Finite Group Databasebeta

← #29699237

Subgroups — #29699237

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D488C211yesmaximal, characteristic
2#21448790C211yesmaximal, characteristic, complemented
3#21448730C211yesmaximal, characteristic, complemented
4#21446516C212yesmaximal, complemented
5#21446391C212yesmaximal, complemented
6#21446166C212yesmaximal
7#21445950C211yesmaximal, characteristic, complemented
8#21445830C211yesmaximal, characteristic, complemented
9#21445830C211yesmaximal, characteristic, complemented
10#21445645C212yesmaximal, complemented
11C61×#1770C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13C2×C4×D244C2212yes—
14C2×C4×D244C2211yescharacteristic
15C2×C4×D244C2211yescharacteristic
16C4×D488C418yescomplemented
17C2×#14519711C2211yescharacteristic
18C2×#14519699C2212yes—
19C2×#14519699C2212yes—
20C2×#14519699C2211yescharacteristic
21C2×#14519699C2211yescharacteristic
22#16839664C2212yes—
23#16839660C2212yes—
24#16839652—22no—
25#16839633—22no—
26#16839628C2211yescharacteristic
27#16839628C2211yescharacteristic
28#16839624—21no—
29#16839621—21no—
30C4×C4⋊Dic61C2212yes—
31#16839609—21no—
32#95×D61—21no—
33#16839525C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839525C2212yes—
37C2×C4⋊C8×C61C2211yescharacteristic
38C2×#97×C61C2212yes—
39C2×C4×C8×C61C2211yescharacteristic
40C4×C4⋊C4×C61C2211yescharacteristic
41C61×#336C2212yes—
42C22×D244C2311yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C4×D61—21no—
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47C2×D488—22no—
48C2×D488—22no—
49C2×D488C2×C414yes—
50C4×D244—22no—
51C4×D244—22no—
52C4×D244—24no—
53C4×D244—42no—
54C4×D244C2×C414yes—
55C4×D244C2×C414yescomplemented
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2312yes—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—22no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×C4×Dic61—22no—
67#14519711C2×C414yescomplemented
68#14519699—24no—
69#14519699—42no—
70#14519699C2×C414yes—
71#14519699C2×C414yescomplemented
72#14519684—21no—
73#14519684—22no—
74#14519684—21no—
75#14519684—22no—
76#14519684—21no—
77#14519684—21no—
78C22×C8×C61C2312yes—
79C22×C8×C61C2311yescharacteristic
80C22×C8×C61C2311yescharacteristic
81C2×C4⋊C4×C61D412yes—
82C2×C42×C61C2312yes—
83C2×C42×C61C2311yescharacteristic
84C4⋊C8×C61—22no—
85#97×C61—42no—
86C4×C8×C61C2×C414yescomplemented
87#95×C61D411yescharacteristic
88#95×C61D411yescharacteristic
89C23×D61—21no—
90C23×D61—21no—
91C22×C4×C61C2×D412yes—
92C22×C4×C61C2×D412yes—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2411yescharacteristic
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—22no—
99C22×Dic61—22no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244—21no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244C22×C412yes—
109C2×D244C22×C412yes—
110C2×D244#4112yes—
111C2×D244#4112yes—
112C2×C4×D61—24no—
113C2×C4×D61—24no—
114C2×C4×D61—44no—
115C2×C4×D61—44no—
116C2×C4×D61—82no—
117C2×C4×D61—82no—
118C2×C4×D61—82no—
119C2×C8×C61—21no—
120C2×C8×C61—21no—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C2×C8×C61C22×C412yes—
127C2×C8×C61C22×C412yes—
128C4⋊C4×C61—42no—
129C42×C61—22no—
130C42×C61—22no—
131C42×C61C22×C412yes—
132C42×C61#4112yes—
133#11758198—24no—
134#11758198—24no—
135#11758198—44no—
136#11758198—44no—
137#11758198—82no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61C22×C412yes—
145C4⋊Dic61C22×C412yes—
146#11758196—82no—
147C4×Dic61—82no—
148D488—28no—
