Finite Group Databasebeta

← #30612405

Subgroups — #30612405

Order 15616.

688 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#21462626C211yesmaximal, characteristic, complemented
2#21462623C211yesmaximal, characteristic, complemented
3#21461447C212yesmaximal, complemented
4#21461443C212yesmaximal, complemented
5#21461439C212yesmaximal, complemented
6#21461435C212yesmaximal, complemented
7#21461368C211yesmaximal, characteristic, complemented
8#21461112C211yesmaximal, characteristic, complemented
9#21461112C211yesmaximal, characteristic, complemented
10C2×#92420⋊C8C211yesmaximal, characteristic, complemented
11#21454590C211yesmaximal, characteristic
12#16840812C2212yes—
13#16840811C2212yes—
14#16840796C2212yes—
15#16840791—22no—
16#16840790C2212yes—
17#16840789—22no—
18#16840781—21no—
19#16840780—21no—
20#16840778C2211yescharacteristic
21#16840778C2211yescharacteristic
22C4×#11758213—21no—
23C4×D61⋊C8—21no—
24C2×C4×#92420C2211yescharacteristic
25C2×C4×#92420C2211yescharacteristic
26#16840626C2212yes—
27#16840624C2212yes—
28#16840612C2212yes—
29#16840612C2212yes—
30#16840610C2212yes—
31#16840610C2212yes—
32#92420⋊C8C2212yes—
33#92420⋊C8C2212yes—
34C2×#14519799C411yescharacteristic
35#16840230C2211yescharacteristic
36#16840230C2211yescharacteristic
37#16839856C411yescharacteristic
38#16839856C2211yescharacteristic
39#16839849C411yescharacteristic
40#16839849C411yescharacteristic
41#16839849C2211yescharacteristic
42#16839849C2211yescharacteristic
43#16839844C411yescharacteristic
44#16839844C2211yescharacteristic
45#16839756C411yescharacteristic
46#16839756C411yescharacteristic
47C2×C8×Dic61C2211yescharacteristic
48C61×#427C411yescharacteristic
49C2×#11758213—22no—
50C2×D61⋊C8—22no—
51C22×#92420—21no—
52C22×#92420D412yes—
53#14519844—22no—
54C4⋊#92420—22no—
55C4⋊#92420C2312yes—
56C4⋊#92420C2312yes—
57#14519834—22no—
58#14519834—22no—
59C22⋊#92420—21no—
60C22⋊#92420—21no—
61#14519832—22no—
62C4×#92420—22no—
63C4×#92420—21no—
64C4×#92420—21no—
65C4×#92420—21no—
66C4×#92420D411yescharacteristic
67C4×#92420D411yescharacteristic
68C4×#92420C2312yes—
69C4×#92420C2312yes—
70C2×#11758222C2×C411yescharacteristic
71C2×#11758222C2×C411yescharacteristic
72#14519799—24no—
73C2×Dic244C2×C411yescharacteristic
74C2×#11758190C2×C411yescharacteristic
75C2×#11758190C2×C411yescharacteristic
76C2×D488C2×C411yescharacteristic
77C22×#92418C2×C411yescharacteristic
78#14519742C2×C411yescharacteristic
79#14519742C2×C411yescharacteristic
80#14519742C2311yescharacteristic
81#14519742C2311yescharacteristic
82#14519739C2×C411yescharacteristic
83#14519739C2×C411yescharacteristic
84#14519739C2311yescharacteristic
85#14519739C2311yescharacteristic
86#14519737—21no—
87#14519737—21no—
88#14519737—21no—
89#14519737—21no—
90#14519734—21no—
91#14519734—21no—
92C2×C4×Dic61C2311yescharacteristic
93C42×D61—21no—
94C42×D61—21no—
95#14519700C2×C411yescharacteristic
96#14519700C2×C411yescharacteristic
