Finite Group Databasebeta

← #30617422

Subgroups — #30617422

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462612C211yesmaximal, characteristic, complemented
2#21462546C211yesmaximal, characteristic
3#21462007C211yesmaximal, characteristic, complemented
4#21461955C211yesmaximal, characteristic
5#21461914C212yesmaximal, complemented
6#21461912C212yesmaximal, complemented
7#21461744C211yesmaximal, characteristic, complemented
8#21461600C212yesmaximal, complemented
9#21461596C212yesmaximal, complemented
10#21461475C211yesmaximal, characteristic, complemented
11#21456143C211yesmaximal, characteristic
12C2×#14519865C2212yes—
13#16840814—22no—
14#16840813—22no—
15#16840792—22no—
16#16840792—22no—
17#16840769C2212yes—
18#16840767C2211yescharacteristic
19#16840767C2211yescharacteristic
20C2×#14519842C2212yes—
21C2×#14519842C2212yes—
22#16840737C2212yes—
23#16840727C2212yes—
24#16840721C2211yescharacteristic
25C2×#14519831C2211yescharacteristic
26C2×#14519831C2211yescharacteristic
27C2×C8×#6923C2211yescharacteristic
28C2×C8×#6923C2211yescharacteristic
29#16840698C2211yescharacteristic
30#16840638C2212yes—
31#16840638C2212yes—
32#16840632C2212yes—
33#16840632C2212yes—
34#16840317C2211yescharacteristic
35#16840313C2211yescharacteristic
36#16840118C412yescomplemented
37#16840118C412yescomplemented
38#16840112C411yescharacteristic, complemented
39#16840110C411yescharacteristic, complemented
40#16840109C411yescharacteristic, complemented
41#16839757C2212yes—
42#16839757C2212yes—
43C4×C8×D61C2211yescharacteristic
44C61×#487C411yescharacteristic, complemented
45#14519865—24no—
46#14519865—24no—
47C2×#11758213—22no—
48C2×#11758213C2312yes—
49C2×#11758213C2312yes—
50C2×D61⋊C8—22no—
51C2×D61⋊C8C2311yescharacteristic
52C2×D61⋊C8C2311yescharacteristic
53C2×C4×#6923C2311yescharacteristic
54C2×C4×#6923C2311yescharacteristic
55#14519844—22no—
56#14519844—21no—
57#14519844—21no—
58#14519844—21no—
59#14519844—21no—
60C4⋊#92420—22no—
61#14519842—24no—
62#14519842—24no—
63#14519834—24no—
64#14519832—22no—
65#14519832D412yes—
66#14519832D412yes—
67#14519831—22no—
68#14519831—22no—
69C4×#92420—22no—
70C8×#6923—22no—
71C8×#6923—22no—
72(C2×#6923)⋊C4—21no—
73(C2×#6923)⋊C4—21no—
74C2×#11758226C2312yes—
75C2×C8×D61C2311yescharacteristic
76C2×C8×D61C2311yescharacteristic
77#14519777C2×C411yescharacteristic
78#14519774C2×C411yescharacteristic
79#14519768C2×C412yes—
80#14519766C2×C412yes—
81#14519757—21no—
82#14519753—21no—
83#14519752—21no—
84#14519752—21no—
85Q8×Dic61—21no—
86#14519742—21no—
87#14519739—21no—
88#14519739—21no—
89#14519735C2312yes—
90C42×D61C2311yescharacteristic
91#14519701C2×C412yes—
92#14519701C2×C412yes—
93#14519699C2×C412yes—
94#14519699C2×C412yes—
95#14519697C2×C412yes—
96#14519697C2×C412yes—
97C4×#92418C2×C411yescharacteristic
98C8×Dic61C2×C411yescharacteristic
99C8×Dic61C2×C411yescharacteristic
100C4⋊Q8×C61C2×C411yescharacteristic
101C4⋊Q8×C61C2×C411yescharacteristic
102#103×C61C2×C412yes—
103#103×C61C2×C412yes—
104C4×C8×C61C2×C411yescharacteristic
105C22×#6923—21no—
106C22×#6923—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61C22×C412yes—
109#11758226—22no—
110#11758226—22no—
111#11758226—22no—
112#11758226—22no—
113C2×D244—22no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244C22×C412yes—
117C2×C4×D61—22no—
118C2×C4×D61#4111yescharacteristic
119C2×C4×D61#4111yescharacteristic
120C2×C4×D61C2411yescharacteristic
121C2×Dic122—21no—
122C2×#92420—22no—
123C2×#92420—22no—
124C2×#92420C2×D412yes—
125C2×#92420C2×D412yes—
126C4×#6923—21no—
127C4×#6923—21no—
128C4×#6923#4112yes—
129C4×#6923#4112yes—
