Finite Group Databasebeta

← #30603301

Subgroups — #30603301

Order 15616.

868 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462472C211yesmaximal, characteristic, complemented
2C2×#16840741C212yesmaximal
3#21461501C212yesmaximal, complemented
4#21461491C211yesmaximal, characteristic, complemented
5#21454262C211yesmaximal, characteristic
6C2×#14519856C2212yes—
7C2×#14519856C2212yes—
8#16840773—24no—
9#16840772—22no—
10#16840741—24no—
11#16840741—24no—
12C2×#14519836—22no—
13C2×#14519835—22no—
14#16840634—24no—
15#16840630—22no—
16C22×#11758222C2211yescharacteristic
17C2×M16×D61C2211yescharacteristic
18C2×M16×D61C2211yescharacteristic
19#16839727—22no—
20#16839727—22no—
21#16839726C411yescharacteristic, complemented
22#16839725C411yescharacteristic, complemented
23#16839725C411yescharacteristic, complemented
24C61×#407C411yescharacteristic, complemented
25#14519856—22no—
26#14519856—22no—
27#14519856—22no—
28#14519856—22no—
29#14519856—22no—
30#14519856—22no—
31#14519856D414yes—
32#14519856D414yes—
33C2×C4⋊#6923—22no—
34C2×C4⋊#6923—22no—
35C2×C22⋊#6923—22no—
36C2×C22⋊#6923—22no—
37C2×C4×#6923—22no—
38C2×C4×#6923—22no—
39#14519846—24no—
40#14519845—24no—
41#14519836—28no—
42#14519835—28no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758222—21no—
48C2×#11758222—21no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C22×Dic122Q811yescharacteristic
52C22×D244D411yescharacteristic
53C22×C22⋊D61D411yescharacteristic
54C22×C22⋊D61Q811yescharacteristic
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C2×#11758194C2×C411yescharacteristic
58C2×#11758194C2×C411yescharacteristic
59C2×#11758189—21no—
60C2×#11758189D412yes—
61M16×D61—22no—
62M16×D61—22no—
63M16×D61—22no—
64M16×D61—22no—
65C2×C8×D61—21no—
66C2×C8×D61D412yes—
67C22⋊#92418—22no—
68#14519689—22no—
69#14519689—22no—
70C23×C4×C61C2×C411yescharacteristic
71C2×M16×C61C2×C411yescharacteristic
72C2×M16×C61C2×C411yescharacteristic
73C22⋊C8×C61—22no—
74C24×C61C4⋊C411yescharacteristic
75C23×D61C4⋊C411yescharacteristic
76C23×D61C2×D411yescharacteristic
77C22×#6923—42no—
78C22×#6923—42no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—22no—
82C22×C4×C61—22no—
83C22×C4×C61C4⋊C411yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C2×C22⋊D61—22no—
86C2×C22⋊D61—21no—
87C2×C22⋊D61—21no—
88C2×C22⋊D61—22no—
89C2×C22⋊D61—22no—
90C2×C22⋊D61—21no—
91C2×C22⋊D61—21no—
92C2×C22⋊D61—22no—
93C22×Dic61C4⋊C411yescharacteristic
94C22×Dic61C2×Q811yescharacteristic
95#11758222—41no—
96#11758222—42no—
97#11758222—42no—
98#11758222—42no—
99#11758222—41no—
100#11758222—42no—
101#11758222—42no—
102#11758222—41no—
103#11758222—42no—
104#11758222—41no—
105C2×D244—22no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—22no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61C2×D412yes—
114C2×C4×D61C2×D411yescharacteristic
115C2×C4×D61C2×D411yescharacteristic
116C2×C4×D61C2×D412yes—
117C2×C4×D61#4111yescharacteristic
118C2×C4×D61#4112yes—
119C2×C4×D61#4112yes—
120C2×C4×D61#4111yescharacteristic
121C2×Dic122—22no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×Dic122—22no—
125C22⋊#6923—24no—
126C22⋊#6923—24no—
127C22⋊#6923—42no—
128C22⋊#6923—42no—
129C4⋊#6923—24no—
130C4⋊#6923—24no—
131C4⋊#6923—42no—
132C4⋊#6923—42no—
133C4×#6923—24no—
134C4×#6923—24no—
135C4×#6923—42no—
136C4×#6923—42no—
137M16×C61—22no—
138M16×C61—22no—
139C2×C8×C61—21no—
140C2×C8×C61C22⋊C412yes—
141#11758194—22no—
142#11758194—22no—
143C2×#92418—21no—
144C2×#92418C22⋊C412yes—
145#11758189—24no—
