Finite Group Databasebeta

← #29686920

Subgroups — #29686920

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
2C4×#14519808C212yesmaximal
3#21454244C211yesmaximal, characteristic
4#21446735C211yesmaximal, characteristic, complemented
5#21446597C211yesmaximal, characteristic, complemented
6#21446471C212yesmaximal, complemented
7#21446337C212yesmaximal
8#21445879C211yesmaximal, characteristic
9#21445725C211yesmaximal, characteristic, complemented
10C4×#14519696C212yesmaximal, complemented
11C2×C61×#441C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519783C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15C4×#11758222C2211yescharacteristic
16#16840034C418yescomplemented
17#16839867—22no—
18#16839860—22no—
19C4×#11758202—22no—
20C4×#11758201—22no—
21C2×#14519713C2211yescharacteristic
22C2×#14519713C2211yescharacteristic
23#16839802C2212yes—
24#16839771C2212yes—
25#16839761C2212yes—
26C2×#14519696C2212yes—
27C2×#14519696C2212yes—
28#16839719—21no—
29#16839719—21no—
30#16839717C2211yescharacteristic
31#16839717C2211yescharacteristic
32C4×#11758194C2212yes—
33C4×#11758194C2211yescharacteristic
34#16839714—21no—
35#97×D61—21no—
36#16839531C2212yes—
37#16839531C2212yes—
38#16839527C2212yes—
39C22×Q16×C61C2211yescharacteristic
40C2×C4×Q8×C61C2212yes—
41C61×#441—21no—
42C61×#441—21no—
43C61×#441—21no—
44C61×#441—21no—
45C2×#106×C61C2211yescharacteristic
46C2×#103×C61C2212yes—
47C2×#97×C61C2211yescharacteristic
48C2×#11758222C2311yescharacteristic
49#14519808—24no—
50#14519808C2×C418yes—
51C2×#11758202—22no—
52C2×#11758201—22no—
53#14519783—22no—
54#14519783—22no—
55#14519783—22no—
56#14519783C2×C414yes—
57C2×#11758194C2312yes—
58C2×#11758194C2311yescharacteristic
59C2×#11758194C2312yes—
60C2×#11758194C2311yescharacteristic
61C2×#11758189—21no—
62C2×#11758189—21no—
63C2×C8×D61—21no—
64C2×C8×D61—21no—
65C4×Dic122D412yes—
66C4×D244D412yes—
67C4×C22⋊D61—22no—
68#14519736C2311yescharacteristic
69#14519734—21no—
70C42×D61—21no—
71#14519713—22no—
72#14519713C2×C414yescomplemented
73#14519710—22no—
74#14519708—22no—
75#14519701—22no—
76#14519701—22no—
77#14519700—22no—
78#14519697—22no—
79#14519696—24no—
80#14519696C2×C418yescomplemented
81#14519688—21no—
82#14519688—22no—
83#14519687—21no—
84#14519687—21no—
85C4×#92418—21no—
86C4×#92418—22no—
87C8×Dic61—21no—
88C8×Dic61—21no—
89C22×Q8×C61C2312yes—
90C2×Q16×C61—21no—
91C2×Q16×C61—21no—
92C2×Q16×C61—22no—
93C2×Q16×C61C2×C414yescomplemented
94C22×C8×C61C2311yescharacteristic
95C22×C8×C61C2311yescharacteristic
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C4×Q8×C61—22no—
99C4×Q8×C61—22no—
100C2×C4⋊C4×C61—22no—
101C2×C4⋊C4×C61C2312yes—
102C2×C42×C61—22no—
103C2×C42×C61C2311yescharacteristic
104#106×C61—21no—
105#106×C61—21no—
106#103×C61—22no—
107#103×C61—22no—
108#97×C61D411yescharacteristic
109#97×C61D411yescharacteristic
110#97×C61D411yescharacteristic
111#97×C61D411yescharacteristic
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×C61—22no—
120C22×C4×C61#4112yes—
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—21no—
