Finite Group Databasebeta

← #30602876

Subgroups — #30602876

Order 15616.

986 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊#6923C211yesmaximal, characteristic
2C2×#16840635C212yesmaximal
3C2×#16840635C212yesmaximal
4#21461480C211yesmaximal, characteristic
5#389×D61C211yesmaximal, characteristic
6C22×D61⋊C8C2212yes—
7C22×C4⋊#6923C2212yes—
8C22×C4×#6923—21no—
9C4×C4⋊#6923—24no—
10C2×(C2×#6923)⋊C4—21no—
11#16840635—24no—
12#16840635C418yescomplemented
13#16840628—21no—
14#16840628—21no—
15C2×C4⋊C4×D61—21no—
16C2×C4⋊C4×D61C2211yescharacteristic
17C2×C4⋊C4×D61C2211yescharacteristic
18C2×C42×D61C2211yescharacteristic
19#16839661C411yescharacteristic, complemented
20#16839660C411yescharacteristic, complemented
21#16839659—21no—
22#16839658C411yescharacteristic, complemented
23#16839657—21no—
24#95×D61—21no—
25C61×#389C411yescharacteristic, complemented
26C23×#6923—22no—
27C2×D61⋊C8—22no—
28C2×D61⋊C8—22no—
29C2×D61⋊C8—22no—
30C2×D61⋊C8—22no—
31C2×D61⋊C8—22no—
32C2×D61⋊C8C2×C414yes—
33C22×#92420—22no—
34C2×C4⋊#6923—22no—
35C2×C4⋊#6923—22no—
36C2×C4⋊#6923—22no—
37C2×C4⋊#6923C2×C414yes—
38C2×C4⋊#6923D414yes—
39C2×C4⋊#6923Q814yes—
40C2×C4×#6923—24no—
41C2×C4×#6923—42no—
42C2×C4×#6923—42no—
43(C2×#6923)⋊C4—24no—
44C22×C4×D61—21no—
45C22×C4×D61—21no—
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61C2311yescharacteristic
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61C2×C411yescharacteristic
51C2×C4⋊Dic61C2×C411yescharacteristic
52C2×#11758196—21no—
53C2×#11758196—21no—
54C2×#11758196—21no—
55C2×#11758196—21no—
56C4⋊C4×D61—22no—
57C4⋊C4×D61—22no—
58C4⋊C4×D61—22no—
59C4⋊C4×D61—44no—
60C4⋊C4×D61C2×C414yes—
61C2×C4×Dic61C2×C411yescharacteristic
62C2×C4×Dic61D411yescharacteristic
63C2×C4×Dic61D411yescharacteristic
64C42×D61—22no—
65C42×D61—22no—
66#14519684—21no—
67#14519684—21no—
68#14519684—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61C2×C411yescharacteristic
71C2×C4⋊C4×C61C2×C411yescharacteristic
72C2×C42×C61C2×C411yescharacteristic
73#95×C61—21no—
74C23×D61C2×D411yescharacteristic
75C22×#6923—24no—
76C22×#6923—24no—
77C22×#6923—24no—
78C22×#6923—42no—
79C22×#6923—42no—
80C22×#6923—42no—
81C22×#6923—42no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22×C411yescharacteristic
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61C22⋊C411yescharacteristic
90C22×Dic61C22⋊C411yescharacteristic
91C22×Dic61C2×D411yescharacteristic
92C2×C4×D61—21no—
93C2×C4×D61—21no—
94C2×C4×D61—21no—
95C2×C4×D61—21no—
96C2×C4×D61—24no—
97C2×C4×D61—21no—
98C2×C4×D61—21no—
99C2×C4×D61—22no—
100C2×C4×D61—22no—
101C2×C4×D61—41no—
102C2×C4×D61—42no—
103C2×C4×D61—41no—
104C2×C4×D61—41no—
105C2×C4×D61—41no—
106C2×C4×D61—42no—
107C2×C4×D61—42no—
108C2×C4×D61SD1612yes—
109C2×C4×D61Q1612yes—
110C2×C4×D61C22×C412yes—
111C2×C4×D61C2×D412yes—
112C2×C4×D61C2×Q812yes—
