Finite Group Databasebeta

← #13059926

Subgroups — #13059926

Order 1344.

660 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66686C211yesmaximal, characteristic, complemented
2C22×D3×Dic7C211yesmaximal, characteristic, complemented
3#10621422C211yesmaximal, characteristic, complemented
4C7×#937C211yesmaximal, characteristic, complemented
5C3×D4×Dic7C211yesmaximal, characteristic, complemented
6#10621138C211yesmaximal, characteristic, complemented
7#10621132C211yesmaximal, characteristic, complemented
8#10621120C211yesmaximal, characteristic, complemented
9#10621117C211yesmaximal, characteristic, complemented
10#10621114C211yesmaximal, characteristic, complemented
11#67×Dic7C211yesmaximal, characteristic, complemented
12#10620998C211yesmaximal, characteristic, complemented
13#10620989C211yesmaximal, characteristic, complemented
14#10620967C211yesmaximal, characteristic, complemented
15#10620960C211yesmaximal, characteristic, complemented
16#90277—31nomaximal, Hall
17C23×D3×C7C2211yescharacteristic
18C2×C3×C7×D4C2211yescharacteristic
19C2×#4268C2211yescharacteristic
20C2×#4268C2211yescharacteristic, complemented
21C22×Dic21C2211yescharacteristic
22C2×C7×#67C2211yescharacteristic
23C2×C7×#67C2211yescharacteristic
24C22×C3×Dic7C2211yescharacteristic
25C22×C3×Dic7C2211yescharacteristic, complemented
26C2×#4248C2211yescharacteristic, complemented
27C2×#4247C2211yescharacteristic
28C2×#4247C2211yescharacteristic
29C2×D21⋊C4C2211yescharacteristic, complemented
30#66686—24no—
31C2×D3×Dic7—24no—
32C2×D3×Dic7C2211yescharacteristic
33C2×D3×Dic7C2211yescharacteristic
34C2×D3×Dic7C2211yescharacteristic
35C2×D3×Dic7C2211yescharacteristic
36#66636C2211yescharacteristic, complemented
37#66631C2211yescharacteristic, complemented
38#66629C2211yescharacteristic, complemented
39C7×#217C2211yescharacteristic
40C7×(C2×D3)⋊C4C2211yescharacteristic
41C7×(C2×D3)⋊C4C2211yescharacteristic
42C3×C22⋊Dic7C2211yescharacteristic
43C3×C22⋊Dic7C2211yescharacteristic, complemented
44C3×C4⋊Dic7C2211yescharacteristic, complemented
45C3×C4×Dic7C2211yescharacteristic, complemented
46Dic3⋊Dic7C2211yescharacteristic, complemented
47#66577C2211yescharacteristic, complemented
48#66577C2211yescharacteristic, complemented
49Dic7⋊Dic3C2211yescharacteristic, complemented
50#66575C2211yescharacteristic, complemented
51(C2×D3)⋊Dic7C2211yescharacteristic
52(C2×D3)⋊Dic7C2211yescharacteristic
53Dic3×Dic7C2211yescharacteristic, complemented
54C23×Dic7—31no—
55C2×#1159—31no—
56C7×#120—31no—
57#6516—31no—
58#6502—31no—
59#6502—31no—
60#6498—31no—
61D4×Dic7—31no—
62D4×Dic7D311yescharacteristic, complemented
63#6449—31no—
64(C22⋊D7)⋊C4—31no—
65(C22⋊D7)⋊C4—31no—
66#6441—31no—
67#6441—31no—
68#6440—31no—
69#5547—71nomaximal, Hall
70C23×C3×C7C2311yescharacteristic
71C23×C3×C7C2311yescharacteristic
72C22×D21C2311yescharacteristic
73C22×D3×C7—22no—
74C22×D3×C7—22no—
75C22×D3×C7—22no—
76C22×D3×C7C2311yescharacteristic
77C22×D3×C7C2311yescharacteristic
78C3×C7×D4—22no—
79C2×C3×C4×C7C2311yescharacteristic
80#4268—22no—
81#4268—22no—
82C2×Dic21—22no—
83C2×Dic21C2311yescharacteristic
84C2×Dic21C2311yescharacteristic
