Finite Group Databasebeta

← #29572124

Subgroups — #29572124

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×#16840472C211yesmaximal, characteristic
2#21446620C211yesmaximal, characteristic, complemented
3#21445816C211yesmaximal, characteristic
4C2×#16839510C211yesmaximal, characteristic, complemented
5C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
6C2×#16839489C211yesmaximal, characteristic
7C61×#1803C211yesmaximal, characteristic, complemented
8#16840472—22no—
9#16840472—22no—
10#16840472—22no—
11#16840472C414yescomplemented
12C2×#14519777C2211yescharacteristic
13C2×#14519769—21no—
14C2×#14519755—22no—
15C2×#14519753—21no—
16C2×C4×D244—21no—
17C2×C4×D244C2211yescharacteristic
18C2×C4⋊C4×D61—21no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519700—22no—
21C2×#14519697—21no—
22C2×C4×#92418C2211yescharacteristic
23#16839524—21no—
24#16839510—22no—
25Q8⋊#92418—22no—
26#16839489—22no—
27C2×C4×Q8×C61C2211yescharacteristic
28C2×C4⋊C8×C61C2211yescharacteristic
29C2×#103×C61—21no—
30C61×#384C2211yescharacteristic
31C61×#340—21no—
32C22×Dic122D412yes—
33C22×D244D411yescharacteristic
34C22×C4×D61—21no—
35C22×C4×D61—22no—
36C22×#92418—21no—
37C22×#92418—21no—
38#14519777—21no—
39#14519777—21no—
40#14519777C2×C412yes—
41#14519777D412yes—
42#14519769—24no—
43#14519769—24no—
44#14519755—28no—
45#14519755—28no—
46#14519753—24no—
47#14519753—24no—
48C4×D244—24no—
49C4×D244—24no—
50C4×D244—21no—
51C4×D244—21no—
52C4×D244C2×C412yes—
53C4×D244D412yes—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61D411yescharacteristic
57C2×#11758196—22no—
58C2×#11758196—22no—
59C4⋊C4×D61—24no—
60C4⋊C4×D61—24no—
61C2×#11758198—22no—
62C2×#11758198—21no—
63C4⋊#92418—21no—
64C4⋊#92418—21no—
65#14519700—44no—
66#14519697—42no—
67C4×#92418—21no—
68C4×#92418—21no—
69C22×Q8×C61D411yescharacteristic
70C22×C8×C61—21no—
71C4×Q8×C61—21no—
72C4×Q8×C61—21no—
73C4×Q8×C61C2×C412yes—
74C4×Q8×C61D412yes—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61D411yescharacteristic
80C2×C42×C61—21no—
81C2×C42×C61C2311yescharacteristic
82C4⋊C8×C61—21no—
83C4⋊C8×C61—21no—
84#103×C61—42no—
85C23×D61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C2×Q8×C61—21no—
89C2×Q8×C61—21no—
90C2×Q8×C61—21no—
91C2×Q8×C61C22⋊C412yes—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—41no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×Dic61—22no—
99C22×Dic61—21no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244—21no—
103C2×D244C22⋊C412yes—
104C2×D244SD1612yes—
105C2×D244Q1612yes—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—28no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61—41no—
112C2×C4×D61—41no—
113C2×C4×D61—82no—
114C2×C4×D61—82no—
115C2×Dic122—22no—
116C2×Dic122—42no—
117C2×Dic122—42no—
118C2×Dic122C22⋊C414yes—
119C2×C8×C61—41no—
120C2×C8×C61—41no—
121C2×C8×C61—41no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—42no—
126C4⋊C4×C61—41no—
127C4⋊C4×C61—41no—
128C4⋊C4×C61—41no—
129C4⋊C4×C61—41no—
130C4⋊C4×C61C22⋊C412yes—
131C42×C61—22no—
132C42×C61—22no—
133C42×C61C22×C411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137#11758198—28no—
138#11758198—22no—
139#11758198—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61C22⋊C412yes—
146#11758196—28no—
147#11758196—28no—
148C2×#92418—22no—
149C2×#92418—41no—