130#11758213—22no—
131#11758213—22no—
132#11758213—22no—
133#11758213—22no—
134#11758213—44no—
135D61⋊C8—21no—
136D61⋊C8—21no—
137D61⋊C8—44no—
138D61⋊C8#4112yes—
139D61⋊C8#4112yes—
140C2×C8×C61C22×C411yescharacteristic
141C2×C8×C61C22×C411yescharacteristic
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61C22×C412yes—
146C42×C61C22×C411yescharacteristic
147#11758198—22no—
148#11758198—22no—
149#11758198—22no—
150C4⋊Dic61—21no—
151C4⋊Dic61—22no—
152C4⋊Dic61C22×C412yes—
153#11758196—22no—
154C4×Dic61—22no—
155C4×Dic61C22×C411yescharacteristic
156C4×Dic61C2×D411yescharacteristic
157C4×Dic61C2×D411yescharacteristic
158C2×#92418C22×C411yescharacteristic
159C2×#92418C22×C411yescharacteristic
160C8×D61—21no—
161C8×D61—21no—
162C8×D61—21no—
163C8×D61—21no—
164C22×D61—22no—
165C22×D61C2×#4111yescharacteristic
166C2×#6923—22no—
167C2×#6923—22no—
168C2×#6923—41no—
169C2×#6923—41no—
170C2×#6923—41no—
171C2×#6923—41no—
172Q8×C61—22no—
173Q8×C61—22no—
174Q8×C61—81no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61C4×D411yescharacteristic
178C2×C4×C61C4×D411yescharacteristic
179C2×C4×C61C23×C411yescharacteristic
180C2×Dic61—22no—
181C2×Dic61C4×D411yescharacteristic
182C2×Dic61C4×D411yescharacteristic
183C2×Dic61C22×D411yescharacteristic
184D244—22no—
185D244—22no—
186D244—42no—
187D244—42no—
188D244—81no—
189D244—81no—
190C4×D61—41no—
191C4×D61—41no—
192C4×D61—42no—
193C4×D61—42no—
194C4×D61C2×#4111yescharacteristic
195C4×D61C2×#4111yescharacteristic
196C4×D61C2×#4111yescharacteristic
197C4×D61C2×#4111yescharacteristic
198Dic122—81no—
199#92420—22no—
200#92420—42no—
201#92420—42no—
202#92420#12414yes—
203C8×C61—21no—
204C8×C61—21no—
205#92418—21no—
206#92418—21no—
207#21659—611nomaximal, Sylow
208C22×C61C2×C4×D411yescharacteristic
209C2×D61—21no—
210C2×D61—21no—
211C2×D61—42no—
212C2×D61—42no—
213C2×D61—42no—
214C2×D61#52911yescharacteristic
215C2×D61#52911yescharacteristic
216#6923—42no—
217#6923—42no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—42no—
221C4×C61—42no—
222C4×C61C4×#4111yescharacteristic
223C4×C61C4×#4111yescharacteristic
224Dic61—21no—
225Dic61—21no—
226Dic61—42no—
227Dic61C2×#12411yescharacteristic
228Dic61#53211yescharacteristic
229#3131—611no—
230#2961—611no—
231#2932—611no—
232#1995—611no—
233#1963—611no—
234#1938—612no—
235#1937—612no—
236#1837—611no—
237#1765—612no—
238#1764—612no—
239#1726—611no—
240C2×C61C4×#12411yescharacteristic
241C2×C61#295711yescharacteristic
242C2×C61#295711yescharacteristic
243D61—21no—
244D61—21no—
245D61—82no—
246C2×#131—612no—
247#533—611no—
248#532—611no—
249#488—612no—
250#488—612no—
251#487—611no—
252#487#692311yescharacteristic, complemented
253#486—611no—
254#486—611no—
255#446—1222no—
256#446—1222no—
257#435—1222no—
258#435—1222no—
259#424—612no—
260#423—611no—
261#423—611no—
262C2×#104—612no—
263C2×#104—612no—
264#416—612no—
265#416—612no—
266#408—612no—
267#405—612no—
268C2×#97—611no—
269C2×#97—611no—
270C2×#97—611no—
271C2×C4×C8—611no—
272C2×C4×C8—611no—
273C2×C4×C8—611no—
274#376—611no—
275#341—612no—
276#341—612no—
277#339—612no—
278#339—612no—
279C61#2165911yescharacteristic, complemented, Sylow
280C2×#41—612no—
281#131—1224no—
282#131—1224no—
283C2×M16—612no—
284C2×M16—612no—
285C2×M16—1222no—
286C22×C8—611no—
287C22×C8—611no—
288C22×C8—611no—
289C22×C8—611no—
290C22×C8—1222no—
291C4⋊Q8—611no—
292C4⋊Q8C2×#692311yescharacteristic
293C4⋊Q8C2×#692311yescharacteristic
294#127—611no—
295#125—612no—
296#124—612no—
297C22⋊Q8—1221no—
298C22⋊Q8—1221no—
299#121—1221no—
300#121—1221no—
301C4×Q8—1221no—
302C4×Q8—1221no—
303C4×D4—1221no—
304C4×D4—1221no—
