Finite Group Databasebeta

← #29696843

Subgroups — #29696843

Order 15616.

1098 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic
2C2×#16839892C211yesmaximal, characteristic, complemented
3#21448731C211yesmaximal, characteristic, complemented
4#21446748C212yesmaximal
5#21446743C211yesmaximal, characteristic, complemented
6#21446632C212yesmaximal
7#21445822C212yesmaximal, complemented
8#21445816C211yesmaximal, characteristic
9#21445651C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic
11C61×#1956C211yesmaximal, characteristic, complemented
12#16840486—24no—
13#16840486C418yescomplemented
14C2×#14519772—22no—
15C2×#14519766C2212yes—
16C2×#14519757—21no—
17C2×Q8×Dic61C2212yes—
18C2×C4×Dic122C2211yescharacteristic
19C2×#14519742—21no—
20C2×C4×D244C2211yescharacteristic
21C2×#14519735—21no—
22#16839892—24no—
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519706C2212yes—
25C2×#14519701C2212yes—
26C2×#14519701C2212yes—
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519697C2212yes—
30C2×#14519687C2212yes—
31#16839658—21no—
32#16839657—22no—
33#16839631C2212yes—
34C4⋊C4×Dic61C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839525C2211yescharacteristic
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×#103×C61C2212yes—
41C2×#103×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C61×#384C2211yescharacteristic
44C22×Dic122D411yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C22×#92418C2312yes—
48#14519772—216no—
49#14519766—24no—
50#14519766C2×C418yes—
51#14519757—28no—
52Q8×Dic61—24no—
53Q8×Dic61C2×C418yes—
54C4×Dic122C2×C414yes—
55C4×Dic122D414yes—
56#14519742—44no—
57C4×D244—22no—
58C4×D244C2×C414yes—
59#14519735—28no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61D411yescharacteristic
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×#11758196D412yes—
68C2×#11758198—21no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×C4×Dic61—22no—
72C2×C4×Dic61—21no—
73C2×C4×Dic61—22no—
74C2×C4×Dic61C2312yes—
75C4⋊#92418—22no—
76#14519706—24no—
77#14519701—24no—
78#14519701—42no—
79#14519699—22no—
80#14519699—22no—
81#14519697—42no—
82#14519687—24no—
83#14519684—21no—
84#14519684—22no—
85C22×Q8×C61C2312yes—
86C22×C8×C61C2311yescharacteristic
87C22×C8×C61C2311yescharacteristic
88C4⋊Q8×C61—22no—
89C4⋊Q8×C61C2×C414yes—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2312yes—
95C2×C42×C61C2311yescharacteristic
96#103×C61—24no—
97#103×C61—42no—
98C4×C8×C61—22no—
99C23×D61—21no—
100C2×Q8×C61—22no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61C22×C414yes—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61C2×D412yes—
115C22×Dic61C2×D412yes—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244C22×C412yes—
122C2×C4×D61—24no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×Dic122—22no—
126C2×Dic122—22no—
127C2×Dic122C22⋊C414yes—
128C2×C8×C61—21no—
129C2×C8×C61—21no—
130C2×C8×C61—21no—
131C2×C8×C61—21no—
132C2×C8×C61—21no—
133C2×C8×C61#4112yes—
134C4⋊C4×C61—24no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61—81no—
140C42×C61C22×C412yes—
141C42×C61C2×D412yes—
142#11758198—24no—
143#11758198—28no—
144#11758198—28no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—28no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61C22⋊C414yes—
151C4⋊Dic61C22×C412yes—
152C4⋊Dic61#4112yes—
153#11758196—44no—
154#11758196—82no—
155#11758196C22⋊C418yes—
156C4×Dic61—28no—
157C4×Dic61—44no—
158C4×Dic61—42no—
159C4×Dic61C22×C414yes—
