Finite Group Databasebeta

← #30622103

Subgroups — #30622103

Order 15616.

708 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462615C211yesmaximal, characteristic, complemented
2#21462465C211yesmaximal, characteristic
3#21462004C211yesmaximal, characteristic, complemented
4#21461934C211yesmaximal, characteristic, complemented
5#21461928C212yesmaximal, complemented
6#21461884C212yesmaximal, complemented
7#21461789C212yesmaximal
8#21461606C211yesmaximal, characteristic, complemented
9#21461600C212yesmaximal
10C4×#14519842C211yesmaximal, characteristic
11#21455610C211yesmaximal, characteristic
12C2×#14519865C2212yes—
13#16840813—22no—
14#16840812—21no—
15#16840811—21no—
16C2×#14519847C2212yes—
17#16840769—22no—
18#16840769C2212yes—
19#16840769C2212yes—
20#16840766—21no—
21#16840766C2211yescharacteristic
22D61⋊(C4⋊C8)—21no—
23D61⋊(C4⋊C8)C2211yescharacteristic
24C2×#14519842C2211yescharacteristic
25C2×#14519842C2211yescharacteristic
26#16840727C2212yes—
27C4×#11758213C2211yescharacteristic
28C2×#14519831C2212yes—
29C42×#6923C2211yescharacteristic
30#16840638C2212yes—
31#16840632C2212yes—
32#16840632C2211yescharacteristic
33#16840632C2211yescharacteristic
34#16840632C2212yes—
35(C2×#6923)⋊C8C2212yes—
36#16840317C2211yescharacteristic
37C4×#11758226C2211yescharacteristic
38#16840071C412yescomplemented
39#16840065C412yescomplemented
40#16840049C411yescharacteristic, complemented
41#16840047C411yescharacteristic, complemented
42#16840043C411yescharacteristic, complemented
43#16839799C2212yes—
44C4⋊C8×D61C2211yescharacteristic
45#16839757C2212yes—
46C61×#477C411yescharacteristic, complemented
47#14519865—24no—
48#14519865—42no—
49C2×#11758213—22no—
50C2×#11758213C2311yescharacteristic
51C2×#11758213C2312yes—
52C2×#11758213C2311yescharacteristic
53C2×D61⋊C8—22no—
54C2×D61⋊C8C2312yes—
55C2×C4×#6923—21no—
56C2×C4×#6923—21no—
57C2×C4×#6923C2311yescharacteristic
58C2×C4×#6923C2311yescharacteristic
59#14519847—42no—
60#14519844—22no—
61#14519844—22no—
62#14519844D412yes—
63C4⋊#92420—22no—
64C4⋊#92420—22no—
65C4⋊#92420—21no—
66C4⋊#92420—21no—
67C4⋊#92420D411yescharacteristic
68C4⋊#92420D411yescharacteristic
69#14519842—22no—
70#14519842Q814yescomplemented
71#14519832—22no—
72#14519832—21no—
73#14519831—42no—
74C4×#92420—22no—
75C4×#92420—21no—
76C2×#11758226C2311yescharacteristic
77C2×C8×D61C2312yes—
78#14519777C2×C411yescharacteristic
79#14519768C2×C412yes—
80#14519757—21no—
81#14519753—21no—
82Q8×Dic61—21no—
83Q8×Dic61C2×C411yescharacteristic
84#14519742—21no—
85C4×D244—21no—
86C4×D244C2×C411yescharacteristic
87#14519739—21no—
88#14519739C2×C411yescharacteristic
89#14519735—21no—
90#14519735C2312yes—
91#14519735C2311yescharacteristic
92C42×D61—21no—
93C42×D61C2311yescharacteristic
94#14519711C2×C412yes—
95#14519708C2×C412yes—
96#14519708C2×C412yes—