85C2×Dic21C2311yescharacteristic
86C7×#67—22no—
87C7×#67—22no—
88C2×C7×Dic3C2311yescharacteristic
89C2×C7×Dic3C2311yescharacteristic
90C2×C3×Dic7—22no—
91C2×C3×Dic7—22no—
92C2×C3×Dic7C2311yescharacteristic
93C2×C3×Dic7C2311yescharacteristic
94C2×C3×Dic7C2311yescharacteristic
95C2×C3×Dic7C2311yescharacteristic
96#4248—22no—
97#4247—22no—
98#4247—22no—
99D21⋊C4—22no—
100D3×Dic7—22no—
101D3×Dic7—22no—
102D3×Dic7—22no—
103D3×Dic7D414yescomplemented
104C24×C7—31no—
105C2×C7×D4—31no—
106C2×C7×D4—31no—
107C2×C7×D4C2×D311yescharacteristic
108C2×C22⋊D7—31no—
109C2×C22⋊D7—31no—
110C2×C22⋊D7—31no—
111C2×C22⋊D7—31no—
112C22×Dic7—31no—
113C22×Dic7—31no—
114C22×Dic7—31no—
115C22×Dic7—31no—
116C22×Dic7—31no—
117C22×Dic7—64no—
118C22×Dic7C2×D311yescharacteristic
119C22×Dic7C2×D311yescharacteristic, complemented
120#1159—64no—
121C2×C4×D7—31no—
122C2×Dic14—31no—
123C7×C22⋊C4—31no—
124C7×C22⋊C4—31no—
125C7×C22⋊C4—31no—
126C22⋊Dic7—31no—
127C22⋊Dic7—31no—
128C22⋊Dic7—31no—
129C22⋊Dic7C2×D311yescharacteristic
130C22⋊Dic7C2×D311yescharacteristic, complemented
131(C2×D7)⋊C4—31no—
132(C2×D7)⋊C4—31no—
133C4⋊Dic7—31no—
134C4⋊Dic7C2×D311yescharacteristic, complemented
135Dic7⋊C4—31no—
136Dic7⋊C4—31no—
137Dic7⋊C4—31no—
138Dic7⋊C4—31no—
139C4×Dic7—31no—
140C4×Dic7C2×D311yescharacteristic, complemented
141C2×#235—71no—
142C22×C4×D3—71no—
143C3×C4×D4—71no—
144#939—71no—
145#937D711yescharacteristic, complemented
146C4×#67—71no—
147C4×#67—71no—
148#896—71no—
149#894—71no—
150#884—71no—
151#883—71no—
152#882—71no—
153#878—71no—
154C4×D12—71no—
155#872—71no—
156C22×C3×C7—21no—
157C22×C3×C7—21no—
158C22×C3×C7—21no—
159C22×C3×C7—21no—
160C22×C3×C7—21no—
161C22×C3×C7#4112yescomplemented
162C22×C3×C7C2411yescharacteristic
163C2×D21—21no—
164C2×D21—21no—
165C2×D21—21no—
166C2×D3×C7—28no—
167C2×D3×C7—21no—
168C2×D3×C7—21no—
169C2×D3×C7—21no—
170C2×D3×C7—21no—
171C2×D3×C7C2×D412yes—
172C2×D3×C7#4112yescomplemented
173C3×C4×C7—21no—
174Dic21—21no—
175Dic21—21no—
176Dic21C2×D412yes—
177C3×Dic7—21no—
178C3×Dic7—21no—
179C3×Dic7—21no—
180C3×Dic7C2×D412yes—
181C7×Dic3—21no—
182C7×Dic3—21no—
183#525—211noSylow
184C23×C7—31no—
185C23×C7—31no—
186C23×C7—62no—
187C23×C7—62no—
188C23×C7—62no—
189C23×C7C22×D311yescharacteristic
190C23×C7C22×D311yescharacteristic
191C22×D7—31no—
192C7×D4—22no—
193C7×D4—62no—
194C7×D4—62no—
195C2×C4×C7—31no—
196C2×C4×C7—31no—
197C2×C4×C7C22×D311yescharacteristic
198C22⋊D7—62no—
199C22⋊D7—62no—
200C22⋊D7—62no—
201C22⋊D7—62no—
202C2×Dic7—22no—
203C2×Dic7—22no—
204C2×Dic7—31no—
205C2×Dic7—34no—
206C2×Dic7—31no—
207C2×Dic7—31no—
208C2×Dic7—62no—
209C2×Dic7—62no—
210C2×Dic7—62no—
211C2×Dic7—62no—
212C2×Dic7C22×D311yescharacteristic
213C2×Dic7C22×D311yescharacteristic
214C2×Dic7C22×D311yescharacteristic
215C2×Dic7C22×D311yescharacteristic
216C4×D7—62no—
217Dic14—62no—
218C23×D3C2×D711yescharacteristic
219C2×C3×D4C2×D711yescharacteristic
220C22×C3×C4—71no—
221C22×C3×C4—71no—
222C2×#67—71no—
223C2×#67—71no—
224C2×#67C2×D711yescharacteristic
225C2×#67C2×D711yescharacteristic
226C22×Dic3—71no—
227#235—144no—
228C2×D12—71no—
229C2×D12—71no—
230C2×C4×D3—71no—
231C2×C4×D3—71no—
232C2×C4×D3—71no—
