Finite Group Databasebeta

← #29686929

Subgroups — #29686929

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×#16840034C211yesmaximal, characteristic
2#21454634C212yesmaximal
3C2×#16839869C211yesmaximal, characteristic, complemented
4#21454246C211yesmaximal, characteristic
5#21446599C211yesmaximal, characteristic, complemented
6#21446477C212yesmaximal, complemented
7#21446331C212yesmaximal
8#21445875C211yesmaximal, characteristic
9#21445727C211yesmaximal, characteristic, complemented
10#101×Dic61C212yesmaximal, complemented
11C61×#1979C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519785C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840230C2211yescharacteristic
16C2×Q8×Dic61C2212yes—
17#16840034C418yescomplemented
18#16839869—21no—
19#16839869—21no—
20#16839869—21no—
21#16839869—21no—
22#16839866—22no—
23#16839865—22no—
24#16839854—22no—
25#16839847—22no—
26C2×#14519713C2211yescharacteristic
27#16839805C2212yes—
28C2×#14519709C2211yescharacteristic
29#16839764C2212yes—
30C2×#14519701C2212yes—
31C2×#14519696C2212yes—
32#16839719—21no—
33#16839718C2211yescharacteristic
34#16839716—21no—
35M16×Dic61C2212yes—
36M16×Dic61C2211yescharacteristic
37C4×#11758189—21no—
38C2×#14519687C2211yescharacteristic
39#97×D61—21no—
40#16839530C2212yes—
41#16839530C2212yes—
42#16839526C2212yes—
43C22×Q16×C61C2211yescharacteristic
44C2×#108×C61C2211yescharacteristic
45C61×#419C2212yes—
46C2×#101×C61C2212yes—
47C61×#405C2211yescharacteristic
48C2×#11758222C2311yescharacteristic
49#14519808—24no—
50#14519808C2×C418yes—
51C2×#11758202—22no—
52C2×#11758201—22no—
53#14519785—22no—
54#14519785—22no—
55#14519783—22no—
56#14519783C2×C414yes—
57C2×#11758194C2311yescharacteristic
58C2×#11758194C2312yes—
59C2×#11758189—21no—
60C2×#11758189—21no—
61C22×#92418C2311yescharacteristic
62C2×C8×D61—21no—
63C2×C8×D61—21no—
64Q8×Dic61—22no—
65Q8×Dic61—22no—
66Q8×Dic61—22no—
67Q8×Dic61—22no—
68#14519742D412yes—
69#14519739D412yes—
70#14519737—22no—
71#14519734—21no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61C2312yes—
74C2×C4×Dic61—22no—
75C2×C4×Dic61C2311yescharacteristic
76C42×D61—21no—
77#14519713C2×C414yescomplemented
78#14519710—22no—
79#14519709—21no—
80#14519709—21no—
81#14519708—22no—
82#14519702—22no—
83#14519701—22no—
84#14519701—22no—
85#14519699—22no—
86#14519696C2×C418yescomplemented
87#14519688—21no—
88#14519687—21no—
89#14519687—22no—
90#14519687D411yescharacteristic
91#14519687D411yescharacteristic
92#14519687D411yescharacteristic
93#14519687D411yescharacteristic
94C4×#92418—21no—
95C8×Dic61—21no—
96C8×Dic61—22no—
97C22×Q8×C61C2312yes—
98C2×Q16×C61—21no—
99C2×Q16×C61—21no—
100C2×Q16×C61—22no—
101C2×Q16×C61C2×C414yescomplemented
102C2×M16×C61C2312yes—
103C2×M16×C61C2311yescharacteristic
104C22×C8×C61C2311yescharacteristic
105#117×C61C2311yescharacteristic
106#108×C61—22no—
107#103×C61—22no—
108#103×C61—22no—
109#101×C61—24no—
110#97×C61—21no—
111C4×C8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61C22×C414yes—
118C22×C4×C61—22no—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—21no—
121C22×Dic61—22no—
122C22×Dic61#4112yes—
123#11758222—21no—
124#11758222—21no—
125#11758222C22×C412yes—
126#11758222#4112yes—
127C2×D244C2×D411yescharacteristic
128C2×C4×D61—21no—
129C2×C4×D61—22no—
130C2×Dic122C2×D411yescharacteristic
131Q16×C61—24no—
132Q16×C61—24no—
133M16×C61—21no—
134M16×C61—22no—
135M16×C61—22no—
136M16×C61#4112yes—
137C2×C8×C61—21no—
138C2×C8×C61—22no—
139C2×C8×C61C22×C412yes—
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61#4112yes—
143C4⋊C4×C61C2×D412yes—
144C22⋊C4×C61—21no—
145C42×C61—21no—
146#11758202—28no—
147#11758201—28no—
148#11758198—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—42no—
152C4⋊Dic61—42no—
153#11758196—22no—
154C4×Dic61—22no—
155C4×Dic61—42no—
156C4×Dic61—42no—
157C4×Dic61C2×D412yes—
158C4×Dic61C2×D411yescharacteristic
159C4×Dic61C2×D411yescharacteristic
160#11758194—22no—
161#11758194—21no—
162#11758194C22×C414yes—
163#11758194C22×C412yes—
164C2×#92418—21no—
165C2×#92418—22no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418C2×D411yescharacteristic
169C2×#92418C2×D411yescharacteristic
170#11758189—24no—
171#11758189—24no—
172C8×D61—24no—
173C8×D61—24no—
174C23×C61C2×#4111yescharacteristic
175C22×D61—21no—
176Q8×C61—24no—
177Q8×C61—24no—
178Q8×C61—42no—
179Q8×C61—42no—
180C2×C4×C61—24no—
181C2×C4×C61—22no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61C23×C411yescharacteristic
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C22⋊D61—22no—
191C22⋊D61—22no—
