Finite Group Databasebeta

← C2×#33916 #7623800

Subgroups — C2×#33916 #7623800

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3342C211yesmaximal, characteristic
2C2×#3336C211yesmaximal, characteristic
3#33916C2116yesmaximal, direct product factor
4C2×#2255C211yesmaximal, characteristic, complemented
5C2×#2236C211yesmaximal, characteristic, complemented
6C2×#2237C211yesmaximal, characteristic, complemented
7C2×#2237C211yesmaximal, characteristic, complemented
8C2×#2237C211yesmaximal, characteristic, complemented
9C2×#2229C211yesmaximal, characteristic, complemented
10C2×#2227C211yesmaximal, characteristic, complemented
11C2×#2219C211yesmaximal, characteristic, complemented
12C2×#2216C211yesmaximal, characteristic, complemented
13C2×#2176C211yesmaximal, characteristic, complemented
14C2×#2168C211yesmaximal, characteristic
15C2×#2159C211yesmaximal, characteristic, complemented
16C2×#2158C211yesmaximal, characteristic, complemented
17C2×#556—21no—
18C2×#552—21no—
19C22×Q32C2211yescharacteristic
20C22×M32C2211yescharacteristic
21C23×C16C2211yescharacteristic
22#3342—24no—
23#3342C2218yes—
24#3336—24no—
25#3336C2218yes—
26C2×#500C2211yescharacteristic
27C2×#499C2211yescharacteristic
28C2×#485—21no—
29C2×#485—21no—
30C2×#485—21no—
31C2×#483—21no—
32C2×#479—21no—
33C2×#475—21no—
34C2×#467C2211yescharacteristic
35C2×#467C2211yescharacteristic
36C2×#467C2211yescharacteristic
37C2×#467C2211yescharacteristic
38C2×C4×Q16C2211yescharacteristic
39C2×C4×Q16C2211yescharacteristic
40C2×C8×D4C2211yescharacteristic
41C2×#427C2211yescharacteristic
42C2×#427C2211yescharacteristic
43C2×#427C2211yescharacteristic
44C22×#107C2211yescharacteristic
45#2255C2218yescomplemented
46#2237C2218yescomplemented
47#2237C2218yescomplemented
48#2237C2218yescomplemented
49#2236C2218yescomplemented
50#2229C2218yescomplemented
51#2227C2218yescomplemented
52#2219C2218yescomplemented
53#2216C2218yescomplemented
54#2176C2218yescomplemented
55#2168C2218yes—
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×#358C2211yescharacteristic
69C2×#358C2211yescharacteristic
70C2×#358C2211yescharacteristic
71C2×#358C2211yescharacteristic
72C2×#358C2211yescharacteristic
73C2×#358C2211yescharacteristic
74C2×C22⋊C16C2211yescharacteristic
75C2×C22⋊C16C2211yescharacteristic
76C2×#346C2211yescharacteristic
77C22×Q16C2311yescharacteristic
78C22×Q16C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81#556—28no—
82#556—44no—
83#552—28no—
84#552—44no—
85C2×C4⋊Q8—21no—
86C2×#126—41no—
87C2×#126—41no—
88C2×#126—41no—
89C2×#125—21no—
90C2×#125—41no—
91C2×#124—41no—
92C2×#123—41no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8—41no—
100C2×C4×D4D411yescharacteristic
101C2×#117—21no—
102C2×#117—21no—
103C2×#117—21no—
104C2×#117—41no—
105C22×C4⋊C4—21no—
106C2×Q32—24no—
107C2×Q32C2314yes—
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#485—28no—
121#485—28no—
122#485—28no—
123#483—28no—
124#479—28no—
125#475—28no—
126#467—22no—
127#467—22no—
128#467—22no—
129#467—22no—
130#467C2314yes—
131#467C2314yes—
132#467C2314yes—
133#467C2314yes—
134C4×Q16—22no—
135C4×Q16—22no—
136C4×Q16C2314yes—
137C4×Q16C2314yes—
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×#103—21no—
162C2×#103—21no—
163C2×#103—21no—
164C2×#103—21no—
165C2×#103—21no—
166C2×#103—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#358C2314yes—
181#358C2314yes—
182#358C2314yes—
183#358C2314yes—
184#358C2314yes—
185#358C2314yes—
186C22⋊C16C2314yes—
187C22⋊C16C2314yes—
188#346C2314yes—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—41no—
192C22×D4D811yescharacteristic
193C23×C4—41no—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C2×Q16—21no—
197C2×Q16—21no—
198C2×Q16—22no—
199C2×Q16—22no—
200C2×Q16—22no—
201C2×Q16—22no—
202C2×Q16C2412yes—
203C2×Q16C2412yes—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D411yescharacteristic
213C22×C8C2412yes—
214C22×C8C2412yes—
215C22×C8C2411yescharacteristic
216C4⋊Q8—24no—
217C4⋊Q8—42no—
218#126—48no—
219#126—48no—
220#126—48no—
221#125—24no—
222#125—48no—
223#125—42no—
224#124—48no—
