Finite Group Databasebeta

← #30612782

Subgroups — #30612782

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462542C211yesmaximal, characteristic, complemented
2C2×#16840797C211yesmaximal, characteristic, complemented
3#21461469C212yesmaximal, complemented
4#21461467C212yesmaximal, complemented
5#21461441C212yesmaximal, complemented
6#21461440C212yesmaximal, complemented
7#21461386C211yesmaximal, characteristic, complemented
8#21461144C211yesmaximal, characteristic, complemented
9#21461124C211yesmaximal, characteristic, complemented
10#21461034C211yesmaximal, characteristic, complemented
11#21455814C211yesmaximal, characteristic
12C2×#14519864—21no—
13C22×#11758217—21no—
14#16840797—21no—
15#16840797—21no—
16#16840797C2212yes—
17#16840797C2212yes—
18#16840797C2211yescharacteristic, complemented
19#16840796C2211yescharacteristic, complemented
20#16840795—21no—
21#16840790C2211yescharacteristic, complemented
22#16840789—21no—
23D4×#92420C2211yescharacteristic, complemented
24#16840779C2211yescharacteristic
25#16840765C2211yescharacteristic
26C2×C4⋊#92420C2211yescharacteristic
27#16840735—21no—
28C2×C22⋊#92420—21no—
29C2×C4×#92420C2211yescharacteristic
30#16840624C2212yescomplemented
31#16840623C2212yescomplemented
32#16840610C2212yescomplemented
33D4⋊#92420C2212yes—
34D4⋊#92420C2212yes—
35D4⋊#92420C2212yescomplemented
36#92420⋊C8C2212yescomplemented
37#16840603C2212yescomplemented
38#16840307C2211yescharacteristic
39C2×D4×Dic61C2211yescharacteristic
40#16840076C411yescharacteristic
41#16840073C411yescharacteristic
42#16840040C412yes—
43#16839955C412yes—
44#16839896C411yescharacteristic
45D8×Dic61C2212yes—
46#16839842C2212yes—
47#16839812C2211yescharacteristic
48C61×#447C411yescharacteristic
49#14519864—44no—
50C2×#11758213—21no—
51C2×#11758217—21no—
52C2×#11758217—21no—
53C2×#11758217—22no—
54C2×#11758217—22no—
55C2×#11758217—44no—
56C2×D61⋊C8—21no—
57C22×#92420—21no—
58C22×#92420—21no—
59C22×#92420D411yescharacteristic
60C22×#92420D411yescharacteristic
61#14519844—21no—
62#14519844—21no—
63C4⋊#92420—21no—
64C4⋊#92420C2312yes—
65C4⋊#92420C2311yescharacteristic
66C4⋊#92420C2311yescharacteristic
67#14519834—21no—
68#14519834—21no—
69C22⋊#92420—21no—
70C22⋊#92420—22no—
71C22⋊#92420—22no—
72C22⋊#92420—21no—
73C22⋊#92420—21no—
74C4×#92420D411yescharacteristic
75C4×#92420D411yescharacteristic
76C4×#92420C2312yes—
77C4×#92420C2311yescharacteristic
78C4×#92420C2311yescharacteristic
79C2×#11758224—21no—
80C23×Dic61—21no—
81C2×#11758199C2×C412yes—
82#14519776C2×C411yescharacteristic
83#14519773C2×C411yescharacteristic
84#14519763—21no—
85#14519751C2×C411yescharacteristic
86#14519749—21no—
87D4×Dic61—21no—
88D4×Dic61—21no—
89D4×Dic61C2×C412yes—
90D4×Dic61C2312yes—
91D4×Dic61C2311yescharacteristic
92#14519739C2311yescharacteristic
93#14519737—21no—
94C2×C4⋊Dic61C2×C411yescharacteristic
95C2×C22⋊Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97#14519711C2×C412yes—
