Finite Group Databasebeta

← C2×#81904 #11675631

Subgroups — C2×#81904 #11675631

Order 768.

1842 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5730C211yesmaximal, characteristic
2C2×#5721C211yesmaximal, characteristic, complemented
3C2×C3×#498C211yesmaximal, characteristic, complemented
4C2×#5091C211yesmaximal, characteristic, complemented
5C2×#5087C211yesmaximal, characteristic
6#81904C2116yesmaximal, direct product factor
7C2×#4857C212yesmaximal, complemented
8C2×#4871C212yesmaximal
9C2×#4858C212yesmaximal, complemented
10C2×#4852C212yesmaximal
11C2×C4×#204C211yesmaximal, characteristic
12C2×C8×Dic6C211yesmaximal, characteristic, complemented
13C2×#3381—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×#946—22no—
17C2×Q8×Dic3—21no—
18C2×#944—21no—
19C22×#204C2211yescharacteristic
20#5730—24no—
21#5730C2218yes—
22#5721—24no—
23#5721C2218yescomplemented
24C2×#897—21no—
25C2×#893C2212yes—
26C2×D3×C4⋊C4—21no—
27C2×#890C2212yes—
28C2×#890C2212yes—
29C2×#889—22no—
30C2×#887C2212yes—
31C2×C4×D12C2211yescharacteristic
32C2×C4×Dic6C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C3×#498C2218yescomplemented
35C2×C3×#106C2212yes—
36C2×C3×#106C2212yes—
37C2×C3×C4×C8C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#817C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×Dic3⋊C8C2212yes—
42C2×C8×Dic3C2212yes—
43#5091C2218yescomplemented
44#5087C2218yes—
45C2×#810C2212yes—
46C2×#809C2212yes—
47C2×#807C2212yes—
48C2×#807C2212yes—
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4871C22116yes—
51#4858C22116yescomplemented
52#4857C22116yescomplemented
53#4852C22116yes—
54C4×#204C2218yes—
55C8×Dic6C2218yescomplemented
56C2×#557—31no—
57C2×#554—31no—
58#3381—316no—
59C2×#499—32no—
60C2×#498D311yescharacteristic, complemented
61C2×#479—32no—
62C2×#477—31no—
63C2×#478—32no—
64C2×#474—31no—
65C2×C8×Q8—31no—
66C2×C4×SD16—32no—
67C2×C4×SD16—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8—21no—
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×C4×D3—22no—
73C22×Dic6D411yescharacteristic
74C22×Dic6C2311yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2311yescharacteristic
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8C2314yes—
80C3×C4⋊Q8C2314yes—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C42C2311yescharacteristic
86#946—216no—
87Q8×Dic3—28no—
88#944—28no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3D411yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4D412yes—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—22no—
103C2×C4×Dic3C2312yes—
104C22×C3⋊C8C2312yes—
105C2×#204Q818yescomplemented
106C2×#204C2314yes—
107#897—28no—
108#893—24no—
109#893C2318yes—
110D3×C4⋊C4—28no—
111#890—24no—
112#890—24no—
113#890C2318yes—
114#890C2318yes—
115#889—216no—
116#887—24no—
117#887C2318yes—
118C4×D12—22no—
119C4×D12C2314yes—
120C4×Dic6—22no—
121C4×Dic6D414yes—
122C4×Dic6C2314yes—
123C4×Dic6C2314yes—
124C3×#106C2318yes—
125C3×#106C2318yes—
126C3×C4×C8C2314yes—
127#821C2314yes—
128#817C2314yes—
129#816C2314yes—
130Dic3⋊C8C2318yes—
131C8×Dic3C2318yes—
132#810C2318yes—
133#809C2318yes—
134#807C2318yes—
135#807C2318yes—
136C4⋊(C3⋊C8)C2314yes—
137C22×SD16—31no—
138#557—38no—
139#557—64no—
140#554—38no—
141#554—64no—
142C2×C4⋊Q8—61no—
143C2×C4⋊Q8C2×D311yescharacteristic
144C2×C4⋊Q8C2×D311yescharacteristic
145C2×#125—32no—
146C2×#125—32no—
147C2×#125—62no—
148C2×C22⋊Q8—61no—
149C2×C22⋊Q8—62no—
150C2×C4×Q8—31no—
151C2×C4×Q8—32no—
152C2×C4×Q8—31no—
153C2×C4×Q8—61no—
154C2×C4×D4—32no—
