Finite Group Databasebeta

← #30610223

Subgroups — #30610223

Order 15616.

952 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462683C211yesmaximal, characteristic, complemented
2#21462673C211yesmaximal, characteristic, complemented
3#21461100C212yesmaximal, complemented
4#21461097C212yesmaximal, complemented
5#21461092C212yesmaximal, complemented
6#21461085C212yesmaximal, complemented
7#21461075C212yesmaximal, complemented
8#21461067C212yesmaximal, complemented
9C2×#16840356C211yesmaximal, characteristic
10#16840795—24no—
11#16840794—24no—
12#16840783C2212yes—
13#16840779C2212yes—
14#16840733C2212yes—
15#16840732C2212yes—
16#16840732C2212yes—
17D61⋊(C22⋊C8)C2212yes—
18#16840607C2214yes—
19#16840607C2214yes—
20#16840607C2214yes—
21#16840605C2214yes—
22#16840356—24no—
23#16840356—24no—
24C2×#14519772C411yescharacteristic
25C2×#14519770C2211yescharacteristic
26C2×#14519768C2211yescharacteristic
27C2×#14519767C411yescharacteristic
28C2×#14519761C411yescharacteristic
29C2×#14519760C411yescharacteristic
30C2×#14519759C2211yescharacteristic
31C2×#14519757C2211yescharacteristic
32C2×#14519754C411yescharacteristic
33C2×#14519754C411yescharacteristic
34C2×#14519749C411yescharacteristic
35C2×D4×Dic61C2211yescharacteristic
36C2×#14519737C2211yescharacteristic
37C22×C4⋊Dic61C2211yescharacteristic
38C2×#123×C61C411yescharacteristic
39C2×#11758213—22no—
40C2×#11758217—22no—
41C2×#11758217—22no—
42C2×D61⋊C8—22no—
43#14519844—22no—
44C4⋊#92420—22no—
45#14519834—22no—
46#14519834—22no—
47C22⋊#92420—22no—
48C22⋊#92420—22no—
49C22⋊#92420C2314yes—
50C22⋊#92420C2314yes—
51#14519832—22no—
52C4×#92420—22no—
53C22×Dic122C2×C411yescharacteristic
54C22×C22⋊D61C2×C411yescharacteristic
55C22×C22⋊D61C2×C411yescharacteristic
56C23×Dic61C2×C411yescharacteristic
57C23×Dic61C2×C411yescharacteristic
58C23×Dic61C2311yescharacteristic
59C22×C4×D61C2311yescharacteristic
60#14519772—24no—
61#14519770—22no—
62#14519770—22no—
63#14519768—22no—
64#14519768—22no—
65#14519767—24no—
66#14519761—24no—
67#14519760—24no—
68#14519759—22no—
69#14519759—22no—
70#14519757—22no—
71#14519757—22no—
72#14519754—24no—
73#14519754—24no—
74#14519749—24no—
75D4×Dic61—22no—
76D4×Dic61—22no—
77#14519737D414yes—
78#14519737D414yes—
79C2×C4⋊Dic61—22no—
80C2×C4⋊Dic61—22no—
81C2×C4⋊Dic61C2×C411yescharacteristic
82C2×C4⋊Dic61C2×C411yescharacteristic
83C2×C4⋊Dic61C2×C411yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196C2×C411yescharacteristic
88C2×#11758196C2×C411yescharacteristic
89C2×#11758196C2×C411yescharacteristic
90C2×#11758196C2×C411yescharacteristic
91C2×#11758196C2×C411yescharacteristic
92C2×#11758196C2×C411yescharacteristic
93C2×C22⋊Dic61C2×C411yescharacteristic
94C2×C22⋊Dic61C2×C411yescharacteristic
95C2×C22⋊Dic61C2311yescharacteristic
96C2×#11758198C2×C411yescharacteristic
97C2×#11758198C2×C411yescharacteristic
98C2×#11758198C2×C411yescharacteristic
99C2×#11758198C2×C411yescharacteristic
100C2×C4×Dic61C2×C411yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C22×D4×C61C2×C411yescharacteristic
103C23×C4×C61C2×C411yescharacteristic
104#123×C61—24no—
105C2×C4⋊C4×C61C2×C411yescharacteristic
106C2×C4⋊C4×C61C2×C411yescharacteristic
107C2×C22⋊C4×C61C2×C411yescharacteristic
108C2×C22⋊C4×C61C2×C411yescharacteristic
109C2×C22⋊C4×C61C2×C411yescharacteristic
110C24×C61C22×C411yescharacteristic
111C24×C61C22×C411yescharacteristic
112C23×D61C22×C411yescharacteristic
113C2×D4×C61—22no—
114C2×D4×C61—42no—
115C22×C4×C61—22no—
116C22×C4×C61—42no—
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C2×C22⋊D61—22no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—81no—
125C2×C22⋊D61C22⋊C414yes—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—81no—
