Finite Group Databasebeta

← #29571955

Subgroups — #29571955

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#21459631C211yesmaximal, characteristic
2#21446742C211yesmaximal, characteristic, complemented
3#21445840C211yesmaximal, characteristic
4#21444509C211yesmaximal, characteristic, complemented
5#21444499C211yesmaximal, characteristic, complemented
6#21444207C211yesmaximal, characteristic
7C61×#1503C211yesmaximal, characteristic, complemented
8Q8×Dic122C412yescomplemented
9#16840490—21no—
10#16840472C412yescomplemented
11#16840470—22no—
12Q8×D244—21no—
13#16840425—22no—
14#16840392—22no—
15#16840390—21no—
16#16840317—21no—
17C2×#14519777C2211yescharacteristic
18C4⋊Q8×D61—21no—
19#16840242—21no—
20C4×#11758222C2211yescharacteristic
21#16839786C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519705—21no—
24C2×#14519700—22no—
25#16839522—21no—
26#16839511—21no—
27#16839510—22no—
28#16839508—21no—
29#16839494—21no—
30#16839491—21no—
31C61×#533C2211yescharacteristic
32C61×#431C2211yescharacteristic
33C4×C4⋊C4×C61C2211yescharacteristic
34C61×#328—22no—
35C2×#11758222D411yescharacteristic
36C22×Dic122D412yes—
37C2×Q8×D61—22no—
38C2×#11758194—21no—
39C22×#92418—21no—
40#14519777—21no—
41#14519777—21no—
42#14519777—22no—
43#14519777C2×C412yes—
44#14519777D412yes—
45#14519776—22no—
46#14519776—24no—
47#14519776—22no—
48#14519775—22no—
49#14519769—22no—
50#14519768—22no—
51#14519767—24no—
52#14519757—24no—
53#14519757—22no—
54#14519755—24no—
55#14519754—24no—
56#14519753—22no—
57Q8×Dic61—22no—
58C4×Dic122—22no—
59C4×Dic122C2×C411yescharacteristic
60C4×Dic122D411yescharacteristic
61#14519742—22no—
62C4×D244—22no—
63C4×D244C2×C411yescharacteristic
64C4×D244D411yescharacteristic
65C4×C22⋊D61—21no—
66C4×C22⋊D61—21no—
67#14519737—24no—
68#14519736D411yescharacteristic
69#14519735—22no—
70#14519734—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C4⋊C4×D61—22no—
74C4⋊C4×D61—22no—
75C42×D61—21no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418—21no—
79C4⋊#92418—21no—
80#14519705—42no—
81#14519700—44no—
82C4⋊Q8×C61D411yescharacteristic
83C4⋊Q8×C61D411yescharacteristic
84#125×C61—21no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61D412yes—
87C4×Q8×C61—21no—
88C4×Q8×C61C2×C412yes—
89#117×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C42×C61—21no—
92C2×C42×C61C2311yescharacteristic
93C4⋊C8×C61—21no—
94C22⋊C8×C61—21no—
95C4×C8×C61—21no—
96#95×C61—21no—
97C2×Q8×C61—21no—
98C2×Q8×C61C22⋊C412yes—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—21no—
104C22×Dic61—22no—
105Q8×D61—82no—
106Q8×D61—82no—
107#11758222—21no—
108#11758222—21no—
109#11758222SD1612yes—
110#11758222Q1612yes—
111C2×D244C22⋊C411yescharacteristic
112C2×C4×D61—21no—
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—24no—
116C2×Dic122—22no—
117C2×Dic122—24no—
118C2×Dic122—42no—
119C2×Dic122—42no—
120C2×Dic122C22⋊C411yescharacteristic
121C2×Dic122C22⋊C414yes—