97C4⋊#92418C2×C411yescharacteristic
98#14519706C2×C412yes—
99#14519701C2×C412yes—
100#14519699C2×C412yes—
101#14519697C2×C412yes—
102C4⋊Q8×C61C2×C411yescharacteristic
103C4×Q8×C61C2×C411yescharacteristic
104#107×C61C2×C412yes—
105C4⋊C8×C61C2×C411yescharacteristic
106#103×C61C2×C412yes—
107C22×#6923—21no—
108C22×#6923—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61C22×C411yescharacteristic
111#11758226—21no—
112#11758226—21no—
113#11758226C2×Q812yes—
114#11758226#4112yes—
115C2×D244—21no—
116C2×D244C22×C411yescharacteristic
117C2×C4×D61—21no—
118C2×C4×D61—21no—
119C2×C4×D61—21no—
120C2×C4×D61#4111yescharacteristic
121C2×C4×D61#4111yescharacteristic
122C2×C4×D61C2411yescharacteristic
123C2×Dic122—21no—
124C2×#92420—21no—
125C2×#92420—21no—
126C2×#92420—22no—
127C2×#92420C2×D412yes—
128C2×#92420C2×D412yes—
129C4×#6923—21no—
130C4×#6923—21no—
131C4×#6923—21no—
132C4×#6923—42no—
133C4×#6923—42no—
134C4×#6923C2×Q812yes—
135#11758213—21no—
136#11758213—21no—
137#11758213—22no—
138#11758213—21no—
139#11758213—22no—
140#11758213—44no—
141#11758213C2×Q812yes—
142D61⋊C8—22no—
143D61⋊C8—22no—
144D61⋊C8—44no—
145C2×C8×C61C22×C412yes—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61C22×C412yes—
149C4⋊C4×C61C22×C411yescharacteristic
150C42×C61—21no—
151C42×C61C22×C411yescharacteristic
152#11758198—21no—
153#11758198—22no—
154#11758198—21no—
155#11758198—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—22no—
159C4⋊Dic61C22×C412yes—
160C4⋊Dic61C22×C411yescharacteristic
161#11758196—22no—
162C4×Dic61—21no—
163C4×Dic61—22no—
164C4×Dic61—21no—
165C4×Dic61C22×C411yescharacteristic
166C4×Dic61C2×D411yescharacteristic
167C4×Dic61C2×D411yescharacteristic
168C2×#92418C22×C412yes—
169C8×D61—22no—
170C8×D61—22no—
171C22×D61—21no—
172C22×D61C2×#4111yescharacteristic
173C2×#6923—22no—
174C2×#6923—22no—
175C2×#6923—41no—
176C2×#6923—41no—
177C2×#6923—41no—
178C2×#6923—41no—
179Q8×C61—21no—
180Q8×C61—81no—
181Q8×C61C4×Q812yes—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—21no—
185C2×C4×C61C4×D411yescharacteristic
186C2×C4×C61C4×D411yescharacteristic
187C2×C4×C61C23×C411yescharacteristic
188C2×Dic61—21no—
189C2×Dic61—22no—
190C2×Dic61C4×D411yescharacteristic
191C2×Dic61C4×D411yescharacteristic
192C2×Dic61C22×D411yescharacteristic
193D244—22no—
194D244—22no—
195D244—21no—
196D244C4×Q812yes—
197C4×D61—22no—
198C4×D61—22no—
199C4×D61—22no—
200C4×D61—41no—
201C4×D61—42no—
202C4×D61—81no—
203C4×D61C22×Q811yescharacteristic
204C4×D61C2×#4111yescharacteristic
205C4×D61H(5,𝔽2)11yescharacteristic
206C4×D612−1+411yescharacteristic
207Dic122—42no—
208#92420—22no—
209#92420—22no—
210#92420—22no—
211#92420—42no—
212#92420—41no—
213C8×C61—22no—
214#92418—22no—
215#22983—611nomaximal, Sylow
216C22×C61C2×C4×D411yescharacteristic
217C2×D61—21no—
218C2×D61—21no—