129C22×Dic61—81no—
130C22×Dic61—81no—
131C22×Dic61C22×C411yescharacteristic
132C22×Dic61C22×C411yescharacteristic
133C22×Dic61C22×C411yescharacteristic
134C22×Dic61C22×C411yescharacteristic
135C22×Dic61C22×C411yescharacteristic
136C22×Dic61C22×C411yescharacteristic
137C22×Dic61C2×D412yes—
138C22×Dic61C2×D412yes—
139C22×Dic61C2411yescharacteristic
140C2×C4×D61—81no—
141C2×C4×D61C2×D412yes—
142C2×C4×D61C2×D412yes—
143C2×Dic122—22no—
144C2×Dic122—81no—
145#11758217—82no—
146#11758217—82no—
147C2×#92420—24no—
148C2×#92420—24no—
149#11758213—82no—
150D61⋊C8—82no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61C22⋊C414yes—
156C22⋊Dic61—22no—
157C22⋊Dic61—22no—
158C22⋊Dic61—21no—
159C22⋊Dic61—21no—
160#11758198—22no—
161#11758198—22no—
162#11758198—22no—
163#11758198C22⋊C414yes—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173#11758196—22no—
174#11758196—22no—
175#11758196—22no—
176#11758196—22no—
177#11758196—22no—
178#11758196C22⋊C414yes—
179C4×Dic61—22no—
180C4×Dic61C2×D412yes—
181C4×Dic61C2×D412yes—
182C23×C61—22no—
183C23×C61—21no—
184C23×C61—22no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61C2×C22⋊C412yes—
190C23×C61C23×C411yescharacteristic
191C22×D61—22no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61C2×C22⋊C412yes—
195D4×C61—44no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—44no—
205C2×C4×C61C2×C22⋊C412yes—
206C2×C4×C61C2×C22⋊C412yes—
207C22⋊D61—44no—
208C22⋊D61—82no—
209C2×Dic61—22no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—82no—
223C2×Dic61—82no—
224C2×Dic61C2×C22⋊C412yes—
225C2×Dic61C2×C22⋊C412yes—
226C2×Dic61C22×D411yescharacteristic
227C2×Dic61C22×D411yescharacteristic
228C4×D61—42no—
229C4×D61—42no—
230Dic122—82no—
231#92420—44no—
232#92420—44no—
233#17406—611nomaximal, Sylow
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—44no—
240C22×C61—41no—
241C22×C61—41no—
242C22×C61—41no—
243C22×C61—82no—
244C22×C61C22×C22⋊C411yescharacteristic
245C22×C61#56812yes—
246C2×D61—22no—
247C2×D61—22no—
248C2×D61—22no—
249C2×D61—82no—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61—41no—
255Dic61—22no—
256Dic61—21no—
257Dic61—21no—
258Dic61—22no—
259Dic61—41no—
260Dic61—41no—
261Dic61—41no—
262Dic61—41no—
263C2×#539—611no—
264#2975—611no—
265#2973—611no—
266#1473—612no—
267#1472—612no—
268#1470—612no—
269#1468—612no—
270#1465—612no—
271#1463—612no—
272C2×C61—22no—
273C2×C61—42no—
274C2×C61—81no—
275C2×C61#287911yescharacteristic
276C2×C61#288311yescharacteristic
277C2×C61#288911yescharacteristic
278D61—42no—
279#539—1224no—
280#539—1224no—
281C2×#126—611no—
282C2×#126—611no—
283C2×#125—611no—
284C2×#125—611no—
285C2×#123—611no—
286C2×#123—611no—
287C2×#123—611no—
288C2×#123#692311yescharacteristic
289C2×C22⋊Q8—611no—
290C2×C22⋊Q8—611no—
291C2×C22⋊Q8—611no—
292C2×#121—611no—
293C2×C4×D4—611no—
294C2×C4×D4—611no—
295C22×C4⋊C4—611no—
296#436—1224no—
297#435—1224no—
298#432—612no—
299#431—612no—
300#407—612no—
301#407—612no—
302#407—612no—
303C2×C22⋊C8—612no—
304#324—614no—
305#324—614no—
306#324—614no—
307#323—614no—
308C61#1740611yescharacteristic, complemented, Sylow
309C22×Q8—611no—
310C22×D4—611no—
311C22×D4—611no—
312C22×D4C2×#692311yescharacteristic
313C23×C4—611no—
314C23×C4—611no—
315C23×C4—611no—
316C23×C4—611no—
317C23×C4C2×#692311yescharacteristic
318C2×M16—1222no—
319C2×M16—1222no—
320C2×M16—1222no—
321C22×C8—1222no—
322#126—1222no—
323#126—1222no—
324#126—1224no—
325#125—1222no—
326#125—1222no—
327#125—1224no—
328#123—24no—
329#123—1224no—
330#123—1222no—
331#123—1222no—
332#123—1224no—
333C22⋊Q8—1224no—
334C22⋊Q8—1222no—
335C22⋊Q8—1222no—
336C22⋊Q8—1224no—
337#121—1224no—
338C4×D4—614no—
339C4×D4—614no—
340C4×D4—1222no—