146#11758189—41no—
147#11758189—41no—
148C8×D61—22no—
149C8×D61—22no—
150C8×D61—42no—
151C23×C61—21no—
152C23×C61—22no—
153C23×C61—81no—
154C23×C61C2×C4⋊C411yescharacteristic
155C22×D61—21no—
156C22×D61—22no—
157C22×D61—81no—
158C22×D61—81no—
159C22×D61#12012yes—
160C22×D61#12111yescharacteristic
161C22×D61#12111yescharacteristic
162C22×D61#12312yes—
163C2×#6923—44no—
164C2×#6923—44no—
165C2×#6923—84no—
166C2×#6923—84no—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—41no—
170C2×C4×C61—42no—
171C2×C4×C61—41no—
172C2×C4×C61C2×C22⋊C411yescharacteristic
173C2×C4×C61#11711yescharacteristic
174C2×C4×C61C4×D412yes—
175C2×C4×C61C4×D412yes—
176C22⋊D61—41no—
177C22⋊D61—42no—
178C22⋊D61—41no—
179C22⋊D61—41no—
180C22⋊D61—42no—
181C22⋊D61—41no—
182C2×Dic61—21no—
183C2×Dic61—22no—
184C2×Dic61C22⋊Q812yes—
185C2×Dic61C22⋊Q811yescharacteristic
186C2×Dic61C22⋊Q811yescharacteristic
187C2×Dic61C22⋊Q812yes—
188D244—41no—
189D244—42no—
190D244—41no—
191C4×D61—41no—
192C4×D61—42no—
193C4×D61—41no—
194C4×D61#12011yescharacteristic
195C4×D61#12112yes—
196C4×D61#12112yes—
197C4×D61#12112yes—
198C4×D61#12311yescharacteristic
199C4×D61#12312yes—
200C4×D61#12412yes—
201C4×D61#12411yescharacteristic
202C4×D61#12612yes—
203C4×D61#12711yescharacteristic
204Dic122—41no—
205Dic122—42no—
206Dic122—41no—
207C8×C61—22no—
208C8×C61—41no—
209#92418—22no—
210#92418—41no—
211#11102—611nomaximal, Sylow
212C22×C61—41no—
213C22×C61—42no—
214C22×C61—41no—
215C22×C61—82no—
216C22×C61#38011yescharacteristic
217C22×C61#38512yes—
218C2×D61—42no—
219C2×D61—42no—
220C2×D61—41no—
221C2×D61—41no—
222C2×D61—42no—
223C2×D61—41no—
224C2×D61—82no—
225C2×D61#39211yescharacteristic
226C2×D61#39412yes—
227C2×D61#39711yescharacteristic
228#6923—84no—
229#6923—84no—
230C4×C61—41no—
231C4×C61#38611yescharacteristic
232C4×C61#38812yes—
233C4×C61#39011yescharacteristic
234Dic61—41no—
235Dic61#39311yescharacteristic
236Dic61#39612yes—
237Dic61#39911yescharacteristic
238#2911—611no—
239C2×#410—612no—
240#2879—611no—
241#1733—612no—
242C2×H(3,ℤ/4)—611no—
243C2×C61—21no—
244C2×C61—22no—
245C2×C61—81no—
246C2×C61#143911yescharacteristic
247D61—21no—
248D61—22no—
249D61—21no—
250D61—81no—
251C22×#41—611no—
252C22×M16—611no—
253C22×M16—611no—
254C2×#117—612no—
255C2×#117—612no—
256#428—1224no—
257#427—1222no—
258#410—1224no—
259#410—1224no—
260C2×#99—1222no—
261C2×#99—1222no—
262#408—1222no—
263#408—1222no—
264#407—611no—
265#407—611no—
266#407—611no—
267#407#692311yescharacteristic, complemented
268#339—1224no—
269H(3,ℤ/4)—1222no—
270C61#1110211yescharacteristic, complemented, Sylow
271C2×#41—1221no—
272C2×#41—1221no—
273C2×#41—1222no—
274C2×#41—1222no—
275C2×#41—1221no—
276C2×#41—1221no—
277C2×#41—1222no—
278C2×#41—1222no—
279C22×Q8—611no—
280C22×D4—611no—
281C22×D4—611no—
282C22×D4—611no—
283C23×C4—611no—
284C23×C4—611no—
285C23×C4C2×#692311yescharacteristic
286C2×M16—611no—
287C2×M16—612no—
288C2×M16—611no—
289C2×M16—1221no—
290C2×M16—1222no—
291C2×M16—1222no—
292C2×M16—1222no—
293C2×M16—1222no—
294C2×M16C2×#692311yescharacteristic
295C2×M16C2×#692311yescharacteristic
296C22×C8—612no—
297C22×C8—1221no—
298#117—614no—
299#117—614no—
300#117—1222no—
301#117—1222no—
302#117—1222no—
303#117—1222no—
304#117—1222no—
305#117—1222no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4—1222no—
308C2×C22⋊C4—1222no—
309C2×C22⋊C4—1222no—
310C2×C42—1222no—
311C2×C42—1222no—
312#107—1224no—
313#106—1224no—
314#99—1228no—
315#99—1228no—
316C22⋊C8—22no—
317C22⋊C8—1222no—
318C22⋊C8—1222no—