113C2×C4×D61#4112yes—
114C2×C4×D61#4112yes—
115C2×C4×D61#4112yes—
116C2×#92420—24no—
117C2×#92420—42no—
118C2×#92420—42no—
119C4⋊#6923—24no—
120C4⋊#6923—24no—
121C4⋊#6923—24no—
122C4⋊#6923C4⋊C418yescomplemented
123C4×#6923—48no—
124D61⋊C8—24no—
125D61⋊C8—24no—
126D61⋊C8—42no—
127D61⋊C8—42no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—42no—
131C4⋊C4×C61C4212yescomplemented
132C42×C61—22no—
133C4⋊Dic61—21no—
134C4⋊Dic61—22no—
135C4⋊Dic61—42no—
136C4⋊Dic61C4212yescomplemented
137#11758196—24no—
138#11758196—42no—
139#11758196—42no—
140#11758196—42no—
141C4×Dic61—22no—
142C4×Dic61—22no—
143C4×Dic61—22no—
144C23×C61C2×C22⋊C411yescharacteristic
145C22×D61C4×D412yes—
146C22×D61#12011yescharacteristic
147C22×D61C22⋊Q812yes—
148C22×D61#12311yescharacteristic
149C22×D61C2×SD1611yescharacteristic
150C22×D61C2×Q1611yescharacteristic
151C22×D61#13512yes—
152C22×D61AGL(1,ℤ/8)12yes—
153C22×D61AGL(1,ℤ/8)11yescharacteristic
154C22×D61#13711yescharacteristic
155C2×#6923—28no—
156C2×#6923—44no—
157C2×#6923—44no—
158C2×#6923—44no—
159C2×#6923—44no—
160C2×#6923—44no—
161C2×#6923—44no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—22no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61#10312yes—
174C2×C4×C61C2×C4211yescharacteristic
175C2×C4×C61#11711yescharacteristic
176C2×C4×C61C4×D412yes—
177C2×C4×C61C4×Q812yes—
178C2×Dic61—21no—
179C2×Dic61—21no—
180C2×Dic61—22no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61#10312yes—
190C2×Dic61C4×D412yes—
191C2×Dic61#12112yes—
192C2×Dic61#12411yescharacteristic
193C2×Dic61C4⋊Q811yescharacteristic
194C4×D61—22no—
195C4×D61—22no—
196C4×D61—22no—
197C4×D61—22no—
198C4×D61—44no—
199C4×D61—44no—
200C4×D61—42no—
201C4×D61—42no—
202C4×D61—82no—
203C4×D61—82no—
204C4×D61C2×C4⋊C412yes—
205C4×D61#11712yes—
206C4×D61C4×D412yes—
207C4×D61C4×Q812yes—
208C4×D61C22⋊Q812yes—
209C4×D61#12312yes—
210C4×D61#12512yes—
211C4×D61#12612yes—
212#92420—44no—
213#10885—611nomaximal, Sylow
214C22×C61C4×C22⋊C411yescharacteristic
215C22×C61#38611yescharacteristic
216C22×C61#39111yescharacteristic
217C22×C61C2×#10311yescharacteristic
218C22×C61#41611yescharacteristic
219C22×C61#41711yescharacteristic
220C22×C61#41811yescharacteristic
221C2×D61#38512yes—
222C2×D61#38712yes—
223C2×D61C4×SD1612yes—
224C2×D61C4×Q1612yes—
225C2×D61#44012yes—
226C2×D61#44212yes—
227C2×D61#44912yes—
228C2×D61#45011yescharacteristic
229C2×D61#45111yescharacteristic
230C2×D61#47612yes—
231C2×D61#47712yes—
232C2×D61#47812yes—
233C2×D61#47912yes—
234C2×D61#48212yes—
235C2×D61#48311yescharacteristic
236C2×D61#48411yescharacteristic
237#6923—48no—
238C4×C61—22no—
239C4×C61—41no—
240C4×C61—42no—
241C4×C61—81no—
242C4×C61C4×C4⋊C412yes—
243C4×C61#38712yes—
244Dic61—22no—
245Dic61—41no—
246Dic61—42no—
247Dic61—81no—
248Dic61#38812yes—
249Dic61#38912yes—
250C2×#389—611no—