97#14519700C2×C411yescharacteristic
98#14519700C2×C411yescharacteristic
99#14519697C2×C411yescharacteristic
100#14519697C2×C411yescharacteristic
101#14519697C2×C411yescharacteristic
102#14519697C2×C411yescharacteristic
103C8×Dic61C2×C411yescharacteristic
104C8×Dic61C2×C411yescharacteristic
105C8×Dic61C2311yescharacteristic
106C8×Dic61C2311yescharacteristic
107C22×C8×C61C2×C411yescharacteristic
108#117×C61C2×C411yescharacteristic
109#117×C61C2×C411yescharacteristic
110#107×C61C2×C411yescharacteristic
111#107×C61C2×C411yescharacteristic
112#106×C61C2×C411yescharacteristic
113#106×C61C2×C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C2×C22⋊D61—21no—
116C2×C22⋊D61—21no—
117C22×Dic61C2×D411yescharacteristic
118C22×Dic61#4111yescharacteristic
119#11758222—22no—
120#11758222—22no—
121#11758222—22no—
122#11758222M1614yescomplemented
123C2×D244C22×C411yescharacteristic
124C2×D244C22×C411yescharacteristic
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×Dic122C22×C411yescharacteristic
132C2×Dic122C22×C411yescharacteristic
133C2×#92420—24no—
134C2×#92420—22no—
135C2×#92420—22no—
136C2×#92420—22no—
137C2×#92420—42no—
138C2×#92420C2×D412yes—
139C2×#92420C2×D412yes—
140#11758213—44no—
141D61⋊C8—44no—
142C2×C8×C61—22no—
143C2×C8×C61C22×C411yescharacteristic
144C2×C8×C61C22×C411yescharacteristic
145C4⋊C4×C61C22×C411yescharacteristic
146C4⋊C4×C61C22×C411yescharacteristic
147C4⋊C4×C61C22×C411yescharacteristic
148C4⋊C4×C61C22×C411yescharacteristic
149C22⋊C4×C61—21no—
150C22⋊C4×C61—21no—
151C42×C61—21no—
152C42×C61—21no—
153#11758198—21no—
154#11758198—21no—
155#11758198—21no—
156#11758198—21no—
157#11758196—21no—
158#11758196—21no—
159#11758196—21no—
160#11758196—21no—
161C4×Dic61—21no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164C4×Dic61—21no—
165C4×Dic61C22×C411yescharacteristic
166C4×Dic61C2×D411yescharacteristic
167C4×Dic61#4111yescharacteristic
168C4×Dic61C2411yescharacteristic
169C2×#92418—22no—
170C2×#92418C22×C411yescharacteristic
171C2×#92418C22×C411yescharacteristic
172Dic244—22no—
173D488—22no—
174#11758190—22no—
175#11758190—22no—
176C23×C61C4×D411yescharacteristic
177C22×D61—21no—
178C22×D61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61C4×D411yescharacteristic
184C2×C4×C61C2×M1612yes—
185C2×C4×C61#13112yes—
186C2×C4×C61C23×C411yescharacteristic
187C22⋊D61—24no—
188C22⋊D61—42no—
189C2×Dic61—21no—
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—21no—
193C2×Dic61C4×D411yescharacteristic
194C2×Dic61C4×D411yescharacteristic
195C2×Dic61C22×D411yescharacteristic
196C2×Dic61C2×#4111yescharacteristic
197D244—21no—
198D244—21no—
199D244—21no—
200D244C2×M1612yes—
201C4×D61—24no—
202C4×D61—42no—
203C4×D61—42no—
204C4×D61—42no—
205C4×D61—42no—
206C4×D61—42no—
207Dic122—21no—
208Dic122—21no—
209Dic122—21no—
210Dic122C2×M1612yes—
211#92420—24no—
212#92420—22no—
213#92420—22no—
214#92420—42no—
215#92420—42no—
216C8×C61—21no—
217C8×C61—21no—
