Finite Group Databasebeta

← #29689714

Subgroups — #29689714

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840297C211yesmaximal, characteristic
2C2×#16839813C211yesmaximal, characteristic, complemented
3#21448225C211yesmaximal, characteristic, complemented
4#21446785C211yesmaximal, characteristic, complemented
5#21446630C211yesmaximal, characteristic, complemented
6#21446628C211yesmaximal, characteristic, complemented
7#21446140C212yesmaximal
8#21446130C212yesmaximal, complemented
9#21445823C212yesmaximal, complemented
10#21445818C212yesmaximal, complemented
11C61×#1991C211yesmaximal, characteristic, complemented
12#16840297—24no—
13#16840297C418yescomplemented
14C2×#14519772—21no—
15C2×#14519765C2211yescharacteristic
16C2×#14519762—21no—
17C2×#14519757C2211yescharacteristic
18C2×#14519752C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×#14519739—21no—
21C2×#14519739C2211yescharacteristic
22C2×#14519735—21no—
23C2×#14519735C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16839813—24no—
26C2×#14519706C2212yes—
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519699C2212yes—
30C2×#14519697C2212yes—
31C2×#14519689C2212yes—
32C2×C4×#92418C2211yescharacteristic
33#16839631C2212yes—
34#16839628C2211yescharacteristic
35#16839621—21no—
36#16839621—21no—
37#16839613—21no—
38#16839613—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#125×C61C2211yescharacteristic
42C2×C4×C8×C61C2211yescharacteristic
43C61×#382C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C22×Dic122C2311yescharacteristic
47C22×D244—21no—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61D411yescharacteristic
51C22×C4×D61D411yescharacteristic
52C22×C4×D61C2311yescharacteristic
53C22×#92418C2312yes—
54#14519772—28no—
55#14519765—22no—
56#14519765C2×C414yes—
57#14519762—28no—
58#14519757—22no—
59#14519757C2×C414yes—
60#14519752—22no—
61#14519752C2×C414yes—
62#14519742—22no—
63#14519742C2×C414yes—
64#14519739—28no—
65#14519739—22no—
66#14519739C2×C414yes—
67#14519735—28no—
68#14519735—42no—
69#14519735—42no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—21no—
73C2×#11758196—22no—
74C2×#11758196—22no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61C2×C414yes—
87C42×D61D414yes—
88#14519706—24no—
89#14519702—42no—
90#14519700—42no—
91#14519699—42no—
92#14519697—42no—
93#14519689—24no—
94C4×#92418—22no—
95#14519684—21no—
96#14519684—21no—
97C22×C8×C61C2312yes—
98#125×C61—22no—
99#125×C61C2×C414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C42×C61C2311yescharacteristic
106C4×C8×C61—22no—
107#95×C61—21no—
108#95×C61—21no—
109C23×D61—21no—
110C23×D61C2×D411yescharacteristic
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—24no—
125C2×D244—21no—
126C2×D244—44no—
127C2×D244C22×C412yes—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—24no—
131C2×C4×D61—24no—
132C2×C4×D61—42no—
133C2×C4×D61—44no—
134C2×C4×D61C22⋊C414yes—
135C2×C4×D61C22⋊C414yes—
136C2×C4×D61C22×C412yes—
137C2×C4×D61C2×D412yes—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—21no—
141C2×Dic122C22×C412yes—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61C22×C412yes—
152C4⋊C4×C61C22×C412yes—
153C42×C61C22×C412yes—
154C42×C61C2×D412yes—
155#11758198—24no—
156#11758198—24no—
157#11758198—24no—
158#11758198—24no—
159#11758198—24no—
160#11758198—44no—