98#14519710C2×C412yes—
99#14519706C2×C412yes—
100#14519699C2×C412yes—
101#14519698C2×C412yes—
102#14519698C2×C412yes—
103C22⋊#92418C2×C411yescharacteristic
104C8×Dic61C2312yes—
105C22×D4×C61C2×C411yescharacteristic
106C2×D8×C61C2×C412yes—
107#121×C61C2×C411yescharacteristic
108#102×C61C2×C412yes—
109C22⋊C8×C61C2×C411yescharacteristic
110C24×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—42no—
114C2×D4×C61C22×C412yes—
115C2×D4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C2×C22⋊D61—21no—
118C2×C22⋊D61—21no—
119C2×C22⋊D61—21no—
120C22×Dic61—21no—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—21no—
124C22×Dic61—42no—
125C22×Dic61—42no—
126C22×Dic61C2×D411yescharacteristic
127C22×Dic61#4111yescharacteristic
128#11758224—42no—
129#11758224—42no—
130C2×D244C22×C411yescharacteristic
131C2×C4×D61—21no—
132C2×Dic122—21no—
133#11758217—44no—
134#11758217—44no—
135#11758217—42no—
136#11758217—42no—
137C2×#92420—22no—
138C2×#92420—22no—
139C2×#92420—22no—
140C2×#92420—21no—
141C2×#92420—21no—
142C2×#92420—22no—
143C2×#92420—42no—
144C2×#92420—42no—
145C2×#92420C2×D411yescharacteristic
146C2×#92420C2×D411yescharacteristic
147C2×#92420C2×D411yescharacteristic
148C2×#92420C2×D411yescharacteristic
149#11758213—42no—
150D61⋊C8—42no—
151D8×C61—42no—
152C2×C8×C61C22×C412yes—
153C4⋊C4×C61C22×C411yescharacteristic
154C22⋊C4×C61—21no—
155C22⋊Dic61—22no—
156C22⋊Dic61—22no—
157C22⋊Dic61—21no—
158#11758199—42no—
159#11758198—21no—
160#11758198—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61C22×C411yescharacteristic
163C4⋊Dic61C22×C412yes—
164#11758196—21no—
165C4×Dic61C22×C411yescharacteristic
166C4×Dic61C2×D411yescharacteristic
167C4×Dic61#4111yescharacteristic
168C4×Dic61C2411yescharacteristic
169C2×#92418C22×C412yes—
170C23×C61—22no—
171C23×C61—21no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61C4×D411yescharacteristic
176C22×D61—21no—
177D4×C61—22no—
178D4×C61—22no—
179D4×C61—22no—
180D4×C61—42no—
181D4×C61—42no—
182C2×C4×C61—22no—
183C2×C4×C61—21no—
184C2×C4×C61C4×D411yescharacteristic
185C2×C4×C61C23×C411yescharacteristic
186C22⋊D61—42no—
187C22⋊D61—42no—
188C22⋊D61—42no—
189C2×Dic61—21no—
190C2×Dic61—22no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—41no—
196C2×Dic61—42no—
197C2×Dic61—44no—
198C2×Dic61C4×D411yescharacteristic
199C2×Dic61C4×D411yescharacteristic
200C2×Dic61AGL(1,ℤ/8)12yes—
201C2×Dic61AGL(1,ℤ/8)12yes—
202C2×Dic61C22×D411yescharacteristic
203C2×Dic61C2×#4111yescharacteristic
204D244—22no—
205C4×D61—41no—
206C4×D61—41no—
207Dic122—42no—
208#92420—21no—
209#92420—21no—
210#92420—21no—
211#92420—21no—
212#92420—42no—
213#92420—41no—
214#92420—41no—
215#92420—42no—
216C8×C61—22no—
217#92418—22no—
218#17935—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—42no—
222C22×C61—41no—
223C22×C61—42no—
224C22×C61—41no—
225C22×C61—41no—
226C22×C61—42no—
