Finite Group Databasebeta

← #72246673

Subgroups — #72246673

Order 2016.

718 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
1D4×C3⋊D21C211yesmaximal, characteristic, complemented
2C3×D4×D21C211yesmaximal, characteristic, complemented
3C3×C7×#237C211yesmaximal, characteristic, complemented
4#11759626C212yesmaximal, complemented
5#67×D21C212yesmaximal, complemented
6C2×#92689C212yesmaximal, complemented
7#11759615C212yesmaximal, complemented
8D3×D84C211yesmaximal, characteristic, complemented
9#11759609C211yesmaximal, characteristic, complemented
10#11759607C211yesmaximal, characteristic, complemented
11#11759602C211yesmaximal, characteristic, complemented
12#10621662—31nomaximal
13#10621573—31nomaximal
14C22×C3⋊D21C2212yescomplemented
15C22×C3×D21C2212yescomplemented
16C2×D3×D21C2212yescomplemented
17C32×C7×D4C2211yescharacteristic, complemented
18#92726C2212yescomplemented
19C4⋊(C3⋊D21)C2211yescharacteristic, complemented
20C4×C3⋊D21C2211yescharacteristic, complemented
21C3×#4268C2212yescomplemented
22C3×D84C2211yescharacteristic, complemented
23C3×C4×D21C2211yescharacteristic, complemented
24C3×C7×#67C2212yescomplemented
25C2×C3×C7×Dic3C2212yescomplemented
26C3×C4×D3×C7C2211yescharacteristic, complemented
27C3×C7×Dic6C2211yescharacteristic, complemented
28#92691C2211yescharacteristic, complemented
29#92690C2211yescharacteristic, complemented
30#92689C2211yescharacteristic, complemented
31#92689C2212yescomplemented
32#92689C2212yescomplemented
33#92689C2212yescomplemented
34#92688C2211yescharacteristic, complemented
35#92687C2212yescomplemented
36Dic3×D21C2212yescomplemented
37C3×C7×#41—31no—
38#66732—31no—
39#66732—31no—
40D4×D21—32no—
41D4×D21—31no—
42D4×D21—31no—
43D4×D21—32no—
44D4×D21—61no—
45D4×D21D311yescharacteristic, complemented
46#66728—32no—
47#66728—31no—
48C2×D84—32no—
49C2×D84—31no—
50C7×#237D311yescharacteristic, complemented
51C3×D4×D7—31no—
52#66694—32no—
53D7×#67—32no—
54C2×#4246—32no—
55#66683—32no—
56D3×D28—31no—
57#66677—31no—
58#66675—31no—
59#66670—31no—
60#64314—71nomaximal, Hall
61C22×C32×C7C2312yes—
62C2×C3⋊D21—42no—
63C2×C3⋊D21C2312yes—
64C2×C3⋊D21C2311yescharacteristic
65C2×C3×D21—42no—
66C2×C3×D21C2312yes—
67C2×C3×D21C2311yescharacteristic
68C2×C3×D3×C7C2311yescharacteristic
69D3×D21—42no—
70C32×C4×C7C2311yescharacteristic
71C3⋊Dic21C2311yescharacteristic
72C3×Dic21C2311yescharacteristic
73C3×C7×Dic3C2312yes—
74C3×C7×Dic3C2311yescharacteristic
75#6553—91noHall
76C22×D21—32no—
77C22×D21—32no—
78C22×D21—32no—
79C22×D21—62no—
80C22×D21C2×D312yescomplemented
81C22×C3×D7—32no—
82C2×D3×D7—32no—
83C3×C7×Q8—31no—
84C3×C7×D4—21no—
85C3×C7×D4—32no—
86C3×C7×D4C2×D311yescharacteristic, complemented
87C3×C7×D4C2×D311yescharacteristic, complemented
88C2×C3×C4×C7—32no—
89C2×C3×C4×C7—31no—
90#4268—32no—
91#4268—32no—
92#4268—31no—
93#4268—31no—
94#4268—62no—
95#4268C2×D312yescomplemented
96D84—32no—
97D84—31no—
98D84—32no—
99D84—32no—
100D84—31no—
101D84—31no—
102D84—61no—
103D84C2×D311yescharacteristic, complemented
104C4×D21—32no—
105C4×D21—31no—
106C4×D21—31no—
107C4×D21—32no—
108C4×D21—61no—
109C4×D21C2×D311yescharacteristic, complemented
