Finite Group Databasebeta

← #30612752

Subgroups — #30612752

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462542C211yesmaximal, characteristic, complemented
2C2×#16840796C211yesmaximal, characteristic, complemented
3#21461451C212yesmaximal, complemented
4#21461450C212yesmaximal, complemented
5#21461419C212yesmaximal, complemented
6#21461418C212yesmaximal, complemented
7#21461388C211yesmaximal, characteristic, complemented
8#21461146C211yesmaximal, characteristic, complemented
9#21461124C211yesmaximal, characteristic, complemented
10#21461024C211yesmaximal, characteristic, complemented
11#21455816C211yesmaximal, characteristic
12C2×#14519864—21no—
13C22×#11758213—21no—
14#16840797C2211yescharacteristic, complemented
15#16840796—21no—
16#16840796—21no—
17#16840796C2212yes—
18#16840796C2212yes—
19#16840796C2211yescharacteristic, complemented
20#16840795—21no—
21#16840790C2211yescharacteristic, complemented
22#16840789—21no—
23D4×#92420C2211yescharacteristic, complemented
24#16840779C2211yescharacteristic
25#16840765C2211yescharacteristic
26C2×C4⋊#92420C2211yescharacteristic
27#16840735—21no—
28C2×#14519834—21no—
29C2×C4×#92420C2211yescharacteristic
30#16840626C2212yescomplemented
31#16840625C2212yescomplemented
32#16840610C2212yes—
33#16840610C2212yes—
34#16840610C2212yescomplemented
35D4⋊#92420C2212yescomplemented
36#92420⋊C8C2212yescomplemented
37C8×#92420C2212yescomplemented
38#16840307C2211yescharacteristic
39C2×#14519739C2211yescharacteristic
40#16840075C411yescharacteristic
41#16840041C412yes—
42#16839956C412yes—
43#16839898C411yescharacteristic
44#16839895C411yescharacteristic
45#16839844C2212yes—
46#16839842C2212yes—
47#16839812C2211yescharacteristic
48C61×#480C411yescharacteristic
49#14519864—44no—
50C2×#11758213—21no—
51C2×#11758213—22no—
52C2×#11758213—22no—
53C2×#11758213—44no—
54C2×#11758217—21no—
55C2×#11758217—21no—
56C2×D61⋊C8—21no—
57C22×#92420—21no—
58C22×#92420—21no—
59C22×#92420D411yescharacteristic
60C22×#92420D411yescharacteristic
61#14519844—21no—
62#14519844—21no—
63C4⋊#92420—21no—
64C4⋊#92420C2312yes—
65C4⋊#92420C2311yescharacteristic
66C4⋊#92420C2311yescharacteristic
67#14519834—22no—
68#14519834—22no—
69#14519834—21no—
70#14519834—21no—
71C22⋊#92420—21no—
72C22⋊#92420—21no—
73C22⋊#92420—21no—
74C4×#92420D411yescharacteristic
75C4×#92420D411yescharacteristic
76C4×#92420C2312yes—
77C4×#92420C2311yescharacteristic
78C4×#92420C2311yescharacteristic
79C2×#11758224—21no—
80C22×D244C2×C411yescharacteristic
81C22×C4×D61—21no—
82C2×D488C2×C412yes—
83C2×#11758199C2×C412yes—
84#14519776C2×C411yescharacteristic
85#14519773C2×C411yescharacteristic
86#14519763—21no—
87#14519751C2×C411yescharacteristic
88#14519749—21no—
89D4×Dic61C2311yescharacteristic
90#14519739—21no—
91#14519739—21no—
92#14519739C2×C412yes—
93#14519739C2312yes—
94#14519739C2311yescharacteristic
95#14519737—21no—
96C2×#11758198—21no—