341C4×D4—1222no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—611no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4C2×#692311yescharacteristic
357C2×C4⋊C4C2×#692311yescharacteristic
358C2×C22⋊C4—611no—
359C2×C22⋊C4—611no—
360C2×C22⋊C4—611no—
361C2×C22⋊C4—611no—
362C2×C22⋊C4—611no—
363C2×C22⋊C4—611no—
364C2×C22⋊C4—611no—
365C2×C22⋊C4C2×#692311yescharacteristic
366C2×C22⋊C4C2×#692311yescharacteristic
367C2×C22⋊C4C2×#692311yescharacteristic
368C2×C42—611no—
369C2×C42—611no—
370C4⋊C8—1222no—
371C4⋊C8—1222no—
372C22⋊C8—614no—
373C22⋊C8—614no—
374C22⋊C8—1222no—
375C22⋊C8—1222no—
376C22⋊C8—1222no—
377C22⋊C8—1222no—
378#97—1222no—
379C4×C8—1222no—
380C24—611no—
381C24C22×#692311yescharacteristic
382C24C22×#692311yescharacteristic
383C2×Q8—1222no—
384C2×Q8—4881no—
385C2×D4—22no—
386C2×D4—42no—
387C2×D4—614no—
388C2×D4—1222no—
389C2×D4—4881no—
390C2×D4—2442no—
391C22×C4—22no—
392C22×C4—42no—
393C22×C4—611no—
394C22×C4—612no—
395C22×C4—612no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—612no—
399C22×C4—612no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—4881no—
407C22×C4—4881no—
408C22×C4—4881no—
409C22×C4—4881no—
410C22×C4C22×#692311yescharacteristic
411C22×C4C22×#692311yescharacteristic
412C22×C4C22×#692311yescharacteristic
413C22×C4C22×#692311yescharacteristic
414C22×C4C22×#692311yescharacteristic
415M16—4882no—
416M16—4882no—
417M16—4882no—
418C2×C8—1224no—
419C2×C8—1224no—
420C2×C8—4882no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—614no—
424C4⋊C4—1222no—
425C4⋊C4—1221no—
426C4⋊C4—1221no—
427C4⋊C4—1222no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1222no—
431C4⋊C4—1222no—
432C4⋊C4—1222no—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—614no—
441C22⋊C4—1222no—
442C22⋊C4—1222no—
443C22⋊C4—1222no—
444C22⋊C4—1221no—
445C22⋊C4—1221no—
446C22⋊C4—1222no—
447C22⋊C4—1222no—
448C22⋊C4C22⋊#692314yes—
449C42—612no—
450C42—612no—
451C42—1222no—
452C23—22no—
453C23—21no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—612no—
460C23—1222no—
461C23—2441no—
462C23—2441no—
463C23C2×C22⋊#692312yes—
464C23C23×#692311yescharacteristic
465Q8—4882no—
466D4—44no—
467D4—4882no—
468D4—2444no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—612no—
479C2×C4—612no—
480C2×C4—611no—
481C2×C4—611no—
482C2×C4—1222no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1221no—
487C2×C4—4882no—
488C2×C4—4882no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4C2×C22⋊#692312yes—
500C2×C4C2×C22⋊#692312yes—
501C8—2444no—
502C8—2444no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—44no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—82no—
513C22—1222no—
514C22—1222no—
515C22—1222no—
516C22—4882no—
517C22C22×C22⋊#692311yescharacteristic
518C22#1684084012yes—
519C4—22no—
520C4—22no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—1222no—
525C4—1221no—
526C4—1221no—
527C4—1222no—
528C4—2441no—
529C4—2441no—
530C4—2441no—
531C4—2441no—
532C2—22no—
533C2—42no—
534C2—81no—
535C2—2442no—
536C2#2146240511yescharacteristic
537C2#2146241911yescharacteristic
538C2#2146243111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#123×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519749C4
cosocleC22×Dic61C24
hypercenterC2×#123#6923
hypocenterC61#17406
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #21462683C2semidirect
2. #21462673C2semidirect
3. #21461100C2semidirect
4. #21461097C2semidirect
5. #21461092C2semidirect
6. #21461085C2semidirect
7. #21461075C2semidirect
8. #21461067C2semidirect
308. C61#17406semidirect

Maximal subgroups

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

subgroupindex
#214626832
#214626732
#214611002
#214610972
#214610922
#214610852
#214610752
#214610672
C2×#168403562
#1740661