110Dic42—31no—
111C7×#67C2×D312yescomplemented
112C2×C7×Dic3C2×D312yescomplemented
113C4×D3×C7C2×D311yescharacteristic, complemented
114C7×Dic6C2×D311yescharacteristic, complemented
115C3×C22⋊D7—32no—
116C3×D28—31no—
117C3×C4×D7—31no—
118#4248—31no—
119#4247—31no—
120#4246—32no—
121#4246—31no—
122#4246—32no—
123#4246—32no—
124#4245—31no—
125D21⋊C4—32no—
126Dic3×D7—32no—
127D4×C3⋊D3—71no—
128C3×#237D711yescharacteristic, complemented
129C3×D3×D4—71no—
130#3811—72no—
131D3×#67—72no—
132C2×#626—72no—
133#3804—72no—
134D3×D12—71no—
135#3799—71no—
136#3798—71no—
137#3796—71no—
138C2×C32×C7—22no—
139C2×C32×C7C2411yescharacteristic
140C3⋊D21—21no—
141C3⋊D21—22no—
142C3×D21—22no—
143C3×D21—21no—
144C3×D3×C7—21no—
145C7×#41—31no—
146#1161—91no—
147#1161—91no—
148D4×D7—31no—
149D4×D7—92no—
150D4×D7—92no—
151D4×D7—91no—
152#1157—91no—
153#1157—92no—
154C2×D28—91no—
155C2×D28—92no—
156#1009—211no—
157#1004—211no—
158C22×C3×C7—22no—
159C22×C3×C7—31no—
160C22×C3×C7C22×D312yes—
161C22×C3×C7C22×D312yes—
162C2×D21—32no—
163C2×D21—31no—
164C2×D21—32no—
165C2×D21—31no—
166C2×D21—42no—
167C2×D21—61no—
168C2×D21—62no—
169C2×D21—122no—
170C2×D21—122no—
171C2×D21—122no—
172C2×D21—122no—
173C2×D21—122no—
174C2×D21C22×D312yes—
175C2×D21C22×D311yescharacteristic
176C2×D3×C7C22×D311yescharacteristic
177C2×C3×D7—32no—
178C2×C3×D7—31no—
179C2×C3×D7—122no—
180D3×D7—122no—
181C3×C4×C7—21no—
182C3×C4×C7—31no—
183C3×C4×C7—32no—
184C3×C4×C7C22×D311yescharacteristic
185C3×C4×C7C22×D311yescharacteristic
186Dic21—31no—
187Dic21—31no—
188Dic21—61no—
189Dic21C22×D311yescharacteristic
190C3×Dic7—31no—
191C7×Dic3C22×D312yes—
192C7×Dic3C22×D311yescharacteristic
193C22×C3⋊D3—72no—
194C22×C3×D3—72no—
195C2×D32—72no—
196C32×D4C2×D711yescharacteristic, complemented
197#638—72no—
198C4⋊(C3⋊D3)—71no—
199C4×C3⋊D3—71no—
200C3×#67—72no—
201C3×#67C2×D712yescomplemented
202C2×C3×Dic3C2×D712yescomplemented
203C3×D12—71no—
204C3×C4×D3—71no—
205C3×C4×D3C2×D711yescharacteristic, complemented
206C3×Dic6C2×D711yescharacteristic, complemented
207#627—71no—
208#626—71no—
209#626—72no—
210#626—71no—
211#626—72no—
212#626—72no—
213#625—71no—
214#624—72no—
215D3×Dic3—72no—
216C32×C7H(5,𝔽2)11yescharacteristic, complemented, Hall
217C22×D7—32no—
218C22×D7—92no—
219C22×D7—92no—
220C7×Q8—31no—
221C7×D4—32no—
222C7×D4D3211yescharacteristic, complemented
223C2×C4×C7—32no—
224C2×C4×C7—31no—
225C22⋊D7—32no—
226C22⋊D7—91no—
227C22⋊D7—91no—
228C22⋊D7—92no—
229D28—31no—
230D28—91no—
231D28—92no—
232D28—92no—
233D28—92no—
234D28—91no—
235C4×D7—31no—
236C4×D7—92no—
237C4×D7—92no—
238C4×D7—91no—
239Dic14—91no—
240C3×#41—31no—
241C2×C3×D4—211no—
242C2×#67—212no—
243C2×#67—212no—
244#239—211no—
245#239—211no—
246#237—212no—
247#237—211no—
248#237D2111yescharacteristic, complemented
249D3×D4—71no—
250D3×D4—212no—
251D3×D4—212no—
252D3×D4—211no—
253D3×D4—211no—
254D3×D4—212no—
255D3×D4—211no—
256D3×D4—421no—
257#235—211no—
258#235—212no—
259#235—211no—
260#235—211no—
261C2×D12—211no—
262C2×D12—212no—
263C2×C3×C7—22no—
264C2×C3×C7—22no—
265C2×C3×C7—21no—
266C2×C3×C7—42no—
267C2×C3×C7—61no—
268C2×C3×C7C23×D311yescharacteristic
