Finite Group Databasebeta

← #29649808

Subgroups — #29649808

Order 15616.

648 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454780C211yesmaximal, characteristic
2C2×#16839889C211yesmaximal, characteristic, complemented
3C4×#14519810C211yesmaximal, characteristic
4#21444727C212yesmaximal, complemented
5#21444720C212yesmaximal, complemented
6#21444715C212yesmaximal, complemented
7#21444708C212yesmaximal, complemented
8#21444213C211yesmaximal, characteristic, complemented
9#21444204C211yesmaximal, characteristic
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#1565C211yesmaximal, characteristic, complemented
12C2×#14519810C411yescharacteristic
13C4×#11758222C2211yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15#16839889—21no—
16#16839889—21no—
17#16839889C2212yes—
18#16839889C2211yescharacteristic
19#16839889C2212yes—
20#16839889C2211yescharacteristic
21#16839870C412yes—
22#16839864C412yes—
23C4×Dic244C2212yes—
24C4×#11758190C2212yes—
25#16839839C2211yescharacteristic
26#16839839C2211yescharacteristic
27#16839830—21no—
28#16839830—21no—
29#16839815C2211yescharacteristic
30#16839813—21no—
31#16839804C411yescharacteristic
32C2×#14519706—21no—
33#16839772C411yescharacteristic
34#16839764C411yescharacteristic
35C4×#11758189—21no—
36C2×#14519687—21no—
37C2×C4×#92418C2211yescharacteristic
38#16839492C2212yes—
39#16839492C2212yes—
40#16839492C2212yes—
41#16839489C2212yes—
42#16839483C2212yes—
43C8×#92418C2212yes—
44C61×#431C2211yescharacteristic
45C2×C4⋊C8×C61C2211yescharacteristic
46C4×M16×C61C2211yescharacteristic
47C61×#335C2212yes—
48C61×#334C2212yes—
49C2×#11758222C2×C411yescharacteristic
50C22×Dic122C2×C411yescharacteristic
51#14519810—24no—
52C2×Dic244C2×C412yes—
53C2×#11758190C2×C412yes—
54C2×#11758189—21no—
55C2×#11758189—21no—
56C22×#92418D411yescharacteristic
57C22×#92418D411yescharacteristic
58C4×Dic122—21no—
59C4×Dic122—21no—
60C4×Dic122C2×C412yes—
61C4×Dic122C2×C411yescharacteristic
62C4×Dic122C2312yes—
63C4×Dic122C2311yescharacteristic
64C4×D244C2×C411yescharacteristic
65C4×D244C2311yescharacteristic
66C4×C22⋊D61—21no—
67C4×C22⋊D61—21no—
68#14519736C2×C411yescharacteristic
69#14519734—21no—
70C2×C4⋊Dic61C2×C411yescharacteristic
71C2×#11758196—21no—
72C2×C4×Dic61—21no—
73C42×D61—21no—
74#14519711C2×C412yes—
75#14519709C2×C412yes—
76C4⋊#92418—21no—
77#14519706—22no—
78#14519706—22no—
79#14519706—21no—
80#14519706—21no—
81#14519702C2×C412yes—
82#14519702C2×C412yes—
83#14519702C2×C412yes—
84#14519699C2×C412yes—
85C22⋊#92418—21no—
86#14519689—21no—
87#14519689—21no—
88#14519687—22no—
89#14519687—22no—
90#14519687—21no—
91#14519687—21no—
92C4×#92418D411yescharacteristic
93C4×#92418D411yescharacteristic
94C4×#92418C2312yes—
95C4×#92418C2311yescharacteristic
96C4×#92418C2311yescharacteristic
97C2×M16×C61C2×C411yescharacteristic
98C2×M16×C61C2×C411yescharacteristic
99C22×C8×C61—21no—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—21no—
102C4⋊C8×C61C2311yescharacteristic
103C4⋊C8×C61C2312yes—
104C4⋊C8×C61C2311yescharacteristic
105C22⋊C8×C61—21no—
106#97×C61C2×C412yes—
107C4×C8×C61—21no—
108C4×C8×C61C2312yes—
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—21no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115#11758222—22no—
116#11758222—22no—
117C2×D244C22×C411yescharacteristic
118C2×C4×D61—21no—
119C2×C4×D61—21no—
120C2×C4×D61—21no—
121C2×Dic122—22no—
122C2×Dic122—22no—
123C2×Dic122—21no—
124C2×Dic122C22×C412yes—
125C2×Dic122C22×C411yescharacteristic
126M16×C61—22no—
127M16×C61M1614yescomplemented
128C2×C8×C61—21no—
129C2×C8×C61—21no—
130C2×C8×C61—22no—
131C2×C8×C61—42no—
132C2×C8×C61C22×C412yes—
133C2×C8×C61C22×C412yes—
134C42×C61C22×C411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61#4111yescharacteristic