218#92418—21no—
219#92418—21no—
220#17868—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61#43512yes—
224C22×C61C2×C4×D411yescharacteristic
225C2×D61—22no—
226C2×D61—41no—
227C2×D61—41no—
228C2×D61—41no—
229C2×D61—41no—
230C2×D61—41no—
231C4×C61—41no—
232C4×C61—41no—
233C4×C61—41no—
234C4×C61—41no—
235C4×C61#43512yes—
236C4×C61C22×M1611yescharacteristic
237C4×C61C2×#13111yescharacteristic
238Dic61—22no—
239Dic61—21no—
240Dic61—21no—
241Dic61—41no—
242Dic61#44312yes—
243Dic61#57512yes—
244#2965—611no—
245C8×#41—611no—
246C4×#135—611no—
247#1588—612no—
248#1587—612no—
249#1586—612no—
250#1585—612no—
251#1564—611no—
252#1476—611no—
253#1476—611no—
254C2×#322—611no—
255C2×C61—22no—
256C2×C61C2×#43511yescharacteristic
257C2×C61#294311yescharacteristic
258C2×C61C2×#44311yescharacteristic
259D61—42no—
260D61—81no—
261C2×#135—611no—
262C4×#41—611no—
263C4×#41—611no—
264#446—612no—
265C8×Q8—612no—
266C4×Q16—611no—
267C4×Q16—611no—
268C4×SD16—611no—
269C4×SD16—611no—
270C4×SD16—611no—
271C4×SD16—611no—
272C4×D8—611no—
273C4×D8—611no—
274#436—612no—
275#435—1222no—
276C8×D4—612no—
277C8×D4—1222no—
278#431—611no—
279#431—611no—
280#431—1221no—
281#431—1221no—
282#427#692311yescharacteristic
283#416—611no—
284#416—611no—
285C4×M16—1221no—
286C2×C4×C8—611no—
287C2×C4×C8—611no—
288C2×C4×C8—611no—
289C2×C4×C8—1221no—
290#335—612no—
291#334—612no—
292Q8⋊C8—612no—
293Q8⋊C8—612no—
294D4⋊C8—612no—
295D4⋊C8—612no—
296#322—612no—
297#322—612no—
298C61#1786811yescharacteristic, complemented, Sylow
299C2×#41—611no—
300C2×#41—611no—
301#135—1224no—
302C2×Q16—611no—
303C2×SD16—611no—
304C2×SD16—611no—
305C2×D8—611no—
306C2×M16—1222no—
307C22×C8—611no—
308C22×C8—612no—
309C22×C8—1222no—
310C22×C8—1221no—
311C22×C8C2×#692311yescharacteristic
312C4×Q8—611no—
313C4×Q8—611no—
314C4×Q8—611no—
315C4×Q8—611no—
316C4×D4—611no—
317C4×D4—611no—
318C4×D4—611no—
319C4×D4—611no—
320C4×D4—1221no—
321C4×D4—1221no—
322C4×D4—1221no—
323C4×D4—1221no—
324#117—1221no—
325#117—1221no—
326#117C2×#692311yescharacteristic
327#117C2×#692311yescharacteristic
328C2×C42—611no—
329C2×C42—1221no—
330C2×C42—1221no—
331#107C2×#692311yescharacteristic
332#107C2×#692311yescharacteristic
333#106C2×#692311yescharacteristic
334#106C2×#692311yescharacteristic
335C4⋊C8—612no—
336C4⋊C8—612no—
337C4⋊C8—1222no—
338C4⋊C8—1222no—
339#103—611no—
340#103—611no—
341#103—611no—
342#103—611no—
343#102—611no—
344#102—611no—
345#102—611no—
346#102—611no—
347C22⋊C8—1222no—
348C22⋊C8—1222no—
349C22⋊C8—1221no—
350C22⋊C8—1221no—
351#97—1222no—
352C4×C8—611no—
353C4×C8—611no—
354C4×C8—612no—
355C4×C8—611no—
356C4×C8—611no—
357C4×C8—611no—
358C4×C8—611no—
359C4×C8—612no—
360C4×C8—1221no—
361C4×C8—1221no—
362C4×C8—1221no—
363C4×C8—1222no—
364#41—614no—
365#41—1222no—
366#41—1222no—
367#41—1222no—
368C2×Q8—611no—
369C2×Q8—611no—
370C2×D4—611no—
371C2×D4—611no—
372C2×D4—1221no—
373C2×D4—1221no—
374C22×C4—611no—
375C22×C4—611no—