251C2×C4×C4⋊C4—611no—
252C2×#340—612no—
253C2×#340—612no—
254C2×#336—611no—
255C2×C61#143611yescharacteristic
256C2×C61C4×#10311yescharacteristic
257C2×C61#176211yescharacteristic
258C2×C61#187211yescharacteristic
259C2×C61#187511yescharacteristic
260C2×C61#198711yescharacteristic
261C2×C61#198911yescharacteristic
262D61#186312yes—
263D61#186412yes—
264D61#192012yes—
265D61#192112yes—
266C23×C8—612no—
267C22×C4⋊C4—612no—
268C22×C4⋊C4—611no—
269C22×C4⋊C4—611no—
270C22×C4⋊C4—1221no—
271C22×C42—611no—
272C22×C42—1221no—
273#389—611no—
274#389—611no—
275#389—611no—
276#389—1221no—
277#389—1221no—
278#389#692311yescharacteristic, complemented
279C4×C4⋊C4—1224no—
280C2×#95—1221no—
281C2×#95—1221no—
282#340—618no—
283#340—1224no—
284#336—1221no—
285#336—1221no—
286C61#1088511yescharacteristic, complemented, Sylow
287C23×C4—611no—
288C23×C4—611no—
289C23×C4—611no—
290C23×C4—1221no—
291C23×C4—1221no—
292C23×C4—1222no—
293C22×C8—614no—
294C22×C8—1222no—
295C22×C8—1222no—
296C22×C8—1222no—
297C22×C8—1222no—
298C22×C8—1222no—
299C22×C8—1222no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—614no—
302C2×C4⋊C4—614no—
303C2×C4⋊C4—614no—
304C2×C4⋊C4—611no—
305C2×C4⋊C4—614no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—1222no—
308C2×C4⋊C4—1222no—
309C2×C4⋊C4—1222no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4—1222no—
313C2×C4⋊C4—1222no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—2444no—
319C2×C4⋊C4C2×#692311yescharacteristic
320C2×C4⋊C4C2×#692311yescharacteristic
321C2×C42—611no—
322C2×C42—611no—
323C2×C42—611no—
324C2×C42—1222no—
325C2×C42—1222no—
326C2×C42—1224no—
327C2×C42—2442no—
328C2×C42—2442no—
329C2×C42C2×#692311yescharacteristic
330#95—21no—
331#95—1221no—
332#95—1221no—
333#95—1224no—
334#95—1221no—
335C24—611no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—611no—
339C22×C4—612no—
340C22×C4—612no—
341C22×C4—612no—
342C22×C4—612no—
343C22×C4—612no—
344C22×C4—612no—
345C22×C4—611no—
346C22×C4—612no—
347C22×C4—612no—
348C22×C4—611no—
349C22×C4—1221no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—1224no—
356C22×C4—1222no—
357C22×C4—1222no—
358C22×C4—1224no—
359C22×C4—1224no—
360C22×C4—1224no—
361C22×C4—1221no—
362C22×C4—1221no—
363C22×C4—2441no—
364C22×C4—2441no—
365C22×C4—2442no—
366C22×C4—2442no—
367C22×C4—2442no—
368C22×C4—2441no—
369C22×C4—2441no—
370C22×C4—2442no—
371C22×C4—2442no—
372C22×C4—2442no—
373C22×C4—2442no—
374C22×C4C22⋊#692311yescharacteristic
375C22×C4C22⋊#692311yescharacteristic
376C22×C4C22×#692311yescharacteristic
377C2×C8—1224no—
378C2×C8—1224no—
379C2×C8—1224no—
380C2×C8—2442no—
381C2×C8—2442no—
382C2×C8—2442no—
383C2×C8—2442no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—42no—
387C4⋊C4—618no—
388C4⋊C4—612no—
389C4⋊C4—1224no—
390C4⋊C4—1224no—
391C4⋊C4—1224no—
392C4⋊C4—1224no—
