Finite Group Databasebeta

← #29571961

Subgroups — #29571961

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459635C211yesmaximal, characteristic
2#21445840C211yesmaximal, characteristic
3#21444507C211yesmaximal, characteristic, complemented
4#21444500C211yesmaximal, characteristic, complemented
5#21444317C211yesmaximal, characteristic, complemented
6#21444204C211yesmaximal, characteristic
7C61×#1981C211yesmaximal, characteristic, complemented
8#16840492C412yescomplemented
9#16840487—21no—
10#16840473C412yescomplemented
11#16840467—22no—
12#16840427—21no—
13Q8×C22⋊D61—22no—
14#16840393—21no—
15#16840389—22no—
16#16840319C2211yescharacteristic
17#16840317—21no—
18C4⋊Q8×D61—21no—
19#16840243—21no—
20C4×#11758222C2211yescharacteristic
21#16839815C2211yescharacteristic
22#16839786C2211yescharacteristic
23C2×#14519705—21no—
24#16839522—21no—
25#16839511—22no—
26#16839510—21no—
27#16839509—21no—
28#16839498—22no—
29#16839492—21no—
30#16839489—21no—
31C2×C4⋊Q8×C61C2211yescharacteristic
32C2×C4⋊C8×C61C2211yescharacteristic
33C2×#103×C61—22no—
34C4×C4⋊C4×C61C2211yescharacteristic
35C2×#11758222D411yescharacteristic
36C2×Q8×D61—22no—
37C2×#11758194—21no—
38#14519777D411yescharacteristic
39#14519777D411yescharacteristic
40#14519776—22no—
41#14519776—22no—
42#14519776—22no—
43#14519775—22no—
44#14519775—24no—
45#14519769—21no—
46#14519768—22no—
47#14519768—22no—
48#14519767—24no—
49#14519757—22no—
50#14519756—24no—
51#14519756—21no—
52#14519756D412yes—
53#14519755—24no—
54#14519753—22no—
55#14519753—24no—
56Q8×Dic61—22no—
57C4×Dic122—21no—
58C4×Dic122C2×C412yes—
59C4×Dic122C2×C411yescharacteristic
60C4×Dic122D411yescharacteristic
61#14519742—22no—
62#14519742—22no—
63C4×D244C2×C411yescharacteristic
64C4×D244D411yescharacteristic
65#14519739—22no—
66C4×C22⋊D61—24no—
67C4×C22⋊D61—21no—
68C4×C22⋊D61—21no—
69#14519736—21no—
70#14519736D411yescharacteristic
71#14519735—22no—
72#14519734—21no—
73C4⋊C4×D61—22no—
74C4⋊C4×D61—22no—
75C42×D61—21no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418—21no—
79#14519705—42no—
80C22⋊#92418—21no—
81C4×#92418—21no—
82C22×Q8×C61D412yes—
83C22×C8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61C2×C412yes—
87C4⋊Q8×C61D412yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C42×C61—21no—
92C2×C42×C61C2311yescharacteristic
93C4⋊C8×C61—21no—
94C4⋊C8×C61—21no—
95#103×C61—44no—
96#95×C61—21no—
97C2×Q8×C61—22no—
98C2×Q8×C61—42no—
99C2×Q8×C61—42no—
100C2×Q8×C61C22⋊C414yes—
101C22×C4×C61—22no—
102C22×C4×C61—41no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C2×C22⋊D61—21no—
107Q8×D61—82no—
108Q8×D61—82no—
109#11758222—21no—
110#11758222—21no—
111#11758222SD1612yes—
112#11758222Q1612yes—
113C2×D244C22⋊C411yescharacteristic
114C2×C4×D61—21no—
115C2×C4×D61—21no—
116C2×C4×D61—21no—
117C2×C4×D61—24no—
118C2×Dic122—21no—
119C2×Dic122—24no—
120C2×Dic122C22⋊C411yescharacteristic
121C2×Dic122C22⋊C412yes—
122C2×C8×C61—42no—
123C2×C8×C61—41no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—41no—
130C42×C61—41no—
131C42×C61—41no—
132C42×C61C22×C411yescharacteristic
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C22⋊Dic61—22no—
137C22⋊Dic61—21no—
138#11758198—24no—
139#11758198—21no—
140#11758198—21no—
141C4⋊Dic61—24no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61C22⋊C412yes—
146C4⋊Dic61C22⋊C411yescharacteristic
147C4⋊Dic61C22⋊C411yescharacteristic
148#11758196—24no—
149#11758196—24no—
150#11758196—24no—
151#11758196—21no—