227C22×C61—84no—
228C22×C61#44212yes—
229C22×C61C2×C4×D411yescharacteristic
230C2×D61—41no—
231C2×D61—41no—
232C2×D61—41no—
233C4×C61—21no—
234C4×C61—41no—
235C4×C61#56812yes—
236Dic61—21no—
237Dic61—21no—
238Dic61—41no—
239Dic61—42no—
240Dic61#44212yes—
241Dic61C2×AGL(1,ℤ/8)11yescharacteristic
242Dic61C2×AGL(1,ℤ/8)11yescharacteristic
243#3100—611no—
244#2931—611no—
245C2×#436—611no—
246#1594—612no—
247#1594—612no—
248#1586—612no—
249#1586—612no—
250#1570—611no—
251#1486—611no—
252#1480—611no—
253#1452—611no—
254C2×C61—22no—
255C2×C61—81no—
256C2×C61—82no—
257C2×C61#293011yescharacteristic
258C2×C61C2×#44211yescharacteristic
259C2×C61#295511yescharacteristic
260D61—81no—
261C2×#131—1221no—
262C22×M16—1221no—
263#532—611no—
264C2×C4×D4—611no—
265#480—611no—
266#480—611no—
267#474—612no—
268#459—612no—
269#447—611no—
270#447#692311yescharacteristic
271C4×D8—612no—
272C4×D8—612no—
273#436—612no—
274#436—611no—
275#436—612no—
276#436—611no—
277#436—1221no—
278#436—1221no—
279#436—1221no—
280C8×D4—611no—
281C8×D4—611no—
282C8×D4—1221no—
283#431—611no—
284#431—611no—
285#423—611no—
286C2×C4⋊C8—611no—
287#408—1221no—
288C2×C22⋊C8—1221no—
289C2×C4×C8—611no—
290#334—612no—
291#334—612no—
292D4⋊C8—612no—
293D4⋊C8—612no—
294D4⋊C8—612no—
295D4⋊C8—612no—
296#322—612no—
297#322—612no—
298C61#1793511yescharacteristic, complemented, Sylow
299C2×#41—1221no—
300C22×D4C2×#692311yescharacteristic
301C23×C4—1221no—
302C2×D8—612no—
303C2×D8C2×#692312yes—
304#131—2444no—
305C2×M16—1222no—
306C2×M16—1222no—
307C2×M16—1221no—
308C2×M16—1221no—
309C2×M16—1221no—
310C2×M16—2444no—
311C22×C8—611no—
312C22×C8—611no—
313C22×C8—1221no—
314C22×C8—1221no—
315C22×C8—1221no—
316C4⋊Q8—611no—
317#127—611no—
318#124—1221no—
319#121—611no—
320#121—1221no—
321#121C2×#692311yescharacteristic
322C4×D4—612no—
323C4×D4—612no—
324C4×D4—611no—
325C4×D4—611no—
326C4×D4—1221no—
327C4×D4—1221no—
328C4×D4—1221no—
329C2×C4⋊C4—611no—
330C2×C22⋊C4—1221no—
331C2×C42—611no—
332#107—612no—
333#107—612no—
334C4⋊C8—612no—
335C4⋊C8—612no—
336C4⋊C8—611no—
337C4⋊C8—611no—
338C4⋊C8—1221no—
339C4⋊C8—1221no—
340C4⋊C8—1221no—
341#102—612no—
342#102—612no—
343#102—612no—
344#102C2×#692312yes—
345C22⋊C8—611no—
346C22⋊C8—1222no—
347C22⋊C8—1222no—
348C22⋊C8—1221no—
349C22⋊C8—1221no—
350C22⋊C8—1221no—
351C22⋊C8—1221no—
352C22⋊C8—1221no—
353C22⋊C8C2×#692311yescharacteristic
354C4×C8—611no—
355C4×C8—611no—
356C4×C8—611no—
357C4×C8—611no—
358C4×C8—612no—
359C4×C8—612no—
360C24—21no—
361#41—2442no—
362#41—2442no—
363C2×Q8—1221no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—42no—
367C2×D4—611no—
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4C22×#692311yescharacteristic
372C2×D4C22×#692312yes—
373C22×C4—611no—
374C22×C4—611no—
375C22×C4—1221no—
376C22×C4—1221no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—1221no—
380C22×C4—2442no—
381C22×C4—2442no—
382C22×C4C22×#692311yescharacteristic