393C4⋊C4—1221no—
394C4⋊C4—1222no—
395C4⋊C4—2442no—
396C4⋊C4—2442no—
397C4⋊C4—2442no—
398C4⋊C4—2442no—
399C4⋊C4C4×#692312yescomplemented
400C42—22no—
401C42—1222no—
402C42—1222no—
403C42—1222no—
404C42—2448no—
405C23—611no—
406C23—611no—
407C23—612no—
408C23—611no—
409C23—611no—
410C23—612no—
411C23—611no—
412C23—611no—
413C23—612no—
414C23—612no—
415C23C2×C22⋊#692311yescharacteristic
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—612no—
428C2×C4—611no—
429C2×C4—611no—
430C2×C4—612no—
431C2×C4—612no—
432C2×C4—612no—
433C2×C4—612no—
434C2×C4—612no—
435C2×C4—612no—
436C2×C4—612no—
437C2×C4—612no—
438C2×C4—612no—
439C2×C4—612no—
440C2×C4—1222no—
441C2×C4—1222no—
442C2×C4—1222no—
443C2×C4—1222no—
444C2×C4—1221no—
445C2×C4—1221no—
446C2×C4—1228no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1222no—
451C2×C4—4882no—
452C2×C4—4882no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2444no—
456C2×C4—2444no—
457C2×C4—2441no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2444no—
461C2×C4—2444no—
462C2×C4—2444no—
463C2×C4—2444no—
464C2×C4—2444no—
465C2×C4—2444no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4#1451984012yes—
469C2×C4C2×C4×#692311yescharacteristic
470C2×C4#1451985611yescharacteristic
471C2×C4D4×#692312yes—
472C2×C4Q8×#692312yes—
473C8—2444no—
474C22—612no—
475C22—612no—
476C22—611no—
477C22—611no—
478C22—611no—
479C22—611no—
480C22—612no—
481C22—612no—
482C22—612no—
483C22—612no—
484C22—612no—
485C22—612no—
486C22—612no—
487C22—612no—
488C22—612no—
489C22—612no—
490C22C4×C22⋊#692311yescharacteristic
491C22#1684071011yescharacteristic
492C22(C2×#6923)⋊Q811yescharacteristic
493C22C2×#1451984011yescharacteristic
494C22#1684075211yescharacteristic
495C22#1684075311yescharacteristic
496C22#1684075411yescharacteristic
497C4—22no—
498C4—41no—
499C4—42no—
500C4—81no—
501C4—612no—
502C4—612no—
503C4—1222no—
504C4—4881no—
505C4—2448no—
506C4—2441no—
507C4—2442no—
508C4C4⋊C4×#692312yes—
509C4#1684071112yes—
510C2—612no—
511C2—612no—
512C2—612no—
513C2—612no—
514C2#2146099811yescharacteristic
515C2C4×#1451984011yescharacteristic
516C2#2146158911yescharacteristic
517C2#2146180411yescharacteristic
518C2#2146180711yescharacteristic
519C2#2146199711yescharacteristic
520C2#2146199911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#389C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#389#6923
hypocenterC61#10885
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
12. #16840635C4semidirect
19. #16839661C4semidirect
20. #16839660C4semidirect
22. #16839658C4semidirect
25. C61×#389C4semidirect
122. C4⋊#6923C4⋊C4semidirect
131. C4⋊C4×C61C42semidirect
136. C4⋊Dic61C42semidirect
278. #389#6923semidirect
286. C61#10885semidirect
399. C4⋊C4C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊#69232
C2×#168406352
#214614802
#389×D612
#1088561