Finite Group Databasebeta

← #10804460

Subgroups — #10804460

Order 768.

1386 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#124C211yesmaximal, characteristic
2#79038C211yesmaximal, characteristic, complemented
3#78517C211yesmaximal, characteristic, complemented
4C3×#1967C211yesmaximal, characteristic, complemented
5#71727C211yesmaximal, characteristic, complemented
6#70523C211yesmaximal, characteristic, complemented
7#70113C212yesmaximal, complemented
8#70099C212yesmaximal, complemented
9#70079C212yesmaximal
10#70065C212yesmaximal, complemented
11#69878C211yesmaximal, characteristic, complemented
12#21798—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C22×D3×D4C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×#947C2211yescharacteristic
17C2×#936C2211yescharacteristic
18D3×#124—22no—
19D3×#124—22no—
20D3×#124—22no—
21D3×#124—22no—
22C2×#885C414yes—
23C2×D3×C22⋊C4C414yes—
24C2×#875C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C3×#423C2211yescharacteristic, complemented
27C2×C3×#101C2212yescomplemented
28C2×C3×#100C2212yes—
29C3×C4×M16C2211yescharacteristic, complemented
30C2×#836C2212yescomplemented
31C2×#833C2212yes—
32C2×#824C2212yescomplemented
33C2×#824C2212yescomplemented
34C2×#823C2212yes—
35C2×#823C2212yes—
36#5126C2212yes—
37#5126C2212yes—
38#5118C2212yescomplemented
39Dic3×M16C2212yescomplemented
40#4924C2211yescharacteristic, complemented
41C4×#208C2211yescharacteristic, complemented
42#4835—22no—
43#4833—22no—
44#4705—22no—
45#4685—22no—
46C22×#124—31no—
47#2920—31no—
48#2920—31no—
49#1967—31no—
50#1967—32no—
51#1967—32no—
52#1967—31no—
53#1967—31no—
54#1967D311yescharacteristic, complemented
55#1784—32no—
56#1783—32no—
57C24×D3C2×C412yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C22×#67C2×C412yes—
61C2×D3×Q8—22no—
62C2×D3×Q8—22no—
63C2×D3×Q8—42no—
64C2×D3×Q8—42no—
65C2×D3×D4—22no—
66C2×D3×D4—22no—
67C2×D3×D4—42no—
68C2×D3×D4—42no—
69C22×D12C2311yescharacteristic
70C22×C4×D3C2×C412yes—
71C22×C4×D3C2×C412yes—
72C22×C4×D3C2311yescharacteristic
73C22×Dic6C2×C412yes—
74C22×Dic6C2311yescharacteristic
75C2×C3×M16C2312yes—
76C2×C3×M16C2312yes—
77C3×#124—21no—
78C3×#124—21no—
79C3×#124—22no—
80C2×C3×C22⋊C4C2×C414yes—
81C2×C3×C42C2311yescharacteristic
82C2×#217C2×C414yes—
83#947—21no—
84#947—22no—
85#947—21no—
86#936—21no—
87#936—22no—
88#936—21no—
89C2×(C2×D3)⋊C4C2×C414yes—
90C2×(C2×D3)⋊C4C2×C414yes—
91C2×C4×Dic3C2×C412yes—
92C2×C4×Dic3C2311yescharacteristic
93C2×#208C2312yes—
94C2×#208C2312yes—
95#885—24no—
96#885—24no—
97#885—24no—
98#885—24no—
99D3×C22⋊C4—24no—
100D3×C22⋊C4—24no—
101D3×C22⋊C4—24no—
102D3×C22⋊C4—24no—
103#875—21no—
104#875—21no—
105#875—22no—
106C42×D3D412yes—
107C42×D3D412yes—
108C42×D3D412yes—
109C42×D3D412yes—
110C3×C4⋊C8—22no—
111C3×#101—42no—
112C3×#100—42no—
113C3×#97—21no—
114C3×C4×C8—21no—
115#836—42no—
116#833—42no—
117#824—42no—
118#824—42no—