233C2×C4×D3—71no—
234C2×C4×D3—71no—
235C2×C4×D3—144no—
236C2×Dic6—71no—
237C3×C4⋊C4—71no—
238C3×C22⋊C4—71no—
239C3×C22⋊C4—71no—
240C3×C42—71no—
241#217—71no—
242#217C2×D711yescharacteristic
243(C2×D3)⋊C4—71no—
244(C2×D3)⋊C4—71no—
245(C2×D3)⋊C4—71no—
246(C2×D3)⋊C4—71no—
247(C2×D3)⋊C4C2×D711yescharacteristic
248(C2×D3)⋊C4C2×D711yescharacteristic
249C4⋊Dic3—71no—
250Dic3⋊C4—71no—
251Dic3⋊C4—71no—
252Dic3⋊C4—71no—
253Dic3⋊C4—71no—
254C4×Dic3—71no—
255C2×C3×C7—22no—
256C2×C3×C7—41no—
257C2×C3×C7C22×D411yescharacteristic
258C2×C3×C7C2×#4111yescharacteristic
259C2×C3×C7C2×#4111yescharacteristic
260D21—41no—
261D3×C7—22no—
262D3×C7—22no—
263C2×#41—211no—
264C23×C4—211no—
265#123—211no—
266#123—211no—
267C22⋊Q8—211no—
268C22⋊Q8—211no—
269#121—211no—
270#121—211no—
271#120—31no—
272#120—211no—
273C4×D4—71no—
274C4×D4—211no—
275C4×D4—211no—
276C4×D4—211no—
277#117—211no—
278C22×C7—21no—
279C22×C7—21no—
280C22×C7—21no—
281C22×C7—21no—
282C22×C7—21no—
283C22×C7—32no—
284C22×C7—32no—
285C22×C7—68no—
286C22×C7—61no—
287C22×C7—61no—
288C22×C7—61no—
289C22×C7—61no—
290C22×C7#23512yescomplemented
291C22×C7C23×D311yescharacteristic
292C2×D7—61no—
293C2×D7—61no—
294C2×D7—61no—
295C4×C7—21no—
296C4×C7—61no—
297C4×C7—61no—
298Dic7—21no—
299Dic7—21no—
300Dic7—21no—
301Dic7—32no—
302Dic7—61no—
303Dic7—61no—
304Dic7D3×D412yes—
305C23×C3C22×D711yescharacteristic
306C23×C3C22×D711yescharacteristic
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—22no—
310C22×D3—71no—
311C22×D3C22×D711yescharacteristic
312C22×D3C22×D711yescharacteristic
313C3×D4—22no—
314C2×C3×C4—71no—
315C2×C3×C4—71no—
316C2×C3×C4—71no—
317C2×C3×C4—71no—
318C2×C3×C4—142no—
319C2×C3×C4—142no—
320C2×C3×C4C22×D711yescharacteristic
321#67—22no—
322#67—22no—
323#67—142no—
324#67—142no—
325C2×Dic3—71no—
326C2×Dic3—71no—
327C2×Dic3—71no—
328C2×Dic3—142no—
329C2×Dic3C22×D711yescharacteristic
330C2×Dic3C22×D711yescharacteristic
331D12—142no—
332D12—142no—
333C4×D3—74no—
334C4×D3—142no—
335C4×D3—142no—
336C4×D3—142no—
337C4×D3—142no—
338Dic6—142no—
339C3×C7#52511yescharacteristic, complemented, Hall
340C24—31no—
341#41—424no—
342C2×Q8—211no—
343C2×D4—31no—
344C2×D4—31no—
345C2×D4—211no—
346C2×D4—211no—
347C2×D4—211no—
348C2×D4—211no—
349C2×D4D3×D711yescharacteristic
350C22×C4—71no—
351C22×C4—71no—
352C22×C4—211no—
353C22×C4—211no—
354C22×C4—211no—
355C22×C4—211no—
356C22×C4—211no—
357C22×C4—211no—
358C22×C4—424no—
359C4⋊C4—71no—
360C4⋊C4—211no—
361C4⋊C4—211no—
362C4⋊C4—211no—
363C4⋊C4—211no—
364C4⋊C4—211no—
365C22⋊C4—31no—
366C22⋊C4—31no—
367C22⋊C4—31no—
368C22⋊C4—71no—
369C22⋊C4—71no—
370C22⋊C4—211no—
371C22⋊C4—211no—
372C22⋊C4—211no—
373C22⋊C4—211no—
374C22⋊C4—211no—
375C42—71no—
376C42—211no—
377C2×C7—22no—
378C2×C7—41no—
379C2×C7—62no—
380C2×C7—62no—
381C2×C7C2×#23511yescharacteristic
382C2×C7C2×D3×D411yescharacteristic
383C2×C7D3×#4111yescharacteristic
384D7—121no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3#115912yescomplemented
391C22×C3C23×D711yescharacteristic
392C2×D3—28no—
393C2×D3—21no—
394C2×D3—21no—
395C2×D3—21no—
396C2×D3—21no—
397C2×D3—141no—
398C2×D3—141no—
399C2×D3—141no—
400C2×D3D4×D712yes—