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—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61C22×C412yes—
136C2×C8×C61C2×D411yescharacteristic
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61#4112yes—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—42no—
144C4⋊C4×C61—42no—
145C42×C61—42no—
146C42×C61—42no—
147C42×C61C2×D412yes—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C22⋊Dic61—21no—
151#11758202—28no—
152#11758201—28no—
153#11758198—22no—
154C4⋊Dic61C2×D412yes—
155#11758196—22no—
156C4×Dic61—22no—
157C4×Dic61—21no—
158#11758194—22no—
159#11758194—21no—
160#11758194—21no—
161#11758194—22no—
162#11758194—22no—
163#11758194C22×C414yes—
164#11758194C22×C412yes—
165#11758194#4112yes—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418—22no—
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—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—82no—
186C2×C4×C61#12112yes—
187C2×C4×C61#12112yes—
188C2×C4×C61C23×C411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C22⋊D61—22no—
195C22⋊D61—22no—
196C2×Dic61—21no—
197C2×Dic61—22no—
198D244—21no—
199D244C4×D412yes—
200C4×D61—22no—
201C4×D61—22no—
202C4×D61—82no—
203Dic122—21no—
204Dic122C4×D412yes—
205C8×C61—21no—
206C8×C61—21no—
207C8×C61C4×D412yes—
208C8×C61C4×D412yes—
209#92418—22no—
210#92418—24no—
211#92418—22no—
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—22no—
221C4×C61—22no—
222C4×C61—44no—
223C4×C61—82no—
224C4×C61C2×C4×D411yescharacteristic
225C4×C61C2×C4×D411yescharacteristic
226C4×C61#52511yescharacteristic
227C4×C61#53211yescharacteristic
228Dic61—22no—
229Dic61—42no—
230C2×#473—611no—
231C4×#137—612no—
232C2×#441D6111yescharacteristic, complemented
233#2875—611no—
234#1979—611no—
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×Q8C2×D6112yes—
252#473—618no—
253#441—21no—
254#441—21no—
255#441—21no—
256#441—21no—
257#441—1222no—
258#440—1222no—
259C4×Q16—1222no—
260C4×SD16—1222no—
261C2×#108—611no—
262C2×#108—611no—
263#428—612no—
264C2×#106C2×D6111yescharacteristic
265#419—612no—
266#417—612no—
267C2×#103C2×D6112yes—
268C2×#101—612no—
269C2×#101—612no—
270#405—611no—
271#405—611no—
272#405—1221no—
273#405—1221no—
274C4×M16—611no—
275C4×M16—612no—
276C4×M16—1221no—
277C2×#97—1221no—
278C2×#97C2×D6111yescharacteristic
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—22no—
288C2×Q16—21no—
289C2×Q16—21no—
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—611no—
299C2×M16—612no—
300C2×M16—612no—
301C2×M16—1221no—
302C2×M16—1221no—
303C22×C8—1221no—
304C22×C8—1221no—
305C22×C8C22×D6111yescharacteristic
306C22×C8C22×D6111yescharacteristic
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—612no—
312C4×D4—612no—
313C4×D4—1222no—
314#117—611no—
315#117—1221no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4C22×D6112yes—
318C2×C42—22no—
319C2×C42—1221no—
320C2×C42C22×D6111yescharacteristic
321#108—614no—
322#108—1222no—
323#107—1222no—
324#106—21no—
325#106—21no—
326#106—1222no—
327#103—22no—
328#103—22no—
329#103—1222no—
330#103—1222no—
331#103—1222no—
332#102—1222no—
333#101—618no—
334#101—1224no—