152#11758196—22no—
153#11758196—21no—
154#11758196—22no—
155C4×Dic61—24no—
156C4×Dic61—21no—
157C4×Dic61—22no—
158C4×Dic61—21no—
159C4×Dic61—21no—
160#11758194—41no—
161#11758194—41no—
162C2×#92418—41no—
163C2×#92418—41no—
164C23×C61#12011yescharacteristic
165C22×D61—21no—
166Q8×C61—44no—
167Q8×C61—42no—
168Q8×C61—42no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—24no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61—82no—
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#13711yescharacteristic
186C2×C4×C61#13711yescharacteristic
187C22⋊D61—22no—
188C22⋊D61—22no—
189C2×Dic61—21no—
190C2×Dic61—22no—
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—22no—
194D244—21no—
195D244#10312yescomplemented
196C4×D61—22no—
197C4×D61—22no—
198C4×D61—42no—
199C4×D61—41no—
200C4×D61—41no—
201C4×D61—84no—
202Dic122—21no—
203Dic122—41no—
204Dic122—41no—
205Dic122—42no—
206Dic122—84no—
207Dic122#10312yescomplemented
208C8×C61—81no—
209#92418—81no—
210#92418—81no—
211#8425—611nomaximal, Sylow
212C22×C61—21no—
213C22×C61—21no—
214C22×C61#37911yescharacteristic
215C22×C61#44811yescharacteristic
216C22×C61#45111yescharacteristic
217C2×D61—22no—
218C2×D61—41no—
219C2×D61—41no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—44no—
224C4×C61—82no—
225C4×C61C2×#10311yescharacteristic
226C4×C61#41911yescharacteristic
227C4×C61#44811yescharacteristic
228C4×C61#45111yescharacteristic
229C4×C61#45311yescharacteristic
230C4×C61#45511yescharacteristic
231Dic61—22no—
232Dic61—42no—
233Dic61—41no—
234Dic61—42no—
235Dic61—41no—
236#3503—611no—
237#1981D6111yescharacteristic, complemented
238#1806—611no—
239#1550—611no—
240#1548—611no—
241#1503—611no—
242#1477—611no—
243C2×C61—21no—
244C2×C61—21no—
245C2×C61#178711yescharacteristic
246C2×C61#179211yescharacteristic
247C2×C61#179611yescharacteristic
248D61—42no—
249Q82—612no—
250#557—1221no—
251#554—612no—
252#554—1222no—
253D4×Q8—1221no—
254D4×Q8—1222no—
255#544—1221no—
256#544—1222no—
257#533—611no—
258#533—1221no—
259C2×C4⋊Q8—1221no—
260C2×C4⋊Q8C2×D6111yescharacteristic
261#520—1221no—
262C4×#41—611no—
263#431—611no—
264#423—611no—
265C2×C4⋊C8C2×D6111yescharacteristic
266C2×#104—1221no—
267C2×#103—22no—
268C4×C4⋊C4C2×D6111yescharacteristic
269#339—1221no—
270#333—1221no—
271#333—1222no—
272#332—1221no—
273#328—1222no—
274Q8⋊C8—1221no—
275D4⋊C8—1221no—
276C61#842511yescharacteristic, complemented, Sylow
277C2×#41—611no—
278C22×Q8—1222no—
279C22×Q8C22⋊D6112yes—
280C2×M16—1221no—
281C22×C8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—611no—
285C4⋊Q8—611no—
286C4⋊Q8—1222no—
287C4⋊Q8—1222no—
288C4⋊Q8—1222no—
289C4⋊Q8—1222no—
290C4⋊Q8—1224no—
291C4⋊Q8C4×D6112yes—
292C4⋊Q8D24412yes—
293#125—1222no—
294#125—1222no—
295#125—1224no—
296#125—1221no—
297C22⋊Q8—612no—
298C22⋊Q8—1222no—
299C22⋊Q8—1224no—
300C22⋊Q8—1224no—
301C22⋊Q8—1222no—
302C22⋊Q8—1224no—
303C22⋊Q8—1221no—
304C4×Q8—611no—
305C4×Q8—612no—
306C4×Q8—611no—
307C4×Q8—1222no—
308C4×Q8—1222no—
309C4×Q8—1222no—
310C4×Q8—1221no—
311C4×D4—611no—
312C4×D4—611no—
313C4×D4—1222no—
314C4×D4—1224no—
315C4×D4—1221no—
316C4×D4—1221no—
317#117—611no—
318#117—1222no—
319#117—1221no—
320#117—1221no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—1222no—
325C2×C4⋊C4—1222no—
326C2×C42—21no—
327C2×C42—1221no—
328C2×C42C22×D6111yescharacteristic
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—1221no—
332C4⋊C8—1221no—
333C4⋊C8—1221no—
334#104—2442no—
335#103—44no—
