Finite Group Databasebeta

← #30620773

Subgroups — #30620773

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462513C211yesmaximal, characteristic
2#21462512C211yesmaximal, characteristic
3#21461926C212yesmaximal, complemented
4#21461887C212yesmaximal, complemented
5#21461884C212yesmaximal, complemented
6#21461727C211yesmaximal, characteristic
7#21461727C211yesmaximal, characteristic
8#21461698C211yesmaximal, characteristic
9#21461622C212yesmaximal, complemented
10C8×C22⋊#6923C211yesmaximal, characteristic
11#427×D61C211yesmaximal, characteristic
12#16840786—22no—
13#16840786—22no—
14#16840785—21no—
15#16840784—21no—
16#16840781—21no—
17#16840780—21no—
18C2×#14519847C2212yes—
19#16840769C2212yes—
20#16840769C2212yes—
21#16840733C2211yescharacteristic
22#16840733C2211yescharacteristic
23D61⋊(C22⋊C8)C2211yescharacteristic
24#16840727C2212yes—
25#16840727C2212yes—
26C2×C8×#6923C2212yes—
27C4×C22⋊#6923C2211yescharacteristic
28#16840638C2212yes—
29#16840632C2212yes—
30#16840632C2212yes—
31#16840632C2212yes—
32#16840632C2212yes—
33(C2×#6923)⋊C8C2212yes—
34C22×C8×D61C2211yescharacteristic
35#117×D61C2211yescharacteristic
36#117×D61C2211yescharacteristic
37#16839800C411yescharacteristic, complemented
38#16839799C2211yescharacteristic
39#16839799C2211yescharacteristic
40#16839798C411yescharacteristic, complemented
41#16839798C411yescharacteristic, complemented
42#16839797C2211yescharacteristic
43#16839797C2211yescharacteristic
44#107×D61C411yescharacteristic, complemented
45#107×D61C411yescharacteristic, complemented
46#106×D61C411yescharacteristic, complemented
47#106×D61C411yescharacteristic, complemented
48C61×#427C411yescharacteristic, complemented
49C2×#11758213C2312yes—
50C2×#11758213C2312yes—
51C2×D61⋊C8C2312yes—
52C2×C22⋊#6923—21no—
53C2×C4×#6923C2312yes—
54#14519847—24no—
55#14519844—22no—
56#14519844—22no—
57C4⋊#92420—22no—
58C4⋊#92420—22no—
59#14519834—21no—
60#14519834—21no—
61#14519834D411yescharacteristic
62#14519834D411yescharacteristic
63C22⋊#92420—21no—
64C22⋊#92420—21no—
65C22⋊#92420D412yes—
66#14519832—22no—
67#14519832—22no—
68C4×#92420—22no—
69C4×#92420—22no—
70C8×#6923—24no—
71(C2×#6923)⋊C4—21no—
72C22×C4×D61C2311yescharacteristic
73C22×#92418C2×C411yescharacteristic
74C2×C8×D61—24no—
75C2×C8×D61C2×C411yescharacteristic
76C2×C8×D61C2×C411yescharacteristic
77C2×C8×D61C2311yescharacteristic
78C2×C8×D61C2311yescharacteristic
79#14519736C2×C411yescharacteristic
80#14519736C2×C411yescharacteristic
81#14519735C2311yescharacteristic
82#14519735C2311yescharacteristic
83#14519735C2311yescharacteristic
84#14519735C2311yescharacteristic
85#14519734—21no—
86#14519734—21no—
87#14519733—21no—
88#14519733—21no—
89C4⋊C4×D61C2×C411yescharacteristic
90C4⋊C4×D61C2×C411yescharacteristic
91C4⋊C4×D61C2×C411yescharacteristic
92C4⋊C4×D61C2×C411yescharacteristic
93C22⋊C4×D61—21no—
94C22⋊C4×D61—21no—
95C42×D61—21no—
96C42×D61—21no—
97#14519711C2×C411yescharacteristic
98#14519711C2×C411yescharacteristic
99#14519710C2×C411yescharacteristic
100#14519710C2×C411yescharacteristic
101#14519710C2×C411yescharacteristic
102#14519710C2×C411yescharacteristic
103#14519709C2×C411yescharacteristic
104#14519709C2×C411yescharacteristic
105#14519708C2×C411yescharacteristic
106#14519708C2×C411yescharacteristic
107#14519708C2×C411yescharacteristic
108#14519708C2×C411yescharacteristic
109C22×C8×C61C2×C411yescharacteristic
110#117×C61C2×C411yescharacteristic
111#117×C61C2×C411yescharacteristic
112#107×C61C2×C411yescharacteristic
113#107×C61C2×C411yescharacteristic
114#106×C61C2×C411yescharacteristic
115#106×C61C2×C411yescharacteristic
116C23×D61#4111yescharacteristic
117C22×#6923—22no—
118C22×C4×C61C22×C411yescharacteristic
119C22×Dic61C2×D411yescharacteristic
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—21no—
123C2×C4×D61—21no—
124C2×C4×D61—21no—
125C2×C4×D61—21no—
126C2×C4×D61C22×C411yescharacteristic
127C2×C4×D61C2×D411yescharacteristic
