Finite Group Databasebeta

← #30619223

Subgroups — #30619223

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462609C211yesmaximal, characteristic, complemented
2#21462544C211yesmaximal, characteristic
3#21462006C211yesmaximal, characteristic, complemented
4#21461955C211yesmaximal, characteristic
5#21461903C212yesmaximal, complemented
6#21461902C212yesmaximal, complemented
7#21461789C212yesmaximal, complemented
8#21461787C212yesmaximal, complemented
9#21461761C211yesmaximal, characteristic, complemented
10#21461539C211yesmaximal, characteristic, complemented
11#21456166C211yesmaximal, characteristic
12C2×#14519865C2212yes—
13#16840814—21no—
14#16840814—21no—
15#16840811—21no—
16#16840811—21no—
17#16840796—21no—
18#16840796—21no—
19#16840790—21no—
20#16840790—21no—
21C2×#14519848C2212yes—
22C2×#14519847C2212yes—
23#16840769C2211yescharacteristic
24#16840769C2211yescharacteristic
25#16840767C2211yescharacteristic
26#16840766C2211yescharacteristic
27C2×#14519842C2212yes—
28C2×#14519842C2212yes—
29#16840737C2212yes—
30#16840727C2211yescharacteristic
31#16840727C2211yescharacteristic
32C4×D61⋊C8C2211yescharacteristic
33#16840698C2211yescharacteristic
34#16840632C2212yes—
35#16840632C2212yes—
36(C2×#6923)⋊C8C2212yes—
37(C2×#6923)⋊C8C2212yes—
38#16840318C2211yescharacteristic
39#16840313C2211yescharacteristic
40#16840130C411yescharacteristic, complemented
41#16840128C411yescharacteristic, complemented
42#16840127C411yescharacteristic, complemented
43#16840123C412yescomplemented
44#16840123C412yescomplemented
45#16839757C2212yes—
46#16839757C2212yes—
47#97×D61C2211yescharacteristic
48C61×#491C411yescharacteristic, complemented
49#14519865—42no—
50#14519865—42no—
51C2×#11758213—22no—
52C2×#11758213C2311yescharacteristic
53C2×#11758213C2311yescharacteristic
54C2×#11758213C2312yes—
55C2×D61⋊C8—22no—
56C2×D61⋊C8C2311yescharacteristic
57C2×D61⋊C8C2311yescharacteristic
58C2×C4×#6923C2312yes—
59#14519848—42no—
60#14519847—42no—
61#14519844—21no—
62#14519844—21no—
63#14519844—21no—
64#14519844—21no—
65C4⋊#92420—21no—
66C4⋊#92420—21no—
67C4⋊#92420—21no—
68C4⋊#92420—21no—
69#14519842—42no—
70#14519842—42no—
71#14519834—22no—
72#14519834—22no—
73#14519832—21no—
74#14519832—21no—
75C4×#92420—21no—
76C4×#92420—21no—
77C4×#92420D411yescharacteristic
78C4×#92420D411yescharacteristic
79C4×#92420D411yescharacteristic
80C4×#92420D411yescharacteristic
81(C2×#6923)⋊C4—21no—
82(C2×#6923)⋊C4—21no—
83C2×#11758226C2312yes—
84C2×C8×D61C2312yes—
85#14519776C2×C411yescharacteristic
86#14519776C2×C411yescharacteristic
87#14519774C2×C411yescharacteristic
88#14519769C2×C411yescharacteristic
89#14519766C2×C412yes—
90#14519755—21no—
91#14519755—21no—
92#14519752—21no—
93#14519752—21no—
94#14519742—21no—
95#14519742—21no—
96#14519739—21no—
97#14519739—21no—
98#14519735C2311yescharacteristic
99#14519735C2311yescharacteristic
100C42×D61C2311yescharacteristic
101#14519701C2×C412yes—
102#14519701C2×C412yes—
103#14519699C2×C412yes—
104#14519699C2×C412yes—
105#14519697C2×C412yes—
106#14519697C2×C412yes—
107#14519688C2×C411yescharacteristic
108#14519687C2×C412yes—
109C4⋊Q8×C61C2×C411yescharacteristic
110#125×C61C2×C411yescharacteristic
111#103×C61C2×C412yes—
112#103×C61C2×C412yes—
113#97×C61C2×C411yescharacteristic
114C22×#6923—22no—
115C2×Q8×C61C22×C412yes—
116#11758226—22no—
117#11758226—22no—
118#11758226—22no—
119#11758226—22no—
120C2×D244—22no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244C22×C412yes—
124C2×C4×D61—22no—
125C2×C4×D61#4111yescharacteristic
126C2×C4×D61#4111yescharacteristic
127C2×C4×D61C2411yescharacteristic
