Finite Group Databasebeta

← #29571957

Subgroups — #29571957

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#21446743C211yesmaximal, characteristic, complemented
3#21445836C211yesmaximal, characteristic
4#21444510C211yesmaximal, characteristic, complemented
5#21444319C211yesmaximal, characteristic, complemented
6#21444207C211yesmaximal, characteristic
7C61×#1548C211yesmaximal, 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
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519703—21no—
23C2×#14519701—22no—
24C4×C4⋊Dic61C2211yescharacteristic
25#16839523—21no—
26#16839510—21no—
27#16839507—21no—
28#16839500—22no—
29#16839494—21no—
30#16839491—21no—
31C2×C4⋊Q8×C61C2211yescharacteristic
32C61×#431C2211yescharacteristic
33C61×#423C2211yescharacteristic
34C61×#333—22no—
35C2×#11758222D411yescharacteristic
36C2×Q8×D61—22no—
37C22×#92418—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—
73C2×C4⋊Dic61—21no—
74C4⋊C4×D61—22no—
75C4⋊C4×D61—22no—
76C2×C4×Dic61—21no—
77C42×D61—21no—
78C4⋊#92418—21no—
79C4⋊#92418—21no—
80#14519703—42no—
81#14519701—44no—
82#14519684—21no—
83C22×Q8×C61D412yes—
84C2×M16×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61C2×C412yes—
88C4⋊Q8×C61D412yes—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C42×C61C2311yescharacteristic
92C4⋊C8×C61—21no—
93C4⋊C8×C61—21no—
94C4⋊C8×C61—21no—
95C22⋊C8×C61—21no—
96C4×C8×C61—21no—
97C2×Q8×C61—22no—
98C2×Q8×C61—42no—
99C2×Q8×C61—42no—
100C2×Q8×C61C22⋊C414yes—
101C22×C4×C61—22no—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C2×C22⋊D61—21no—
106C22×Dic61—41no—
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—
122M16×C61—41no—
123M16×C61—41no—
124C2×C8×C61—41no—
125C2×C8×C61—41no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C42×C61C22×C411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61C2×D411yescharacteristic
133C42×C61C2×D411yescharacteristic
134C22⋊Dic61—22no—
135C22⋊Dic61—21no—
136#11758198—24no—
137#11758198—21no—
138#11758198—21no—
139C4⋊Dic61—24no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—22no—
144C4⋊Dic61—41no—
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—
160C4×Dic61—41no—
161C4×Dic61—41no—
162C2×#92418—42no—
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—82no—
174C2×C4×C61C2×C22⋊C411yescharacteristic
175C2×C4×C61C2×C22⋊C411yescharacteristic
176C2×C4×C61C2×C22⋊C411yescharacteristic
177C2×C4×C61#12011yescharacteristic
178C2×C4×C61#12011yescharacteristic
179C2×C4×C61#12011yescharacteristic
180C2×C4×C61C2×SD1611yescharacteristic
181C2×C4×C61C2×Q1611yescharacteristic
182C2×C4×C61#13711yescharacteristic
183C2×C4×C61#13711yescharacteristic
184C22⋊D61—22no—
185C22⋊D61—22no—
186C2×Dic61—21no—
187C2×Dic61—22no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—22no—
191C2×Dic61—42no—
192C2×Dic61—81no—
193C2×Dic61—81no—
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—
209C8×C61—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×C61C2×#10311yescharacteristic
225C4×C61#41911yescharacteristic
226C4×C61#44811yescharacteristic
227C4×C61#45111yescharacteristic
228C4×C61#45311yescharacteristic
229C4×C61#45511yescharacteristic
230Dic61—22no—
231Dic61—42no—
232Dic61—41no—
233Dic61—42no—
234Dic61—41no—
235Dic61—82no—
236#3503—611no—
237#1981—611no—
238#1806—611no—
239#1550—611no—
240#1548D6111yescharacteristic, complemented
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#431C2×D6111yescharacteristic
264#423C2×D6111yescharacteristic
265C2×C4⋊C8—611no—
266C2×#104—1221no—
267C2×#103—1222no—
268C4×C4⋊C4—611no—
