Finite Group Databasebeta

← #29572122

Subgroups — #29572122

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840473C211yesmaximal, characteristic
2#21446622C211yesmaximal, characteristic, complemented
3#21445825C211yesmaximal, characteristic, complemented
4#21445820C211yesmaximal, characteristic
5C2×#16839492C211yesmaximal, characteristic, complemented
6C2×#16839489C211yesmaximal, characteristic
7C2×C61×#333C211yesmaximal, characteristic, complemented
8#16840473—22no—
9#16840473—22no—
10#16840473—22no—
11#16840473C414yescomplemented
12C2×#14519768—21no—
13C2×#14519757—21no—
14C2×#14519753—22no—
15C2×C4×Dic122C2211yescharacteristic
16C2×C4×D244C2211yescharacteristic
17C2×#14519739—21no—
18C2×C4⋊C4×D61—21no—
19C2×#14519702—21no—
20C2×#14519701—22no—
21C2×#14519699—21no—
22C2×C4×#92418C2211yescharacteristic
23#16839632C2211yescharacteristic
24#16839525—21no—
25#16839525—21no—
26#16839492—22no—
27#16839489—22no—
28C2×C4⋊Q8×C61C2211yescharacteristic
29C2×C4⋊C8×C61C2211yescharacteristic
30C2×C4⋊C8×C61C2211yescharacteristic
31C61×#333—22no—
32C22×Dic122D411yescharacteristic
33C22×D244D411yescharacteristic
34C22×C4×D61—21no—
35C22×C4×D61—22no—
36C22×#92418—21no—
37#14519768—24no—
38#14519768—24no—
39#14519757—24no—
40#14519757—24no—
41#14519753—28no—
42#14519753—28no—
43C4×Dic122—21no—
44C4×Dic122—21no—
45C4×Dic122C2×C412yes—
46C4×Dic122D412yes—
47C4×D244—21no—
48C4×D244—21no—
49C4×D244C2×C412yes—
50C4×D244D412yes—
51#14519739—24no—
52#14519739—24no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61D411yescharacteristic
58C2×C4⋊Dic61D411yescharacteristic
59C2×#11758196—22no—
60C2×#11758196—21no—
61C4⋊C4×D61—24no—
62C4⋊C4×D61—24no—
63C2×#11758198—22no—
64C2×#11758198—21no—
65C2×C4×Dic61—21no—
66#14519702—42no—
67#14519701—44no—
68#14519699—42no—
69C4×#92418—21no—
70C4×#92418—21no—
71C22×Q8×C61D412yes—
72C22×C8×C61—21no—
73C22×C8×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61C2×C412yes—
77C4⋊Q8×C61D412yes—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—21no—
80C2×C42×C61C2311yescharacteristic
81C4⋊C8×C61—21no—
82C4⋊C8×C61—21no—
83C4⋊C8×C61—21no—
84C4⋊C8×C61—21no—
85C23×D61—21no—
86C2×Q8×C61—22no—
87C2×Q8×C61—42no—
88C2×Q8×C61—42no—
89C2×Q8×C61C22⋊C414yes—
90C22×C4×C61—22no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—41no—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244C22⋊C412yes—
102C2×D244SD1612yes—
103C2×D244Q1612yes—
104C2×C4×D61—22no—
105C2×C4×D61—22no—
106C2×C4×D61—28no—
107C2×C4×D61—41no—
108C2×C4×D61—41no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61—82no—
112C2×C4×D61—82no—
113C2×Dic122—21no—
114C2×Dic122—21no—
115C2×Dic122—21no—
116C2×Dic122—21no—
117C2×Dic122—21no—
118C2×Dic122C22⋊C412yes—
119C2×C8×C61—41no—
120C2×C8×C61—41no—
121C2×C8×C61—41no—
122C2×C8×C61—41no—
123C2×C8×C61—41no—
124C2×C8×C61—41no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C42×C61C22×C411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61C2×D411yescharacteristic
133#11758198—28no—
134#11758198—22no—
135#11758198—22no—
136C4⋊Dic61—28no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—42no—
140C4⋊Dic61—41no—
141C4⋊Dic61—41no—
142C4⋊Dic61—41no—
143C4⋊Dic61—41no—
144C4⋊Dic61C22⋊C412yes—
145C4⋊Dic61C22⋊C412yes—
146#11758196—28no—
147#11758196—22no—
148#11758196—22no—
149C4×Dic61—22no—
150C4×Dic61—22no—
151C2×#92418—22no—
152C2×#92418—41no—
153C2×#92418—41no—
154C23×C61#12011yescharacteristic
155C22×D61—22no—
156C22×D61—22no—
157C22×D61—22no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162Q8×C61—44no—
163Q8×C61—82no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—24no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61C2×C22⋊C411yescharacteristic