150C2×#92418—41no—
151C2×#92418—41no—
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—22no—
163Q8×C61—22no—
164Q8×C61—41no—
165Q8×C61—41no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—81no—
175C2×C4×C61—81no—
176C2×C4×C61—81no—
177C2×C4×C61C2×C22⋊C411yescharacteristic
178C2×C4×C61C2×C22⋊C411yescharacteristic
179C2×C4×C61C2×C22⋊C411yescharacteristic
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61#12011yescharacteristic
182C2×C4×C61#12011yescharacteristic
183C2×C4×C61C2×SD1611yescharacteristic
184C2×C4×C61C2×Q1611yescharacteristic
185C2×C4×C61#13511yescharacteristic
186C2×C4×C61#13511yescharacteristic
187C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C61#13711yescharacteristic
189C2×C4×C61#13711yescharacteristic
190C2×C4×C61#13711yescharacteristic
191C2×Dic61—22no—
192C2×Dic61—24no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197D244—22no—
198D244—22no—
199D244—22no—
200D244#10314yescomplemented
201C4×D61—42no—
202C4×D61—42no—
203C4×D61—42no—
204C4×D61—42no—
205C4×D61—88no—
206Dic122—44no—
207Dic122—82no—
208C8×C61—81no—
209#92418—42no—
210#92418—81no—
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—42no—
229C4×C61—42no—
230C4×C61—161no—
231C4×C61C2×#10311yescharacteristic
232C4×C61#41611yescharacteristic
233C4×C61#41711yescharacteristic
234C4×C61#41911yescharacteristic
235C4×C61#44811yescharacteristic
236C4×C61#44911yescharacteristic
237C4×C61#45111yescharacteristic
238C4×C61#45211yescharacteristic
239Dic61—44no—
240Dic61—42no—
241C2×#554—611no—
242#1956—611no—
243#1803D6111yescharacteristic, complemented
244#1802—611no—
245C2×#333—611no—
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⋊Q8—611no—
261C2×#125—1221no—
262C2×C22⋊Q8—1222no—
263C2×C22⋊Q8—1221no—
264C2×C4×Q8C2×D6111yescharacteristic
265C2×C4×D4—611no—
266C2×C4×D4—1221no—
267C22×C4⋊C4—1221no—
268C2×C4⋊C8—611no—
269C2×C4⋊C8C2×D6111yescharacteristic
270C2×#103—21no—
271C2×#103—1222no—
272C2×#102—1221no—
273C2×C4×C8—611no—
274#384C2×D6111yescharacteristic
275#340—21no—
276#340—1221no—
277#333—1222no—
278Q8⋊C8—1222no—
279D4⋊C8—1222no—
280C61#845011yescharacteristic, complemented, Sylow
281C22×Q8—612no—
282C22×Q8C22⋊D6111yescharacteristic
283C22×D4—611no—
284C23×C4—1222no—
285C23×C4—1221no—
286C22×C8—21no—
287C22×C8—1221no—
288C22×C8—1221no—
289C4⋊Q8—612no—
290C4⋊Q8—612no—
291C4⋊Q8—1221no—
292C4⋊Q8—1221no—
293#125—1224no—
294#125—1224no—
295C22⋊Q8—1228no—
296C22⋊Q8—1228no—
297C22⋊Q8—1224no—
298C22⋊Q8—1224no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8C4×D6112yes—
302C4×Q8D24412yes—
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—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—611no—
314C2×C4⋊C4—1224no—
315C2×C4⋊C4—1222no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1224no—
320C2×C4⋊C4C22⋊D6111yescharacteristic
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1221no—
323C2×C42—21no—
324C2×C42C22×D6111yescharacteristic
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—1221no—
328C4⋊C8—1221no—
329#103—42no—
330#103—2444no—
331#102—2442no—
332C4×C8—1221no—
333C4×C8—1221no—
334C24—1221no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—614no—
341C2×Q8—1222no—
342C2×Q8—2442no—
343C2×Q8—2442no—
344C2×Q8#1175819812yes—
345C2×D4—612no—
346C2×D4—612no—
347C2×D4—612no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—41no—
