Finite Group Databasebeta

← #10831737

Subgroups — #10831737

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×#5722C211yesmaximal, characteristic, complemented
2C2×#5087C211yesmaximal, characteristic
3C2×#4710C211yesmaximal, characteristic
4#74190C211yesmaximal, characteristic
5C3×#1993C211yesmaximal, characteristic, complemented
6#69838C211yesmaximal, characteristic, complemented
7#69559C211yesmaximal, characteristic, complemented
8#68909C212yesmaximal
9#68906C212yesmaximal, complemented
10#68898C212yesmaximal
11#68826C212yesmaximal, complemented
12#24341—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#944—22no—
15C22×#204C2211yescharacteristic
16#5722—24no—
17#5722—24no—
18C2×#888—21no—
19C2×#888C2212yes—
20C2×#887—21no—
21C2×#887C2212yes—
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×#382C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#816C2211yescharacteristic
30#5087—24no—
31#4988C2212yes—
32#4984C2212yes—
33C2×#810C2212yes—
34C2×#809C2212yes—
35C2×#807C2212yes—
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37#4710—24no—
38#4677—21no—
39#4673—21no—
40#4668—22no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×Q82—31no—
47C2×#474—31no—
48C2×#440—31no—
49#2580—31no—
50#2057—32no—
51#2023—32no—
52#1993D311yescharacteristic, complemented
53#1985—31no—
54#1880—32no—
55#1863—32no—
56#1805—31no—
57C22×C3×Q8—21no—
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6—22no—
61C22×Dic6D411yescharacteristic
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C42C2311yescharacteristic
71#944—216no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3D411yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×Dic3⋊C4—21no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4D412yes—
83C2×C4×Dic3—21no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3C2312yes—
86C22×C3⋊C8C2312yes—
87C2×#204—22no—
88C2×#204—22no—
89C2×#204Q814yescomplemented
90#888—28no—
91#888—24no—
92#888—24no—
93#887—28no—
94#887—24no—
95#887—24no—
96C4×D12—22no—
97C4×D12—22no—
98C4×Dic6—22no—
99C4×Dic6—22no—
100C4×Dic6—22no—
101C4×Dic6D414yes—
102C3×#97—22no—
103C3×#95—21no—
104C3×#95—21no—
105C3⋊#95—22no—
106#821—22no—
107#818—22no—
108#816—22no—
109#810—24no—
110#809—24no—
111#807—24no—
112C4⋊(C3⋊C8)—22no—
113C22×SD16—31no—
114Q82—64no—
115Q82—64no—
116C2×C4⋊Q8—32no—
117C2×C4⋊Q8—61no—
118C2×C4⋊Q8—62no—
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×C4×Q8—31no—
121C2×C4×Q8—32no—
122C2×C4×Q8—31no—
123C2×C4×Q8—61no—
124C2×C4×D4—31no—
125#474—64no—
126#440—64no—
127C2×#107—31no—
128C2×#107—32no—
129C2×C4⋊C8—31no—
130C2×#103—31no—
131C2×#103—32no—
132C2×#102—31no—
133C2×#102—32no—
134C2×#97C2×D311yescharacteristic
135#399—61no—
136#393—61no—
137#390—32no—
138#389—32no—
139#385—62no—
140#382C2×D311yescharacteristic
141#340—32no—
142#340C2×D312yes—
143#340C2×D312yes—
144#336—32no—
145#336—32no—
146C23×D3—21no—
147C2×C3×Q8—24no—
148C2×C3×Q8—82no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3C2×D412yes—
158C22×Dic3C2×D412yes—
159C2×D12—21no—
160C2×D12—21no—
161C2×D12—21no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12C2×Q812yes—
165C2×C4×D3—42no—
166C2×C4×D3—82no—
167C2×Dic6—22no—
168C2×Dic6—21no—
169C2×Dic6—21no—
170C2×Dic6—21no—
171C2×Dic6—24no—
172C2×Dic6—28no—
173C2×Dic6—82no—
174C2×Dic6—82no—
175C2×Dic6C2×D412yes—
176C2×Dic6C2×Q812yes—
177C2×Dic6#4112yes—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8C2×Q812yes—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4C2×D414yes—
189C3×C42C2×D412yes—
190C3×C42C2×D412yes—
191(C2×D3)⋊C4—42no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—82no—
194C4⋊Dic3—22no—
195C4⋊Dic3—21no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—82no—
199C4⋊Dic3C2×D412yes—
200Dic3⋊C4—28no—
201Dic3⋊C4—28no—
202Dic3⋊C4—24no—
203Dic3⋊C4—24no—
204C4×Dic3—28no—
205C4×Dic3—24no—
206C4×Dic3—22no—
207C4×Dic3—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211#204—28no—
212C22×Q8—21no—
213C22×Q8—31no—
214C22×Q8—31no—
215C22×Q8—62no—
216C22×D4—31no—
217C23×C4—61no—
218C2×SD16—34no—
219C2×SD16—62no—
220C2×SD16—62no—
221C22×C8—32no—
222C22×C8C22×D311yescharacteristic
223C22×C8C22×D311yescharacteristic
224C4⋊Q8—22no—
225C4⋊Q8—64no—
226C4⋊Q8—68no—
227C4⋊Q8—616no—
228C4⋊Q8—64no—
229C4⋊Q8D1214yes—
230C4×Q8—34no—
231C4×Q8—64no—
232C4×Q8—68no—