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×#92420—21no—
131C2×#92420—21no—
132C2×#92420—22no—
133C2×#92420C2×D411yescharacteristic
134C2×#92420C2×D411yescharacteristic
135C2×#92420C2×D411yescharacteristic
136C2×#92420C2×D411yescharacteristic
137C4×#6923—22no—
138C4×#6923—22no—
139#11758213—22no—
140#11758213—22no—
141#11758213—22no—
142#11758213—22no—
143#11758213—44no—
144D61⋊C8—22no—
145D61⋊C8—22no—
146D61⋊C8—44no—
147C2×C8×C61C22×C412yes—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61C22×C411yescharacteristic
153C4⋊C4×C61C22×C411yescharacteristic
154C42×C61C22×C411yescharacteristic
155#11758198—21no—
156#11758198—21no—
157#11758198—22no—
158#11758198—22no—
159C4⋊Dic61C22×C411yescharacteristic
160C4⋊Dic61C22×C411yescharacteristic
161#11758196—21no—
162#11758196—21no—
163#11758196—21no—
164#11758196—21no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61C22×C411yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169C4×Dic61C2×D411yescharacteristic
170C2×#92418C22×C412yes—
171C8×D61—22no—
172C8×D61—22no—
173C22×D61—22no—
174C22×D61C2×#4111yescharacteristic
175C2×#6923—42no—
176C2×#6923—42no—
177C2×#6923—42no—
178Q8×C61—22no—
179Q8×C61—22no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—22no—
183C2×C4×C61C4×D411yescharacteristic
184C2×C4×C61C4×D411yescharacteristic
185C2×C4×C61C23×C411yescharacteristic
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61C4×D411yescharacteristic
189C2×Dic61C4×D411yescharacteristic
190C2×Dic61C22×D411yescharacteristic
191D244—22no—
192D244—22no—
193D244—42no—
194D244—42no—
195D244—42no—
196D244—42no—
197C4×D61—41no—
198C4×D61—41no—
199C4×D61—42no—
200C4×D61—42no—
201C4×D61H(5,𝔽2)12yes—
202C4×D612−1+412yes—
203Dic122—42no—
204Dic122—42no—
205#92420—21no—
206#92420—21no—
207#92420—21no—
208#92420—21no—
209#92420—42no—
210#92420—41no—
211#92420—41no—
212C8×C61—22no—
213#92418—22no—
214#22235—611nomaximal, Sylow
215C22×C61C2×C4×D411yescharacteristic
216C2×D61—21no—
217C2×D61—21no—
218C2×D61—42no—
219C2×D61—42no—
220C2×D61—42no—
221C2×D61#54212yes—
222#6923—82no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—42no—
226C4×C61—41no—
227C4×C61—41no—
228C4×C61#52012yes—
229Dic61—21no—
230Dic61—21no—
231Dic61—41no—
232Dic61—41no—
233Dic61#53511yescharacteristic
234Dic61#53611yescharacteristic
235#3135—611no—
236#2960—611no—
237#2931—611no—
238#1995—611no—
239#1963—611no—
240#1929—612no—
241#1929—612no—
242#1866—612no—
243#1866—612no—
244#1849—611no—
245#1746—611no—
246C2×C61#232611yescharacteristic
247C2×C61#295511yescharacteristic
248C2×C61#295511yescharacteristic
249D61—22no—
250D61—82no—
251C2×#131—612no—
252#533—611no—
253#532—611no—
254#491—611no—
255#491#692311yescharacteristic, complemented
256#490—611no—
257#490—611no—
258#489—612no—
259#489—612no—
260#446—1221no—
261#446—1221no—
262C8×Q8—1221no—
263C8×Q8—1221no—
264#436—1221no—
265#436—1221no—
266C8×D4—1221no—
267C8×D4—1221no—
268C2×#108—612no—
269C2×#108—612no—
270#424—611no—
271#424—611no—
272#423—611no—
273#423—611no—
274C2×#104—612no—
275C2×#104—612no—
276#416—612no—
277#416—612no—
278#408—612no—
279#405—611no—
280#405—611no—
281C2×#97—611no—
282C2×C4×C8—611no—
283#376—611no—
284#339—612no—
285#339—612no—
286#336—612no—
287#336—612no—
288C61#2223511yescharacteristic, complemented, Sylow
289C2×#41—612no—
290#131—2442no—
291#131—2442no—
292C2×M16—612no—
293C2×M16—611no—
294C2×M16—611no—
295C2×M16—1222no—
296C22×C8—611no—
297C22×C8—611no—
298C22×C8—612no—
299C22×C8—1222no—
300C4⋊Q8—611no—
301C4⋊Q8—611no—
302C4⋊Q8C2×#692311yescharacteristic
303#127—611no—
304#125—611no—
305#125C2×#692311yescharacteristic
306#124—612no—
307C22⋊Q8—1221no—