225#123—48no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—42no—
231C22⋊Q8—42no—
232C22⋊Q8—42no—
233C22⋊Q8—42no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—48no—
237C4×Q8—42no—
238C4×Q8—42no—
239C4×D4D814yes—
240C4×D4C2×D414yes—
241#117—24no—
242#117—24no—
243#117—24no—
244#117—42no—
245#117—42no—
246#117—42no—
247#117—48no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
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⋊C4D811yescharacteristic
275C2×C22⋊C4D811yescharacteristic
276C2×C42—41no—
277C2×C42—41no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42C2×D411yescharacteristic
282Q32—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#103—24no—
313#103—24no—
314#103—24no—
315#103—24no—
316#103—24no—
317#103—24no—
318C22⋊C8C2×D414yes—
319C22⋊C8C2×D414yes—
320C4×C8C2×D412yes—
321C4×C8C2412yes—
322C24C2×D811yescharacteristic
323C24C2×D811yescharacteristic
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—44no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—84no—
334C2×D4—42no—
335C2×D4C2×D814yes—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—44no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—84no—
350C22×C4C2×D812yes—
351C22×C4C2×D812yes—
352C22×C4C2×D811yescharacteristic
353C22×C4C2×D811yescharacteristic
354C22×C4C2×D811yescharacteristic
355C22×C4C2×D811yescharacteristic
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359Q16—22no—
360Q16—22no—
361Q16—22no—
362Q16—22no—
363Q16—42no—
364Q16—42no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8C22×D412yes—
375C2×C8C22×D412yes—
376C2×C8C22×D412yes—
377C2×C8C22×D411yescharacteristic
378C2×C8C2×#4112yes—
379C2×C82−1+412yes—
380C2×C8C2511yescharacteristic
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—44no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4C2×D814yes—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4C2×D814yes—
412C22⋊C4C2×D814yes—
413C42—44no—
414C42—44no—
415C42—44no—
416C42—44no—
417C42—44no—
418C42C2×D812yes—
419C42C22×D412yes—
420C16—22no—
421C16—22no—
422C16—22no—
423C16C2×#4114yes—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23C22×D812yes—
431C23C22×D812yes—
432C23C22×D811yescharacteristic
433Q8—42no—
434Q8—42no—
435Q8—42no—
436Q8—42no—
437Q8—88no—
438Q8—82no—
439Q8—82no—
440D4—44no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—88no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4#54712yes—
469C2×C4C22×D812yes—
470C2×C4C22×D812yes—
471C2×C4C22×D812yes—
472C2×C4C22×D812yes—
473C2×C4C22×D811yescharacteristic
474C2×C4#57812yes—
475C2×C4C23×D411yescharacteristic
476C8—22no—
477C8—22no—
478C8—22no—
479C8#54712yes—
480C8C22×#4111yescharacteristic
481C8C2×2−1+411yescharacteristic
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—44no—
489C22—44no—
490C22#333412yes—
491C22#341412yes—
492C22#341712yes—
493C22C23×D811yescharacteristic
494C4—22no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—82no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4#333412yes—
507C4C2×#54711yescharacteristic
508C4C2×#57811yescharacteristic
509C2—42no—
510C2—42no—
511C2#3391614yesdirect product factor
512C2C2×#333411yescharacteristic
513C2C2×#341411yescharacteristic
514C2C2×#341711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC22×D4D8
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
3. #33916C2direct (N × H)
4. C2×#2255C2semidirect
5. C2×#2236C2semidirect
6. C2×#2237C2semidirect
7. C2×#2237C2semidirect
8. C2×#2237C2semidirect
9. C2×#2229C2semidirect
10. C2×#2227C2semidirect
11. C2×#2219C2semidirect
12. C2×#2216C2semidirect
13. C2×#2176C2semidirect
15. C2×#2159C2semidirect
16. C2×#2158C2semidirect
45. #2255C22semidirect
46. #2237C22semidirect
47. #2237C22semidirect
48. #2237C22semidirect
49. #2236C22semidirect
50. #2229C22semidirect
51. #2227C22semidirect
52. #2219C22semidirect
53. #2216C22semidirect
54. #2176C22semidirect
56. #2159C22semidirect
57. #2158C22semidirect
511. C2#33916direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33422
C2×#33362
#339162
C2×#22552
C2×#22362
C2×#22372
C2×#22292
C2×#22272
C2×#22192
C2×#22162
C2×#21762
C2×#21682
C2×#21592
C2×#21582