155C2×C4×D4—31no—
156C22×C4⋊C4—61no—
157#499—316no—
158#498C2×D318yescomplemented
159#479—316no—
160#478—316no—
161#477—38no—
162#474—38no—
163C8×Q8—38no—
164C4×SD16—316no—
165C4×SD16—38no—
166C2×#107—32no—
167C2×#107—32no—
168C2×#106—31no—
169C2×#106C2×D312yes—
170C2×#106C2×D312yes—
171C2×C4⋊C8—32no—
172C2×C4⋊C8—31no—
173C2×#103—31no—
174C2×#103—32no—
175C2×#102—31no—
176C2×#102—32no—
177C2×C4×C8—32no—
178C2×C4×C8C2×D311yescharacteristic
179C23×D3—21no—
180C2×C3×Q8—24no—
181C2×C3×Q8—24no—
182C2×C3×Q8—44no—
183C2×C3×Q8—82no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3C2×D412yes—
193C22×Dic3C2×D412yes—
194C2×D12—21no—
195C2×D12—22no—
196C2×D12C2×Q814yes—
197C2×D12C2412yes—
198C2×C4×D3—24no—
199C2×C4×D3—28no—
200C2×C4×D3—44no—
201C2×C4×D3—84no—
202C2×Dic6—24no—
203C2×Dic6—21no—
204C2×Dic6—22no—
205C2×Dic6C2×D414yes—
206C2×Dic6C2×Q814yes—
207C2×Dic6C2412yes—
208C2×C3×C8—22no—
209C2×C3×C8C2×Q814yes—
210C2×C3×C8C2412yes—
211C2×C3×C8C2412yes—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4C2414yes—
217C3×C4⋊C4C2414yes—
218C3×C42C2×D412yes—
219C3×C42C2412yes—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—28no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—28no—
225C4⋊Dic3—28no—
226C4⋊Dic3C2×D414yes—
227C4⋊Dic3C2412yes—
228C4⋊Dic3C2412yes—
229Dic3⋊C4—24no—
230Dic3⋊C4—28no—
231Dic3⋊C4—28no—
232Dic3⋊C4—28no—
233Dic3⋊C4C2×D418yes—
234C4×Dic3—24no—
235C4×Dic3—28no—
236C4×Dic3C2×D414yes—
237C4×Dic3C2414yes—
238C2×C3⋊C8—24no—
239C2×C3⋊C8C2414yes—
240#204C2×Q8116yescomplemented
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—31no—
244C22×Q8—31no—
245C22×D4—31no—
246C23×C4—62no—
247C23×C4—61no—
248C2×SD16—38no—
249C2×SD16—34no—
250C22×C8—32no—
251C22×C8C22×D311yescharacteristic
252C22×C8C22×D311yescharacteristic
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—68no—
256C4⋊Q8C22×D314yes—
257C4⋊Q8C22×D314yes—
258#125—38no—
259#125—38no—
260#125—64no—
261#125—64no—
262#125—616no—
263C22⋊Q8—68no—
264C22⋊Q8—616no—
265C4×Q8—34no—
266C4×Q8—34no—
267C4×Q8—38no—
268C4×Q8—34no—
269C4×Q8—64no—
270C4×Q8—68no—
271C4×Q8—62no—
272C4×D4—38no—
273C4×D4—34no—
274C4×D4—64no—
275C4×D4—62no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—32no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—68no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D312yes—
290C2×C4⋊C4C22×D312yes—
291C2×C22⋊C4—62no—
292C2×C22⋊C4—61no—
293C2×C42—32no—
294C2×C42—62no—
295C2×C42—61no—
296C2×C42C22×D311yescharacteristic
297#107—38no—
298#107—38no—
299#106—34no—
300#106C22×D318yes—
301#106C22×D318yes—
302C4⋊C8—38no—
303C4⋊C8—34no—
304#103—34no—
305#103—38no—
306#102—34no—
307#102—38no—
308C4×C8—38no—
309C4×C8C22×D314yes—
310C23×C3C22×D411yescharacteristic
311C22×D3—24no—
312C22×D3—22no—
313C22×D3—42no—
314C22×D3—42no—
315C3×Q8—48no—
316C3×Q8—84no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—44no—
321C2×C3×C4—44no—
322C2×C3×C4#13514yes—
323C2×C3×C4C22×D412yes—
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D412yes—
326C2×C3×C4C22×Q812yes—
327C2×C3×C42−1+412yes—
328C2×C3×C4C2511yescharacteristic
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3C22×D414yes—
338C2×Dic3C22×D414yes—
339D12—24no—
340D12—22no—
341D12C22×Q814yes—
342C4×D3—48no—
343C4×D3—88no—
344Dic6—24no—
345Dic6—28no—
346Dic6—22no—
347Dic6C22×Q814yes—
348C3×C8—22no—
349C3×C8C22×Q814yes—
350C3⋊C8—24no—
351C24—61no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—34no—
355C2×Q8—34no—
356C2×Q8—32no—
357C2×Q8—44no—
358C2×Q8—64no—