308C22⋊Q8—1221no—
309#121—1221no—
310#121—1221no—
311C4×Q8—1221no—
312C4×Q8—1221no—
313C4×D4—1221no—
314C4×D4—1221no—
315#117—611no—
316#117—611no—
317C2×C42—611no—
318C2×C42—612no—
319#108—2442no—
320#108—2442no—
321C4⋊C8—1221no—
322C4⋊C8—1221no—
323C4⋊C8—1221no—
324C4⋊C8—1221no—
325C4⋊C8—1221no—
326C4⋊C8—1221no—
327C4⋊C8—1221no—
328C4⋊C8—1221no—
329#104—2442no—
330#104—2442no—
331#103—612no—
332#103—612no—
333#103C2×#692312yes—
334#103C2×#692312yes—
335#102—612no—
336#102—612no—
337#102—612no—
338#102—612no—
339C22⋊C8—1222no—
340C22⋊C8—1222no—
341#97—612no—
342#97—611no—
343#97—1221no—
344#97—1221no—
345#97C2×#692311yescharacteristic
346C4×C8—611no—
347C4×C8—611no—
348C4×C8—611no—
349C4×C8—611no—
350C4×C8—1221no—
351C4×C8—1221no—
352#95—1221no—
353#95—1221no—
354#41—1222no—
355#41—1222no—
356#41—1222no—
357#41—1222no—
358C2×Q8—1221no—
359C2×Q8—1221no—
360C2×Q8C22×#692312yes—
361C2×D4—612no—
362C2×D4—1222no—
363C2×D4—1221no—
364C2×D4—1221no—
365C22×C4—611no—
366C22×C4—611no—
367C22×C4—611no—
368C22×C4—1222no—
369C22×C4—1222no—
370M16—1222no—
371M16—1222no—
372M16—1222no—
373M16—1222no—
374M16—2444no—
375C2×C8—611no—
376C2×C8—611no—
377C2×C8—611no—
378C2×C8—611no—
379C2×C8—612no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8—1222no—
383C2×C8—1222no—
384C2×C8—1221no—
385C2×C8—1221no—
386C2×C8—1222no—
387C2×C8—2444no—
388C2×C8C22×#692312yes—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—611no—
394C4⋊C4—611no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4C22×#692311yescharacteristic
400C4⋊C4C22×#692311yescharacteristic
401C22⋊C4—1222no—
402C22⋊C4—1222no—
403C22⋊C4—1221no—
404C22⋊C4—1221no—
405C42—611no—
406C42—611no—
407C42—611no—
408C42—1221no—
409C42—1221no—
410C42—1222no—
411C42—1222no—
412C42C22×#692311yescharacteristic
413C23—611no—
414C23—1222no—
415Q8—22no—
416Q8—22no—
417Q8—2442no—
418Q8—2442no—
419D4—1222no—
420D4—1222no—
421D4—2442no—
422D4—2442no—
423D4—2442no—
424D4—2442no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—611no—
429C2×C4—612no—
430C2×C4—612no—
431C2×C4—611no—
432C2×C4—611no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4D4×#692311yescharacteristic
443C2×C4D4×#692311yescharacteristic
444C2×C4C23×#692311yescharacteristic
445C8—22no—
446C8—1221no—
447C8—1221no—
448C8—1221no—
449C8—1221no—
450C8—1222no—
451C8—2442no—
452C8—2441no—
453C8—2441no—
454C22—612no—
455C22—1221no—
456C22—1221no—
457C22—2442no—
458C22—2442no—
459C22—2442no—
460C22C2×D4×#692311yescharacteristic
461C4—21no—
462C4—21no—
463C4—42no—
464C4—41no—
465C4—41no—
466C4—611no—
467C4—611no—
468C4—1221no—
469C4—1221no—
470C4—4882no—
471C4—2441no—
472C4—2441no—
473C4#1684083112yes—
474C2—1222no—
475C2—4882no—
476C2#2146230911yescharacteristic
477C2#2146259211yescharacteristic
478C2#2146259211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#491C4
socleC22×C61C2×C4×D4
prime radical#16840313C22
cosocleC2×C4×D61C24
hypercenter#491#6923
hypocenterC61#22235
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #21462609C2semidirect
3. #21462006C2semidirect
5. #21461903C2semidirect
6. #21461902C2semidirect
7. #21461789C2semidirect
8. #21461787C2semidirect
9. #21461761C2semidirect
10. #21461539C2semidirect
40. #16840130C4semidirect
41. #16840128C4semidirect
42. #16840127C4semidirect
43. #16840123C4semidirect
44. #16840123C4semidirect
48. C61×#491C4semidirect
255. #491#6923semidirect
288. C61#22235semidirect

Maximal subgroups

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

subgroupindex
#214626092
#214625442
#214620062
#214619552
#214619032
#214619022
#214617892
#214617872
#214617612
#214615392
#214561662
#2223561