Finite Group Databasebeta

← C2×#81137 #11675675

Subgroups — C2×#81137 #11675675

Order 768.

1854 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5730C211yesmaximal, characteristic
2C2×Q8×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#477C211yesmaximal, characteristic, complemented
4C2×#5078C211yesmaximal, characteristic, complemented
5C2×C4×#215C211yesmaximal, characteristic
6#81137C2116yesmaximal, direct product factor
7C2×#4890C212yesmaximal
8C2×#4876C212yesmaximal, complemented
9C2×#4842C211yesmaximal, characteristic
10C2×#4831C211yesmaximal, characteristic, complemented
11C2×#4799C212yesmaximal, complemented
12C2×#4817C212yesmaximal
13C2×#3385—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#946—22no—
17C2×Q8×Dic3C2212yes—
18C22×#215C2211yescharacteristic
19#5730—24no—
20#5730C2218yes—
21Q8×Dic6—24no—
22Q8×Dic6C2218yescomplemented
23C2×#897—21no—
24C2×#893C2212yes—
25C2×D3×C4⋊C4—21no—
26C2×#890C2212yes—
27C2×#888—22no—
28C2×#888C2212yes—
29C2×C4×D12C2211yescharacteristic
30C2×#869—21no—
31C2×#869C2211yescharacteristic
32C2×C4×Dic6—21no—
33C2×C4×Dic6C2211yescharacteristic
34C3×#477C2218yescomplemented
35C2×C3×#107C2212yes—
36C2×C3×C4⋊C8C2211yescharacteristic
37C2×C3×#103C2212yes—
38C2×#835C2211yescharacteristic
39C2×#821C2212yes—
40C2×#817C2212yes—
41C2×#815C2212yes—
42C2×Dic3⋊C8C2212yes—
43#5078C2218yescomplemented
44C4×#215C2218yes—
45C2×#809C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2212yes—
48C2×#807C2212yes—
49C2×C4×C3⋊C8C2211yescharacteristic
50#4890C22116yes—
51#4876C22116yescomplemented
52#4842C2218yes—
53#4831C2218yescomplemented
54#4817C22116yes—
55#4799C22116yescomplemented
56C2×Q82—31no—
57C2×#554—31no—
58#3385—316no—
59C2×#501—32no—
60C2×#500—31no—
61C2×#479—32no—
62C2×#477D311yescharacteristic, complemented
63C2×#476—31no—
64C2×#474—32no—
65C2×#446—31no—
66C2×#440—32no—
67C2×C4×SD16—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8C2311yescharacteristic
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×C4×D3—22no—
73C22×Dic6—21no—
74C22×Dic6—22no—
75C22×Dic6D411yescharacteristic
76C22×C3×C8C2312yes—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8C2314yes—
79C3×C4×Q8—22no—
80C3×C4×Q8C2314yes—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87#946—216no—
88Q8×Dic3—24no—
89Q8×Dic3C2318yes—
90C2×#215Q818yescomplemented
91C2×#215C2314yes—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3D411yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4D412yes—
103C2×C4×Dic3—22no—
104C2×C4×Dic3C2312yes—
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107#897—28no—
108#893—24no—
109#893C2318yes—
110D3×C4⋊C4—28no—
111#890—24no—
112#890C2318yes—
113#888—216no—
114#888—24no—
115#888C2318yes—
116C4×D12—22no—
117C4×D12C2314yes—
118#869—22no—
119#869—28no—
120#869C2314yes—
121C4×Dic6—28no—
122C4×Dic6D414yes—
123C4×Dic6C2314yes—
124C3×#107C2318yes—
125C3×C4⋊C8C2314yes—
126C3×#103C2318yes—
127#835C2314yes—
128#821C2318yes—
129#817C2318yes—
130#815C2318yes—
131Dic3⋊C8C2318yes—
132#809C2314yes—
133#808C2314yes—
134#807C2318yes—
135#807C2318yes—
136C4×C3⋊C8C2314yes—
137C22×SD16—31no—
138Q82—38no—
139Q82—64no—
140#554—38no—
141#554—64no—
142C2×C4⋊Q8—32no—
143C2×C4⋊Q8—31no—
144C2×C4⋊Q8—61no—
145C2×C4⋊Q8—62no—
146C2×C4⋊Q8C2×D311yescharacteristic
147C2×#125—32no—
148C2×C22⋊Q8—61no—
149C2×C22⋊Q8—62no—
150C2×C4×Q8—31no—
151C2×C4×Q8—32no—
152C2×C4×Q8—61no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—32no—
155C2×C4×D4—31no—
156C22×C4⋊C4—61no—
157#501—316no—
