Finite Group Databasebeta

← #10831740

Subgroups — #10831740

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic6C211yesmaximal, characteristic, complemented
2C2×#5026C211yesmaximal, characteristic
3C2×#4846C211yesmaximal, characteristic
4#74190C211yesmaximal, characteristic
5C3×#1805C211yesmaximal, characteristic, complemented
6#70473C211yesmaximal, characteristic, complemented
7#70240C212yesmaximal
8#70236C212yesmaximal, complemented
9#69555C211yesmaximal, characteristic, complemented
10#68898C212yesmaximal
11#68826C212yesmaximal, complemented
12#24341—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×Q8×Dic3C2212yes—
15C22×#215C2211yescharacteristic
16Q8×Dic6—24no—
17Q8×Dic6—24no—
18C2×#888—22no—
19C2×#888C2212yes—
20C2×C4×D12C2211yescharacteristic
21C2×#869—21no—
22C2×#869C2211yescharacteristic
23C2×C4×Dic6—21no—
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2212yes—
27C3×#382C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#821C2212yes—
30C2×#818C2212yes—
31#5026—24no—
32#4988C2212yes—
33#4984C2212yes—
34C2×#809C2211yescharacteristic
35C2×#807C2211yescharacteristic
36C2×#803C2211yescharacteristic
37#4846—24no—
38#4677—21no—
39#4673—21no—
40#4668—22no—
41C3×#340C2212yes—
42#4550C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×Q82—31no—
47C2×#474—31no—
48C2×#440—31no—
49#2580—31no—
50#2057—32no—
51#2023—32no—
52#1993—31no—
53#1985—31no—
54#1880—32no—
55#1863—32no—
56#1805D311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6—21no—
61C22×Dic6—22no—
62C22×Dic6D411yescharacteristic
63C22×C3×C8C2312yes—
64C3×C4×Q8—22no—
65C3×C4×Q8D414yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71Q8×Dic3—24no—
72Q8×Dic3—24no—
73C2×#215—22no—
74C2×#215—22no—
75C2×#215Q814yescomplemented
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4D412yes—
87C2×C4×Dic3—22no—
88C2×C4×Dic3C2312yes—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2311yescharacteristic
91#888—216no—
92#888—24no—
93#888—24no—
94C4×D12—22no—
95C4×D12—22no—
96#869—28no—
97#869—22no—
98#869D414yes—
99C4×Dic6—28no—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C3×C4⋊C8—22no—
103C3×#103—24no—
104C3×#95—21no—
105C3×#95—21no—
106#835—22no—
107C3⋊#95—22no—
108#821—24no—
109#818—24no—
110#809—22no—
111#807—22no—
112#803—22no—
113C22×SD16—31no—
114Q82—64no—
115Q82—64no—
116C2×C4⋊Q8—32no—
117C2×C4⋊Q8—31no—
118C2×C4⋊Q8—61no—
119C2×C4⋊Q8—62no—
120C2×C4×Q8—31no—
121C2×C4×Q8—32no—
122C2×C4×Q8—61no—
123C2×C4×Q8C2×D311yescharacteristic
124C2×C4×D4—31no—
125#474—64no—
126#440—64no—
127C2×#107—32no—
128C2×#107—31no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#103—31no—
131C2×#103C2×D312yes—
132C2×#102—32no—
133C2×#102—31no—
134C2×#97—31no—
135#399—61no—
136#393—61no—
137#390—32no—
138#389—32no—
139#385—62no—
140#382C2×D311yescharacteristic
141#340—32no—
142#340—32no—
143#340C2×D312yes—
144#336—32no—
145#336—32no—
146C23×D3—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8C2×D412yes—
151C2×C3×Q8C2×Q812yes—
152C2×C3×Q8#4112yes—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—22no—
160C22×Dic3—21no—
