Finite Group Databasebeta

← #29734124

Subgroups — #29734124

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic
2C2×#16840130C211yesmaximal, characteristic, complemented
3C2×#16840063C211yesmaximal, characteristic
4#21451202C211yesmaximal, characteristic, complemented
5#21447057C212yesmaximal
6#21447057C212yesmaximal
7#21446758C211yesmaximal, characteristic, complemented
8#21446089C212yesmaximal, complemented
9#21446075C212yesmaximal, complemented
10#21445647C211yesmaximal, characteristic
11C61×#1981C211yesmaximal, characteristic, complemented
12#16840486—24no—
13#16840486—24no—
14C2×#14519772—22no—
15C2×#14519769C2211yescharacteristic
16C2×#14519766C2212yes—
17C2×#14519757—21no—
18C2×Q8×Dic61C2212yes—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519735—21no—
22#16840130—24no—
23#16840063—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519701C2212yes—
27C2×#14519701C2212yes—
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×#14519688C2211yescharacteristic
32#16839694—21no—
33#16839683—21no—
34#16839661C2212yes—
35#16839661C2212yes—
36#16839660—22no—
37#16839524C2212yes—
38#16839516C2212yes—
39#16839516C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#103×C61C2212yes—
44C4×C4⋊C4×C61C2211yescharacteristic
45C22×Dic122D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×#92418C2312yes—
49#14519772—216no—
50#14519769—22no—
51#14519769—22no—
52#14519766—24no—
53#14519766—24no—
54#14519757—28no—
55Q8×Dic61—24no—
56Q8×Dic61—24no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59C4×D244—22no—
60C4×D244D414yes—
61#14519735—28no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758196D412yes—
72C2×#11758198—21no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61C2312yes—
77#14519711—22no—
78#14519709—22no—
79#14519701—24no—
80#14519701—24no—
81#14519699—22no—
82#14519699—22no—
83#14519697—24no—
84#14519688—22no—
85#14519684—22no—
86#14519684—22no—
87C22×Q8×C61C2312yes—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61D414yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2312yes—
95C2×C42×C61—21no—
96C2×C42×C61C2311yescharacteristic
97C4⋊C8×C61—22no—
98#103×C61—24no—
99#103×C61—24no—
100#95×C61—21no—
101C23×D61—21no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61C2×D414yes—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61C2×D412yes—
117C22×Dic61C2×D412yes—
118C2×D244—22no—
119C2×D244—22no—
120C2×D244—21no—
121C2×D244C2×D412yes—
122C2×C4×D61—24no—
123C2×C4×D61—44no—
124C2×Dic122—22no—
125C2×Dic122—42no—
126C2×C8×C61—21no—
127C2×C8×C61—21no—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C4⋊C4×C61—24no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61#4114yes—
134C42×C61—81no—
135C42×C61C2×D412yes—
136C42×C61C2×D412yes—
137#11758198—28no—
138#11758198—28no—
139#11758198—24no—
140C4⋊Dic61—28no—
141C4⋊Dic61—22no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—44no—
148C4⋊Dic61—82no—
149C4⋊Dic61C2×D412yes—
150#11758196—24no—
151#11758196—42no—
152#11758196—42no—
153C4×Dic61—28no—
154C4×Dic61—22no—
155C4×Dic61—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C23×C61C22×D411yescharacteristic
160C22×D61—22no—
161C22×D61—42no—
162C22×D61—42no—
163C22×D61—41no—
164C22×D61—41no—
165Q8×C61—24no—
166Q8×C61—24no—
167Q8×C61—42no—
168Q8×C61—42no—
169C2×C4×C61—22no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—24no—
173C2×C4×C61—24no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61#12012yes—
179C2×C4×C61#12012yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C2×#4112yes—
183C2×C4×C612−1+412yes—
184C2×Dic61—22no—
185C2×Dic61—22no—
186C2×Dic61—22no—
187C2×Dic61—22no—
188C2×Dic61—22no—
189C2×Dic61—22no—
190C2×Dic61—22no—
191C2×Dic61—42no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197D244—22no—
198D244—22no—
199D244—42no—
200C4×D61—48no—
201Dic122—44no—
202C8×C61—41no—
203C8×C61—41no—
204#92418—42no—
205#25656—611nomaximal, Sylow
206C22×C61C2×#12011yescharacteristic
207C22×C61#54812yes—
208C22×C61#57512yes—
209C22×C61#57711yescharacteristic
210C22×C61#57811yescharacteristic
211C2×D61—42no—
212C2×D61—42no—
213C2×D61—42no—
214C2×D61—82no—
215C2×D61—82no—
216C4×C61—22no—
217C4×C61—22no—
218C4×C61—44no—
219C4×C61—44no—
220C4×C61#53211yescharacteristic
221C4×C61#53611yescharacteristic
222C4×C61#54712yes—
223Dic61—42no—
224Dic61—42no—
225Dic61—42no—
226Dic61—81no—
227Dic61—81no—
228C2×#556—611no—
229C2×#491—611no—
230C2×#478—611no—
231#2584—611no—
232#2042—612no—
233#2042—612no—
234#1985—611no—
235#1981D6111yescharacteristic, complemented
236#1865—612no—
237#1863—612no—
238#1762—611no—
239C2×C61#243211yescharacteristic
240C2×C61#300511yescharacteristic