336C22⋊C8—1221no—
337C4×C8—1221no—
338#95—21no—
339#41—612no—
340#41—612no—
341#41—1221no—
342#41—1221no—
343C2×Q8—22no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—611no—
347C2×Q8—612no—
348C2×Q8—1224no—
349C2×Q8—1221no—
350C2×Q8—4882no—
351C2×Q8—4882no—
352C2×Q8#1175819814yes—
353C2×D4—611no—
354C2×D4—1221no—
355C22×C4—22no—
356C22×C4—41no—
357C22×C4—1224no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4D4×D6111yescharacteristic
362C22×C4D4×D6111yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364M16—2441no—
365M16—2441no—
366C2×C8—42no—
367C2×C8—41no—
368C2×C8—2441no—
369C2×C8—2441no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—41no—
376C4⋊C4—612no—
377C4⋊C4—611no—
378C4⋊C4—611no—
379C4⋊C4—1224no—
380C4⋊C4—1224no—
381C4⋊C4—1224no—
382C4⋊C4—1224no—
383C4⋊C4—1221no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1222no—
389C4⋊C4—1221no—
390C22⋊C4—1224no—
391C22⋊C4—1221no—
392C22⋊C4—1222no—
393C22⋊C4—1221no—
394C22⋊C4—1221no—
395C42—41no—
396C42—41no—
397C42—1224no—
398C42—1221no—
399C42—1222no—
400C42—1221no—
401C42—1221no—
402C42C2×C4×D6111yescharacteristic
403C42C2×D24411yescharacteristic
404C42D4×D6111yescharacteristic
405C42D4×D6111yescharacteristic
406C23—1221no—
407C23#1451974511yescharacteristic
408Q8—44no—
409Q8—42no—
410Q8—42no—
411Q8—612no—
412Q8—1221no—
413Q8—4884no—
414Q8—2442no—
415Q8—2441no—
416Q8—2441no—
417D4—612no—
418D4—1222no—
419D4—1222no—
420D4—1221no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—24no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—1221no—
430C2×C4—1222no—
431C2×C4—1222no—
432C2×C4—1222no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—1222no—
436C2×C4—4884no—
437C2×C4—2442no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4C22⋊C4×D6111yescharacteristic
441C2×C4C22⋊C4×D6111yescharacteristic
442C2×C4C2×#1175819811yescharacteristic
443C2×C4#1451974511yescharacteristic
444C2×C4#1451974711yescharacteristic
445C2×C4#1451974711yescharacteristic
446C2×C4SD16×D6111yescharacteristic
447C2×C4Q16×D6111yescharacteristic
448C2×C4#1451980611yescharacteristic
449C2×C4#1451980911yescharacteristic
450C8—81no—
451C8—4881no—
452C8—4881no—
453C22—21no—
454C22—21no—
455C22—1222no—
456C22—2441no—
457C22—2441no—
458C22#1683961711yescharacteristic
459C22#1683989911yescharacteristic
460C22#1683991611yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—44no—
465C4—82no—
466C4—1222no—
467C4—2442no—
468C4—2441no—
469C4—2441no—
470C4—2442no—
471C4#103×D6111yescharacteristic
472C4#1683976911yescharacteristic
473C4#1683990211yescharacteristic
474C4#1683991911yescharacteristic
475C4#1683992911yescharacteristic
476C4#1683993911yescharacteristic
477C2—21no—
478C2—21no—
479C2—2442no—
480C2#2144575211yescharacteristic
481C2#2144577211yescharacteristic
482C2#2144579111yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1981C2
socleC22×C61#379
prime radicalC2×#11758222D4
cosocleC2×C42×C61C23
hypercenter#1981D61
hypocenterC61#8425
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
3. #21444507C2semidirect
4. #21444500C2semidirect
5. #21444317C2semidirect
7. C61×#1981C2semidirect
8. #16840492C4semidirect
10. #16840473C4semidirect
195. D244#103semidirect
207. Dic122#103semidirect
237. #1981D61semidirect
276. C61#8425semidirect

Maximal subgroups

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

subgroupindex
#214596352
#214458402
#214445072
#214445002
#214443172
#214442042
C61×#19812
#842561