161C22×Dic3C2×D412yes—
162C22×Dic3C2×D412yes—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12C2×Q812yes—
169C2×C4×D3—42no—
170C2×C4×D3—82no—
171C2×Dic6—22no—
172C2×Dic6—24no—
173C2×Dic6—28no—
174C2×Dic6—24no—
175C2×Dic6—82no—
176C2×Dic6—82no—
177C2×Dic6—82no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4C2×D412yes—
186C3×C42—24no—
187C3×C42C2×D412yes—
188C3×C42C2×D412yes—
189#215—28no—
190(C2×D3)⋊C4—42no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—82no—
193C4⋊Dic3—28no—
194C4⋊Dic3—22no—
195C4⋊Dic3—24no—
196C4⋊Dic3—22no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—82no—
200C4⋊Dic3C2×D414yes—
201Dic3⋊C4—28no—
202Dic3⋊C4—24no—
203C4×Dic3—28no—
204C4×Dic3—22no—
205C4×Dic3—22no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8C2×Q812yes—
212C22×Q8—31no—
213C22×Q8—62no—
214C22×Q8—61no—
215C22×Q8C22×D311yescharacteristic
216C22×D4—31no—
217C23×C4—61no—
218C2×SD16—34no—
219C2×SD16—62no—
220C2×SD16—62no—
221C22×C8—31no—
222C22×C8—31no—
223C22×C8C22×D312yes—
224C4⋊Q8—34no—
225C4⋊Q8—68no—
226C4⋊Q8—64no—
227C4⋊Q8—616no—
228C4⋊Q8—64no—
229C4⋊Q8—62no—
230C4×Q8—22no—
231C4×Q8—68no—
232C4×Q8—64no—
233C4×Q8—64no—
234C4×Q8—62no—
235C4×Q8—62no—
236C4×Q8D1214yes—
237C4×D4—62no—
238C4×D4—62no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—32no—
241C2×C4⋊C4—32no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—61no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4C22×D312yes—
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C22⋊C4—62no—
251C2×C22⋊C4—61no—
252C2×C22⋊C4—62no—
253C2×C42—21no—
254C2×C42—32no—
255C2×C42—62no—
256C2×C42C22×D311yescharacteristic
257#107—64no—
258#107—62no—
259C4⋊C8—22no—
260#103—24no—
261#103—62no—
262#102—64no—
263#102—62no—
264#97—62no—
265#95—21no—
266#95—21no—
267#95—62no—
268C23×C3C22×D411yescharacteristic
269C22×D3—22no—
270C22×D3—41no—
271C22×D3—41no—
272C22×D3—42no—
273C22×D3—42no—
274C3×Q8—22no—
275C3×Q8—22no—
276C3×Q8—22no—
277C3×Q8C22⋊Q814yes—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4#12012yes—
289C2×C3×C4#12012yes—
290C2×C3×C4C22×D411yescharacteristic
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C22×Q811yescharacteristic
293C2×C3×C4C2×#4111yescharacteristic
294C2×C3×C42−1+411yescharacteristic
295C2×C3×C42−1+411yescharacteristic
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—24no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—42no—
306C2×Dic3#13714yes—
307D12—22no—
308D12—22no—
309D12—41no—
310D12—41no—
311C4×D3—84no—
312Dic6—28no—
313Dic6—84no—
314Dic6—88no—
315C3×C8—42no—
316C3⋊C8—22no—
317C3⋊C8—41no—
318C24—61no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—64no—
323C2×Q8—68no—
324C2×Q8—62no—
325C2×Q8—64no—
326C2×Q8—242no—
327C2×Q8—242no—
328C2×Q8—242no—
329C2×Q8C2×D1212yes—
330C2×Q8#23712yes—
331C2×Q8D3×Q812yes—
332C2×D4—32no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4—61no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—32no—
342C22×C4—32no—
343C22×C4—62no—
344C22×C4—61no—
345C22×C4—122no—
346C22×C4—242no—
347C22×C4D3×D411yescharacteristic
348C22×C4D3×D411yescharacteristic