158#500—38no—
159#479—316no—
160#477C2×D318yescomplemented
161#476—38no—
162#474—316no—
163#446—38no—
164#440—316no—
165C4×SD16—38no—
166C2×#107—32no—
167C2×#107—32no—
168C2×#107C2×D312yes—
169C2×#106—32no—
170C2×#106—31no—
171C2×C4⋊C8—32no—
172C2×C4⋊C8C2×D311yescharacteristic
173C2×#103—31no—
174C2×#103C2×D312yes—
175C2×#102—32no—
176C2×#102—31no—
177C2×#97—32no—
178C2×C4×C8—31no—
179C23×D3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—24no—
182C2×C3×Q8—22no—
183C2×C3×Q8—82no—
184C2×C3×Q8C2×Q814yes—
185C2×C3×Q8C2412yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3C2×D412yes—
195C22×Dic3C2×D412yes—
196C2×D12—21no—
197C2×D12—22no—
198C2×D12C2×Q814yes—
199C2×D12C2412yes—
200C2×C4×D3—24no—
201C2×C4×D3—28no—
202C2×C4×D3—44no—
203C2×C4×D3—84no—
204C2×Dic6—24no—
205C2×Dic6—24no—
206C2×Dic6—28no—
207C2×Dic6—84no—
208C2×Dic6—82no—
209C2×Dic6C2×D414yes—
210C2×C3×C8—24no—
211C2×C3×C8C2414yes—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4C2414yes—
217C3×C4⋊C4C2412yes—
218C3×C42—24no—
219C3×C42C2×D412yes—
220C3×C42C2412yes—
221#215C2×Q8116yescomplemented
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—28no—
224C4⋊Dic3—24no—
225C4⋊Dic3—22no—
226C4⋊Dic3—21no—
227C4⋊Dic3—28no—
228C4⋊Dic3—28no—
229C4⋊Dic3C2×D414yes—
230C4⋊Dic3C2414yes—
231C4⋊Dic3C2412yes—
232Dic3⋊C4—28no—
233Dic3⋊C4—28no—
234Dic3⋊C4C2×D418yes—
235C4×Dic3—28no—
236C4×Dic3C2×D414yes—
237C4×Dic3C2414yes—
238C2×C3⋊C8—22no—
239C2×C3⋊C8C2×Q814yes—
240C2×C3⋊C8C2412yes—
241C2×C3⋊C8C2412yes—
242C22×Q8—21no—
243C22×Q8—31no—
244C22×Q8—62no—
245C22×Q8—61no—
246C22×Q8C22×D311yescharacteristic
247C22×D4—31no—
248C23×C4—62no—
249C23×C4—61no—
250C2×SD16—38no—
251C2×SD16—34no—
252C22×C8—31no—
253C22×C8—31no—
254C22×C8C22×D312yes—
255C4⋊Q8—22no—
256C4⋊Q8—38no—
257C4⋊Q8—34no—
258C4⋊Q8—68no—
259C4⋊Q8—64no—
260C4⋊Q8—616no—
261C4⋊Q8—62no—
262C4⋊Q8C22×D314yes—
263#125—38no—
264#125—64no—
265C22⋊Q8—68no—
266C22⋊Q8—616no—
267C4×Q8—22no—
268C4×Q8—34no—
269C4×Q8—34no—
270C4×Q8—38no—
271C4×Q8—68no—
272C4×Q8—64no—
273C4×Q8C22×D314yes—
274C4×D4—38no—
275C4×D4—34no—
276C4×D4—64no—
277C4×D4—62no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4—68no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4C22×D312yes—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—62no—
293C2×C22⋊C4—61no—
294C2×C42—21no—
295C2×C42—32no—
296C2×C42—62no—
297C2×C42C22×D311yescharacteristic
298#107—38no—
299#107—38no—
300#107C22×D318yes—
301#106—38no—
302#106—34no—
303C4⋊C8—38no—
304C4⋊C8C22×D314yes—
305#103—34no—
306#103C22×D318yes—
307#102—38no—
308#102—34no—
309#97—38no—
310C4×C8—34no—
311C23×C3C22×D411yescharacteristic
312C22×D3—24no—
313C22×D3—22no—
314C22×D3—42no—
315C22×D3—42no—
316C3×Q8—24no—
317C3×Q8—22no—
318C3×Q8—84no—
319C3×Q8C22×Q814yes—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—42no—
326C2×C3×C4—44no—
327C2×C3×C4#13714yes—
328C2×C3×C4C22×D412yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D412yes—
331C2×C3×C4C22×Q812yes—
332C2×C3×C42−1+412yes—
333C2×C3×C4C2511yescharacteristic
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—22no—
338C2×Dic3—42no—
339C2×Dic3—44no—
340C2×Dic3C22×D414yes—
341C2×Dic3C22×D414yes—
342D12—24no—
343D12—22no—
344D12C22×Q814yes—
345C4×D3—48no—
346C4×D3—88no—
347Dic6—28no—
348Dic6—88no—
349Dic6—84no—
350C3×C8—24no—
351C3⋊C8—22no—
352C3⋊C8C22×Q814yes—
353C24—61no—
354C2×Q8—21no—
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—34no—
358C2×Q8—68no—
359C2×Q8—64no—
360C2×Q8—64no—
361C2×Q8—82no—
362C2×Q8—242no—
363C2×Q8—244no—