383D8—42no—
384D8—2442no—
385M16—2444no—
386M16—2444no—
387M16—2442no—
388M16—2442no—
389M16—2442no—
390C2×C8—611no—
391C2×C8—611no—
392C2×C8—611no—
393C2×C8—611no—
394C2×C8—612no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—1222no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—2442no—
402C2×C8—2442no—
403C2×C8—2442no—
404C2×C8C22×#692312yes—
405C4⋊C4—612no—
406C4⋊C4—611no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4C22×#692311yescharacteristic
410C22⋊C4—21no—
411C22⋊C4—1221no—
412C22⋊C4—1222no—
413C22⋊C4—1222no—
414C22⋊C4—1221no—
415C22⋊C4—1221no—
416C42—611no—
417C42—611no—
418C42—611no—
419C42—611no—
420C23—21no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—1221no—
426C23D4×#692311yescharacteristic
427Q8—2442no—
428D4—22no—
429D4—22no—
430D4—22no—
431D4—42no—
432D4—42no—
433D4—1222no—
434D4—2442no—
435D4—2442no—
436D4—2442no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—612no—
440C2×C4—612no—
441C2×C4—611no—
442C2×C4—611no—
443C2×C4—611no—
444C2×C4—611no—
445C2×C4—1222no—
446C2×C4—1221no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2444no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2442no—
456C2×C4D4×#692311yescharacteristic
457C2×C4C23×#692311yescharacteristic
458C8—22no—
459C8—1221no—
460C8—1221no—
461C8—1221no—
462C8—1221no—
463C8—1222no—
464C8—2441no—
465C8—2441no—
466C8—2442no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—42no—
471C22—41no—
472C22—42no—
473C22—41no—
474C22—41no—
475C22—42no—
476C22—84no—
477C22—2441no—
478C22—2441no—
479C22—2441no—
480C22#1684080312yes—
481C22C2×D4×#692311yescharacteristic
482C4—21no—
483C4—41no—
484C4—611no—
485C4—611no—
486C4—612no—
487C4—1221no—
488C4—1221no—
489C4—2442no—
490C4—2441no—
491C4#1684084012yes—
492C2—22no—
493C2—82no—
494C2—81no—
495C2—4881no—
496C2#2146253811yescharacteristic
497C2C2×#1684080311yescharacteristic
498C2#2146259111yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#447C4
socleC22×C61C2×C4×D4
prime radical#21455814C2
cosocleC4×Dic61C24
hypercenter#447#6923
hypocenterC61#17935
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #21462542C2semidirect
2. C2×#16840797C2semidirect
3. #21461469C2semidirect
4. #21461467C2semidirect
5. #21461441C2semidirect
6. #21461440C2semidirect
7. #21461386C2semidirect
8. #21461144C2semidirect
9. #21461124C2semidirect
10. #21461034C2semidirect
18. #16840797C22semidirect
19. #16840796C22semidirect
21. #16840790C22semidirect
23. D4×#92420C22semidirect
30. #16840624C22semidirect
31. #16840623C22semidirect
32. #16840610C22semidirect
35. D4⋊#92420C22semidirect
36. #92420⋊C8C22semidirect
37. #16840603C22semidirect
298. C61#17935semidirect

Maximal subgroups

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

subgroupindex
#214625422
C2×#168407972
#214614692
#214614672
#214614412
#214614402
#214613862
#214611442
#214611242
#214610342
#214558142
#1793561