219C2×D61—22no—
220C2×D61—41no—
221C2×D61—41no—
222C2×D61#55211yescharacteristic
223C2×D61#55411yescharacteristic
224#6923—42no—
225#6923—42no—
226C4×C61—22no—
227C4×C61—21no—
228C4×C61—41no—
229C4×C61—42no—
230C4×C61#44412yes—
231C4×C61C2×C4×Q811yescharacteristic
232C4×C61#52011yescharacteristic
233Dic61—21no—
234Dic61—21no—
235Dic61—42no—
236Dic61—41no—
237Dic61#54611yescharacteristic
238Dic61D4×Q811yescharacteristic
239#3076—611no—
240#2962—611no—
241#2906—611no—
242#1994—611no—
243#1950—611no—
244#1946—612no—
245#1922—612no—
246#1866—612no—
247#1767—611no—
248#1765—612no—
249C4×#104—611no—
250C2×C61#236011yescharacteristic
251C2×C61C2×#44411yescharacteristic
252C2×C61#295611yescharacteristic
253D61—21no—
254D61—21no—
255D61—42no—
256C2×#131—612no—
257#533—611no—
258C4×#41—611no—
259#479—612no—
260#478—612no—
261#477#692311yescharacteristic, complemented
262#476—611no—
263#475—611no—
264#474—611no—
265#446—1221no—
266#446—1222no—
267C8×Q8—1221no—
268C2×#108—612no—
269#427—612no—
270#424—612no—
271#424—612no—
272#424—1222no—
273#423—611no—
274#423—1221no—
275C2×C4⋊C8—611no—
276C2×C4⋊C8—611no—
277C2×C4⋊C8—1221no—
278C2×#104—611no—
279C2×#104—611no—
280#416—612no—
281#405—612no—
282C4×M16—611no—
283C2×#97—612no—
284C43—611no—
285#341—612no—
286#339—612no—
287#339—611no—
288#339—611no—
289#339—612no—
290#336—612no—
291C61#2298311yescharacteristic, complemented, Sylow
292C2×#41—611no—
293#131—1224no—
294#131—2442no—
295C2×M16—611no—
296C2×M16—612no—
297C2×M16—611no—
298C2×M16—1222no—
299C22×C8—612no—
300C22×C8—612no—
301C22×C8—1222no—
302C4⋊Q8—611no—
303C4⋊Q8C2×#692311yescharacteristic
304#125—612no—
305C22⋊Q8—1221no—
306C22⋊Q8—1221no—
307C4×Q8—611no—
308C4×Q8—1221no—
309C4×Q8—1221no—
310C4×Q8C2×#692311yescharacteristic
311C4×D4—611no—
312C4×D4—611no—
313C4×D4—1221no—
314C4×D4—1221no—
315#117—611no—
316#117—612no—
317#117—1221no—
318C2×C42—611no—
319C2×C42—611no—
320C2×C42—611no—
321C2×C42—1221no—
322C2×C42—1221no—
323C2×C42—1221no—
324#108—2442no—
325#107—612no—
326#107C2×#692312yes—
327#106—612no—
328#106—612no—
329C4⋊C8—611no—
330C4⋊C8—611no—
331C4⋊C8—612no—
332C4⋊C8—612no—
333C4⋊C8—611no—
334C4⋊C8—1221no—
335C4⋊C8—1221no—
336C4⋊C8—1222no—
337C4⋊C8—1222no—
338C4⋊C8—1222no—
339C4⋊C8—1222no—
340C4⋊C8C2×#692311yescharacteristic
341#104—614no—
342#104—1222no—
343#103—612no—
344#103C2×#692312yes—
345#102—612no—
346#102—612no—
347#97—1221no—
348#97—1222no—
349#97—2442no—
350C4×C8—1221no—
351C4×C8—1222no—
352#41—612no—
353#41—612no—
354#41—1221no—
355#41—1221no—
356C2×Q8—21no—
357C2×Q8—1221no—
358C2×Q8C22×#692311yescharacteristic
359C2×D4—611no—
360C2×D4—1221no—
361C22×C4—611no—
362C22×C4—611no—
363C22×C4—611no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—1221no—
369M16—612no—
370M16—1221no—
371M16—1221no—
372M16—1221no—
373M16—1222no—