97C2×C4×Dic61C2311yescharacteristic
98#14519710C2×C412yes—
99#14519706C2×C412yes—
100#14519699C2×C412yes—
101#14519697C2×C412yes—
102#14519697C2×C412yes—
103C22⋊#92418C2×C411yescharacteristic
104C8×Dic61C2312yes—
105#121×C61C2×C411yescharacteristic
106C2×C4⋊C4×C61C2×C411yescharacteristic
107#107×C61C2×C412yes—
108#102×C61C2×C412yes—
109C22⋊C8×C61C2×C411yescharacteristic
110C23×D61—21no—
111C2×D4×C61—21no—
112C2×D4×C61C22×C411yescharacteristic
113C22×C4×C61—21no—
114C22×C4×C61C22×C411yescharacteristic
115C2×C22⋊D61—21no—
116C2×C22⋊D61—21no—
117C2×C22⋊D61—21no—
118C22×Dic61—21no—
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61#4111yescharacteristic
121#11758224—42no—
122#11758224—42no—
123C2×D244—21no—
124C2×D244—42no—
125C2×D244C22×C412yes—
126C2×D244C22×C411yescharacteristic
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—21no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×Dic122—21no—
133#11758217—42no—
134#11758217—42no—
135C2×#92420—22no—
136C2×#92420—21no—
137C2×#92420—21no—
138C2×#92420—22no—
139C2×#92420—22no—
140C2×#92420—42no—
141C2×#92420—42no—
142C2×#92420D814yescomplemented
143C2×#92420C2×D411yescharacteristic
144C2×#92420C2×D411yescharacteristic
145C2×#92420C2×D411yescharacteristic
146C2×#92420C2×D411yescharacteristic
147#11758213—44no—
148#11758213—44no—
149#11758213—42no—
150D61⋊C8—42no—
151C2×C8×C61C22×C412yes—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C22×C412yes—
154C4⋊C4×C61C22×C411yescharacteristic
155C22⋊C4×C61—21no—
156C22⋊Dic61—21no—
157#11758199—42no—
158#11758198—21no—
159#11758198—22no—
160#11758198—22no—
161#11758198—21no—
162C4⋊Dic61C22×C411yescharacteristic
163#11758196—21no—
164C4×Dic61C22×C411yescharacteristic
165C4×Dic61C2×D411yescharacteristic
166C4×Dic61#4111yescharacteristic
167C4×Dic61C2411yescharacteristic
168C2×#92418C22×C412yes—
169D488—42no—
170C23×C61—21no—
171C23×C61C4×D411yescharacteristic
172C22×D61—22no—
173C22×D61—21no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177D4×C61—22no—
178D4×C61—42no—
179C2×C4×C61—22no—
180C2×C4×C61—21no—
181C2×C4×C61—22no—
182C2×C4×C61—42no—
183C2×C4×C61C4×D411yescharacteristic
184C2×C4×C61C23×C411yescharacteristic
185C22⋊D61—42no—
186C22⋊D61—42no—
187C22⋊D61—42no—
188C2×Dic61—21no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61C4×D411yescharacteristic
193C2×Dic61C4×D411yescharacteristic
194C2×Dic61C2×D812yes—
195C2×Dic61#13512yes—
196C2×Dic61C22×D411yescharacteristic
197C2×Dic61C2×#4111yescharacteristic
198D244—22no—
199D244—22no—
200D244—22no—
201D244—42no—
202C4×D61—41no—
203C4×D61—42no—
204C4×D61—42no—
205C4×D61—41no—
206C4×D61—44no—
207Dic122—42no—
208#92420—21no—
209#92420—21no—
210#92420—42no—
211#92420—41no—
212#92420—42no—
213#92420—41no—
214#92420C2×D812yes—
215#92420C2×D812yes—
216C8×C61—22no—
217#92418—22no—
218#17928—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—41no—
222C22×C61—41no—