137C42×C61C2411yescharacteristic
138C22⋊Dic61—21no—
139#11758198—21no—
140#11758198—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61C22×C412yes—
143C4⋊Dic61C22×C411yescharacteristic
144C4⋊Dic61C22×C411yescharacteristic
145#11758196—22no—
146#11758196—21no—
147#11758196—22no—
148#11758196—21no—
149C4×Dic61—21no—
150C4×Dic61—21no—
151C4×Dic61—22no—
152C4×Dic61—21no—
153C4×Dic61—22no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—41no—
157C2×#92418C2×D411yescharacteristic
158C2×#92418C2×D411yescharacteristic
159C2×#92418C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161Dic244—24no—
162#11758190—24no—
163#11758189—42no—
164#11758189—42no—
165C23×C61C4×D411yescharacteristic
166C22×D61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—41no—
170C2×C4×C61C4×D411yescharacteristic
171C2×C4×C61C4×D411yescharacteristic
172C2×C4×C61C4×D411yescharacteristic
173C2×C4×C61C2×M1612yes—
174C2×C4×C61#13112yes—
175C2×C4×C61C23×C411yescharacteristic
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C2×#4111yescharacteristic
178C22⋊D61—42no—
179C2×Dic61—22no—
180C2×Dic61—22no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—42no—
185C2×Dic61—42no—
186D244—21no—
187D244—21no—
188C4×D61—42no—
189C4×D61—41no—
190C4×D61—41no—
191C4×D61—42no—
192Dic122—22no—
193Dic122—21no—
194Dic122—22no—
195Dic122—21no—
196Dic122—42no—
197C8×C61—22no—
198C8×C61—41no—
199C8×C61—41no—
200C8×C61—42no—
201C8×C61C2×M1614yes—
202#92418—21no—
203#92418—21no—
204#92418—21no—
205#92418—42no—
206#92418#13712yescomplemented
207#17892—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61#43612yes—
211C22×C61C2×C4×D411yescharacteristic
212C2×D61—41no—
213C2×D61—41no—
214C2×D61—41no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61#43612yes—
219C4×C61C22×M1611yescharacteristic
220C4×C61C2×#13111yescharacteristic
221C4×C61C2×#13711yescharacteristic
222C4×C61#57511yescharacteristic
223Dic61—42no—
224Dic61—41no—
225Dic61—41no—
226Dic61—41no—
227Dic61—42no—
228#2965—611no—
229C2×#446—611no—
230C4×#137—611no—
231#1593—612no—
232#1592—612no—
233#1591—612no—
234#1590—612no—
235#1565D6111yescharacteristic, complemented
236#1479—611no—
237#1477—611no—
238C8×M16—611no—
239C2×C61—22no—
240C2×C61C2×#43611yescharacteristic
241C2×C61C4×#13711yescharacteristic
242C2×C61#295511yescharacteristic
243D61—81no—
244C2×#137—611no—
245C4×#41—611no—
246C2×C4×Q8—611no—
247#446—612no—
248#446—611no—
249#446—612no—
250#446—611no—
251#446—1221no—
252#446—1221no—
253#441—612no—
254#440—612no—
255C4×Q16—612no—
256C4×SD16—612no—
257#436—611no—
258#436—611no—
259#435—1221no—
260#435—1221no—
261#431—611no—
262#431—1221no—
263#431C2×D6111yescharacteristic
264#428—611no—
265C2×C4⋊C8—1221no—
266C2×C4⋊C8C2×D6111yescharacteristic
267#419—611no—
268#417—611no—
269C4×M16—1221no—
270C4×M16C2×D6111yescharacteristic
271C2×#97—1221no—
272C2×C4×C8—611no—
273#335C2×D6112yes—
274#334C2×D6112yes—
275Q8⋊C8—612no—
276Q8⋊C8—612no—
277Q8⋊C8—612no—
278D4⋊C8—612no—
279#322—612no—
280C82—612no—
281C61#1789211yescharacteristic, complemented, Sylow
282C2×#41—611no—
283C22×Q8—611no—
284#137—1224no—
285C2×Q16—612no—
286C2×SD16—612no—
287C2×M16—1221no—
288C2×M16—1221no—
289C2×M16C4×D6111yescharacteristic
290C2×M16C4×D6111yescharacteristic
291C22×C8—21no—
292C22×C8—611no—
293C22×C8—611no—
294C4×Q8—612no—
295C4×Q8—611no—
296C4×Q8—612no—
297C4×Q8—611no—
298C4×Q8—1221no—
299C4×Q8—1221no—
300C4×D4—611no—
301C4×D4—611no—
302C4×D4—1221no—
303C4×D4—1221no—
304#117—611no—
305#117—1221no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—1221no—
308C2×C42—1221no—
309C2×C42—1221no—
310C2×C42C22×D6111yescharacteristic
311#107—612no—
312#106—612no—
