Finite Group Databasebeta

← #10831726

Subgroups — #10831726

Order 768.

1036 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5721C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic
3C2×#4710C211yesmaximal, characteristic
4#74190C211yesmaximal, characteristic
5C3×#1993C211yesmaximal, characteristic, complemented
6#69838C211yesmaximal, characteristic, complemented
7#69559C211yesmaximal, characteristic, complemented
8#68909C212yesmaximal
9#68907C212yesmaximal, complemented
10#68899C212yesmaximal
11#68829C212yesmaximal, complemented
12#24340—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×Q8×Dic3—21no—
15C2×#944—21no—
16C22×#204C2211yescharacteristic
17#5721—24no—
18#5721—24no—
19C2×#890C2212yes—
20C2×#889—22no—
21C2×#887C2212yes—
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×#382C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#816C2211yescharacteristic
30#5084—24no—
31#4988C2212yes—
32#4984C2212yes—
33C2×#810C2212yes—
34C2×#809C2212yes—
35C2×#808C2212yes—
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×#557—31no—
47C2×#476—31no—
48C2×#440—31no—
49#2580—31no—
50#2059—32no—
51#2025—32no—
52#1993D311yescharacteristic, complemented
53#1985—31no—
54#1880—32no—
55#1865—32no—
56#1805—31no—
57C22×C3×Q8—21no—
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6D411yescharacteristic
61C22×Dic6C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C42C2311yescharacteristic
70Q8×Dic3—28no—
71#944—28no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3D411yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—21no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4D412yes—
84C2×C4×Dic3—21no—
85C2×C4×Dic3—22no—
86C2×C4×Dic3C2312yes—
87C22×C3⋊C8C2312yes—
88C2×#204—22no—
89C2×#204—22no—
90C2×#204Q814yescomplemented
91#890—24no—
92#890—24no—
93#889—216no—
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#808—24no—
112C4⋊(C3⋊C8)—22no—
113C22×SD16—31no—
114#557—64no—
115#557—64no—
116C2×C4⋊Q8—61no—
117C2×C4⋊Q8C2×D311yescharacteristic
118C2×#125—32no—
119C2×#125—62no—
120C2×C4×Q8—31no—
121C2×C4×Q8—32no—
122C2×C4×Q8—31no—
123C2×C4×Q8—61no—
124C2×C4×D4—31no—
125#476—64no—
126#440—64no—
127C2×#107—31no—
128C2×#106—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—44no—
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—21no—
168C2×Dic6—21no—
169C2×Dic6—21no—
170C2×Dic6—22no—
171C2×Dic6—42no—
172C2×Dic6C2×D412yes—
173C2×Dic6C2×Q812yes—
174C2×Dic6#4112yes—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8C2×Q812yes—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4C2×D414yes—
186C3×C42C2×D412yes—
187C3×C42C2×D412yes—
188(C2×D3)⋊C4—44no—
189(C2×D3)⋊C4—42no—
190(C2×D3)⋊C4—82no—
191C4⋊Dic3—28no—
192C4⋊Dic3—22no—
193C4⋊Dic3—21no—
194C4⋊Dic3—21no—
195C4⋊Dic3—21no—
196C4⋊Dic3—82no—
197C4⋊Dic3C2×D412yes—
198Dic3⋊C4—28no—
199Dic3⋊C4—28no—
200Dic3⋊C4—24no—
201Dic3⋊C4—24no—
202C4×Dic3—28no—
203C4×Dic3—24no—
204C4×Dic3—22no—
205C4×Dic3—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209#204—28no—
210C22×Q8—21no—
211C22×Q8—31no—
212C22×Q8—31no—
213C22×D4—31no—
214C23×C4—61no—
215C2×SD16—34no—
216C2×SD16—62no—
217C2×SD16—62no—
218C22×C8—32no—
219C22×C8C22×D311yescharacteristic
220C22×C8C22×D311yescharacteristic
221C4⋊Q8—22no—
222C4⋊Q8—68no—
223C4⋊Q8D1214yes—
224#125—64no—
225#125—616no—
226#125—64no—
227C4×Q8—34no—
228C4×Q8—64no—
229C4×Q8—68no—
230C4×Q8—64no—
231C4×Q8—62no—
232C4×Q8—62no—
233C4×Q8—62no—
234C4×D4—62no—
235C4×D4—62no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—61no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4C22×D312yes—
247C2×C4⋊C4C22×D312yes—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—61no—
250C2×C22⋊C4—62no—
251C2×C42—32no—
252C2×C42—62no—
253C2×C42—61no—
254C2×C42C22×D311yescharacteristic
255#107—62no—
256#106—64no—
257C4⋊C8—62no—
258#103—64no—
259#103—62no—
260#102—62no—
261#102—64no—
262#97—22no—
263#95—21no—
264#95—21no—
265#95—62no—
266C23×C3C22×D411yescharacteristic