269C2×C3×C7C23×D311yescharacteristic
270D21—21no—
271D21—22no—
272D21—61no—
273D21—61no—
274D21—61no—
275D21—62no—
276D21—62no—
277D21—61no—
278D21—122no—
279C3×D7—62no—
280C3×D7—61no—
281D3×C7—21no—
282C22×C32C22×D712yes—
283C2×C3⋊D3—72no—
284C2×C3⋊D3—71no—
285C2×C3⋊D3—282no—
286C2×C3×D3—72no—
287C2×C3×D3—71no—
288C2×C3×D3—282no—
289C2×C3×D3C22×D711yescharacteristic
290D32—282no—
291C32×C4C22×D711yescharacteristic
292C3⋊Dic3—71no—
293C3×Dic3—71no—
294C3×Dic3C22×D712yes—
295C3×Dic3C22×D711yescharacteristic
296H(5,𝔽2)—631noSylow
297C22×C7—31no—
298C22×C7C2×D3212yes—
299C2×D7—31no—
300C2×D7—32no—
301C2×D7—91no—
302C2×D7—92no—
303C2×D7—122no—
304C2×D7—362no—
305C2×D7—362no—
306C4×C7—31no—
307C4×C7—32no—
308C4×C7C2×D3211yescharacteristic
309Dic7—31no—
310Dic7—91no—
311C23×C3—212no—
312C22×D3—72no—
313C22×D3—212no—
314C22×D3—212no—
315C22×D3—212no—
316C22×D3—212no—
317C22×D3—422no—
318C3×Q8—31no—
319C3×D4—21no—
320C3×D4—32no—
321C3×D4—212no—
322C3×D4—211no—
323C3×D4D3×D711yescharacteristic, complemented
324C3×D4C2×D2111yescharacteristic, complemented
325C2×C3×C4—32no—
326C2×C3×C4—31no—
327C2×C3×C4—211no—
328#67—72no—
329#67—211no—
330#67—211no—
331#67—212no—
332#67—212no—
333#67—211no—
334#67—211no—
335#67—212no—
336#67—212no—
337#67—212no—
338#67—422no—
339#67C2×D2112yescomplemented
340C2×Dic3—212no—
341C2×Dic3C2×D2112yescomplemented
342D12—71no—
343D12—211no—
344D12—212no—
345D12—212no—
346D12—212no—
347D12—211no—
348D12—211no—
349D12—211no—
350D12—421no—
351C4×D3—71no—
352C4×D3—212no—
353C4×D3—212no—
354C4×D3—211no—
355C4×D3—211no—
356C4×D3—212no—
357C4×D3—421no—
358C4×D3C2×D2111yescharacteristic, complemented
359Dic6—211no—
360Dic6—211no—
361Dic6C2×D2111yescharacteristic, complemented
362C3×C7—21no—
363C3×C7#100411yescharacteristic, complemented
364C3×C7#100911yescharacteristic, complemented
365C2×C32—22no—
366C2×C32C23×D711yescharacteristic
367C3⋊D3—141no—
368C3⋊D3—142no—
369C3×D3—21no—
370C3×D3—142no—
371C3×D3—141no—
372#41—31no—
373#41—631no—
374#41—632no—
375#41—631no—
376#41—631no—
377C2×D4—211no—
378C2×D4—631no—
379C2×D4—632no—
380C2×D4—632no—
381C2×D4—632no—
382C2×D4—631no—
383C2×C7—22no—
384C2×C7—61no—
385C2×C7C22×D3211yescharacteristic
386D7—61no—
387D7—62no—
388D7—181no—
389D7—182no—
390C22×C3—22no—
391C22×C3—31no—
392C22×C3—212no—
393C22×C3—211no—
394C22×C3—842no—
395C22×C3C2×D3×D712yes—
396C22×C3C22×D2112yes—
397C2×D3—71no—
398C2×D3—72no—
399C2×D3—212no—
400C2×D3—211no—
401C2×D3—211no—
402C2×D3—212no—
403C2×D3—282no—
404C2×D3—421no—
405C2×D3—422no—
406C2×D3—842no—
407C2×D3—842no—
408C2×D3—842no—
409C2×D3—842no—
410C2×D3—842no—
411C2×D3—842no—
412C2×D3C22×D2111yescharacteristic
413C3×C4—21no—
414C3×C4—31no—
415C3×C4—32no—
416C3×C4—211no—
417C3×C4C2×D3×D711yescharacteristic
418C3×C4C22×D2111yescharacteristic
419Dic3—71no—
420Dic3—211no—
421Dic3—211no—
422Dic3—421no—
423Dic3C22×D2111yescharacteristic
424Dic3C22×D2112yes—
425C32#655311yescharacteristic, complemented, Sylow
426C23—212no—
427C23—632no—
428C23—632no—
429Q8—31no—
430Q8—631no—
431D4—32no—
432D4—212no—
433D4—211no—
434D4—631no—
435D4—632no—
436D4—632no—
437D4—632no—
438D4—631no—
439D4—631no—