192C2×Dic61—21no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—82no—
196C2×Dic61#12112yes—
197C2×Dic61#12112yes—
198D244—21no—
199D244C4×D412yes—
200C4×D61—22no—
201C4×D61—22no—
202C4×D61—82no—
203Dic122—21no—
204Dic122C4×D412yes—
205C8×C61—22no—
206C8×C61—42no—
207C8×C61C4×D412yes—
208C8×C61C4×D412yes—
209#92418—22no—
210#92418—24no—
211#92418—21no—
212#92418—21no—
213#21556—611nomaximal, Sylow
214C22×C61—22no—
215C22×C61C4×#4111yescharacteristic
216C22×C61C2×#12111yescharacteristic
217C22×C61#52911yescharacteristic
218C2×D61—22no—
219C2×D61—42no—
220C4×C61—44no—
221C4×C61—42no—
222C4×C61C2×C4×D411yescharacteristic
223C4×C61C2×C4×D411yescharacteristic
224C4×C61#52511yescharacteristic
225C4×C61#53211yescharacteristic
226Dic61—22no—
227Dic61—22no—
228Dic61—22no—
229Dic61—82no—
230C2×#473—611no—
231C4×#137—612no—
232C2×#441—611no—
233#2875—611no—
234#1979D6111yescharacteristic, complemented
235#1951—611no—
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×Q16C2×D6111yescharacteristic
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#428—612no—
264C2×#106—611no—
265#419C2×D6112yes—
266#417—612no—
267C2×#103—612no—
268C2×#101—612no—
269C2×#101C2×D6112yes—
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#343—612no—
281#341—612no—
282C61#2155611yescharacteristic, complemented, Sylow
283C2×#41—611no—
284C22×Q8C22×D6112yes—
285#137—618no—
286#137—1224no—
287C2×Q16—21no—
288C2×Q16—21no—
289C2×Q16—22no—
290C2×Q16—1222no—
291C2×Q16C4×D6114yescomplemented
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—612no—
317C2×C4⋊C4—1222no—
318C2×C42—611no—
319C2×C42—1221no—
320C2×C42—1222no—
321#108—22no—
322#108—614no—
323#107—1222no—
324#106—1221no—
325#106—1221no—
326#106—1222no—
327#103—22no—
328#103—22no—
329#103—1222no—
330#103—1222no—
331#103—1222no—
332#102—1222no—
333#101—24no—
334#101—618no—
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—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—611no—
357C2×Q8C2×C4×D6114yes—
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—24no—
367Q16—24no—
368Q16—1228no—
369SD16—1228no—
370M16—21no—
371M16—22no—
372M16—22no—
373M16—612no—
374M16—614no—
375M16—1224no—
376M16—1222no—
377M16—1221no—
378M16—1224no—
379M16#1175822412yes—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—611no—
383C2×C8—611no—
384C2×C8—1224no—
385C2×C8—1221no—
386C2×C8—1222no—
387C2×C8—1221no—
388C2×C8—1221no—
389C2×C8—1224no—
390C2×C8C2×C4×D6112yes—
391C2×C8D4×D6111yescharacteristic
392C2×C8D4×D6111yescharacteristic
393C2×C8#1175822412yes—
394C4⋊C4—1222no—
395C4⋊C4—1222no—
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—24no—
412Q8—24no—
413Q8—42no—
414Q8—42no—
415Q8—612no—
416Q8—1221no—
417D4—612no—
418D4—1222no—
419D4—1222no—
420D4—1221no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—612no—
426C2×C4—612no—
427C2×C4—1221no—
428C2×C4—1222no—
429C2×C4—1222no—
430C2×C4—4882no—
431C2×C4—4882no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4C22×C4×D6111yescharacteristic
435C2×C4C2×D4×D6111yescharacteristic
436C2×C4C2×D4×D6111yescharacteristic
437C2×C4#41×D6111yescharacteristic
438C2×C4C2×#1175822411yescharacteristic
439C2×C4C2×#1175822411yescharacteristic
440C8—22no—
441C8—42no—
442C8—1224no—
443C8—1222no—
444C8—1221no—
445C8—1221no—
446C8#1451973712yes—
447C8#1451973712yes—
448C22—22no—
449C22—1222no—
450C22—2442no—
451C22C4×#1175822411yescharacteristic
452C22#121×D6111yescharacteristic
453C22#1684029111yescharacteristic
454C4—44no—
455C4—42no—
456C4—1222no—
457C4—1222no—
458C4—1222no—
459C4—4882no—
460C4C4×D4×D6111yescharacteristic
461C4C2×#1451973711yescharacteristic
462C4#1684027211yescharacteristic
463C4#1684031011yescharacteristic
464C2—21no—
465C2—21no—
466C2—2442no—
467C2#2144833511yescharacteristic
468C2#2145555412yes—
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1979C2
socleC22×C61C2×#121
prime radicalC2×#11758222C23
cosocleC22×C4×C61C24
hypercenter#1979D61
hypocenterC61#21556
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1979

Complements

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

NH ≅ G/Ntype
3. C2×#16839869C2semidirect
5. #21446599C2semidirect
6. #21446477C2semidirect
9. #21445727C2semidirect
10. #101×Dic61C2semidirect
11. C61×#1979C2semidirect
17. #16840034C4semidirect
77. #14519713C2×C4semidirect
86. #14519696C2×C4semidirect
101. C2×Q16×C61C2×C4semidirect
234. #1979D61semidirect
282. C61#21556semidirect
291. C2×Q16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400342
#214546342
C2×#168398692
#214542462
#214465992
#214464772
#214463312
#214458752
#214457272
#101×Dic612
C61×#19792
#2155661