267C22×D3—22no—
268C22×D3—41no—
269C22×D3—41no—
270C22×D3—42no—
271C22×D3—42no—
272C3×Q8—48no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—44no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4#12012yes—
281C2×C3×C4#12012yes—
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C22×Q811yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C42−1+411yescharacteristic
287C2×C3×C42−1+411yescharacteristic
288C2×Dic3—22no—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—42no—
301D12—22no—
302D12—22no—
303D12—41no—
304D12—41no—
305C4×D3—84no—
306Dic6—22no—
307Dic6—22no—
308Dic6—22no—
309Dic6—44no—
310Dic6C22⋊Q814yes—
311C3×C8—22no—
312C3×C8—41no—
313C3⋊C8—42no—
314C24—61no—
315C2×Q8—24no—
316C2×Q8—32no—
317C2×Q8—32no—
318C2×Q8—32no—
319C2×Q8—44no—
320C2×Q8—62no—
321C2×Q8—61no—
322C2×Q8—61no—
323C2×Q8—61no—
324C2×Q8—122no—
325C2×D4—32no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—61no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—122no—
339C22×C4—242no—
340C22×C4D3×D411yescharacteristic
341C22×C4D3×D411yescharacteristic
342C22×C4C23×D311yescharacteristic
343SD16—68no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—62no—
350C2×C8—62no—
351C2×C8—62no—
352C2×C8D3×Q812yes—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—32no—
358C4⋊C4—68no—
359C4⋊C4—68no—
360C4⋊C4—68no—
361C4⋊C4—62no—
362C4⋊C4—64no—
363C4⋊C4—64no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—242no—
368C4⋊C4C2×D1214yes—
369C22⋊C4—124no—
370C22⋊C4—122no—
371C22⋊C4—242no—
372C42—68no—
373C42—64no—
374C42—62no—
375C42—62no—
376C42C2×D1212yes—
377C42D3×D412yes—
378C22×C3C2×#12011yescharacteristic
379C22×C3#54811yescharacteristic
380C22×C3D4×Q811yescharacteristic
381C22×C3#57512yes—
382C22×C3#57712yes—
383C2×D3—44no—
384C2×D3—42no—
385C2×D3—82no—
386C2×D3—81no—
387C2×D3—81no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—44no—
391C3×C4—82no—
392C3×C4C2×C22⋊Q811yescharacteristic
393C3×C4#53611yescharacteristic
394C3×C4#54811yescharacteristic
395C3×C4D4×Q811yescharacteristic
396Dic3—24no—
397Dic3—42no—
398Dic3—42no—
399Dic3—42no—
400Dic3—81no—
401C23—62no—
402C23—121no—
403C23—121no—
404C23—122no—
405C23—122no—
406C23C2×D3×D411yescharacteristic
407Q8—34no—
408Q8—48no—
409Q8—62no—
410Q8—62no—
411Q8—62no—
412Q8—124no—
413D4—62no—
414D4—62no—
415D4—121no—
416D4—121no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4—122no—
437C2×C4—244no—
438C2×C4#88212yes—
439C2×C4#88212yes—
440C2×C4C22×D1211yescharacteristic
441C2×C4C2×D3×D411yescharacteristic
442C2×C4C2×D3×Q811yescharacteristic
443C2×C4#100711yescharacteristic
444C2×C4D3×#4111yescharacteristic
445C2×C4#101011yescharacteristic
446C8—22no—
447C8—41no—
448C8—122no—
449C2×C3#240911yescharacteristic
450C2×C3#300511yescharacteristic
451C2×C3#301011yescharacteristic
452C2×C3#337112yes—
453C2×C3#338812yes—
454D3—82no—
455C22—124no—
456C22—122no—
457C22—242no—
458C22—241no—
459C22—241no—
460C22C2×#88211yescharacteristic
461C22#567711yescharacteristic
462C22#572711yescharacteristic
463C22#577312yes—
464C22#577812yes—
465C4—22no—
466C4—22no—
467C4—44no—
468C4—64no—
469C4—82no—
470C4—122no—
471C4—122no—
472C4—122no—
473C4—241no—
474C4Q8×D1211yescharacteristic
475C4#557611yescharacteristic
476C4D3×C22⋊Q811yescharacteristic
477C4#562811yescharacteristic
478C3#2434011yescharacteristic, complemented, Sylow
479C2—242no—
480C2#7661911yescharacteristic
481C2#7892711yescharacteristic
482C2#7896011yescharacteristic
483C2#8167312yes—
484C2#8196812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1993D3
hypocenterC3#24340
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×#5721C2semidirect
5. C3×#1993C2semidirect
6. #69838C2semidirect
7. #69559C2semidirect
9. #68907C2semidirect
11. #68829C2semidirect
52. #1993D3semidirect
90. C2×#204Q8semidirect
478. C3#24340semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C2×#50842
C2×#47102
#741902
C3×#19932
#698382
#695592
#689092
#689072
#688992
#688292
#243403