128C2×C4×D61#4111yescharacteristic
129C2×C4×D61C2411yescharacteristic
130C22⋊#6923—24no—
131C2×#92420—22no—
132C2×#92420—22no—
133C2×#92420C2×D412yes—
134C2×#92420C2×D412yes—
135C4×#6923—22no—
136C4×#6923—22no—
137#11758213—22no—
138#11758213—22no—
139#11758213—22no—
140#11758213—22no—
141D61⋊C8—22no—
142D61⋊C8—22no—
143C2×C8×C61—22no—
144C2×C8×C61C22×C411yescharacteristic
145C2×C8×C61C22×C411yescharacteristic
146C4⋊C4×C61C22×C411yescharacteristic
147C4⋊C4×C61C22×C411yescharacteristic
148C4⋊C4×C61C22×C411yescharacteristic
149C4⋊C4×C61C22×C411yescharacteristic
150C22⋊C4×C61—21no—
151C22⋊C4×C61—21no—
152C42×C61—21no—
153C42×C61—21no—
154C22⋊Dic61—21no—
155C22⋊Dic61—21no—
156#11758198—21no—
157#11758198—21no—
158#11758198—21no—
159#11758198—21no—
160C4⋊Dic61C22×C411yescharacteristic
161C4⋊Dic61C22×C411yescharacteristic
162C4⋊Dic61C22×C411yescharacteristic
163C4⋊Dic61C22×C411yescharacteristic
164#11758196—21no—
165#11758196—21no—
166#11758196—21no—
167#11758196—21no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C4×Dic61—21no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C2×#92418—22no—
175C2×#92418C22×C411yescharacteristic
176C2×#92418C22×C411yescharacteristic
177C8×D61—22no—
178C8×D61—22no—
179C8×D61—22no—
180C8×D61#4114yes—
181C23×C61C4×D411yescharacteristic
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—41no—
185C22×D61C4×D411yescharacteristic
186C22×D61C2×#4111yescharacteristic
187C2×#6923—24no—
188C2×#6923—42no—
189C2×#6923—42no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61C4×D411yescharacteristic
197C2×C4×C61C23×C411yescharacteristic
198C2×Dic61—22no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61C4×D411yescharacteristic
204C2×Dic61C22×D411yescharacteristic
205C4×D61—22no—
206C4×D61—21no—
207C4×D61—21no—
208C4×D61—22no—
209C4×D61—42no—
210C4×D61—42no—
211C4×D61—81no—
212C4×D61—81no—
213C4×D61C2×#4112yes—
214C4×D612−1+412yes—
215#92420—22no—
216#92420—22no—
217#92420—42no—
218#92420—42no—
219C8×C61—21no—
220C8×C61#11712yes—
221#92418—21no—
222#92418#11712yes—
223#22606—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61C2×C4×D411yescharacteristic
228C2×D61—21no—
229C2×D61—21no—
230C2×D61—21no—
231C2×D61—21no—
232C2×D61—44no—
233C2×D61#44312yes—
234C2×D61#55212yes—
235#6923—44no—
236C4×C61—21no—
237C4×C61—41no—
238C4×C61—41no—
239C4×C61—41no—
240C4×C61—41no—
241C4×C61C2×#11711yescharacteristic
242C4×C61#51911yescharacteristic
243Dic61—21no—
244Dic61—41no—
245Dic61—41no—
246Dic61—41no—
247Dic61—41no—
248Dic61#54712yes—
249#2924—611no—
250#2924—611no—
251C2×#427—611no—
252#1945—612no—
253#1922—612no—
254#1922—612no—
255#1826—611no—
256#1826—611no—
257#1810—611no—
258#1772—612no—
259C8×C22⋊C4—611no—
260C2×C61—41no—
261C2×C61#234411yescharacteristic
262C2×C61C2×#44311yescharacteristic
263C2×C61#295711yescharacteristic
264D61—22no—
265D61—22no—
266C23×C8—611no—
267C2×#117—611no—
268C2×#117—611no—
269#433—1222no—
270#433—1222no—
271#432—1221no—
272#432—1221no—
273#431—1221no—
274#431—1221no—
275C2×#108—612no—
276#427—611no—
277#427—611no—
278#427—611no—
279#427—611no—
280#427—611no—
281#427—611no—
282#427—611no—
283#427#692311yescharacteristic, complemented
284C2×#107—611no—
285C2×#107—611no—
286C2×#106—611no—
287C2×#106—611no—
288#424—612no—
289#424—612no—
290#407—611no—
291#407—611no—
292C2×C22⋊C8—611no—
293#405—612no—
294#405—612no—
295C2×C4×C8—612no—
296C4×C22⋊C4—611no—
297#341—612no—
298#339—612no—
299#339—612no—
300#339—612no—
301#339—612no—
302#336—612no—
303C61#2260611yescharacteristic, complemented, Sylow
304C23×C4—611no—
305C2×M16—612no—
306C2×M16—612no—
307C22×C8—611no—
308C22×C8—612no—
309C22×C8—611no—
310C22×C8—611no—
311C22×C8—611no—
312C22×C8—611no—
313C22×C8—1224no—
314C22×C8C2×#692311yescharacteristic
315#117—611no—
316#117—611no—
317#117—611no—
318#117—611no—
319#117—611no—
320#117—611no—
321#117—1221no—
322#117—1221no—
323#117—1221no—
324#117—1221no—