241C2×C61#301311yescharacteristic
242C2×C61#337612yes—
243C2×C61#337712yes—
244D61—82no—
245#556—1224no—
246#556—1224no—
247C2×C4⋊Q8C2×D6111yescharacteristic
248C2×#126—1222no—
249C2×#125—611no—
250C2×#124—612no—
251C2×C22⋊Q8—1221no—
252C2×C4×Q8—611no—
253C2×C4×Q8—612no—
254C2×C4×D4—611no—
255C2×#117—1221no—
256#491—1224no—
257#478—1224no—
258C2×#107—611no—
259C2×#106—611no—
260C2×C4⋊C8C2×D6111yescharacteristic
261C2×#103—612no—
262C2×#103—612no—
263C2×#103C2×D6112yes—
264C2×#103C2×D6112yes—
265C2×#102—611no—
266C2×#102—611no—
267C2×#102—612no—
268C2×#97—611no—
269#398—1221no—
270#395—1221no—
271#389—612no—
272#389—612no—
273#389—1222no—
274C4×C4⋊C4C2×D6111yescharacteristic
275#340—612no—
276#336—612no—
277#336—612no—
278C61#2565611yescharacteristic, complemented, Sylow
279C22×Q8—611no—
280C22×Q8C22×D6112yes—
281C22×D4—611no—
282C23×C4—1221no—
283C22×C8—612no—
284C22×C8C22×D6111yescharacteristic
285C22×C8C22×D6111yescharacteristic
286C4⋊Q8—22no—
287C4⋊Q8D24414yes—
288#126—12216no—
289#125—1222no—
290#125—1222no—
291#124—1224no—
292#124—1224no—
293C22⋊Q8—1228no—
294C4×Q8—1224no—
295C4×Q8—1224no—
296C4×Q8—1222no—
297C4×Q8—1222no—
298C4×D4—614no—
299C4×D4—1222no—
300#117—1228no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—611no—
305C2×C4⋊C4—612no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—1222no—
309C2×C4⋊C4—1222no—
310C2×C4⋊C4—1222no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4C22×D6112yes—
314C2×C22⋊C4—1222no—
315C2×C22⋊C4—1222no—
316C2×C22⋊C4—1221no—
317C2×C42—21no—
318C2×C42—612no—
319C2×C42—1222no—
320C2×C42C22×D6111yescharacteristic
321#107—1222no—
322#106—1222no—
323C4⋊C8—22no—
324#103—24no—
325#103—24no—
326#103—1224no—
327#103—1224no—
328#102—1222no—
329#102—1222no—
330#102—1224no—
331#97—1222no—
332#95—21no—
333#95—1222no—
334#95—1222no—
335C24—1221no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—1222no—
342C2×Q8—2442no—
343C2×Q8C2×D24414yes—
344C2×D4—612no—
345C2×D4—1221no—
346C2×D4—1222no—
347C2×D4—1222no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—612no—
351C22×C4—612no—
352C22×C4—1221no—
353C22×C4—1224no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—2444no—
357C22×C4D4×D6111yescharacteristic
358C22×C4D4×D6111yescharacteristic
359C22×C4C23×D6111yescharacteristic
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—1222no—
365C2×C8—1222no—
366C2×C8—1222no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—612no—
371C4⋊C4—1228no—
372C4⋊C4—1221no—
373C4⋊C4—1224no—
374C4⋊C4—1222no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C4⋊C4—4882no—
380C4⋊C4—2444no—
381C4⋊C4—2442no—
382C4⋊C4—2442no—
383C4⋊C4#1175822414yes—
384C22⋊C4—1228no—
385C22⋊C4—1228no—
386C22⋊C4—1224no—
387C42—81no—
388C42—1228no—
389C42—1222no—
390C42—1222no—
391C42C2×D24412yes—
392C42D4×D6112yes—
393C23—1222no—
394C23—2442no—
395C23—2441no—
396C23—2441no—
397C23—2442no—
398C23C2×D4×D6111yescharacteristic
399Q8—24no—
400Q8—24no—
401Q8—42no—
402Q8—42no—
403Q8—2444no—
404D4—1222no—
405D4—1222no—
406D4—2442no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—1222no—
417C2×C4—1222no—
418C2×C4—1222no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—1222no—
423C2×C4—2448no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4—2442no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2442no—
430C2×C4#1451974712yes—
431C2×C4#1451974712yes—
432C2×C4C2×D4×D6111yescharacteristic
433C2×C4C22×D24411yescharacteristic
434C2×C4C2×#1175822412yes—
435C2×C4#1451982412yes—
436C8—41no—
437C8—41no—
438C8—2442no—
439C22—4882no—
440C22—4882no—
441C22—2442no—
442C22—2442no—
443C22—2442no—
444C22C2×#1451974711yescharacteristic
445C22#1684041812yes—
446C22#1684056112yes—
447C22#1684056811yescharacteristic
448C22#1684057511yescharacteristic
449C4—22no—
450C4—22no—
451C4—44no—
452C4—44no—
453C4—4881no—
454C4—4881no—
455C4—2442no—
456C4—2442no—
457C4—2442no—
458C4#1684031011yescharacteristic
459C4#1684033211yescharacteristic
460C4#1684041612yes—
461C2—4882no—
462C2#2144944911yescharacteristic
463C2#2145506511yescharacteristic
464C2#2145513911yescharacteristic
465C2#2145863912yes—
466C2#2145865412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1981C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1981D61
hypocenterC61#25656
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
2. C2×#16840130C2semidirect
4. #21451202C2semidirect
7. #21446758C2semidirect
8. #21446089C2semidirect
9. #21446075C2semidirect
11. C61×#1981C2semidirect
235. #1981D61semidirect
278. C61#25656semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×#168401302
C2×#168400632
#214512022
#214470572
#214467582
#214460892
#214460752
#214456472
C61×#19812
#2565661