349C22×C4C23×D311yescharacteristic
350SD16—68no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—32no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8—61no—
358C2×C8—61no—
359C2×C8—61no—
360C4⋊C4—24no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—34no—
365C4⋊C4—68no—
366C4⋊C4—68no—
367C4⋊C4—64no—
368C4⋊C4—62no—
369C4⋊C4—64no—
370C4⋊C4—62no—
371C4⋊C4—61no—
372C4⋊C4—61no—
373C4⋊C4—242no—
374C4⋊C4C2×D1212yes—
375C22⋊C4—124no—
376C22⋊C4—122no—
377C22⋊C4—242no—
378C42—24no—
379C42—68no—
380C42—62no—
381C42—62no—
382C42C2×D1212yes—
383C42D3×D412yes—
384C22×C3C2×#12011yescharacteristic
385C22×C3#54811yescharacteristic
386C22×C3D4×Q811yescharacteristic
387C22×C3C2×#13712yes—
388C22×C3#57612yes—
389C2×D3—44no—
390C2×D3—42no—
391C2×D3—82no—
392C2×D3—81no—
393C2×D3—81no—
394C3×C4—24no—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—42no—
398C3×C4—82no—
399C3×C4C2×C22⋊Q811yescharacteristic
400C3×C4#53611yescharacteristic
401C3×C4#54811yescharacteristic
402C3×C4D4×Q811yescharacteristic
403Dic3—24no—
404Dic3—42no—
405Dic3—44no—
406Dic3—81no—
407C23—62no—
408C23—121no—
409C23—122no—
410C23—122no—
411C23—121no—
412C23C2×D3×D411yescharacteristic
413Q8—22no—
414Q8—22no—
415Q8—22no—
416Q8—68no—
417Q8—244no—
418Q8—248no—
419Q8#89714yes—
420D4—62no—
421D4—62no—
422D4—121no—
423D4—121no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—34no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—64no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—121no—
444C2×C4—122no—
445C2×C4—244no—
446C2×C4#88212yes—
447C2×C4#88212yes—
448C2×C4C22×D1211yescharacteristic
449C2×C4C2×D3×D411yescharacteristic
450C2×C4C2×#23711yescharacteristic
451C2×C4C2×D3×Q811yescharacteristic
452C2×C4#101011yescharacteristic
453C2×C4#101011yescharacteristic
454C8—42no—
455C8—62no—
456C8—121no—
457C2×C3#240911yescharacteristic
458C2×C3#300411yescharacteristic
459C2×C3#300811yescharacteristic
460C2×C3#337512yes—
461C2×C3#338512yes—
462D3—82no—
463C22—122no—
464C22—124no—
465C22—241no—
466C22—242no—
467C22—241no—
468C22C2×#88211yescharacteristic
469C22#559811yescharacteristic
470C22#563211yescharacteristic
471C22#576912yes—
472C22D3×#13712yes—
473C4—24no—
474C4—22no—
475C4—22no—
476C4—42no—
477C4—64no—
478C4—82no—
479C4—122no—
480C4—124no—
481C4—241no—
482C4C2×#89711yescharacteristic
483C4#552611yescharacteristic
484C4#559811yescharacteristic
485C4#563211yescharacteristic
486C3#2434111yescharacteristic, complemented, Sylow
487C2—242no—
488C2#7401711yescharacteristic
489C2#7883411yescharacteristic
490C2#7895811yescharacteristic
491C2#8058512yes—
492C2#8122312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1805D3
hypocenterC3#24341
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic6C2semidirect
5. C3×#1805C2semidirect
6. #70473C2semidirect
8. #70236C2semidirect
9. #69555C2semidirect
11. #68826C2semidirect
56. #1805D3semidirect
75. C2×#215Q8semidirect
486. C3#24341semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic62
C2×#50262
C2×#48462
#741902
C3×#18052
#704732
#702402
#702362
#695552
#688982
#688262
#243413