401C2×D3#115912yescomplemented
402C3×C4—21no—
403C3×C4—72no—
404C3×C4—141no—
405C3×C4—141no—
406C3×C4—141no—
407Dic3—21no—
408Dic3—21no—
409Dic3—72no—
410Dic3—141no—
411Dic3—141no—
412C23—31no—
413C23—31no—
414C23—62no—
415C23—62no—
416C23—62no—
417C23—211no—
418C23C2×D3×D711yescharacteristic
419C23C2×D3×D711yescharacteristic
420Q8—422no—
421D4—22no—
422D4—62no—
423D4—62no—
424D4—422no—
425D4—422no—
426D4—422no—
427D4—422no—
428C2×C4—31no—
429C2×C4—31no—
430C2×C4—71no—
431C2×C4—71no—
432C2×C4—71no—
433C2×C4—71no—
434C2×C4—142no—
435C2×C4—142no—
436C2×C4—214no—
437C2×C4—211no—
438C2×C4—211no—
439C2×C4—211no—
440C2×C4—422no—
441C2×C4—422no—
442C2×C4—422no—
443C2×C4—422no—
444C2×C4—422no—
445C2×C4C2×D3×D711yescharacteristic
446C7#554711yescharacteristic, complemented, Sylow
447C2×C3—22no—
448C2×C3—41no—
449C2×C3C2×D4×D711yescharacteristic
450C2×C3C2×#115911yescharacteristic
451C2×C3C2×#115911yescharacteristic
452D3—22no—
453D3—22no—
454D3—281no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—32no—
461C22—32no—
462C22—68no—
463C22—61no—
464C22—61no—
465C22—61no—
466C22—61no—
467C22—421no—
468C22—421no—
469C22—421no—
470C22#6668612yescomplemented
471C22C22×D3×D711yescharacteristic
472C4—21no—
473C4—61no—
474C4—61no—
475C4—72no—
476C4—141no—
477C4—141no—
478C4—141no—
479C4—212no—
480C4—421no—
481C4—421no—
482C3#9027711yescharacteristic, complemented, Sylow
483C2—22no—
484C2—41no—
485C2—62no—
486C2—62no—
487C2—841no—
488C2D3×D4×D711yescharacteristic
489C2D3×#115911yescharacteristic
490C2C2×#6668611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#525
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#66686C2semidirect
2. C22×D3×Dic7C2semidirect
3. #10621422C2semidirect
4. C7×#937C2semidirect
5. C3×D4×Dic7C2semidirect
6. #10621138C2semidirect
7. #10621132C2semidirect
8. #10621120C2semidirect
9. #10621117C2semidirect
10. #10621114C2semidirect
11. #67×Dic7C2semidirect
12. #10620998C2semidirect
13. #10620989C2semidirect
14. #10620967C2semidirect
15. #10620960C2semidirect
20. C2×#4268C22semidirect
25. C22×C3×Dic7C22semidirect
26. C2×#4248C22semidirect
29. C2×D21⋊C4C22semidirect
36. #66636C22semidirect
37. #66631C22semidirect
38. #66629C22semidirect
43. C3×C22⋊Dic7C22semidirect
44. C3×C4⋊Dic7C22semidirect
45. C3×C4×Dic7C22semidirect
46. Dic3⋊Dic7C22semidirect
47. #66577C22semidirect
48. #66577C22semidirect
49. Dic7⋊Dic3C22semidirect
50. #66575C22semidirect
53. Dic3×Dic7C22semidirect
62. D4×Dic7D3semidirect
103. D3×Dic7D4semidirect
119. C22×Dic7C2×D3semidirect
130. C22⋊Dic7C2×D3semidirect
134. C4⋊Dic7C2×D3semidirect
140. C4×Dic7C2×D3semidirect
145. #937D7semidirect
161. C22×C3×C7#41semidirect
172. C2×D3×C7#41semidirect
290. C22×C7#235semidirect
339. C3×C7#525semidirect
390. C22×C3#1159semidirect
401. C2×D3#1159semidirect
446. C7#5547semidirect
470. C22#66686semidirect
482. C3#90277semidirect

Maximal subgroups

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

subgroupindex
C2×#666862
C22×D3×Dic72
#106214222
C7×#9372
C3×D4×Dic72
#106211382
#106211322
#106211202
#106211172
#106211142
#67×Dic72
#106209982
#106209892
#106209672
#106209602
#902773
#55477