223C22×C61—41no—
224C22×C61C4×D812yes—
225C22×C61C2×C4×D411yescharacteristic
226C2×D61—42no—
227C2×D61—41no—
228C2×D61—42no—
229C2×D61—41no—
230C2×D61—41no—
231C2×D61—42no—
232C2×D61—84no—
233C4×C61—21no—
234C4×C61—41no—
235C4×C61—42no—
236C4×C61#56812yes—
237Dic61—21no—
238Dic61—21no—
239Dic61—41no—
240Dic61C4×D812yes—
241Dic61C22×D811yescharacteristic
242Dic61C2×#13511yescharacteristic
243#3100—611no—
244#2931—611no—
245C2×#436—611no—
246#1589—612no—
247#1589—612no—
248#1581—612no—
249#1581—612no—
250#1571—611no—
251#1486—611no—
252#1480—611no—
253#1450—611no—
254C2×C61—22no—
255C2×C61—81no—
256C2×C61#293011yescharacteristic
257C2×C61C2×C4×D811yescharacteristic
258C2×C61#295511yescharacteristic
259D61—81no—
260D61—82no—
261C2×#131—1221no—
262C22×M16—1221no—
263#532—611no—
264C2×C4×D4—611no—
265#480—611no—
266#480#692311yescharacteristic
267#474—612no—
268#459—612no—
269#447—611no—
270#447—611no—
271C4×D8—612no—
272C4×D8—612no—
273#436—612no—
274#436—612no—
275#436—611no—
276#436—611no—
277#436—1221no—
278#436—1221no—
279#436—1221no—
280C8×D4—611no—
281C8×D4—611no—
282C8×D4—1221no—
283#431—611no—
284#431—611no—
285#423—611no—
286C2×C4⋊C8—611no—
287#408—1221no—
288C2×C22⋊C8—1221no—
289C2×C4×C8—611no—
290#335—612no—
291#335—612no—
292D4⋊C8—612no—
293D4⋊C8—612no—
294D4⋊C8—612no—
295D4⋊C8—612no—
296#322—612no—
297C82—612no—
298C61#1792811yescharacteristic, complemented, Sylow
299C2×#41—1221no—
300C22×D4—611no—
301C23×C4—1221no—
302C2×D8—612no—
303C2×D8—612no—
304#131—2444no—
305C2×M16—1221no—
306C2×M16—1222no—
307C2×M16—1222no—
308C2×M16—1221no—
309C2×M16—1221no—
310C2×M16—2444no—
311C22×C8—611no—
312C22×C8—611no—
313C22×C8—1221no—
314C22×C8—1221no—
315C22×C8—1221no—
316C4⋊Q8—611no—
317#127—611no—
318#124—1221no—
319#121—611no—
320#121—1221no—
321#121C2×#692311yescharacteristic
322C4×D4—611no—
323C4×D4—612no—
324C4×D4—612no—
325C4×D4—611no—
326C4×D4—1221no—
327C4×D4—1221no—
328C4×D4—1221no—
329C2×C4⋊C4C2×#692311yescharacteristic
330C2×C22⋊C4—1221no—
331C2×C42—611no—
332#107—612no—
333#107C2×#692312yes—
334C4⋊C8—612no—
335C4⋊C8—612no—
336C4⋊C8—611no—
337C4⋊C8—611no—
338C4⋊C8—1221no—
339C4⋊C8—1221no—
340C4⋊C8—1221no—
341#102—612no—
342#102—612no—
343#102—612no—
344#102C2×#692312yes—
345C22⋊C8—611no—
346C22⋊C8—1221no—
347C22⋊C8—1221no—
348C22⋊C8—1222no—
349C22⋊C8—1222no—
350C22⋊C8—1221no—
351C22⋊C8—1221no—
352C22⋊C8—1221no—
353C22⋊C8C2×#692311yescharacteristic
354C4×C8—611no—
355C4×C8—611no—
356C4×C8—611no—
357C4×C8—611no—
358C4×C8—612no—
359C4×C8—612no—
360C24—1221no—
361#41—2442no—
362#41—2442no—
363C2×Q8—1221no—
364C2×D4—21no—
365C2×D4—612no—
366C2×D4—611no—
367C2×D4—1221no—
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—2442no—
372C2×D4C22×#692311yescharacteristic
373C22×C4—21no—
374C22×C4—611no—
375C22×C4—611no—
376C22×C4—1221no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—1221no—
380C22×C4—2442no—
381C22×C4—2442no—