359C2×Q8—61no—
360C2×Q8—62no—
361C2×Q8—82no—
362C2×D4—34no—
363C2×D4—32no—
364C2×D4—61no—
365C2×D4—62no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—68no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4—64no—
375C22×C4—124no—
376C22×C4—244no—
377C22×C4D3×D411yescharacteristic
378C22×C4D3×D411yescharacteristic
379C22×C4C23×D311yescharacteristic
380SD16—316no—
381C2×C8—22no—
382C2×C8—34no—
383C2×C8—64no—
384C2×C8D3×Q814yes—
385C2×C8C23×D312yes—
386C2×C8C23×D312yes—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—34no—
392C4⋊C4—38no—
393C4⋊C4—32no—
394C4⋊C4—32no—
395C4⋊C4—68no—
396C4⋊C4—68no—
397C4⋊C4—68no—
398C4⋊C4—68no—
399C4⋊C4—68no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—64no—
403C4⋊C4C23×D314yes—
404C4⋊C4C23×D314yes—
405C22⋊C4—68no—
406C22⋊C4—64no—
407C42—34no—
408C42—34no—
409C42—68no—
410C42—64no—
411C42D3×D412yes—
412C42C23×D312yes—
413C22×C3D4×Q812yes—
414C22×C3C2×#13512yes—
415C22×C3#57712yes—
416C22×C3C23×D411yescharacteristic
417C2×D3—24no—
418C2×D3—48no—
419C2×D3—42no—
420C2×D3—42no—
421C3×C4—22no—
422C3×C4—44no—
423C3×C4—44no—
424C3×C4D4×Q812yes—
425C3×C4C2×#13514yes—
426C3×C4C23×Q811yescharacteristic
427C3×C4C2×2−1+411yescharacteristic
428Dic3—24no—
429Dic3—24no—
430Dic3—24no—
431Dic3—42no—
432Dic3—42no—
433C23—62no—
434C23—64no—
435C23—122no—
436C23—122no—
437C23C2×D3×D411yescharacteristic
438Q8—34no—
439Q8—48no—
440Q8—62no—
441Q8—64no—
442Q8—68no—
443Q8—84no—
444D4—34no—
445D4—62no—
446D4—64no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—34no—
451C2×C4—34no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—62no—
455C2×C4—64no—
456C2×C4—64no—
457C2×C4—64no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—128no—
463C2×C4—248no—
464C2×C4#91614yes—
465C2×C4C2×D3×D412yes—
466C2×C4C2×D3×D411yescharacteristic
467C2×C4C2×D3×D412yes—
468C2×C4C2×D3×Q812yes—
469C2×C4#100712yes—
470C2×C4C24×D311yescharacteristic
471C8—22no—
472C8—64no—
473C8C2×D3×Q814yes—
474C2×C3#338114yescomplemented
475C2×C3C2×D4×Q811yescharacteristic
476C2×C3C22×#13511yescharacteristic
477C2×C3C2×#57711yescharacteristic
478D3—44no—
479C22—64no—
480C22—122no—
481C22—122no—
482C22—128no—
483C22#572712yes—
484C22C2×#91612yes—
485C22#576212yes—
486C22C22×D3×D411yescharacteristic
487C4—22no—
488C4—44no—
489C4—44no—
490C4—64no—
491C4—64no—
492C4—64no—
493C4—122no—
494C4—122no—
495C4#572712yes—
496C4C2×#91614yes—
497C4C22×D3×Q811yescharacteristic
498C4C2×#100711yescharacteristic
499C3C2×#338111yescharacteristic, complemented, Sylow
500C2—124no—
501C2#8190414yesdirect product factor
502C2C2×#572711yescharacteristic
503C2C22×#91611yescharacteristic
504C2C2×#576211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#498C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#498D3
hypocenterC3C2×#3381
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
2. C2×#5721C2semidirect
3. C2×C3×#498C2semidirect
4. C2×#5091C2semidirect
6. #81904C2direct (N × H)
7. C2×#4857C2semidirect
9. C2×#4858C2semidirect
12. C2×C8×Dic6C2semidirect
23. #5721C22semidirect
34. C3×#498C22semidirect
43. #5091C22semidirect
51. #4858C22semidirect
52. #4857C22semidirect
55. C8×Dic6C22semidirect
60. C2×#498D3semidirect
105. C2×#204Q8semidirect
158. #498C2×D3semidirect
240. #204C2×Q8semidirect
474. C2×C3#3381semidirect
499. C3C2×#3381semidirect
501. C2#81904direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57302
C2×#57212
C2×C3×#4982
C2×#50912
C2×#50872
#819042
C2×#48572
C2×#48712
C2×#48582
C2×#48522
C2×C4×#2042
C2×C8×Dic62
C2×#33813