160C4×Dic61C2×D414yes—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164C23×C61C22×D411yescharacteristic
165C22×D61—22no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172Q8×C61—24no—
173Q8×C61—24no—
174Q8×C61—42no—
175Q8×C61—42no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—24no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61C2×C22⋊C412yes—
185C2×C4×C61C2×C22⋊C412yes—
186C2×C4×C61AGL(1,ℤ/8)12yes—
187C2×C4×C61#13712yes—
188C2×C4×C61C23×C411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61H(5,𝔽2)11yescharacteristic
193C2×C4×C612−1+411yescharacteristic
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61C2×C22⋊C414yes—
208C2×Dic61C2×C22⋊C414yes—
209D244—22no—
210D244—22no—
211D244—41no—
212D244—41no—
213C4×D61—48no—
214Dic122—28no—
215C8×C61—22no—
216C8×C61—41no—
217#92418—42no—
218#22544—611nomaximal, Sylow
219C22×C61C22×C22⋊C411yescharacteristic
220C22×C61#54811yescharacteristic
221C22×C61#55011yescharacteristic
222C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
223C22×C61C2×#13711yescharacteristic
224C22×C61#57711yescharacteristic
225C22×C61#57811yescharacteristic
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—42no—
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—81no—
232C2×D61—81no—
233C4×C61—22no—
234C4×C61—44no—
235C4×C61—82no—
236C4×C61#41714yes—
237C4×C61C4×#4111yescharacteristic
238C4×C61#52011yescharacteristic
239C4×C61#54811yescharacteristic
240C4×C61#55011yescharacteristic
241Dic61—24no—
242Dic61—24no—
243Dic61—42no—
244Dic61—42no—
245Dic61—81no—
246C2×#556—611no—
247C2×#446—611no—
248Q8×C4⋊C4—611no—
249#1983—612no—
250#1981—611no—
251#1959—612no—
252#1956D6111yescharacteristic, complemented
253#1803—612no—
254#1802—611no—
255#1763—612no—
256C4×#102—611no—
257C2×C61#234111yescharacteristic
258C2×C61C2×#41711yescharacteristic
259C2×C61#289811yescharacteristic
260C2×C61#336711yescharacteristic
261C2×C61#336911yescharacteristic
262C2×C61#339711yescharacteristic
263C2×C61#339811yescharacteristic
264D61—82no—
265#556—618no—
266#556—1224no—
267C2×C4⋊Q8C2×D6111yescharacteristic
268C2×#126—1222no—
269C2×#124—612no—
270C2×C22⋊Q8—1221no—
271C2×C4×Q8—611no—
272C2×C4×Q8—612no—
273C2×C4×Q8—1221no—
274C2×C4×D4—611no—
275C2×#117—1221no—
276#446—1224no—
277C2×C4⋊C8—612no—
278C2×C4⋊C8—611no—
279C2×#103—612no—
280C2×#103—612no—
281C2×#103C2×D6112yes—
282C2×#103C2×D6112yes—
283C2×#102—611no—
284C2×#102—611no—
285C2×#102—612no—
286C2×#97—612no—
287C2×C4×C8C2×D6111yescharacteristic
288#389—1222no—
289#389—1221no—
290#384—612no—
291#384C2×D6111yescharacteristic
292C4×C4⋊C4—612no—
293C4×C4⋊C4—611no—
294#340—611no—
295#340—611no—
296#340—612no—
297C61#2254411yescharacteristic, complemented, Sylow
298C22×Q8—611no—
299C22×Q8C22×D6112yes—
300C22×D4—611no—
301C23×C4—1221no—
302C22×C8—612no—
303C22×C8C22×D6111yescharacteristic
304C22×C8C22×D6111yescharacteristic
305C4⋊Q8—22no—
306C4⋊Q8C4×D6114yes—
307#126—12216no—
308#124—618no—
309#124—1224no—
310C22⋊Q8—1228no—
311C4×Q8—614no—
312C4×Q8—614no—
313C4×Q8—618no—
314C4×Q8—1224no—
315C4×Q8—2444no—
316C4×D4—614no—
317C4×D4—1222no—
318#117—1228no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—611no—
324C2×C4⋊C4—612no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—1222no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4C22×D6112yes—
332C2×C22⋊C4—1222no—
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1221no—
335C2×C42—612no—
336C2×C42—1222no—
337C2×C42—1222no—
338C2×C42—1221no—
339C2×C42C22×D6111yescharacteristic
340C4⋊C8—1224no—
341C4⋊C8—1222no—
342#103—24no—
343#103—42no—