119#823—42no—
120#823—42no—
121(C2×D3)⋊C8—24no—
122(C2×D3)⋊C8—24no—
123#815—22no—
124Dic3⋊C8—24no—
125C8×Dic3—22no—
126C4⋊(C3⋊C8)—22no—
127#803—21no—
128C4×C3⋊C8—21no—
129C23×Q8—31no—
130C23×D4—31no—
131C2×#124—31no—
132C2×#124—31no—
133C2×#124—34no—
134C2×#124—31no—
135C2×#124—62no—
136C2×#124—62no—
137C2×#124—62no—
138C2×#124—62no—
139C2×#124C2×D311yescharacteristic
140C22×C22⋊C4—34no—
141C22×C42—31no—
142#432—62no—
143#432—62no—
144#431—62no—
145#431—62no—
146#423—32no—
147#423—31no—
148#423C2×D311yescharacteristic, complemented
149C2×#101—32no—
150C2×#101—32no—
151C2×#101—32no—
152C2×#101C2×D312yescomplemented
153C2×#100—32no—
154C2×#100—32no—
155C2×#100—32no—
156C2×#100C2×D312yes—
157#407—32no—
158#407—32no—
159C4×M16—32no—
160C4×M16—31no—
161C4×M16C2×D311yescharacteristic, complemented
162C24×C3C22×C412yes—
163C23×D3—24no—
164C23×D3—44no—
165C23×D3—48no—
166C23×D3C22×C411yescharacteristic
167C23×D3C22×C412yes—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C22×C3×C4C22×C412yes—
177C22×C3×C4C22×C412yes—
178C22×C3×C4C2411yescharacteristic
179C2×#67—24no—
180C2×#67—44no—
181C22×Dic3C22×C412yes—
182C22×Dic3C22×C411yescharacteristic
183C22×Dic3C22×C412yes—
184D3×Q8—44no—
185D3×Q8—44no—
186D3×Q8—48no—
187D3×D4—44no—
188D3×D4—44no—
189D3×D4—48no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—22no—
193C2×D12—22no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—24no—
197C2×C4×D3—44no—
198C2×C4×D3—44no—
199C2×C4×D3C22⋊C412yes—
200C2×C4×D3C22⋊C412yes—
201C2×C4×D3C22⋊C412yes—
202C2×C4×D3C22⋊C412yes—
203C2×C4×D3C2×D412yes—
204C2×C4×D3C2×D412yes—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—22no—
208C2×Dic6—22no—
209C2×Dic6—24no—
210C2×Dic6—44no—
211C3×M16—22no—
212C3×M16—22no—
213C3×M16—22no—
214C3×M16—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C42C2×D411yescharacteristic
220C3×C42C2×D411yescharacteristic
221C3×C42C2×D412yes—
222#217—24no—
223#217—24no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—24no—
228C4×Dic3C22⋊C412yes—
229C4×Dic3C22⋊C412yes—
230C4×Dic3C22⋊C412yes—
231C4×Dic3C22⋊C412yes—
232C4×Dic3C2×D411yescharacteristic
233C4×Dic3C2×D412yes—
234C4×Dic3C2×D411yescharacteristic
235#208—22no—
236#208—22no—
237#208—22no—
238#208—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C25—32no—
242C22×Q8—32no—
243C22×Q8—31no—
244C22×Q8—62no—
245C22×Q8—62no—
246C22×Q8—122no—
247C22×Q8—122no—
248C22×Q8C22×D311yescharacteristic
249C22×D4—32no—
250C22×D4—31no—
251C22×D4—62no—
252C22×D4—62no—
253C22×D4—122no—
254C22×D4—122no—
255C22×D4C22×D311yescharacteristic
256C23×C4—31no—
257C23×C4—32no—
258C23×C4—32no—
259C2×M16—32no—
260C2×M16—32no—
261C2×M16C22×D312yes—
262C2×M16C22×D312yes—
263#124—21no—
264#124—21no—
265#124—22no—
266#124—64no—
267#124—64no—
268#124—64no—
269#124—64no—
270#124—61no—
271#124—61no—
272#124—62no—
273#124—61no—
274#124—61no—
275#124—62no—
276#124—61no—