149C23×C61C22×D411yescharacteristic
150C22×D61—22no—
151C22×D61—22no—
152C22×D61—22no—
153C22×D61—22no—
154C22×D61—42no—
155C22×D61—42no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—42no—
159C22×D61—42no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61C4×D412yes—
169C2×C4×C61C4×D412yes—
170C2×C4×C61C23×C411yescharacteristic
171C2×C4×C61C2×#4111yescharacteristic
172C2×C4×C61C2×#4111yescharacteristic
173C2×C4×C61C2×#4111yescharacteristic
174C2×C4×C61H(5,𝔽2)11yescharacteristic
175C2×C4×C612−1+411yescharacteristic
176C2×Dic61—22no—
177C2×Dic61—22no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185C2×Dic61—42no—
186D244—22no—
187D244—22no—
188D244—22no—
189D244—22no—
190D244—22no—
191D244—41no—
192D244—41no—
193D244#11714yes—
194C4×D61—84no—
195C4×D61—84no—
196C4×D61—88no—
197C8×C61—22no—
198C8×C61—22no—
199C8×C61—42no—
200#22764—611nomaximal, Sylow
201C22×C61C2×C4×D411yescharacteristic
202C22×C61#54611yescharacteristic
203C22×C61#54811yescharacteristic
204C22×C61#57512yes—
205C22×C61#57612yes—
206C2×D61—24no—
207C2×D61—44no—
208C2×D61—42no—
209C2×D61—42no—
210C2×D61—44no—
211C2×D61—42no—
212C2×D61—42no—
213C2×D61—42no—
214C2×D61—82no—
215C2×D61—82no—
216C4×C61—22no—
217C4×C61—22no—
218C4×C61—42no—
219C4×C61—42no—
220C4×C61C2×#11711yescharacteristic
221C4×C61#52011yescharacteristic
222C4×C61#55211yescharacteristic
223C4×C61#55311yescharacteristic
224Dic61—42no—
225Dic61—42no—
226Dic61—82no—
227Dic61—82no—
228C2×C4×D8—611no—
229#2357—611no—
230#2349—611no—
231#1936—612no—
232#1918—612no—
233#1879—612no—
234#1835—611no—
235#1804—611no—
236#1804—611no—
237#1770D6111yescharacteristic, complemented
238#1762—612no—
239C2×C61#235211yescharacteristic
240C2×C61#294311yescharacteristic
241C2×C61#294811yescharacteristic
242C2×C61#332212yes—
243C2×C61#337012yes—
244D61—44no—
245D61—82no—
246C22×D8—611no—
247C2×C4×D4—611no—
248C2×C4×D4—611no—
249C2×C4×D4—612no—
250C4×D8—618no—
251C2×#107—611no—
252C2×C4⋊C8C2×D6111yescharacteristic
253C2×#102—611no—
254C2×#102—611no—
255C2×#102—612no—
256C2×#102—612no—
257C2×#97C2×D6112yes—
258C2×C4×C8C2×D6111yescharacteristic
259#390—612no—
260#389—612no—
261#388—1222no—
262#385—1222no—
263#382—611no—
264#382—611no—
265#381—1221no—
266#380—1221no—
267C4×C4⋊C4—612no—
268C4×C4⋊C4C2×D6111yescharacteristic
269C4×C22⋊C4—1221no—
270C2×#95—1221no—
271#340—612no—
272#340—612no—
273#340—612no—
274#340—612no—
275#336C2×D6112yes—
276C61#2276411yescharacteristic, complemented, Sylow
277C22×D4—611no—
278C22×D4—611no—
279C23×C4—1221no—
280C23×C4—1221no—
281C23×C4—1222no—
282C2×D8—614no—
283C2×D8—1222no—
284C2×D8—1222no—
285C22×C8C22×D6112yes—
286C22×C8C22×D6111yescharacteristic
287C22×C8C22×D6111yescharacteristic
288C4×D4—614no—
289C4×D4—614no—
290C4×D4—1224no—
291C4×D4—1222no—
292C4×D4—1222no—
293C4×D4—2442no—
294C2×C4⋊C4—611no—
295C2×C4⋊C4—611no—
296C2×C4⋊C4—612no—
297C2×C4⋊C4—612no—
298C2×C4⋊C4—1222no—
299C2×C4⋊C4D24412yes—
300C2×C22⋊C4—1221no—
301C2×C22⋊C4—1221no—
302C2×C22⋊C4—1222no—
303C2×C22⋊C4—1222no—
304C2×C22⋊C4—1222no—
305C2×C42—1222no—
306C2×C42C22×D6112yes—
307C2×C42C22×D6111yescharacteristic
308#107—614no—
309C4⋊C8—22no—
310#102—614no—
311#102—614no—
312#102—1224no—
313#102—2442no—
314#97—42no—
315C4×C8C4×D6114yescomplemented
316#95—1221no—
317#95—1222no—
318#95—1221no—
319#95—1222no—
320#95—1221no—
321#95—1221no—
322#95D24411yescharacteristic
323#95D24411yescharacteristic
324C24—1221no—
325C24—1221no—
326C2×D4—612no—
327C2×D4—612no—