354C22×C4—1228no—
355C22×C4—1221no—
356C22×C4—1222no—
357C22×C4—1222no—
358C22×C4—1222no—
359C22×C4—4882no—
360C22×C4—4882no—
361C22×C4—2441no—
362C22×C4—2441no—
363C22×C4—2441no—
364C22×C4—2441no—
365C22×C4D4×D6111yescharacteristic
366C22×C4D4×D6111yescharacteristic
367C22×C4C2×C22⋊D6111yescharacteristic
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—1222no—
372C2×C8—2441no—
373C2×C8—2441no—
374C2×C8—2441no—
375C2×C8—2441no—
376C2×C8—2441no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—612no—
386C4⋊C4—1228no—
387C4⋊C4—1228no—
388C4⋊C4—1222no—
389C4⋊C4—1222no—
390C4⋊C4—1222no—
391C4⋊C4—1221no—
392C4⋊C4—1222no—
393C4⋊C4#1175819812yes—
394C22⋊C4—1228no—
395C22⋊C4—1222no—
396C22⋊C4—1222no—
397C42—22no—
398C42—22no—
399C42C2×C4×D6111yescharacteristic
400C42C2×D24411yescharacteristic
401C42D4×D6111yescharacteristic
402C42D4×D6111yescharacteristic
403C23—1222no—
404C23—1222no—
405C23—1222no—
406C23—2441no—
407C23—2441no—
408C23—2441no—
409C23—2441no—
410C23#1451974511yescharacteristic
411Q8—22no—
412Q8—22no—
413Q8—41no—
414Q8—41no—
415Q8—4882no—
416Q8—2444no—
417D4—614no—
418D4—1222no—
419D4—1222no—
420D4—1222no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—1222no—
433C2×C4—1224no—
434C2×C4—4888no—
435C2×C4—2442no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2442no—
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#1451974711yescharacteristic
448C2×C4#1451974711yescharacteristic
449C2×C4SD16×D6111yescharacteristic
450C2×C4Q16×D6111yescharacteristic
451C2×C4#1451979711yescharacteristic
452C2×C4#1451979711yescharacteristic
453C2×C4#1451980311yescharacteristic
454C2×C4#1451980611yescharacteristic
455C2×C4#1451980811yescharacteristic
456C2×C4#1451980911yescharacteristic
457C8—81no—
458C8—4881no—
459C8—2442no—
460C22—1224no—
461C22—2442no—
462C22—2442no—
463C22—2442no—
464C22—2442no—
465C22—2442no—
466C22—2442no—
467C22#1683961711yescharacteristic
468C22#1683989911yescharacteristic
469C22#1683990511yescharacteristic
470C22#1683991611yescharacteristic
471C22#1683992211yescharacteristic
472C22#1683992811yescharacteristic
473C22#1683993811yescharacteristic
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—161no—
479C4—2442no—
480C4—2444no—
481C4#103×D6111yescharacteristic
482C4#1683975611yescharacteristic
483C4#1683976111yescharacteristic
484C4#1683976911yescharacteristic
485C4#1683990211yescharacteristic
486C4#1683990511yescharacteristic
487C4#1683991911yescharacteristic
488C4#1683992211yescharacteristic
489C2—2444no—
490C2#2144489011yescharacteristic
491C2#2144495211yescharacteristic
492C2#2144502511yescharacteristic
493C2#2144503111yescharacteristic
494C2#2144575211yescharacteristic
495C2#2144575711yescharacteristic
496C2#2144577111yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1803C2
socleC23×C61#120
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1803D61
hypocenterC61#8450
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. #21446620C2semidirect
4. C2×#16839510C2semidirect
5. C2×Q8⋊#92418C2semidirect
7. C61×#1803C2semidirect
11. #16840472C4semidirect
200. D244#103semidirect
243. #1803D61semidirect
280. C61#8450semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
#214466202
#214458162
C2×#168395102
C2×Q8⋊#924182
C2×#168394892
C61×#18032
#845061