170C2×C4×C61C2×C22⋊C411yescharacteristic
171C2×C4×C61C2×C22⋊C411yescharacteristic
172C2×C4×C61#12011yescharacteristic
173C2×C4×C61#12011yescharacteristic
174C2×C4×C61#12011yescharacteristic
175C2×C4×C61C2×SD1611yescharacteristic
176C2×C4×C61C2×Q1611yescharacteristic
177C2×C4×C61#13511yescharacteristic
178C2×C4×C61#13511yescharacteristic
179C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C61#13711yescharacteristic
181C2×C4×C61#13711yescharacteristic
182C2×C4×C61#13711yescharacteristic
183C2×Dic61—22no—
184C2×Dic61—22no—
185C2×Dic61—22no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—81no—
193C2×Dic61—81no—
194C2×Dic61—81no—
195D244—22no—
196D244—22no—
197D244—22no—
198D244#10314yescomplemented
199C4×D61—42no—
200C4×D61—42no—
201C4×D61—42no—
202C4×D61—42no—
203C4×D61—88no—
204Dic122—22no—
205Dic122—22no—
206Dic122—41no—
207Dic122—41no—
208C8×C61—81no—
209C8×C61—81no—
210#92418—42no—
211#8450—611nomaximal, Sylow
212C22×C61#37911yescharacteristic
213C22×C61#44811yescharacteristic
214C22×C61#44911yescharacteristic
215C22×C61#45111yescharacteristic
216C22×C61#45211yescharacteristic
217C22×C61#45311yescharacteristic
218C22×C61#45511yescharacteristic
219C2×D61—24no—
220C2×D61—42no—
221C2×D61—42no—
222C2×D61—42no—
223C2×D61—42no—
224C2×D61—42no—
225C2×D61—42no—
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—44no—
229C4×C61C2×#10311yescharacteristic
230C4×C61#41611yescharacteristic
231C4×C61#41711yescharacteristic
232C4×C61#41911yescharacteristic
233C4×C61#44811yescharacteristic
234C4×C61#44911yescharacteristic
235C4×C61#45111yescharacteristic
236C4×C61#45211yescharacteristic
237Dic61—42no—
238Dic61—42no—
239Dic61—42no—
240Dic61—161no—
241C2×#554—611no—
242#1956—611no—
243#1803—611no—
244#1802—611no—
245C2×#333D6111yescharacteristic, complemented
246C2×Q8⋊C8—611no—
247C2×D4⋊C8—611no—
248C2×C61#162211yescharacteristic
249C2×C61#163211yescharacteristic
250C2×C61#164311yescharacteristic
251C2×C61#164411yescharacteristic
252C2×C61#178711yescharacteristic
253C2×C61#178811yescharacteristic
254C2×C61#179211yescharacteristic
255D61—44no—
256#554—614no—
257#554—1222no—
258#554—1222no—
259#554—1222no—
260C2×C4⋊Q8C2×D6111yescharacteristic
261C2×#125—1221no—
262C2×C22⋊Q8—1221no—
263C2×C22⋊Q8—1222no—
264C2×C4×Q8—611no—
265C2×C4×D4—611no—
266C2×C4×D4—1221no—
267C22×C4⋊C4—1221no—
268C2×C4⋊C8C2×D6111yescharacteristic
269C2×C4⋊C8C2×D6111yescharacteristic
270C2×#103—1221no—
271C2×#103—1222no—
272C2×#102—1221no—
273C2×C4×C8—611no—
274#384—611no—
275#340—1221no—
276#340—1221no—
277#333—22no—
278Q8⋊C8—1222no—
279D4⋊C8—1222no—
280C61#845011yescharacteristic, complemented, Sylow
281C22×Q8—611no—
282C22×Q8D24412yes—
283C22×D4—611no—
284C23×C4—1222no—
285C23×C4—1221no—
286C22×C8—21no—
287C22×C8—21no—
288C22×C8—1221no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8C4×D6112yes—
292C4⋊Q8C22⋊D6112yes—
293#125—1224no—
294#125—1224no—
295C22⋊Q8—1224no—
296C22⋊Q8—1224no—
297C22⋊Q8—1228no—
298C22⋊Q8—1228no—
299C4×Q8—612no—
300C4×Q8—612no—
301C4×Q8—1221no—
302C4×Q8—1221no—
303C4×D4—612no—
304C4×D4—612no—
305C4×D4—1224no—
306C4×D4—1224no—
307C4×D4—1221no—
308C4×D4—1221no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—1224no—
314C2×C4⋊C4—1224no—
315C2×C4⋊C4—1222no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1221no—
323C2×C42—1221no—
324C2×C42C22×D6111yescharacteristic
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329#103—2442no—
330#103—2444no—
331#102—2442no—
332C4×C8—1221no—
333C4×C8—1221no—
334C24—1221no—
335C2×Q8—22no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—612no—
339C2×Q8—1221no—
340C2×Q8—1221no—
341C2×Q8—1221no—
342C2×Q8—1221no—
343C2×Q8—1221no—