382C22×C4C22×#692311yescharacteristic
383D8—2442no—
384D8—2442no—
385M16—2444no—
386M16—2444no—
387M16—2442no—
388M16—2442no—
389M16—2442no—
390C2×C8—611no—
391C2×C8—611no—
392C2×C8—614no—
393C2×C8—611no—
394C2×C8—611no—
395C2×C8—612no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8—1222no—
401C2×C8—2442no—
402C2×C8—2442no—
403C2×C8—2442no—
404C2×C8C22×#692312yes—
405C4⋊C4—21no—
406C4⋊C4—611no—
407C4⋊C4—1221no—
408C4⋊C4C22×#692311yescharacteristic
409C4⋊C4C22×#692312yes—
410C22⋊C4—21no—
411C22⋊C4—1222no—
412C22⋊C4—1222no—
413C22⋊C4—1221no—
414C22⋊C4—1221no—
415C22⋊C4—1221no—
416C42—611no—
417C42—611no—
418C42—611no—
419C42—611no—
420C23—21no—
421C23—1222no—
422C23—1221no—
423C23—2442no—
424C23—2442no—
425C23—2442no—
426C23D4×#692311yescharacteristic
427Q8—2442no—
428D4—22no—
429D4—42no—
430D4—1222no—
431D4—1222no—
432D4—1222no—
433D4—2442no—
434D4—2442no—
435D4—2442no—
436D4—2442no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—612no—
442C2×C4—612no—
443C2×C4—611no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—611no—
447C2×C4—1221no—
448C2×C4—2441no—
449C2×C4—2442no—
450C2×C4—2442no—
451C2×C4—2444no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4D4×#692311yescharacteristic
457C2×C4C23×#692311yescharacteristic
458C8—22no—
459C8—612no—
460C8—612no—
461C8—1221no—
462C8—1221no—
463C8—1222no—
464C8—2441no—
465C8—2442no—
466C8—2441no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—4884no—
474C22—2442no—
475C22—2441no—
476C22—2442no—
477C22—2441no—
478C22—2441no—
479C22—2442no—
480C22D8×#692312yes—
481C22C2×D4×#692311yescharacteristic
482C4—21no—
483C4—42no—
484C4—41no—
485C4—611no—
486C4—611no—
487C4—612no—
488C4—1221no—
489C4—1221no—
490C4—2441no—
491C4#1684084112yes—
492C2—22no—
493C2—81no—
494C2—4882no—
495C2—4881no—
496C2#2146254011yescharacteristic
497C2C2×D8×#692311yescharacteristic
498C2#2146259211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#480C4
socleC22×C61C2×C4×D4
prime radical#21455816C2
cosocleC4×Dic61C24
hypercenter#480#6923
hypocenterC61#17928
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #21462542C2semidirect
2. C2×#16840796C2semidirect
3. #21461451C2semidirect
4. #21461450C2semidirect
5. #21461419C2semidirect
6. #21461418C2semidirect
7. #21461388C2semidirect
8. #21461146C2semidirect
9. #21461124C2semidirect
10. #21461024C2semidirect
14. #16840797C22semidirect
19. #16840796C22semidirect
21. #16840790C22semidirect
23. D4×#92420C22semidirect
30. #16840626C22semidirect
31. #16840625C22semidirect
34. #16840610C22semidirect
35. D4⋊#92420C22semidirect
36. #92420⋊C8C22semidirect
37. C8×#92420C22semidirect
142. C2×#92420D8semidirect
298. C61#17928semidirect

Maximal subgroups

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

subgroupindex
#214625422
C2×#168407962
#214614512
#214614502
#214614192
#214614182
#214613882
#214611462
#214611242
#214610242
#214558162
#1792861