161C4⋊Dic61—24no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164#11758196—24no—
165#11758196—44no—
166#11758196—44no—
167C4×Dic61—24no—
168C4×Dic61—44no—
169C4×Dic61C22⋊C414yes—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22×C412yes—
172C4×Dic61C2×D412yes—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61C22×D411yescharacteristic
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61C2×C22⋊C412yes—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61C2×C22⋊C412yes—
197C2×C4×C61C2×C22⋊C412yes—
198C2×C4×C61C23×C411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61H(5,𝔽2)12yes—
201C2×C4×C612−1+412yes—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61C2×C22⋊C412yes—
212C2×Dic61C2×C22⋊C412yes—
213C2×Dic61C2×C22⋊C412yes—
214C2×Dic61#13514yes—
215D244—24no—
216D244—48no—
217D244—42no—
218C4×D61—24no—
219C4×D61—24no—
220C4×D61—28no—
221C4×D61—44no—
222C4×D61—48no—
223Dic122—24no—
224Dic122—42no—
225C8×C61—42no—
226#92418—42no—
227#21862—611nomaximal, Sylow
228C22×C61C22×C22⋊C411yescharacteristic
229C22×C61#53511yescharacteristic
230C22×C61#53611yescharacteristic
231C22×C61C2×#13512yes—
232C22×C61#57512yes—
233C2×D61—22no—
234C2×D61—22no—
235C2×D61—22no—
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—82no—
242C2×D61—82no—
243C4×C61—22no—
244C4×C61—22no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—82no—
248C4×C61#52012yes—
249C4×C61#53712yes—
250Dic61—22no—
251Dic61—22no—
252Dic61—42no—
253Dic61—82no—
254Dic61#41614yes—
255C2×#529—611no—
256C2×#431—611no—
257#2285—611no—
258#1991—611no—
259#1991D6111yescharacteristic, complemented
260#1958—611no—
261#1958—611no—
262#1875—612no—
263#1873—612no—
264#1803—612no—
265#1802—612no—
266C2×C61#231111yescharacteristic
267C2×C61C2×#41611yescharacteristic
268C2×C61#289611yescharacteristic
269C2×C61#321212yes—
270C2×C61#323312yes—
271D61—42no—
272D61—82no—
273#529—618no—
274#529—1224no—
275C2×#126—1221no—
276C2×#125C2×D6111yescharacteristic
277C2×#124—611no—
278C2×#123—1221no—
279C2×C22⋊Q8—611no—
280C2×#121—611no—
281C2×C4×Q8—611no—
282C2×C4×D4—611no—
283C2×C4×D4—1221no—
284C2×#117—612no—
285C2×#117—1221no—
286C22×C42—611no—
287#431—1224no—
288C2×C4⋊C8—612no—
289C2×#103—612no—
290C2×#103—612no—
291C2×#102—612no—
292C2×#102—612no—
293C2×C22⋊C8—612no—
294C2×C4×C8—611no—
295C2×C4×C8C2×D6111yescharacteristic
296#384—612no—
297#382—611no—
298#382C2×D6111yescharacteristic
299#380—1221no—
300#380—1221no—
301C4×C4⋊C4—1221no—
302C4×C4⋊C4—1221no—
303#340—612no—
304#340—612no—
305#340C2×D6112yes—
306#340C2×D6112yes—
307C61#2186211yescharacteristic, complemented, Sylow
308C22×Q8—611no—
309C22×D4—611no—
310C22×D4—1221no—
311C23×C4—611no—
312C23×C4—611no—
313C23×C4—611no—
314C23×C4—1221no—
315C22×C8—612no—
316C22×C8C22×D6112yes—
317#126—1228no—
318#125—22no—
319#125C4×D6114yes—
320#124—614no—
321#124—1222no—
322#123—1228no—
323C22⋊Q8—614no—
324C22⋊Q8—1222no—
325#121—614no—
326#121—1222no—
327C4×Q8—614no—
328C4×Q8—1222no—
329C4×D4—614no—
330C4×D4—1222no—
331C4×D4—1228no—
332#117—1228no—
333#117—2442no—
334#117—2442no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4C22×D6112yes—
343C2×C4⋊C4C22×D6111yescharacteristic
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C22⋊C4—1221no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1222no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—614no—