305#117—612no—
306C2×C42—611no—
307C2×C42—611no—
308C2×C42—611no—
309C4⋊C8—1221no—
310C4⋊C8—1221no—
311C4⋊C8—1221no—
312C4⋊C8—1221no—
313C4⋊C8—1222no—
314C4⋊C8—1222no—
315#104—1224no—
316#104—1224no—
317#103—612no—
318#103—612no—
319#103C2×#692312yes—
320#103C2×#692312yes—
321#102—612no—
322#102—612no—
323#102—612no—
324#102—612no—
325C22⋊C8—1224no—
326#97—612no—
327#97—612no—
328#97—1222no—
329#97—1222no—
330#97—1222no—
331C4×C8—611no—
332C4×C8—611no—
333C4×C8—611no—
334C4×C8—1222no—
335C4×C8—1222no—
336C4×C8—1222no—
337C4×C8C2×#692311yescharacteristic
338#95—1221no—
339#95—1221no—
340#41—1222no—
341#41—1222no—
342#41—1222no—
343#41—1222no—
344C2×Q8—21no—
345C2×Q8—1221no—
346C2×Q8C22×#692312yes—
347C2×D4—612no—
348C2×D4—1222no—
349C2×D4—1221no—
350C2×D4—1221no—
351C22×C4—611no—
352C22×C4—611no—
353C22×C4—611no—
354C22×C4—1222no—
355C22×C4—1221no—
356C22×C4—1221no—
357M16—1222no—
358M16—1222no—
359M16—1222no—
360M16—1222no—
361M16—2444no—
362C2×C8—612no—
363C2×C8—612no—
364C2×C8—612no—
365C2×C8—612no—
366C2×C8—611no—
367C2×C8—611no—
368C2×C8—1221no—
369C2×C8—1221no—
370C2×C8—1221no—
371C2×C8—1221no—
372C2×C8—1221no—
373C2×C8—1221no—
374C2×C8—1222no—
375C2×C8—1222no—
376C2×C8—2444no—
377C2×C8C22×#692311yescharacteristic
378C2×C8C22×#692311yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—612no—
383C4⋊C4—1222no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4C22×#692312yes—
387C22⋊C4—1222no—
388C22⋊C4—1222no—
389C22⋊C4—1222no—
390C42—611no—
391C42—611no—
392C42—611no—
393C42—612no—
394C42—612no—
395C42—1222no—
396C42—1221no—
397C42—1221no—
398C42C22×#692311yescharacteristic
399C23—611no—
400C23—1222no—
401Q8—22no—
402Q8—22no—
403Q8—81no—
404Q8—4881no—
405D4—1222no—
406D4—1222no—
407D4—4881no—
408D4—4881no—
409D4—2442no—
410D4—2442no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—611no—
414C2×C4—611no—
415C2×C4—611no—
416C2×C4—611no—
417C2×C4—611no—
418C2×C4—611no—
419C2×C4—611no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—1222no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4D4×#692311yescharacteristic
432C2×C4D4×#692311yescharacteristic
433C2×C4C23×#692311yescharacteristic
434C8—21no—
435C8—21no—
436C8—614no—
437C8—1222no—
438C8—1221no—
439C8—1221no—
440C8—2442no—
441C8—2442no—
442C22—611no—
443C22—611no—
444C22—1221no—
445C22—1221no—
446C22—2442no—
447C22—2442no—
448C22—2442no—
449C22C2×D4×#692311yescharacteristic
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—611no—
455C4—611no—
456C4—1221no—
457C4—1221no—
458C4—2442no—
459C4—2442no—
460C4—2442no—
461C4#41×#692311yescharacteristic
462C4#41×#692311yescharacteristic
463C2—1221no—
464C2—1221no—
465C2—4882no—
466C2#124×#692311yescharacteristic
467C2#2146259711yescharacteristic
468C2#2146259911yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#487C4
socleC22×C61C2×C4×D4
prime radical#16840313C22
cosocleC2×C4×D61C24
hypercenter#487#6923
hypocenterC61#21659
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #21462612C2semidirect
3. #21462007C2semidirect
5. #21461914C2semidirect
6. #21461912C2semidirect
7. #21461744C2semidirect
8. #21461600C2semidirect
9. #21461596C2semidirect
10. #21461475C2semidirect
36. #16840118C4semidirect
37. #16840118C4semidirect
38. #16840112C4semidirect
39. #16840110C4semidirect
40. #16840109C4semidirect
44. C61×#487C4semidirect
252. #487#6923semidirect
279. C61#21659semidirect

Maximal subgroups

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

subgroupindex
#214626122
#214625462
#214620072
#214619552
#214619142
#214619122
#214617442
#214616002
#214615962
#214614752
#214561432
#2165961