Finite Group Databasebeta

← #29686921

Subgroups — #29686921

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840038C211yesmaximal, characteristic
2#21454637C212yesmaximal
3C2×#16839868C211yesmaximal, characteristic, complemented
4#21454246C211yesmaximal, characteristic
5#21446734C211yesmaximal, characteristic, complemented
6#21446473C212yesmaximal, complemented
7#21446335C212yesmaximal
8#21445879C211yesmaximal, characteristic
9#21445727C211yesmaximal, characteristic, complemented
10#21445614C212yesmaximal, complemented
11C61×#1951C211yesmaximal, characteristic, complemented
12C2×#14519810C2212yes—
13C22×Dic244C2211yescharacteristic
14C2×#14519785C2211yescharacteristic
15C2×#14519783C2211yescharacteristic
16#16840230C2211yescharacteristic
17C2×#14519742C2212yes—
18#16840038C418yescomplemented
19#16839868—22no—
20#16839868—21no—
21#16839868—21no—
22#16839868—21no—
23#16839868—21no—
24#16839861—22no—
25#16839856—22no—
26#16839849—22no—
27C2×#14519713C2211yescharacteristic
28#16839770C2212yes—
29#16839761C2212yes—
30C2×#14519700C2212yes—
31C2×#14519695C2212yes—
32C2×#14519695C2212yes—
33#16839719—21no—
34#16839718C2211yescharacteristic
35#16839716—21no—
36M16×Dic61C2212yes—
37M16×Dic61C2211yescharacteristic
38C4×#11758189—21no—
39C2×#14519687C2211yescharacteristic
40#97×D61—21no—
41#16839531C2212yes—
42#16839526C2212yes—
43C2×#108×C61C2211yescharacteristic
44C61×#428C2212yes—
45C2×#106×C61C2211yescharacteristic
46C61×#405C2211yescharacteristic
47C61×#343C2212yes—
48C2×#11758222C2311yescharacteristic
49C22×Dic122C2312yes—
50#14519810—24no—
51#14519810C2×C418yes—
52C2×Dic244—22no—
53C2×Dic244—21no—
54C2×Dic244—21no—
55C2×Dic244—22no—
56C2×Dic244C2×C414yescomplemented
57C2×#11758190—22no—
58#14519785—22no—
59#14519785—22no—
60#14519783—22no—
61#14519783C2×C414yes—
62C2×#11758194C2312yes—
63C2×#11758194C2311yescharacteristic
64C2×#11758189—21no—
65C2×#11758189—21no—
66C22×#92418C2311yescharacteristic
67C2×C8×D61—21no—
68C2×C8×D61—21no—
69#14519742—22no—
70#14519742—22no—
71#14519742—22no—
72#14519742—22no—
73#14519742D412yes—
74#14519739D412yes—
75#14519737—22no—
76#14519734—21no—
77C2×#11758196—22no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61C2311yescharacteristic
80C42×D61—21no—
81#14519713—22no—
82#14519700—22no—
83#14519700—22no—
84#14519700—22no—
85#14519700—22no—
86#14519700—22no—
87#14519697—22no—
88#14519695—24no—
89#14519695C2×C418yescomplemented
90#14519688—21no—
91#14519687—21no—
92#14519687—22no—
93#14519687D411yescharacteristic
94#14519687D411yescharacteristic
95#14519687D411yescharacteristic
96#14519687D411yescharacteristic
97C4×#92418—21no—
98C8×Dic61—21no—
99C8×Dic61—22no—
100C2×M16×C61C2312yes—
101C2×M16×C61C2311yescharacteristic
102C22×C8×C61C2311yescharacteristic
103#117×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2312yes—
105#108×C61C2×C414yescomplemented
106#107×C61—22no—
107#106×C61—21no—
108#106×C61—21no—
109#106×C61—22no—
110#97×C61—21no—
111C4×C8×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61C2411yescharacteristic
114C2×C22⋊D61—21no—
115C22×Dic61—22no—
116C22×Dic61#4112yes—
117#11758222—21no—
118#11758222—21no—
119#11758222C22×C412yes—
120#11758222#4112yes—
121C2×D244C2×D411yescharacteristic
122C2×C4×D61—21no—
123C2×C4×D61—22no—
124C2×Dic122—22no—
125C2×Dic122—22no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122C22×C414yes—
130C2×Dic122C2×D411yescharacteristic
131M16×C61—22no—
132M16×C61—21no—
133M16×C61C22×C414yes—
134M16×C61#4112yes—
135C2×C8×C61—21no—
136C2×C8×C61—22no—
137C2×C8×C61C22×C412yes—