374M16—1222no—
375M16—2444no—
376C2×C8—612no—
377C2×C8—612no—
378C2×C8—612no—
379C2×C8—1222no—
380C2×C8—1222no—
381C2×C8—1221no—
382C2×C8—1222no—
383C2×C8—1222no—
384C2×C8—1221no—
385C2×C8—1222no—
386C2×C8—2444no—
387C2×C8C22×#692312yes—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—612no—
391C4⋊C4—611no—
392C4⋊C4—1222no—
393C4⋊C4—1221no—
394C4⋊C4—1222no—
395C4⋊C4—1221no—
396C4⋊C4C22×#692312yes—
397C4⋊C4C22×#692311yescharacteristic
398C22⋊C4—1221no—
399C22⋊C4—1221no—
400C22⋊C4—1222no—
401C22⋊C4—1221no—
402C42—21no—
403C42—611no—
404C42—611no—
405C42—611no—
406C42—612no—
407C42—1222no—
408C42—1221no—
409C42—1221no—
410C42—1221no—
411C42—1221no—
412C42—1221no—
413C42—2442no—
414C42—2442no—
415C42C22×#692311yescharacteristic
416C23—611no—
417C23—1221no—
418Q8—21no—
419Q8—81no—
420Q8—2442no—
421Q8Q8×#692312yes—
422D4—612no—
423D4—1222no—
424D4—1222no—
425D4—1221no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—611no—
430C2×C4—611no—
431C2×C4—611no—
432C2×C4—611no—
433C2×C4—611no—
434C2×C4—611no—
435C2×C4—611no—
436C2×C4—1222no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—1222no—
440C2×C4—1222no—
441C2×C4—1221no—
442C2×C4—1222no—
443C2×C4—4881no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2442no—
450C2×C4D4×#692311yescharacteristic
451C2×C4D4×#692311yescharacteristic
452C2×C4C23×#692311yescharacteristic
453C8—22no—
454C8—1222no—
455C8—1222no—
456C8—1222no—
457C8—1222no—
458C8—2442no—
459C8—2441no—
460C22—611no—
461C22—611no—
462C22—1221no—
463C22—1221no—
464C22—1222no—
465C22—2441no—
466C22—2441no—
467C22C2×D4×#692311yescharacteristic
468C4—21no—
469C4—22no—
470C4—41no—
471C4—42no—
472C4—611no—
473C4—611no—
474C4—1221no—
475C4—1221no—
476C4—2442no—
477C4—2442no—
478C4—2441no—
479C4—2442no—
480C4#1684080812yes—
481C4C2×Q8×#692311yescharacteristic
482C4#1684083111yescharacteristic
483C2—1221no—
484C2—1221no—
485C2—2442no—
486C2#2146234311yescharacteristic
487C2C2×#1684080811yescharacteristic
488C2#2146259311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#477C4
socleC22×C61C2×C4×D4
prime radicalC2×#11758226C23
cosocleC2×C4×D61C24
hypercenter#477#6923
hypocenterC61#22983
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #21462615C2semidirect
3. #21462004C2semidirect
4. #21461934C2semidirect
5. #21461928C2semidirect
6. #21461884C2semidirect
8. #21461606C2semidirect
38. #16840071C4semidirect
39. #16840065C4semidirect
40. #16840049C4semidirect
41. #16840047C4semidirect
42. #16840043C4semidirect
46. C61×#477C4semidirect
70. #14519842Q8semidirect
261. #477#6923semidirect
291. C61#22983semidirect

Maximal subgroups

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

subgroupindex
#214626152
#214624652
#214620042
#214619342
#214619282
#214618842
#214617892
#214616062
#214616002
C4×#145198422
#214556102
#2298361