122C2×C8×C61—41no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—41no—
130C4⋊C4×C61C22⋊C412yes—
131C22⋊C4×C61—22no—
132C42×C61—22no—
133C42×C61—41no—
134C42×C61—41no—
135C42×C61C22×C411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C22⋊Dic61—21no—
140#11758198—24no—
141#11758198—21no—
142#11758198—21no—
143C4⋊Dic61—24no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146C4⋊Dic61—22no—
147C4⋊Dic61—22no—
148C4⋊Dic61C22⋊C411yescharacteristic
149C4⋊Dic61C22⋊C411yescharacteristic
150#11758196—24no—
151#11758196—24no—
152#11758196—24no—
153#11758196—21no—
154#11758196—21no—
155C4×Dic61—24no—
156C4×Dic61—21no—
157C4×Dic61—21no—
158C4×Dic61—21no—
159#11758194—41no—
160#11758194—41no—
161C2×#92418—42no—
162C2×#92418—41no—
163C2×#92418—41no—
164C23×C61#12011yescharacteristic
165C22×D61—21no—
166Q8×C61—42no—
167Q8×C61—41no—
168Q8×C61—41no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
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—21no—
191C2×Dic61—21no—
192C2×Dic61—24no—
193C2×Dic61—82no—
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—42no—
204Dic122—42no—
205Dic122—44no—
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—42no—
224C4×C61—42no—
225C4×C61—82no—
226C4×C61C2×#10311yescharacteristic
227C4×C61#41911yescharacteristic
228C4×C61#44811yescharacteristic
229C4×C61#45111yescharacteristic
230C4×C61#45311yescharacteristic
231C4×C61#45511yescharacteristic
232Dic61—22no—
233Dic61—44no—
234Dic61—41no—
235Dic61—41no—
236#3503—611no—
237#1981—611no—
238#1806—611no—
239#1550—611no—
240#1548—611no—
241#1503D6111yescharacteristic, complemented
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—1221no—
258#533C2×D6111yescharacteristic
259C2×C4⋊Q8—611no—
260C2×C4⋊Q8—1221no—
261#520—1221no—
262C4×#41—611no—
263#431C2×D6111yescharacteristic
264#423—611no—
265C2×C4⋊C8—611no—
266C2×#104—1221no—
267C2×#103—1222no—
268C4×C4⋊C4C2×D6111yescharacteristic
269#339—1221no—
270#333—1222no—
271#333—1221no—
272#332—1221no—
273#328—22no—
274Q8⋊C8—1221no—
275D4⋊C8—1221no—
276C61#842511yescharacteristic, complemented, Sylow
277C2×#41—611no—
278C22×Q8—612no—
279C22×Q8—1222no—
280C2×M16—1221no—
281C22×C8—1221no—
282C4⋊Q8—612no—
283C4⋊Q8—612no—
284C4⋊Q8—1222no—
285C4⋊Q8—1224no—
286C4⋊Q8—1222no—
287C4⋊Q8—1222no—
288C4⋊Q8—1222no—
289C4⋊Q8—1221no—
290C4⋊Q8—1221no—
291C4⋊Q8D24411yescharacteristic
292C4⋊Q8D24411yescharacteristic
293#125—21no—
294#125—1222no—
295#125—1222no—
296#125—1224no—
297C22⋊Q8—21no—
298C22⋊Q8—1224no—
299C22⋊Q8—1224no—
300C22⋊Q8—1222no—
301C22⋊Q8—1224no—
302C22⋊Q8—1222no—
303C22⋊Q8C22⋊D6112yes—
304C4×Q8—21no—
305C4×Q8—611no—
306C4×Q8—611no—
307C4×Q8—1222no—
308C4×Q8—1222no—
309C4×Q8—1222no—
310C4×Q8C4×D6112yes—
311C4×D4—611no—
312C4×D4—611no—
313C4×D4—1222no—
314C4×D4—1224no—
315C4×D4—1221no—
316C4×D4—1221no—
317#117—21no—
318#117—611no—
319#117—1222no—
320#117—1221no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—1222no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4—1221no—