376C22×C4—1221no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4C22×#692311yescharacteristic
383Q16—1222no—
384SD16—1222no—
385SD16—1222no—
386D8—1222no—
387M16—2444no—
388C2×C8—22no—
389C2×C8—612no—
390C2×C8—612no—
391C2×C8—611no—
392C2×C8—611no—
393C2×C8—1222no—
394C2×C8—1224no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—2442no—
399C2×C8—2444no—
400C2×C8C22×#692311yescharacteristic
401C2×C8C22×#692311yescharacteristic
402C4⋊C4—1221no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4C22×#692311yescharacteristic
407C4⋊C4C22×#692311yescharacteristic
408C4⋊C4C22×#692311yescharacteristic
409C4⋊C4C22×#692311yescharacteristic
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—1221no—
413C22⋊C4—1221no—
414C22⋊C4—1221no—
415C22⋊C4—1221no—
416C42—21no—
417C42—21no—
418C42—611no—
419C42—611no—
420C42—611no—
421C42—611no—
422C42—1221no—
423C42—1221no—
424C42—1221no—
425C42—1221no—
426C23—1221no—
427C23—1221no—
428C23D4×#692311yescharacteristic
429Q8—612no—
430Q8—1221no—
431Q8—1221no—
432Q8—1221no—
433D4—612no—
434D4—1224no—
435D4—1221no—
436D4—1221no—
437D4—1221no—
438D4—2442no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—611no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—611no—
447C2×C4—1222no—
448C2×C4—1222no—
449C2×C4—1221no—
450C2×C4—1224no—
451C2×C4—1221no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4D4×#692311yescharacteristic
458C2×C4D61⋊M1612yes—
459C2×C4#1451986512yes—
460C2×C4C23×#692311yescharacteristic
461C8—21no—
462C8—21no—
463C8—1222no—
464C8—1222no—
465C8—1224no—
466C8—1221no—
467C8—1221no—
468C8—2442no—
469C8—2442no—
470C22—21no—
471C22—21no—
472C22—1222no—
473C22—2441no—
474C22—2441no—
475C22—2441no—
476C22—2441no—
477C22—2441no—
478C22#1684079212yes—
479C22C2×D4×#692311yescharacteristic
480C4—41no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—612no—
485C4—612no—
486C4—1221no—
487C4—1221no—
488C4—1222no—
489C4—2441no—
490C4#1684079212yes—
491C4C2×D61⋊M1611yescharacteristic
492C4C2×#1451986511yescharacteristic
493C2—22no—
494C2—4881no—
495C2—2442no—
496C2C2×#1684079211yescharacteristic
497C2#2146257311yescharacteristic
498C2D61⋊#44311yescharacteristic
subgroupG/H
centerC2×C4D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#427C4
socleC22×C61C2×C4×D4
prime radicalC2×#14519799C4
cosocleC4×Dic61C24
hypercenter#427#6923
hypocenterC61#17868
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #21462626C2semidirect
2. #21462623C2semidirect
3. #21461447C2semidirect
4. #21461443C2semidirect
5. #21461439C2semidirect
6. #21461435C2semidirect
7. #21461368C2semidirect
8. #21461112C2semidirect
9. #21461112C2semidirect
10. C2×#92420⋊C8C2semidirect
122. #11758222M16semidirect
298. C61#17868semidirect

Maximal subgroups

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

subgroupindex
#214626262
#214626232
#214614472
#214614432
#214614392
#214614352
#214613682
#214611122
C2×#92420⋊C82
#214545902
#1786861