328C2×D4—612no—
329C2×D4—612no—
330C2×D4—1221no—
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4—1221no—
334C2×D4—1221no—
335C2×D4—1221no—
336C2×D4—1221no—
337C2×D4—1221no—
338C22×C4—1221no—
339C22×C4—1221no—
340C22×C4—1224no—
341C22×C4—1224no—
342C22×C4—1222no—
343C22×C4—1222no—
344C22×C4—4882no—
345C22×C4—4882no—
346C22×C4—4882no—
347C22×C4—2444no—
348C22×C4—2444no—
349C22×C4C2×D24412yes—
350C22×C4C2×D24412yes—
351C22×C4C2×D24411yescharacteristic
352C22×C4C2×D24411yescharacteristic
353C22×C4C23×D6111yescharacteristic
354D8—1228no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8C2×C4×D6112yes—
363C2×C8C2×C4×D6112yes—
364C4⋊C4—42no—
365C4⋊C4—612no—
366C4⋊C4—612no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C4⋊C4—1222no—
370C4⋊C4—1222no—
371C4⋊C4—1222no—
372C4⋊C4—1222no—
373C4⋊C4—4882no—
374C22⋊C4—1224no—
375C22⋊C4—1224no—
376C22⋊C4—4882no—
377C22⋊C4—2444no—
378C22⋊C4—2444no—
379C42—22no—
380C42—22no—
381C42—4882no—
382C42C2×C4×D6112yes—
383C42#1175822212yes—
384C23—1222no—
385C23—1222no—
386C23—1222no—
387C23—1222no—
388C23—2442no—
389C23—2442no—
390C23—2441no—
391C23—2441no—
392C23—2442no—
393C23—2442no—
394C23C22×D24411yescharacteristic
395D4—614no—
396D4—1222no—
397D4—1222no—
398D4—1222no—
399D4—1222no—
400D4—1222no—
401D4—2441no—
402D4—2441no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—1222no—
412C2×C4—1222no—
413C2×C4—4884no—
414C2×C4—4884no—
415C2×C4—4888no—
416C2×C4—2442no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4—2442no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4C4×D24412yes—
425C2×C4C4×D24412yes—
426C2×C4C22×C4×D6111yescharacteristic
427C2×C4#41×D6111yescharacteristic
428C2×C4#41×D6111yescharacteristic
429C2×C4C2×#1175822211yescharacteristic
430C2×C4#1451982211yescharacteristic
431C2×C4#1451982511yescharacteristic
432C8—22no—
433C8—22no—
434C8—42no—
435C22—1224no—
436C22—4882no—
437C22—4882no—
438C22—2442no—
439C22—2442no—
440C22—2442no—
441C22—2442no—
442C22—2442no—
443C22—2444no—
444C22—2444no—
445C22C2×C4×D24411yescharacteristic
446C22#1684040711yescharacteristic
447C22#1684042411yescharacteristic
448C22#1684056212yes—
449C22#1684056712yes—
450C4—22no—
451C4—22no—
452C4—42no—
453C4—42no—
454C4—4882no—
455C4—4882no—
456C4—2442no—
457C4—2442no—
458C4#117×D6111yescharacteristic
459C4#1684024311yescharacteristic
460C4#1684045311yescharacteristic
461C4#1684046211yescharacteristic
462C2—4882no—
463C2—2444no—
464C2#2144875111yescharacteristic
465C2#2145462111yescharacteristic
466C2#2145467211yescharacteristic
467C2#2145802212yes—
468C2#2145857412yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1770C2
socleC23×C61C22×D4
prime radicalC22×D488C22
cosocleC22×C4×C61C24
hypercenter#1770D61
hypocenterC61#22764
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1770

Complements

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

NH ≅ G/Ntype
2. #21448790C2semidirect
3. #21448730C2semidirect
4. #21446516C2semidirect
5. #21446391C2semidirect
7. #21445950C2semidirect
8. #21445830C2semidirect
9. #21445830C2semidirect
10. #21445645C2semidirect
11. C61×#1770C2semidirect
16. C4×D488C4semidirect
55. C4×D244C2×C4semidirect
67. #14519711C2×C4semidirect
71. #14519699C2×C4semidirect
86. C4×C8×C61C2×C4semidirect
237. #1770D61semidirect
276. C61#22764semidirect
315. C4×C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4882
#214487902
#214487302
#214465162
#214463912
#214461662
#214459502
#214458302
#214456452
C61×#17702
#2276461