277#124—61no—
278#124—62no—
279C2×C22⋊C4—34no—
280C2×C22⋊C4—34no—
281C2×C22⋊C4—34no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4C4×D314yes—
287C2×C42—32no—
288C2×C42—31no—
289C2×C42—32no—
290C2×C42—32no—
291C2×C42—32no—
292C2×C42—32no—
293C2×C42C22×D311yescharacteristic
294C4⋊C8—22no—
295C4⋊C8—64no—
296C4⋊C8—62no—
297#101—42no—
298#101—122no—
299#101—122no—
300#101—122no—
301#100—42no—
302#100—122no—
303#100—122no—
304#100—122no—
305C22⋊C8—64no—
306C22⋊C8—64no—
307#97—21no—
308#97—62no—
309#97—61no—
310C4×C8—21no—
311C4×C8—62no—
312C4×C8—61no—
313C23×C3—22no—
314C23×C3—24no—
315C23×C3—44no—
316C23×C3C23×C411yescharacteristic
317C22×D3—22no—
318C22×D3—24no—
319C22×D3—22no—
320C22×D3—24no—
321C22×D3—48no—
322C22×D3—48no—
323C22×D3—48no—
324C22×D3—44no—
325C22×D3—816no—
326C22×D3C2×C22⋊C412yes—
327C3×Q8—41no—
328C3×Q8—41no—
329C3×Q8—42no—
330C3×D4—41no—
331C3×D4—41no—
332C3×D4—42no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—42no—
336C2×C3×C4C2×C22⋊C412yes—
337C2×C3×C4C2×C22⋊C412yes—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4H(5,𝔽2)12yes—
341C2×C3×C4H(5,𝔽2)12yes—
342#67—48no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—42no—
347C2×Dic3C2×C22⋊C412yes—
348C2×Dic3C2×C22⋊C412yes—
349C2×Dic3C2×C22⋊C412yes—
350D12—41no—
351D12—41no—
352D12—42no—
353C4×D3—24no—
354C4×D3—24no—
355C4×D3—22no—
356C4×D3—22no—
357C4×D3—24no—
358C4×D3—42no—
359C4×D3—42no—
360C4×D3—48no—
361Dic6—48no—
362Dic6—41no—
363Dic6—41no—
364Dic6—42no—
365C3×C8—42no—
366C3×C8—42no—
367C3⋊C8—42no—
368C3⋊C8—42no—
369C24—31no—
370C24—32no—
371C24—64no—
372C24—124no—
373C24—128no—
374C24C2×C4×D312yes—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—64no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—62no—
383C2×Q8—62no—
384C2×Q8—124no—
385C2×Q8—124no—
386C2×Q8—124no—
387C2×Q8—128no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—64no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—62no—
396C2×D4—62no—
397C2×D4—124no—
398C2×D4—124no—
399C2×D4—128no—
400C2×D4—124no—
401C22×C4—31no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—32no—
410C22×C4—64no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—124no—
414C22×C4—124no—
415C22×C4C2×C4×D312yes—
416C22×C4C2×C4×D312yes—
417C22×C4C23×D311yescharacteristic
418M16—22no—
419M16—22no—
420M16—22no—
421M16—22no—
422M16—62no—
423M16—62no—
424M16—62no—
425M16—62no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—62no—
429C2×C8—62no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—64no—
433C22⋊C4—64no—
434C22⋊C4—64no—
435C22⋊C4—64no—
436C22⋊C4—64no—
437C22⋊C4—64no—
438C42—32no—
439C42—32no—
440C42—32no—
441C42—32no—
442C42—31no—
443C42—31no—
444C42—32no—
445C42D3×D411yescharacteristic
446C42D3×D411yescharacteristic
447C42D3×D412yes—
448C22×C3—22no—
449C22×C3—42no—
450C22×C3—44no—
451C22×C3—84no—
452C22×C3C22×C22⋊C411yescharacteristic
