yes—
179#366C2314yes—
180#358C2314yes—
181#358C2314yes—
182#358C2314yes—
183#358C2314yes—
184#358C2314yes—
185#358C2314yes—
186#357C2314yes—
187#357C2314yes—
188#346C2314yes—
189#346C2314yes—
190C4×C16C2314yes—
191C4×C16C2314yes—
192C22×Q8—21no—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8—41no—
196C22×D4—21no—
197C22×D4—41no—
198C23×C4—41no—
199C23×C4C2×D411yescharacteristic
200C2×Q16—24no—
201C2×Q16—21no—
202C2×Q16—21no—
203C2×Q16—21no—
204C2×Q16—22no—
205C2×Q16—22no—
206C2×Q16—22no—
207C2×Q16—22no—
208C2×Q16—22no—
209C2×Q16—22no—
210C2×Q16—44no—
211C2×Q16C2412yes—
212C2×Q16C2412yes—
213C2×Q16C2412yes—
214C2×SD16—24no—
215C2×SD16—44no—
216C2×D8—21no—
217C2×D8—22no—
218C2×D8—22no—
219C2×D8C2412yes—
220C2×M16—24no—
221C2×M16C2×D414yes—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D412yes—
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2412yes—
229C22×C8C2411yescharacteristic
230C4⋊Q8—24no—
231C4⋊Q8—24no—
232C4⋊Q8—42no—
233C4⋊Q8—42no—
234#126—48no—
235#126—48no—
236#126—48no—
237#125—24no—
238#125—48no—
239#125—48no—
240#125—42no—
241#124—24no—
242#124—48no—
243#124—42no—
244#123—48no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—42no—
249C22⋊Q8—42no—
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—42no—
262#117D814yes—
263#117C2×D414yes—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4D811yescharacteristic
280C2×C4⋊C4D811yescharacteristic
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4D811yescharacteristic
289C2×C22⋊C4D811yescharacteristic
290C2×C42—41no—
291C2×C42—41no—
292C2×C42—41no—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295Q32—28no—
296Q32—28no—
297SD32—28no—
298SD32—28no—
299M32—28no—
300C2×C16—22no—
301C2×C16—22no—
302C2×C16—28no—
303C2×C16#4114yes—
304C2×C16#4114yes—
305C2×C16C2412yes—
306C2×C16C2412yes—
307C2×C16C2412yes—
308C2×C16C2412yes—
309#107—24no—
310#107—21no—
311#107—21no—
312#107—21no—
313#107—21no—
314#107C2412yes—
315#107C2412yes—
316#107C2412yes—
317#107C2412yes—
318#106—24no—
319#103—24no—
320#103—24no—
321#103—24no—
322#103—24no—
323#103—24no—
324#103—24no—
325#102—24no—
326#102—24no—
327#97C2×D414yes—
328#97C2×D414yes—
329C4×C8C2×D412yes—
330C4×C8C2×D412yes—
331C4×C8C2412yes—
332C4×C8C2412yes—
333C24—41no—
334C24C2×D811yescharacteristic
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—44no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—162no—
349C2×D4—22no—
350C2×D4—44no—
351C2×D4—42no—
352C2×D4—41no—
353C2×D4—42no—
354C2×D4—162no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—44no—
365C22×C4—162no—
366C22×C4C2×D812yes—
367C22×C4C2×D811yescharacteristic
368C22×C4C2×D811yescharacteristic
369C22×C4C2×D811yescharacteristic
370C22×C4C2×D811yescharacteristic
371C22×C4C2×D811yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×D411yescharacteristic
374Q16—22no—
375Q16—22no—
376Q16—22no—
377Q16—22no—
378Q16—22no—
379Q16—22no—
380Q16—42no—
381Q16—42no—
382Q16—42no—
383Q16—48no—
384SD16—48no—
385D8—22no—
386D8—22no—
387D8—42no—
388M16—28no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—24no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8#13514yes—
395C2×C8C22×D412yes—
396C2×C8C22×D412yes—
397C2×C8C22×D411yescharacteristic
398C2×C8C22×D412yes—
399C2×C8C2×#4112yes—
400C2×C8C2×#4112yes—
401C2×C8C2511yescharacteristic
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4C2×D814yes—
422C4⋊C4C2×D814yes—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4C2×D814yes—
431C22⋊C4C2×D814yes—
432C42—44no—
433C42—44no—
434C42—44no—
435C42C2×D812yes—
436C42C2×D812yes—
437C42C22×D412yes—
438C42C22×D412yes—
439C16—22no—
440C16—22no—
441C16C2×#4114yes—
442C16C2×#4114yes—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—42no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23C22×D812yes—
451C23C22×D811yescharacteristic
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—42no—
456Q8—42no—
457Q8—42no—
458Q8—82no—
459Q8—82no—
460Q8—82no—
461Q8—164no—
462D4—42no—
463D4—42no—
464D4—82no—
465D4—164no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—164no—
489C2×C4#53212yes—
490C2×C4C22×D812yes—
491C2×C4C22×D812yes—
492C2×C4C22×D812yes—
493C2×C4C22×D812yes—
494C2×C4C22×D812yes—
495C2×C4C22×D811yescharacteristic
496C2×C4C2×#13512yes—
497C2×C4C23×D411yescharacteristic
498C8—22no—
499C8—22no—
500C8#53212yes—
501C8C2×#13514yes—
502C8C22×#4111yescharacteristic
503C8C22×#4111yescharacteristic
504C22—22no—
505C22—22no—
506C22—22no—
507C22—44no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—164no—
512C22#316912yes—
513C22#341612yes—
514C22#341712yes—
515C22C23×D811yescharacteristic
516C4—22no—
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—82no—
522C4—82no—
523C4—82no—
524C4—82no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4#316912yes—
529C4C2×#53211yescharacteristic
530C4C22×#13511yescharacteristic
531C2—42no—
532C2—162no—
533C2#3374514yesdirect product factor
534C2C2×#316911yescharacteristic
535C2C2×#341611yescharacteristic
536C2C2×#341711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC2×#466C22
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×#3177C2semidirect
3. #33745C2direct (N × H)
4. C2×#2255C2semidirect
5. C2×#2251C2semidirect
6. C2×#2242C2semidirect
7. C2×#2241C2semidirect
9. C2×#2219C2semidirect
11. C2×#2216C2semidirect
12. C2×#2176C2semidirect
13. C2×#2175C2semidirect
14. C2×C4×Q32C2semidirect
15. C2×C4×SD32C2semidirect
16. C2×#2108C2semidirect
26. #3177C22semidirect
48. #2255C22semidirect
49. #2251C22semidirect
50. #2242C22semidirect
51. #2241C22semidirect
53. #2219C22semidirect
55. #2216C22semidirect
56. #2176C22semidirect
57. #2175C22semidirect
58. C4×Q32C22semidirect
59. C4×SD32C22semidirect
72. #2108C22semidirect
533. C2#33745direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31772
C2×#31752
#337452
C2×#22552
C2×#22512
C2×#22422
C2×#22412
C2×#22202
C2×#22192
C2×#22172
C2×#22162
C2×#21762
C2×#21752
C2×C4×Q322
C2×C4×SD322
C2×#21082