325#117C2×#692311yescharacteristic
326#117C2×#692311yescharacteristic
327C2×C4⋊C4—611no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—611no—
331C2×C22⋊C4—1221no—
332C2×C22⋊C4—1221no—
333C2×C22⋊C4—1221no—
334C2×C42—612no—
335C2×C42—1221no—
336C2×C42—1221no—
337#108—1224no—
338#107—611no—
339#107—611no—
340#107—611no—
341#107—611no—
342#107—611no—
343#107—611no—
344#107C2×#692311yescharacteristic
345#107C2×#692311yescharacteristic
346#106—611no—
347#106—611no—
348#106—611no—
349#106—611no—
350#106—611no—
351#106—611no—
352#106C2×#692311yescharacteristic
353#106C2×#692311yescharacteristic
354C4⋊C8—1222no—
355C4⋊C8—1222no—
356C4⋊C8—1222no—
357C4⋊C8—1222no—
358C22⋊C8—612no—
359C22⋊C8—611no—
360C22⋊C8—611no—
361C22⋊C8—1221no—
362C22⋊C8—1221no—
363C22⋊C8—1221no—
364C22⋊C8—1221no—
365#97—1222no—
366#97—1222no—
367C4×C8—1224no—
368C4×C8—1222no—
369C4×C8—1222no—
370#95—1221no—
371C24—611no—
372C22×C4—611no—
373C22×C4—611no—
374C22×C4—611no—
375C22×C4—611no—
376C22×C4—611no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1222no—
384C22×C4C22×#692311yescharacteristic
385M16—1222no—
386M16—1222no—
387M16—1222no—
388M16—1222no—
389C2×C8—22no—
390C2×C8—611no—
391C2×C8—614no—
392C2×C8—612no—
393C2×C8—612no—
394C2×C8—611no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8C22×#692311yescharacteristic
404C2×C8C22×#692311yescharacteristic
405C4⋊C4—611no—
406C4⋊C4—611no—
407C4⋊C4—611no—
408C4⋊C4—611no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4C22×#692311yescharacteristic
414C4⋊C4C22×#692311yescharacteristic
415C4⋊C4C22×#692311yescharacteristic
416C4⋊C4C22×#692311yescharacteristic
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—1221no—
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C22⋊C4—1221no—
423C22⋊C4—1224no—
424C22⋊C4—1221no—
425C22⋊C4—1221no—
426C42—21no—
427C42—21no—
428C42—1221no—
429C42—1221no—
430C42—1221no—
431C42—1221no—
432C42—1222no—
433C42—1222no—
434C42—1221no—
435C42—1221no—
436C23—611no—
437C23—611no—
438C23—1222no—
439C23—1222no—
440C23—2441no—
441C23D4×#692311yescharacteristic
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—611no—
449C2×C4—612no—
450C2×C4—612no—
451C2×C4—611no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—1222no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—1224no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—4881no—
463C2×C4—4881no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4D4×#692311yescharacteristic
469C2×C4C23×#692311yescharacteristic
470C8—21no—
471C8—612no—
472C8—1222no—
473C8—1222no—
474C8—1221no—
475C8—2442no—
476C8—2442no—
477C8#1451985612yes—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—612no—
482C22—612no—
483C22—1221no—
484C22—1221no—
485C22—1221no—
486C22—1221no—
487C22—2444no—
488C22C2×D4×#692311yescharacteristic
489C4—21no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—612no—
495C4—1221no—
496C4—2444no—
497C4—2441no—
498C4—2441no—
499C4—2441no—
500C4—2441no—
501C4C2×#1451985611yescharacteristic
502C4#1684083011yescharacteristic
503C2—41no—
504C2—1222no—
505C2—1222no—
506C2#2146232711yescharacteristic
507C2D61⋊#44311yescharacteristic
508C2#2146259811yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#427C4
socleC22×C61C2×C4×D4
prime radicalC23×D61#41
cosocleC2×C4×D61C24
hypercenter#427#6923
hypocenterC61#22606
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #21461926C2semidirect
4. #21461887C2semidirect
5. #21461884C2semidirect
9. #21461622C2semidirect
37. #16839800C4semidirect
40. #16839798C4semidirect
41. #16839798C4semidirect
44. #107×D61C4semidirect
45. #107×D61C4semidirect
46. #106×D61C4semidirect
47. #106×D61C4semidirect
48. C61×#427C4semidirect
283. #427#6923semidirect
303. C61#22606semidirect

Maximal subgroups

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

subgroupindex
#214625132
#214625122
#214619262
#214618872
#214618842
#214617272
#214616982
#214616222
C8×C22⋊#69232
#427×D612
#2260661