364C2×Q8D3×Q814yes—
365C2×Q8C23×D312yes—
366C2×D4—34no—
367C2×D4—32no—
368C2×D4—61no—
369C2×D4—62no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—32no—
374C22×C4—32no—
375C22×C4—68no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—64no—
379C22×C4—124no—
380C22×C4—244no—
381C22×C4D3×D411yescharacteristic
382C22×C4D3×D411yescharacteristic
383C22×C4C23×D311yescharacteristic
384SD16—316no—
385C2×C8—24no—
386C2×C8—34no—
387C2×C8—32no—
388C2×C8—32no—
389C2×C8—62no—
390C2×C8C23×D314yes—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—34no—
396C4⋊C4—38no—
397C4⋊C4—34no—
398C4⋊C4—32no—
399C4⋊C4—68no—
400C4⋊C4—68no—
401C4⋊C4—68no—
402C4⋊C4—68no—
403C4⋊C4—61no—
404C4⋊C4—62no—
405C4⋊C4—64no—
406C4⋊C4C23×D312yes—
407C4⋊C4C23×D314yes—
408C22⋊C4—68no—
409C22⋊C4—64no—
410C42—24no—
411C42—34no—
412C42—34no—
413C42—68no—
414C42D3×D412yes—
415C42C23×D312yes—
416C22×C3D4×Q812yes—
417C22×C3C2×#13712yes—
418C22×C3#57612yes—
419C22×C3C23×D411yescharacteristic
420C2×D3—24no—
421C2×D3—48no—
422C2×D3—42no—
423C2×D3—42no—
424C3×C4—24no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—44no—
428C3×C4D4×Q812yes—
429C3×C4C2×#13714yes—
430C3×C4C23×Q811yescharacteristic
431C3×C4C2×2−1+411yescharacteristic
432Dic3—24no—
433Dic3—24no—
434Dic3—42no—
435Dic3—44no—
436C23—64no—
437C23—62no—
438C23—122no—
439C23—122no—
440C23C2×D3×D411yescharacteristic
441Q8—24no—
442Q8—22no—
443Q8—68no—
444Q8—84no—
445Q8—244no—
446Q8—248no—
447Q8C2×D3×Q814yes—
448D4—34no—
449D4—64no—
450D4—62no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—34no—
457C2×C4—34no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—64no—
461C2×C4—64no—
462C2×C4—64no—
463C2×C4—62no—
464C2×C4—122no—
465C2×C4—124no—
466C2×C4—128no—
467C2×C4—248no—
468C2×C4#91814yes—
469C2×C4C2×D3×D412yes—
470C2×C4C2×D3×D411yescharacteristic
471C2×C4C2×D3×D412yes—
472C2×C4C2×D3×Q812yes—
473C2×C4#101012yes—
474C2×C4C24×D311yescharacteristic
475C8—24no—
476C8—34no—
477C8—62no—
478C2×C3#338514yescomplemented
479C2×C3C2×D4×Q811yescharacteristic
480C2×C3C22×#13711yescharacteristic
481C2×C3C2×#57611yescharacteristic
482D3—44no—
483C22—64no—
484C22—122no—
485C22—122no—
486C22—128no—
487C22#563212yes—
488C22C2×#91812yes—
489C22#577412yes—
490C22C22×D3×D411yescharacteristic
491C4—22no—
492C4—24no—
493C4—44no—
494C4—42no—
495C4—64no—
496C4—64no—
497C4—122no—
498C4—124no—
499C4#563212yes—
500C4C2×#91814yes—
501C4C22×D3×Q811yescharacteristic
502C4C2×#101011yescharacteristic
503C3C2×#338511yescharacteristic, complemented, Sylow
504C2—124no—
505C2#8113714yesdirect product factor
506C2C2×#563211yescharacteristic
507C2C22×#91811yescharacteristic
508C2C2×#577411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#477C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#477D3
hypocenterC3C2×#3385
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic6C2semidirect
3. C2×C3×#477C2semidirect
4. C2×#5078C2semidirect
6. #81137C2direct (N × H)
8. C2×#4876C2semidirect
10. C2×#4831C2semidirect
11. C2×#4799C2semidirect
22. Q8×Dic6C22semidirect
34. C3×#477C22semidirect
43. #5078C22semidirect
51. #4876C22semidirect
53. #4831C22semidirect
55. #4799C22semidirect
62. C2×#477D3semidirect
90. C2×#215Q8semidirect
160. #477C2×D3semidirect
221. #215C2×Q8semidirect
478. C2×C3#3385semidirect
503. C3C2×#3385semidirect
505. C2#81137direct (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×Q8×Dic62
C2×C3×#4772
C2×#50782
C2×C4×#2152
#811372
C2×#48902
C2×#48762
C2×#48422
C2×#48312
C2×#47992
C2×#48172
C2×#33853