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61#4112yes—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61C2×D412yes—
144C22⋊C4×C61—21no—
145C42×C61—21no—
146#11758198—22no—
147#11758196—22no—
148#11758196—42no—
149#11758196—42no—
150C4×Dic61—22no—
151C4×Dic61—42no—
152C4×Dic61—42no—
153C4×Dic61C2×D412yes—
154C4×Dic61C2×D411yescharacteristic
155C4×Dic61C2×D411yescharacteristic
156#11758194—21no—
157#11758194—22no—
158#11758194—22no—
159#11758194C22×C412yes—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—22no—
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166Dic244—28no—
167Dic244—24no—
168Dic244—24no—
169#11758190—28no—
170#11758189—24no—
171#11758189—24no—
172C8×D61—24no—
173C8×D61—24no—
174C23×C61C2×#4111yescharacteristic
175C22×D61—21no—
176C2×C4×C61—22no—
177C2×C4×C61—42no—
178C2×C4×C61—82no—
179C2×C4×C61C23×C411yescharacteristic
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C2×C4×C61C2×#4111yescharacteristic
185C22⋊D61—22no—
186C22⋊D61—22no—
187C2×Dic61—21no—
188C2×Dic61—24no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61#12112yes—
193C2×Dic61#12112yes—
194D244—21no—
195D244C4×D412yes—
196C4×D61—22no—
197C4×D61—22no—
198C4×D61—82no—
199Dic122—21no—
200Dic122—24no—
201Dic122—24no—
202Dic122—42no—
203Dic122—42no—
204Dic122C4×D412yes—
205C8×C61—24no—
206C8×C61—22no—
207C8×C61C4×D412yes—
208C8×C61C4×D412yes—
209#92418—22no—
210#92418—21no—
211#92418—21no—
212#92418—42no—
213#21556—611nomaximal, Sylow
214C22×C61—22no—
215C22×C61C4×#4111yescharacteristic
216C22×C61C2×#12111yescharacteristic
217C22×C61#52911yescharacteristic
218C2×D61—22no—
219C2×D61—42no—
220C4×C61—42no—
221C4×C61—82no—
222C4×C61C2×C4×D411yescharacteristic
223C4×C61C2×C4×D411yescharacteristic
224C4×C61#52511yescharacteristic
225C4×C61#53211yescharacteristic
226Dic61—22no—
227Dic61—22no—
228Dic61—22no—
229Dic61—44no—
230C2×#473—611no—
231C4×#137—612no—
232C2×#441—611no—
233#2875—611no—
234#1979—611no—
235#1951D6111yescharacteristic, complemented
236#1930—612no—
237#1908—612no—
238#1815—611no—
239#1778—611no—
240C4×#101—612no—
241C2×C61—21no—
242C2×C61—21no—
243C2×C61C4×#12111yescharacteristic
244C2×C61#306912yes—
245D61—42no—
246C2×#137—612no—
247C22×Q16—611no—
248C2×#131—611no—
249C2×#131—611no—
250C4×#41—611no—
251C2×C4×Q8—612no—
252#473—618no—
253#441—1221no—
254#441—1221no—
255#441—1221no—
256#441—1221no—
257#441—1222no—
258#440—1222no—
259C4×Q16—1222no—
260C4×SD16—1222no—
261C2×#108—611no—
262C2×#108C2×D6111yescharacteristic
263#428C2×D6112yes—
264C2×#106C2×D6111yescharacteristic
265#419—612no—
266#417—612no—
267C2×#103—612no—
268C2×#101—612no—
269C2×#101—612no—
270#405—611no—
271#405—1221no—
272#405—1221no—
273#405C2×D6111yescharacteristic
274C4×M16—611no—
275C4×M16—612no—
276C4×M16—1221no—
277C2×#97—611no—
278C2×#97—1221no—
279#343—612no—
280#343C2×D6112yes—
281#341—612no—
282C61#2155611yescharacteristic, complemented, Sylow
283C2×#41—611no—
284C22×Q8—612no—
285#137—618no—
286#137—1224no—
287C2×Q16—614no—
288C2×Q16—1222no—
289C2×Q16—1221no—
290C2×Q16—1221no—
291C2×Q16—1222no—
292C2×SD16—1222no—
293#131—614no—
294#131—1222no—
295#131—1222no—
296#131—1222no—
297C2×M16—611no—
298C2×M16—612no—
299C2×M16—1221no—
300C2×M16—1221no—
301C2×M16C22×D6112yes—
302C2×M16C22×D6111yescharacteristic
303C22×C8—611no—
304C22×C8—1221no—
305C22×C8—1221no—
306C22×C8C22×D6111yescharacteristic
307C4×Q8—612no—
308C4×Q8—1222no—
309C4×Q8—1222no—
310C4×Q8—1222no—
311C4×Q8—1222no—
312C4×D4—612no—
313C4×D4—1222no—
314#117—1221no—
315#117C22×D6111yescharacteristic
316C2×C4⋊C4—1222no—
317C2×C4⋊C4C22×D6112yes—
318C2×C42—611no—