344C2×Q8#1175819814yes—
345C2×D4—612no—
346C2×D4—612no—
347C2×D4—612no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C22×C4—22no—
352C22×C4—1228no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—1222no—
357C22×C4—1222no—
358C22×C4—4882no—
359C22×C4—4882no—
360C22×C4—2441no—
361C22×C4—2441no—
362C22×C4—2441no—
363C22×C4—2441no—
364C22×C4—2441no—
365C22×C4C2×D24411yescharacteristic
366C22×C4D4×D6111yescharacteristic
367C22×C4D4×D6111yescharacteristic
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—1222no—
375C2×C8—2441no—
376C2×C8—2441no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—612no—
382C4⋊C4—612no—
383C4⋊C4—1228no—
384C4⋊C4—1228no—
385C4⋊C4—1222no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1222no—
389C4⋊C4—2442no—
390C4⋊C4—2441no—
391C4⋊C4—2441no—
392C4⋊C4—2441no—
393C4⋊C4—2441no—
394C22⋊C4—1228no—
395C22⋊C4—1222no—
396C22⋊C4—1222no—
397C42—1222no—
398C42—1222no—
399C42C2×C4×D6111yescharacteristic
400C42D4×D6111yescharacteristic
401C42D4×D6111yescharacteristic
402C42C2×C22⋊D6111yescharacteristic
403C23—1222no—
404C23—1222no—
405C23—1222no—
406C23—2441no—
407C23—2441no—
408C23—2441no—
409C23—2441no—
410C23#1451974711yescharacteristic
411Q8—44no—
412Q8—82no—
413Q8—1222no—
414Q8—1222no—
415Q8—2441no—
416Q8—2441no—
417D4—614no—
418D4—1222no—
419D4—1222no—
420D4—1222no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—24no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—1222no—
427C2×C4—1222no—
428C2×C4—1222no—
429C2×C4—4888no—
430C2×C4—4881no—
431C2×C4—4881no—
432C2×C4—4881no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4C22⋊C4×D6111yescharacteristic
444C2×C4C22⋊C4×D6111yescharacteristic
445C2×C4C2×#1175819811yescharacteristic
446C2×C4#1451974511yescharacteristic
447C2×C4#1451974511yescharacteristic
448C2×C4#1451974711yescharacteristic
449C2×C4SD16×D6111yescharacteristic
450C2×C4Q16×D6111yescharacteristic
451C2×C4#1451979811yescharacteristic
452C2×C4#1451980011yescharacteristic
453C2×C4#1451980411yescharacteristic
454C2×C4#1451980611yescharacteristic
455C2×C4#1451980911yescharacteristic
456C2×C4#1451980911yescharacteristic
457C8—81no—
458C8—81no—
459C8—2442no—
460C22—1224no—
461C22—2442no—
462C22—2442no—
463C22—2442no—
464C22—2442no—
465C22—2442no—
466C22—2442no—
467C22#1683961711yescharacteristic
468C22#1683990211yescharacteristic
469C22#1683991011yescharacteristic
470C22#1683991911yescharacteristic
471C22#1683992711yescharacteristic
472C22#1683993211yescharacteristic
473C22#1683994211yescharacteristic
474C4—21no—
475C4—21no—
476C4—44no—
477C4—9761no—
478C4—2442no—
479C4—2442no—
480C4—2442no—
481C4#103×D6111yescharacteristic
482C4#1683975711yescharacteristic
483C4#1683976211yescharacteristic
484C4#1683976911yescharacteristic
485C4#1683989911yescharacteristic
486C4#1683990611yescharacteristic
487C4#1683991611yescharacteristic
488C4#1683992311yescharacteristic
489C2—2444no—
490C2#2144489211yescharacteristic
491C2#2144495411yescharacteristic
492C2#2144502711yescharacteristic
493C2#2144503411yescharacteristic
494C2#2144575211yescharacteristic
495C2#2144575811yescharacteristic
496C2#2144577511yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#333C2
socleC23×C61#120
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenterC2×#333D61
hypocenterC61#8450
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
2. #21446622C2semidirect
3. #21445825C2semidirect
5. C2×#16839492C2semidirect
7. C2×C61×#333C2semidirect
11. #16840473C4semidirect
198. D244#103semidirect
245. C2×#333D61semidirect
280. C61#8450semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
#214466222
#214458252
#214458202
C2×#168394922
C2×#168394892
C2×C61×#3332
#845061