344#103—1224no—
345#103—2442no—
346#102—1222no—
347#102—1222no—
348#102—2442no—
349#97—1224no—
350C4×C8—22no—
351#95—1222no—
352#95—1221no—
353C24—1221no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—614no—
360C2×Q8—1222no—
361C2×Q8—1222no—
362C2×Q8C2×C4×D6114yes—
363C2×D4—612no—
364C2×D4—1221no—
365C2×D4—1221no—
366C2×D4—1221no—
367C2×D4—1221no—
368C2×D4—1221no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—612no—
372C22×C4—612no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—1224no—
376C22×C4—1221no—
377C22×C4—2442no—
378C22×C4—2442no—
379C22×C4D4×D6111yescharacteristic
380C22×C4D4×D6111yescharacteristic
381C22×C4C23×D6111yescharacteristic
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—1222no—
388C2×C8—1222no—
389C2×C8—1222no—
390C2×C8#1175822412yes—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—42no—
396C4⋊C4—81no—
397C4⋊C4—618no—
398C4⋊C4—614no—
399C4⋊C4—612no—
400C4⋊C4—612no—
401C4⋊C4—1228no—
402C4⋊C4—1221no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—4882no—
407C4⋊C4—2444no—
408C22⋊C4—1228no—
409C22⋊C4—1228no—
410C22⋊C4—1224no—
411C42—614no—
412C42—614no—
413C42—1228no—
414C42—2444no—
415C42—2442no—
416C42C2×C4×D6112yes—
417C42D4×D6112yes—
418C23—1222no—
419C23—2441no—
420C23—2441no—
421C23—2441no—
422C23—2441no—
423C23—2441no—
424C23—2441no—
425C23C2×D4×D6111yescharacteristic
426Q8—24no—
427Q8—24no—
428Q8—42no—
429Q8—42no—
430Q8—1228no—
431D4—1222no—
432D4—1222no—
433D4—2441no—
434D4—2441no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—614no—
444C2×C4—614no—
445C2×C4—1222no—
446C2×C4—1222no—
447C2×C4—1222no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4—2441no—
458C2×C4—2448no—
459C2×C4C22⋊C4×D6112yes—
460C2×C4C22⋊C4×D6112yes—
461C2×C4#1451980412yes—
462C2×C4#1451980912yes—
463C2×C4C22×C4×D6111yescharacteristic
464C2×C4C2×D4×D6111yescharacteristic
465C2×C4#41×D6111yescharacteristic
466C2×C4C2×#1175822411yescharacteristic
467C2×C4#1451982211yescharacteristic
468C2×C4#1451982411yescharacteristic
469C8—22no—
470C8—41no—
471C8—2442no—
472C22—4881no—
473C22—4881no—
474C22—4881no—
475C22—4881no—
476C22—2442no—
477C22—2442no—
478C22—2442no—
479C22C2×C22⋊C4×D6111yescharacteristic
480C22#1684042011yescharacteristic
481C22#1684043211yescharacteristic
482C22C2×#1451980411yescharacteristic
483C22C2×#1451980911yescharacteristic
484C22#1684056811yescharacteristic
485C22#1684057511yescharacteristic
486C4—22no—
487C4—44no—
488C4—82no—
489C4—1224no—
490C4—1224no—
491C4—4881no—
492C4—2442no—
493C4—2442no—
494C4#1683976214yes—
495C4C4×#1175822411yescharacteristic
496C4#1684024111yescharacteristic
497C4#1684041811yescharacteristic
498C4#1684043011yescharacteristic
499C2—4882no—
500C2#2144865811yescharacteristic
501C2C2×#1683976211yescharacteristic
502C2#2145434711yescharacteristic
503C2#2145854211yescharacteristic
504C2#2145856411yescharacteristic
505C2#2145884411yescharacteristic
506C2#2145885511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1956C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1956D61
hypocenterC61#22544
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. C2×#16839892C2semidirect
3. #21448731C2semidirect
5. #21446743C2semidirect
7. #21445822C2semidirect
9. #21445651C2semidirect
11. C61×#1956C2semidirect
13. #16840486C4semidirect
252. #1956D61semidirect
297. C61#22544semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×#168398922
#214487312
#214467482
#214467432
#214466322
#214458222
#214458162
#214456512
C4×#145196992
C61×#19562
#2254461