325C2×C4⋊C4—1222no—
326C2×C42—21no—
327C2×C42—1221no—
328C2×C42C22×D6111yescharacteristic
329C4⋊C8—21no—
330C4⋊C8—1221no—
331C4⋊C8—1221no—
332C4⋊C8—1221no—
333C4⋊C8—1221no—
334#104—2442no—
335#103—2444no—
336C22⋊C8—21no—
337C4×C8—21no—
338#95—21no—
339#41—612no—
340#41—612no—
341#41—1221no—
342#41—1221no—
343C2×Q8—21no—
344C2×Q8—611no—
345C2×Q8—614no—
346C2×Q8—1224no—
347C2×Q8—1222no—
348C2×Q8—4882no—
349C2×Q8—4882no—
350C2×Q8—2442no—
351C2×Q8—2442no—
352C2×Q8#1175819812yes—
353C2×D4—611no—
354C2×D4—1221no—
355C22×C4—41no—
356C22×C4—1224no—
357C22×C4—1221no—
358C22×C4—1222no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4D4×D6111yescharacteristic
362C22×C4D4×D6111yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364M16—2441no—
365M16—2441no—
366C2×C8—41no—
367C2×C8—2442no—
368C2×C8—2441no—
369C2×C8—2441no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—41no—
377C4⋊C4—611no—
378C4⋊C4—611no—
379C4⋊C4—1224no—
380C4⋊C4—1224no—
381C4⋊C4—1224no—
382C4⋊C4—1224no—
383C4⋊C4—1222no—
384C4⋊C4—1221no—
385C4⋊C4—1222no—
386C4⋊C4—1222no—
387C4⋊C4—1221no—
388C4⋊C4—1222no—
389C4⋊C4#1175819812yes—
390C22⋊C4—22no—
391C22⋊C4—1224no—
392C22⋊C4—1221no—
393C22⋊C4—1221no—
394C22⋊C4—1221no—
395C42—22no—
396C42—41no—
397C42—41no—
398C42—1224no—
399C42—1221no—
400C42—1221no—
401C42—1221no—
402C42C2×C4×D6111yescharacteristic
403C42C2×D24411yescharacteristic
404C42D4×D6111yescharacteristic
405C42D4×D6111yescharacteristic
406C23—1221no—
407C23#1451974511yescharacteristic
408Q8—42no—
409Q8—41no—
410Q8—41no—
411Q8—612no—
412Q8—1221no—
413Q8—4884no—
414Q8—2444no—
415Q8—2442no—
416Q8—2442no—
417D4—612no—
418D4—1222no—
419D4—1222no—
420D4—1221no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—1221no—
430C2×C4—1222no—
431C2×C4—1222no—
432C2×C4—1221no—
433C2×C4—1221no—
434C2×C4—1224no—
435C2×C4—4884no—
436C2×C4—4882no—
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—42no—
465C4—42no—
466C4—82no—
467C4—1222no—
468C4—2444no—
469C4—2441no—
470C4—2441no—
471C4#103×D6111yescharacteristic
472C4#1683976911yescharacteristic
473C4#1683990211yescharacteristic
474C4#1683991911yescharacteristic
475C4#1683993111yescharacteristic
476C4#1683994111yescharacteristic
477C2—21no—
478C2—21no—
479C2—2442no—
480C2#2144575211yescharacteristic
481C2#2144577411yescharacteristic
482C2#2144579211yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1503C2
socleC22×C61#379
prime radicalC2×#11758222D4
cosocleC2×C42×C61C23
hypercenter#1503D61
hypocenterC61#8425
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. #21446742C2semidirect
4. #21444509C2semidirect
5. #21444499C2semidirect
7. C61×#1503C2semidirect
8. Q8×Dic122C4semidirect
10. #16840472C4semidirect
195. D244#103semidirect
207. Dic122#103semidirect
241. #1503D61semidirect
276. C61#8425semidirect

Maximal subgroups

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

subgroupindex
#214596312
#214467422
#214458402
#214445092
#214444992
#214442072
C61×#15032
#842561