233C4×Q8—64no—
234C4×Q8—62no—
235C4×Q8—62no—
236C4×Q8—62no—
237C4×D4—62no—
238C4×D4—62no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—31no—
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×D312yes—
250C2×C22⋊C4—62no—
251C2×C22⋊C4—61no—
252C2×C22⋊C4—62no—
253C2×C42—32no—
254C2×C42—62no—
255C2×C42—61no—
256C2×C42C22×D311yescharacteristic
257#107—62no—
258#107—64no—
259C4⋊C8—62no—
260#103—64no—
261#103—62no—
262#102—62no—
263#102—64no—
264#97—22no—
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—84no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4#12012yes—
284C2×C3×C4#12012yes—
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×Q811yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C42−1+411yescharacteristic
290C2×C3×C42−1+411yescharacteristic
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—42no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3#13714yes—
304D12—22no—
305D12—22no—
306D12—41no—
307D12—41no—
308C4×D3—84no—
309Dic6—28no—
310Dic6—22no—
311Dic6—22no—
312Dic6—22no—
313Dic6—88no—
314Dic6C22⋊Q814yes—
315C3×C8—22no—
316C3×C8—41no—
317C3⋊C8—42no—
318C24—61no—
319C2×Q8—24no—
320C2×Q8—32no—
321C2×Q8—32no—
322C2×Q8—32no—
323C2×Q8—64no—
324C2×Q8—68no—
325C2×Q8—62no—
326C2×Q8—61no—
327C2×Q8—61no—
328C2×Q8—61no—
329C2×Q8—82no—
330C2×Q8—242no—
331C2×Q8—242no—
332C2×D4—32no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4—61no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—32no—
341C22×C4—32no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—122no—
346C22×C4—242no—
347C22×C4D3×D411yescharacteristic
348C22×C4D3×D411yescharacteristic
349C22×C4C23×D311yescharacteristic
350SD16—68no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—62no—
357C2×C8—62no—
358C2×C8—62no—
359C2×C8D3×Q812yes—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—32no—
365C4⋊C4—68no—
366C4⋊C4—68no—
367C4⋊C4—64no—
368C4⋊C4—62no—
369C4⋊C4—64no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—61no—
373C4⋊C4—242no—
374C4⋊C4C2×D1214yes—
375C22⋊C4—124no—
376C22⋊C4—122no—
377C22⋊C4—242no—
378C42—68no—
379C42—64no—
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—22no—
395C3×C4—22no—
396C3×C4—44no—
397C3×C4—82no—
398C3×C4C2×C22⋊Q811yescharacteristic
399C3×C4#53611yescharacteristic
400C3×C4#54811yescharacteristic
401C3×C4D4×Q811yescharacteristic
402Dic3—24no—
403Dic3—24no—
404Dic3—42no—
405Dic3—42no—
406Dic3—81no—
407C23—62no—
408C23—121no—
409C23—121no—
410C23—122no—
411C23—122no—
412C23C2×D3×D411yescharacteristic
413Q8—34no—
414Q8—68no—
415Q8—62no—
416Q8—62no—
417Q8—62no—
418Q8—84no—
419Q8—248no—
420D4—62no—
421D4—62no—
422D4—121no—
423D4—121no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—34no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—62no—
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×D3×Q811yescharacteristic
451C2×C4#100711yescharacteristic
452C2×C4D3×#4111yescharacteristic
453C2×C4#101011yescharacteristic
454C8—22no—
455C8—41no—
456C8—122no—
457C2×C3#240911yescharacteristic
458C2×C3#300411yescharacteristic
459C2×C3#300811yescharacteristic
460C2×C3#337512yes—
461C2×C3#338512yes—
462D3—82no—
463C22—124no—
464C22—122no—
465C22—242no—
466C22—241no—
467C22—241no—
468C22C2×#88211yescharacteristic
469C22#567711yescharacteristic
470C22#572711yescharacteristic
471C22#577412yes—
472C22D3×#13712yes—
473C4—22no—
474C4—22no—
475C4—44no—
476C4—64no—
477C4—64no—
478C4—82no—
479C4—122no—
480C4—122no—
481C4—241no—
482C4Q8×D1211yescharacteristic
483C4#557611yescharacteristic
484C4D3×C22⋊Q811yescharacteristic
485C4#562811yescharacteristic
486C3#2434111yescharacteristic, complemented, Sylow
487C2—242no—
488C2#7661911yescharacteristic
489C2#7892811yescharacteristic
490C2#7895811yescharacteristic
491C2#8167512yes—
492C2#8196512yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1993D3
hypocenterC3#24341
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
1. C2×#5722C2semidirect
5. C3×#1993C2semidirect
6. #69838C2semidirect
7. #69559C2semidirect
9. #68906C2semidirect
11. #68826C2semidirect
52. #1993D3semidirect
89. C2×#204Q8semidirect
486. C3#24341semidirect

Maximal subgroups

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

subgroupindex
C2×#57222
C2×#50872
C2×#47102
#741902
C3×#19932
#698382
#695592
#689092
#689062
#688982
#688262
#243413