319C22⋊C8—1222no—
320C24—611no—
321C24—611no—
322C24C4⋊#692311yescharacteristic
323#41—2441no—
324#41—2442no—
325#41—2441no—
326#41—2442no—
327#41—2442no—
328#41—2442no—
329#41—2442no—
330#41—2442no—
331#41—2441no—
332#41—2441no—
333C2×Q8—1221no—
334C2×Q8—1221no—
335C2×Q8—1222no—
336C2×Q8—1222no—
337C2×D4—1222no—
338C2×D4—1221no—
339C2×D4—1221no—
340C2×D4—1222no—
341C2×D4—1221no—
342C2×D4—1221no—
343C2×D4—1222no—
344C2×D4—1222no—
345C2×D4—1222no—
346C2×D4—1221no—
347C2×D4—1221no—
348C2×D4—1222no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—611no—
354C22×C4—611no—
355C22×C4—611no—
356C22×C4—611no—
357C22×C4—611no—
358C22×C4—612no—
359C22×C4—612no—
360C22×C4—612no—
361C22×C4—612no—
362C22×C4—611no—
363C22×C4—1221no—
364C22×C4—1221no—
365C22×C4—1222no—
366C22×C4—1222no—
367C22×C4—2442no—
368C22×C4—2442no—
369C22×C4C4⋊#692311yescharacteristic
370C22×C4C22×#692311yescharacteristic
371M16—22no—
372M16—22no—
373M16—1224no—
374M16—1222no—
375M16—1222no—
376M16—2441no—
377M16—2441no—
378C2×C8—21no—
379C2×C8—612no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8—1221no—
383C2×C8—2442no—
384C2×C8C22⋊#692312yes—
385C4⋊C4—1224no—
386C4⋊C4—1224no—
387C4⋊C4—2442no—
388C4⋊C4—2442no—
389C22⋊C4—1224no—
390C22⋊C4—1224no—
391C22⋊C4—2442no—
392C22⋊C4—2442no—
393C42—1224no—
394C42—1224no—
395C42—2442no—
396C42—2442no—
397C23—21no—
398C23—22no—
399C23—81no—
400C23—611no—
401C23—611no—
402C23—612no—
403C23—612no—
404C23—1222no—
405C23—1221no—
406C23—4881no—
407C23—4881no—
408C23C2×C4⋊#692311yescharacteristic
409Q8—2441no—
410Q8—2442no—
411Q8—2441no—
412D4—2441no—
413D4—2442no—
414D4—2441no—
415D4—2441no—
416D4—2442no—
417D4—2441no—
418D4—2441no—
419D4—2442no—
420D4—2441no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—611no—
427C2×C4—611no—
428C2×C4—612no—
429C2×C4—612no—
430C2×C4—611no—
431C2×C4—612no—
432C2×C4—611no—
433C2×C4—612no—
434C2×C4—612no—
435C2×C4—612no—
436C2×C4—612no—
437C2×C4—612no—
438C2×C4—611no—
439C2×C4—611no—
440C2×C4—1221no—
441C2×C4—1222no—
442C2×C4—4884no—
443C2×C4—4884no—
444C2×C4—2444no—
445C2×C4—2444no—
446C2×C4—2441no—
447C2×C4—2442no—
448C2×C4—2441no—
449C2×C4C2×C22⋊#692311yescharacteristic
450C2×C4#1451985611yescharacteristic
451C2×C4D4×#692312yes—
452C2×C4D4×#692312yes—
453C8—22no—
454C8—41no—
455C8—1222no—
456C8—2441no—
457C22—41no—
458C22—42no—
459C22—41no—
460C22—82no—
461C22—611no—
462C22—612no—
463C22—611no—
464C22—4882no—
465C22—2442no—
466C22—2442no—
467C22—2441no—
468C22—2441no—
469C22—2442no—
470C22—2441no—
471C22#1684070411yescharacteristic
472C22#1684070912yes—
473C4—41no—
474C4—611no—
475C4—612no—
476C4—611no—
477C4—4884no—
478C4—4884no—
479C4—2441no—
480C4#1684071011yescharacteristic
481C4#1684071212yes—
482C4#1684071411yescharacteristic
483C2—21no—
484C2—22no—
485C2—81no—
486C2—1221no—
487C2—1222no—
488C2—1221no—
489C2—4881no—
490C2#2146100211yescharacteristic
subgroupG/H
centerC4#16840714
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#407C4
socleC2×C61#1439
prime radicalC22×#11758222C22
cosocleC22×C4×D61C23
hypercenter#407#6923
hypocenterC61#11102
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #21462472C2semidirect
3. #21461501C2semidirect
4. #21461491C2semidirect
21. #16839726C4semidirect
22. #16839725C4semidirect
23. #16839725C4semidirect
24. C61×#407C4semidirect
267. #407#6923semidirect
270. C61#11102semidirect

Maximal subgroups

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

subgroupindex
#214624722
C2×#168407412
#214615012
#214614912
#214542622
#1110261