319C2×C42—1221no—
320C2×C42—1222no—
321#108—1222no—
322#108C4×D6114yescomplemented
323#107—22no—
324#106—21no—
325#106—21no—
326#106—22no—
327#103—1222no—
328#103—1222no—
329#103—1222no—
330#103—1222no—
331#103—1222no—
332#102—1222no—
333#101—618no—
334#101—1224no—
335#97—21no—
336#97—611no—
337#97—611no—
338#97—611no—
339#97—611no—
340#97—1221no—
341#97—1221no—
342#97—1222no—
343C4×C8—21no—
344C4×C8—1221no—
345C4×C8—1221no—
346C4×C8—1222no—
347#41—612no—
348#41—612no—
349#41—1221no—
350#41—1221no—
351C2×Q8—614no—
352C2×Q8—611no—
353C2×Q8—1222no—
354C2×Q8—1222no—
355C2×Q8—1222no—
356C2×Q8—1222no—
357C2×Q8—1222no—
358C2×D4—611no—
359C2×D4—1221no—
360C22×C4—22no—
361C22×C4—612no—
362C22×C4—1222no—
363C22×C4—1221no—
364C22×C4—1222no—
365C22×C4C23×D6111yescharacteristic
366Q16—1228no—
367Q16—1224no—
368Q16—1224no—
369SD16—1228no—
370M16—21no—
371M16—22no—
372M16—612no—
373M16—1224no—
374M16—1221no—
375M16—1224no—
376M16—1222no—
377M16—1222no—
378M16C2×C4×D6114yes—
379M16#1175822612yes—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—611no—
383C2×C8—611no—
384C2×C8—1224no—
385C2×C8—1221no—
386C2×C8—1221no—
387C2×C8—1221no—
388C2×C8—1224no—
389C2×C8—1222no—
390C2×C8C2×C4×D6112yes—
391C2×C8D4×D6111yescharacteristic
392C2×C8D4×D6111yescharacteristic
393C2×C8#1175822612yes—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—1222no—
397C4⋊C4—2442no—
398C4⋊C4—2442no—
399C4⋊C4D4×D6112yes—
400C22⋊C4—21no—
401C22⋊C4—1222no—
402C42—21no—
403C42—611no—
404C42—612no—
405C42—611no—
406C42—1222no—
407C42—2442no—
408C42—2442no—
409C23—1221no—
410C23#41×D6111yescharacteristic
411Q8—612no—
412Q8—1221no—
413Q8—1224no—
414Q8—1224no—
415Q8—2442no—
416Q8—2442no—
417D4—612no—
418D4—1222no—
419D4—1222no—
420D4—1221no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—612no—
425C2×C4—612no—
426C2×C4—1221no—
427C2×C4—1222no—
428C2×C4—1222no—
429C2×C4—1224no—
430C2×C4—4882no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4C22×C4×D6111yescharacteristic
435C2×C4C2×D4×D6111yescharacteristic
436C2×C4C2×D4×D6111yescharacteristic
437C2×C4#41×D6111yescharacteristic
438C2×C4C2×#1175822611yescharacteristic
439C2×C4C2×#1175822611yescharacteristic
440C8—24no—
441C8—22no—
442C8—1222no—
443C8—1221no—
444C8—1221no—
445C8—2442no—
446C8#1451973912yes—
447C8#1451973912yes—
448C22—22no—
449C22—1222no—
450C22—2442no—
451C22C4×#1175822611yescharacteristic
452C22#121×D6111yescharacteristic
453C22#1684029711yescharacteristic
454C4—42no—
455C4—82no—
456C4—1222no—
457C4—1222no—
458C4—1222no—
459C4—2444no—
460C4C4×D4×D6111yescharacteristic
461C4C2×#1451973911yescharacteristic
462C4#1684027511yescharacteristic
463C4#1684031311yescharacteristic
464C2—21no—
465C2—21no—
466C2—2442no—
467C2#2144832711yescharacteristic
468C2#2145555112yes—
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1951C2
socleC22×C61C2×#121
prime radicalC2×#11758222C23
cosocleC22×C4×C61C24
hypercenter#1951D61
hypocenterC61#21556
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1951

Complements

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

NH ≅ G/Ntype
3. C2×#16839868C2semidirect
5. #21446734C2semidirect
6. #21446473C2semidirect
9. #21445727C2semidirect
10. #21445614C2semidirect
11. C61×#1951C2semidirect
18. #16840038C4semidirect
56. C2×Dic244C2×C4semidirect
89. #14519695C2×C4semidirect
105. #108×C61C2×C4semidirect
235. #1951D61semidirect
282. C61#21556semidirect
322. #108C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400382
#214546372
C2×#168398682
#214542462
#214467342
#214464732
#214463352
#214458792
#214457272
#214456142
C61×#19512
#2155661