440D4—632no—
441D4—631no—
442D4D3×D2111yescharacteristic, complemented
443C2×C4—32no—
444C2×C4—31no—
445C2×C4—211no—
446C2×C4—632no—
447C2×C4—632no—
448C2×C4—631no—
449C7#6431411yescharacteristic, complemented, Sylow
450C2×C3—22no—
451C2×C3—22no—
452C2×C3—21no—
453C2×C3—42no—
454C2×C3—61no—
455C2×C3—422no—
456C2×C3—421no—
457C2×C3C22×D3×D711yescharacteristic
458C2×C3C23×D2111yescharacteristic
459D3—21no—
460D3—141no—
461D3—142no—
462D3—421no—
463D3—421no—
464D3—421no—
465D3—422no—
466D3—421no—
467D3—422no—
468D3—842no—
469C22—31no—
470C22—211no—
471C22—212no—
472C22—631no—
473C22—632no—
474C22—842no—
475C22—2522no—
476C22—2522no—
477C22C2×D3×D2112yes—
478C4—31no—
479C4—32no—
480C4—211no—
481C4—631no—
482C4C2×D3×D2111yescharacteristic
483C3—21no—
484C3#1062157311yescharacteristic, complemented
485C3#1062166211yescharacteristic, complemented
486C2—22no—
487C2—61no—
488C2—421no—
489C2—422no—
490C2—1261no—
491C2—1262no—
492C2C22×D3×D2111yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×D21
commutator, socle, cosocleC2×C32×C7C24
FittingC32×C7×D4C22
hypercenterD4D3×D21
hypocenterC32×C7H(5,𝔽2)
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. D4×C3⋊D21C2semidirect
2. C3×D4×D21C2semidirect
3. C3×C7×#237C2semidirect
4. #11759626C2semidirect
5. #67×D21C2semidirect
6. C2×#92689C2semidirect
7. #11759615C2semidirect
8. D3×D84C2semidirect
9. #11759609C2semidirect
10. #11759607C2semidirect
11. #11759602C2semidirect
14. C22×C3⋊D21C22semidirect
15. C22×C3×D21C22semidirect
16. C2×D3×D21C22semidirect
17. C32×C7×D4C22semidirect
18. #92726C22semidirect
19. C4⋊(C3⋊D21)C22semidirect
20. C4×C3⋊D21C22semidirect
21. C3×#4268C22semidirect
22. C3×D84C22semidirect
23. C3×C4×D21C22semidirect
24. C3×C7×#67C22semidirect
25. C2×C3×C7×Dic3C22semidirect
26. C3×C4×D3×C7C22semidirect
27. C3×C7×Dic6C22semidirect
28. #92691C22semidirect
29. #92690C22semidirect
30. #92689C22semidirect
31. #92689C22semidirect
32. #92689C22semidirect
33. #92689C22semidirect
34. #92688C22semidirect
35. #92687C22semidirect
36. Dic3×D21C22semidirect
45. D4×D21D3semidirect
50. C7×#237D3semidirect
80. C22×D21C2×D3semidirect
86. C3×C7×D4C2×D3semidirect
87. C3×C7×D4C2×D3semidirect
95. #4268C2×D3semidirect
103. D84C2×D3semidirect
109. C4×D21C2×D3semidirect
111. C7×#67C2×D3semidirect
112. C2×C7×Dic3C2×D3semidirect
113. C4×D3×C7C2×D3semidirect
114. C7×Dic6C2×D3semidirect
128. C3×#237D7semidirect
196. C32×D4C2×D7semidirect
201. C3×#67C2×D7semidirect
202. C2×C3×Dic3C2×D7semidirect
205. C3×C4×D3C2×D7semidirect
206. C3×Dic6C2×D7semidirect
216. C32×C7H(5,𝔽2)semidirect
222. C7×D4D32semidirect
248. #237D21semidirect
323. C3×D4D3×D7semidirect
324. C3×D4C2×D21semidirect
339. #67C2×D21semidirect
341. C2×Dic3C2×D21semidirect
358. C4×D3C2×D21semidirect
361. Dic6C2×D21semidirect
363. C3×C7#1004semidirect
364. C3×C7#1009semidirect
425. C32#6553semidirect
442. D4D3×D21semidirect
449. C7#64314semidirect
484. C3#10621573semidirect
485. C3#10621662semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊D212
C3×D4×D212
C3×C7×#2372
#117596262
#67×D212
C2×#926892
#117596152
D3×D842
#117596092
#117596072
#117596022
#106216623
#106215733
#643147