313C4⋊C8—21no—
314C4⋊C8—1222no—
315C4⋊C8—1222no—
316C4⋊C8—1221no—
317C4⋊C8—1221no—
318C4⋊C8—1221no—
319C4⋊C8C22×D6112yes—
320C4⋊C8C22×D6111yescharacteristic
321C4⋊C8C22×D6111yescharacteristic
322#103—612no—
323#103—612no—
324#103—612no—
325#102—612no—
326C22⋊C8—21no—
327C22⋊C8—1221no—
328C22⋊C8—1221no—
329C22⋊C8—1221no—
330#97—1222no—
331#97—1222no—
332#97—1221no—
333#97—1221no—
334#97C4×D6112yes—
335C4×C8—21no—
336C4×C8—611no—
337C4×C8—611no—
338C4×C8—611no—
339C4×C8—611no—
340C4×C8—612no—
341C4×C8C22×D6112yes—
342#41—1222no—
343#41—1222no—
344C2×Q8—612no—
345C2×Q8—611no—
346C2×Q8—1221no—
347C2×Q8—1222no—
348C2×Q8—1222no—
349C2×D4—611no—
350C2×D4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4C2×C4×D6111yescharacteristic
357C22×C4D4×D6111yescharacteristic
358C22×C4#1175822611yescharacteristic
359Q16—1224no—
360SD16—1224no—
361M16—22no—
362M16—2442no—
363M16—2442no—
364M16#1175818914yescomplemented
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—22no—
368C2×C8—42no—
369C2×C8—611no—
370C2×C8—611no—
371C2×C8—611no—
372C2×C8—611no—
373C2×C8—1222no—
374C2×C8—1222no—
375C2×C8—2441no—
376C2×C8C2×C4×D6112yes—
377C2×C8C2×C4×D6112yes—
378C4⋊C4—612no—
379C4⋊C4—611no—
380C4⋊C4—611no—
381C4⋊C4—1222no—
382C4⋊C4—1221no—
383C4⋊C4—1221no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C22⋊C4—1221no—
387C22⋊C4—1221no—
388C22⋊C4—1221no—
389C42—1221no—
390C42—1222no—
391C42—1221no—
392C42—1221no—
393C42—1222no—
394C42C2×C4×D6111yescharacteristic
395C42D4×D6111yescharacteristic
396C42#1175822611yescharacteristic
397C42C23×D6111yescharacteristic
398C23—1221no—
399C23#1451973911yescharacteristic
400Q8—1222no—
401Q8—1221no—
402Q8—1222no—
403Q8—1221no—
404Q8—2442no—
405D4—1221no—
406D4—1221no—
407D4—2442no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—1221no—
412C2×C4—1221no—
413C2×C4—1222no—
414C2×C4—1221no—
415C2×C4—1222no—
416C2×C4—2442no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4—2441no—
420C2×C4—2441no—
421C2×C4—2442no—
422C2×C4#1451973911yescharacteristic
423C2×C4#1451973911yescharacteristic
424C2×C4#1451973911yescharacteristic
425C2×C4C2×#1175818912yes—
426C2×C4#1451978512yes—
427C2×C4C22×C4×D6111yescharacteristic
428C2×C4C2×D4×D6111yescharacteristic
429C2×C4C2×#1175822611yescharacteristic
430C8—22no—
431C8—41no—
432C8—42no—
433C8—41no—
434C8—612no—
435C8—1221no—
436C8—1221no—
437C8—1221no—
438C8—2442no—
439C8C2×#1175818914yes—
440C22—21no—
441C22—21no—
442C22—2441no—
443C22—2441no—
444C22—2441no—
445C22#1683983912yes—
446C22C2×#1451973911yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—2441no—
451C4—2441no—
452C4—2442no—
453C4—2441no—
454C4—2442no—
455C4#1683983912yes—
456C4C22×#1175818911yescharacteristic
457C4C2×#1451978511yescharacteristic
458C4#137×D6111yescharacteristic
459C4#1684055811yescharacteristic
460C2—22no—
461C2—4881no—
462C2C2×#1683983911yescharacteristic
463C2#2145463711yescharacteristic
464C2#2145472111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1565C2
socleC22×C61C2×C4×D4
prime radicalC2×#11758222C2×C4
cosocleC42×C61C24
hypercenter#1565D61
hypocenterC61#17892
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
2. C2×#16839889C2semidirect
4. #21444727C2semidirect
5. #21444720C2semidirect
6. #21444715C2semidirect
7. #21444708C2semidirect
8. #21444213C2semidirect
10. M16×#92418C2semidirect
11. C61×#1565C2semidirect
127. M16×C61M16semidirect
206. #92418#137semidirect
235. #1565D61semidirect
281. C61#17892semidirect
364. M16#11758189semidirect

Maximal subgroups

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

subgroupindex
#214547802
C2×#168398892
C4×#145198102
#214447272
#214447202
#214447152
#214447082
#214442132
#214442042
M16×#924182
C61×#15652
#1789261