453C22×C3#51812yes—
454C2×D3—28no—
455C2×D3—22no—
456C2×D3—22no—
457C2×D3—42no—
458C2×D3—44no—
459C2×D3—816no—
460C2×D3—84no—
461C2×D3#41314yes—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—42no—
465C3×C4#53511yescharacteristic
466C3×C4#53511yescharacteristic
467C3×C4#53512yes—
468Dic3—22no—
469Dic3—22no—
470Dic3—22no—
471Dic3—42no—
472C23—22no—
473C23—24no—
474C23—32no—
475C23—44no—
476C23—64no—
477C23—62no—
478C23—62no—
479C23—64no—
480C23—124no—
481C23—128no—
482C23—128no—
483C23—128no—
484C23—2416no—
485C23C22×C4×D311yescharacteristic
486Q8—41no—
487Q8—41no—
488Q8—42no—
489Q8—128no—
490Q8—121no—
491Q8—121no—
492Q8—122no—
493D4—41no—
494D4—41no—
495D4—42no—
496D4—128no—
497D4—121no—
498D4—121no—
499D4—122no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—32no—
503C2×C4—32no—
504C2×C4—32no—
505C2×C4—42no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—64no—
512C2×C4—62no—
513C2×C4—64no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—128no—
517C2×C4—122no—
518C2×C4D3×C22⋊C412yes—
519C2×C4D3×C22⋊C412yes—
520C2×C4C2×D3×D411yescharacteristic
521C2×C4C2×D3×D411yescharacteristic
522C2×C4#100412yes—
523C2×C4#100912yes—
524C8—42no—
525C8—42no—
526C8—122no—
527C8—122no—
528C2×C3—22no—
529C2×C3—82no—
530C2×C3#230911yescharacteristic
531C2×C3C2×#41312yes—
532D3—24no—
533D3—82no—
534C22—22no—
535C22—34no—
536C22—44no—
537C22—42no—
538C22—68no—
539C22—62no—
540C22—62no—
541C22—84no—
542C22—124no—
543C22—122no—
544C22—244no—
545C22—2416no—
546C22C2×D3×C22⋊C411yescharacteristic
547C22#554512yes—
548C4—22no—
549C4—22no—
550C4—42no—
551C4—62no—
552C4—62no—
553C4—62no—
554C4—122no—
555C4#560712yes—
556C4#567411yescharacteristic
557C4#571511yescharacteristic
558C3#2179811yescharacteristic, complemented, Sylow
559C2—22no—
560C2—64no—
561C2—82no—
562C2—242no—
563C2#7642811yescharacteristic
564C2D3×#41312yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1967C2
socleC22×C3C22×C22⋊C4
prime radicalC22×D3×D4C22
cosocleC22×C3×C4C24
hypercenter#1967D3
hypocenterC3#21798
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1967

Complements

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

NH ≅ G/Ntype
2. #79038C2semidirect
3. #78517C2semidirect
4. C3×#1967C2semidirect
5. #71727C2semidirect
6. #70523C2semidirect
7. #70113C2semidirect
8. #70099C2semidirect
10. #70065C2semidirect
11. #69878C2semidirect
26. C3×#423C22semidirect
27. C2×C3×#101C22semidirect
29. C3×C4×M16C22semidirect
30. C2×#836C22semidirect
32. C2×#824C22semidirect
33. C2×#824C22semidirect
38. #5118C22semidirect
39. Dic3×M16C22semidirect
40. #4924C22semidirect
41. C4×#208C22semidirect
54. #1967D3semidirect
148. #423C2×D3semidirect
152. C2×#101C2×D3semidirect
161. C4×M16C2×D3semidirect
558. C3#21798semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1242
#790382
#785172
C3×#19672
#717272
#705232
#701132
#700992
#700792
#700652
#698782
#217983