335#97—1221no—
336#97—1221no—
337#97—1222no—
338#97—1221no—
339#97C22⋊D6111yescharacteristic
340#97C22⋊D6111yescharacteristic
341#97C22⋊D6111yescharacteristic
342#97C22⋊D6111yescharacteristic
343C4×C8—1221no—
344C4×C8—1221no—
345C4×C8—1222no—
346C4×C8—1221no—
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—22no—
362C22×C4—1221no—
363C22×C4—1222no—
364C22×C4#1175822412yes—
365C22×C4C23×D6111yescharacteristic
366Q16—24no—
367Q16—24no—
368Q16—1228no—
369SD16—1228no—
370M16—612no—
371M16—614no—
372M16—612no—
373M16—1224no—
374M16—1221no—
375M16—1222no—
376M16—1221no—
377M16—1224no—
378M16—1222no—
379M16—1222no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—1224no—
383C2×C8—1222no—
384C2×C8—1221no—
385C2×C8—1221no—
386C2×C8—1224no—
387C2×C8—1222no—
388C2×C8C2×C4×D6112yes—
389C2×C8#1175822212yes—
390C2×C8C2×C22⋊D6111yescharacteristic
391C2×C8C2×C22⋊D6111yescharacteristic
392C2×C8C2×C22⋊D6111yescharacteristic
393C2×C8C2×C22⋊D6111yescharacteristic
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—612no—
399C4⋊C4—1222no—
400C22⋊C4—1221no—
401C22⋊C4—1222no—
402C42—42no—
403C42—42no—
404C42—1221no—
405C42—1222no—
406C42D4×D6112yes—
407C42C2×C22⋊D6111yescharacteristic
408C42C2×C22⋊D6111yescharacteristic
409C23—1221no—
410C23C2×#1175822411yescharacteristic
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—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—1221no—
428C2×C4—1222no—
429C2×C4—1222no—
430C2×C4—1222no—
431C2×C4—4882no—
432C2×C4#1451974812yes—
433C2×C4#1451974812yes—
434C2×C4C22×C4×D6111yescharacteristic
435C2×C4C2×D4×D6111yescharacteristic
436C2×C4C22×C22⋊D6111yescharacteristic
437C2×C4#41×D6111yescharacteristic
438C2×C4#41×D6111yescharacteristic
439C2×C4C2×#1175822211yescharacteristic
440C8—21no—
441C8—21no—
442C8—1224no—
443C8—1222no—
444C8—1222no—
445C8—2442no—
446C8C4×C22⋊D6112yes—
447C8C4×C22⋊D6112yes—
448C22—22no—
449C22—1222no—
450C22—2442no—
451C22C4×#1175822411yescharacteristic
452C22C2×#1451974811yescharacteristic
453C22#1684029411yescharacteristic
454C4—22no—
455C4—22no—
456C4—44no—
457C4—82no—
458C4—1222no—
459C4—2442no—
460C4C4×D4×D6111yescharacteristic
461C4C2×C4×C22⋊D6111yescharacteristic
462C4#1684026911yescharacteristic
463C4#1684030911yescharacteristic
464C2—21no—
465C2—21no—
466C2—2442no—
467C2C4×#1451974811yescharacteristic
468C2#2145555212yes—
subgroupG/H
centerC22C2×#14519748
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#441C2
socleC22×C61C2×#121
prime radicalC2×#11758222C23
cosocleC22×C4×C61C24
hypercenterC2×#441D61
hypocenterC61#21556
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
4. #21446735C2semidirect
5. #21446597C2semidirect
6. #21446471C2semidirect
9. #21445725C2semidirect
10. C4×#14519696C2semidirect
11. C2×C61×#441C2semidirect
16. #16840034C4semidirect
72. #14519713C2×C4semidirect
80. #14519696C2×C4semidirect
93. C2×Q16×C61C2×C4semidirect
232. C2×#441D61semidirect
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
C4×#145198082
#214542442
#214467352
#214465972
#214464712
#214463372
#214458792
#214457252
C4×#145196962
C2×C61×#4412
#2155661