355C2×C42—614no—
356C2×C42—1221no—
357C2×C42—1222no—
358C2×C42C22×D6111yescharacteristic
359C4⋊C8—1224no—
360#103—2442no—
361#103—2442no—
362#102—2442no—
363#102—2442no—
364C22⋊C8—1224no—
365C4×C8—22no—
366C4×C8—1222no—
367#95—21no—
368#95—21no—
369#95—1221no—
370#95—1221no—
371C24—611no—
372C24—1221no—
373C2×Q8—612no—
374C2×Q8—1221no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×D4—612no—
378C2×D4—1224no—
379C2×D4—1221no—
380C2×D4—1222no—
381C2×D4—1222no—
382C2×D4—2444no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—612no—
390C22×C4—612no—
391C22×C4—614no—
392C22×C4—614no—
393C22×C4—1224no—
394C22×C4—1224no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—1221no—
398C22×C4—1222no—
399C22×C4—2442no—
400C22×C4—2444no—
401C22×C4D4×D6111yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—1224no—
417C4⋊C4—1224no—
418C4⋊C4—1222no—
419C4⋊C4—1222no—
420C4⋊C4—2444no—
421C4⋊C4—2444no—
422C4⋊C4C2×C4×D6112yes—
423C4⋊C4C2×C4×D6112yes—
424C22⋊C4—1224no—
425C22⋊C4—1224no—
426C22⋊C4—1224no—
427C22⋊C4—1224no—
428C22⋊C4—1224no—
429C22⋊C4—2444no—
430C42—612no—
431C42—612no—
432C42—614no—
433C42—614no—
434C42—1224no—
435C42—2444no—
436C42C2×C4×D6112yes—
437C42D4×D6112yes—
438C23—612no—
439C23—1222no—
440C23—1222no—
441C23—1221no—
442C23—1221no—
443C23—1222no—
444C23—2442no—
445C23—2442no—
446C23—2441no—
447C23—2441no—
448C23C2×D4×D6111yescharacteristic
449Q8—1224no—
450Q8—2442no—
451D4—1224no—
452D4—2448no—
453D4—2442no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—612no—
464C2×C4—614no—
465C2×C4—612no—
466C2×C4—612no—
467C2×C4—1228no—
468C2×C4—1222no—
469C2×C4—1221no—
470C2×C4—1221no—
471C2×C4—1224no—
472C2×C4—1224no—
473C2×C4—1222no—
474C2×C4—2444no—
475C2×C4—2448no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4C22⋊C4×D6112yes—
482C2×C4C22⋊C4×D6112yes—
483C2×C4C22×C4×D6111yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4#1451982312yes—
486C2×C4#1451982512yes—
487C8—42no—
488C8—2442no—
489C22—1222no—
490C22—1222no—
491C22—1222no—
492C22—4882no—
493C22—4882no—
494C22—2442no—
495C22—2442no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22C2×C22⋊C4×D6111yescharacteristic
500C22#1684033011yescharacteristic
501C22#1684033711yescharacteristic
502C22#135×D6112yes—
503C22#1684055912yes—
504C4—22no—
505C4—22no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—614no—
510C4—1222no—
511C4—1222no—
512C4—4882no—
513C4—2442no—
514C4#1684024412yes—
515C4#1684034212yes—
516C2—4882no—
517C2—2442no—
518C2#2144841011yescharacteristic
519C2#416×D6111yescharacteristic
520C2#2145433411yescharacteristic
521C2#2145679612yes—
522C2#2145699912yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1991C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1991D61
hypocenterC61#21862
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
2. C2×#16839813C2semidirect
3. #21448225C2semidirect
4. #21446785C2semidirect
5. #21446630C2semidirect
6. #21446628C2semidirect
8. #21446130C2semidirect
9. #21445823C2semidirect
10. #21445818C2semidirect
11. C61×#1991C2semidirect
13. #16840297C4semidirect
259. #1991D61semidirect
307. C61#21862semidirect

Maximal subgroups

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

subgroupindex
C2×#168402972
C2×#168398132
#214482252
#214467852
#214466302
#214466282
#214461402
#214461302
#214458232
#214458182
C61×#19912
#2186261