269#339—1221no—
270#333—22no—
271#333—1221no—
272#332—1221no—
273#328—1222no—
274Q8⋊C8—1221no—
275D4⋊C8—1221no—
276C61#842511yescharacteristic, complemented, Sylow
277C2×#41—611no—
278C22×Q8—1222no—
279C22×Q8D24412yes—
280C2×M16—21no—
281C22×C8—1221no—
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⋊Q8C22⋊D6112yes—
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—1222no—
324C2×C4⋊C4—1222no—
325C2×C4⋊C4—1221no—
326C2×C42—1221no—
327C2×C42—1221no—
328C2×C42C22×D6111yescharacteristic
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—1221no—
333C4⋊C8—1221no—
334#104—2442no—
335#103—2444no—
336C22⋊C8—21no—
337C4×C8—21no—
338#95—1221no—
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—1224no—
357C22×C4—1221no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—2441no—
361C22×C4C2×D24411yescharacteristic
362C22×C4D4×D6111yescharacteristic
363C22×C4D4×D6111yescharacteristic
364M16—41no—
365M16—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—2442no—
369C2×C8—2441no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—612no—
375C4⋊C4—611no—
376C4⋊C4—611no—
377C4⋊C4—1224no—
378C4⋊C4—1224no—
379C4⋊C4—1224no—
380C4⋊C4—1224no—
381C4⋊C4—1222no—
382C4⋊C4—1221no—
383C4⋊C4—1222no—
384C4⋊C4—1221no—
385C4⋊C4—1221no—
386C4⋊C4—1222no—
387C4⋊C4—1221no—
388C4⋊C4—1221no—
389C4⋊C4—2441no—
390C22⋊C4—1224no—
391C22⋊C4—1221no—
392C22⋊C4—1222no—
393C22⋊C4—1221no—
394C22⋊C4—1221no—
395C42—1224no—
396C42—1221no—
397C42—1222no—
398C42—1221no—
399C42—1221no—
400C42—2441no—
401C42—2441no—
402C42C2×C4×D6111yescharacteristic
403C42D4×D6111yescharacteristic
404C42D4×D6111yescharacteristic
405C42C2×C22⋊D6111yescharacteristic
406C23—1221no—
407C23#1451974711yescharacteristic
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—82no—
426C2×C4—1221no—
427C2×C4—1222no—
428C2×C4—1222no—
429C2×C4—1222no—
430C2×C4—1221no—
431C2×C4—1221no—
432C2×C4—1222no—
433C2×C4—4884no—
434C2×C4—4881no—
435C2×C4—4881no—
436C2×C4—2442no—
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#1451974511yescharacteristic
445C2×C4#1451974711yescharacteristic
446C2×C4SD16×D6111yescharacteristic
447C2×C4Q16×D6111yescharacteristic
448C2×C4#1451980611yescharacteristic
449C2×C4#1451980911yescharacteristic
450C8—81no—
451C8—81no—
452C8—4881no—
453C22—21no—
454C22—21no—
455C22—1222no—
456C22—2441no—
457C22—2441no—
458C22#1683961711yescharacteristic
459C22#1683990211yescharacteristic
460C22#1683991911yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—44no—
465C4—1222no—
466C4—4882no—
467C4—2442no—
468C4—2441no—
469C4—2441no—
470C4—2442no—
471C4#103×D6111yescharacteristic
472C4#1683976911yescharacteristic
473C4#1683989911yescharacteristic
474C4#1683991611yescharacteristic
475C4#1683993211yescharacteristic
476C4#1683994211yescharacteristic
477C2—21no—
478C2—21no—
479C2—2442no—
480C2#2144575211yescharacteristic
481C2#2144577511yescharacteristic
482C2#2144579111yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1548C2
socleC22×C61#379
prime radicalC2×#11758222D4
cosocleC2×C42×C61C23
hypercenter#1548D61
hypocenterC61#8425
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
2. #21446743C2semidirect
4. #21444510C2semidirect
5. #21444319C2semidirect
7. C61×#1548C2semidirect
8. #16840492C4semidirect
10. #16840473C4semidirect
195. D244#103semidirect
207. Dic122#103semidirect
240. #1548D61semidirect
276. C61#8425semidirect

Maximal subgroups

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

subgroupindex
#214596352
#214467432
#214458362
#214445102
#214443192
#214442072
C61×#15482
#842561