Finite Group Databasebeta

← #30619328

Subgroups — #30619328

Order 15616.

846 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462544C211yesmaximal, characteristic, complemented
2#21462541C211yesmaximal, characteristic, complemented
3#21461953C211yesmaximal, characteristic, complemented
4#21461952C211yesmaximal, characteristic, complemented
5#21461811C214yesmaximal, complemented
6#21461808C214yesmaximal, complemented
7#21461743C211yesmaximal, characteristic, complemented
8#21461532C211yesmaximal, characteristic, complemented
9#21456242C211yesmaximal, characteristic
10C2×#14519865C2212yes—
11C2×#14519864C2212yes—
12#16840797—21no—
13#16840797—21no—
14#16840796—21no—
15#16840796—21no—
16#16840790—21no—
17#16840790—21no—
18D4×#92420—21no—
19D4×#92420—21no—
20#16840766C2211yescharacteristic
21#16840766C2211yescharacteristic
22C2×#14519842C2214yes—
23C2×#14519841C2214yes—
24#16840737C2212yes—
25#16840736C2212yes—
26C4×D61⋊C8C2211yescharacteristic
27C42×#6923C2211yescharacteristic
28#16840638C2214yes—
29(C2×#6923)⋊C8C2212yes—
30(C2×#6923)⋊C8C2212yes—
31C2×#14519798C2214yes—
32#16840313C2211yescharacteristic
33#16840310C2211yescharacteristic
34#16840143C414yescomplemented
35#16840141C411yescharacteristic, complemented
36#16840138C411yescharacteristic, complemented
37#16840135C411yescharacteristic, complemented
38C4×C8×D61C2211yescharacteristic
39C61×#492C411yescharacteristic, complemented
40#14519865—44no—
41#14519864—44no—
42C2×#11758213—22no—
43C2×#11758213C2312yes—
44C2×#11758213C2312yes—
45C2×#11758217—22no—
46C2×D61⋊C8—22no—
47C2×D61⋊C8C2311yescharacteristic
48C2×D61⋊C8C2311yescharacteristic
49C22×#92420—22no—
50C2×C4×#6923—21no—
51C2×C4×#6923—21no—
52C2×C4×#6923C2312yes—
53C4⋊#92420—21no—
54C4⋊#92420—21no—
55C4⋊#92420—21no—
56C4⋊#92420—21no—
57#14519842—44no—
58#14519841—44no—
59#14519834—22no—
60#14519834—22no—
61C22⋊#92420—22no—
62C22⋊#92420—22no—
63C4×#92420D411yescharacteristic
64C4×#92420D411yescharacteristic
65C4×#92420D411yescharacteristic
66C4×#92420D411yescharacteristic
67C2×#11758226C2312yes—
68C2×#11758224C2312yes—
69#14519798—44no—
70#14519798—44no—
71C2×#11758202C2×C414yes—
72C2×#11758190C2×C414yes—
73C2×#11758199C2×C414yes—
74C2×C8×D61C2312yes—
75#14519777C2×C411yescharacteristic
76#14519774C2×C411yescharacteristic
77#14519766C2×C412yes—
78#14519764C2×C412yes—
79#14519752—21no—
80#14519752—21no—
81#14519748—21no—
82#14519748—21no—
83D4×Dic61—21no—
84D4×Dic61—21no—
85#14519739—21no—
86#14519739—21no—
87C42×D61C2311yescharacteristic
88C4×#92418C2×C411yescharacteristic
89C8×Dic61C2×C412yes—
90C2×SD16×C61C2×C414yes—
91C4⋊Q8×C61C2×C411yescharacteristic
92#127×C61C2×C411yescharacteristic
93C4×C8×C61C2×C411yescharacteristic
94C22×#6923—22no—
95C2×Q8×C61C22×C412yes—
96C2×D4×C61—21no—
97C2×D4×C61—21no—
98C2×D4×C61C22×C412yes—
99C2×C22⋊D61—22no—
100C22×Dic61—22no—
101#11758226—24no—
102#11758226—24no—
103#11758224—24no—
104#11758224—24no—
105C2×D244—22no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244C22×C412yes—
109C2×C4×D61—22no—
110C2×C4×D61#4111yescharacteristic
111C2×C4×D61#4111yescharacteristic
112C2×C4×D61C2411yescharacteristic
113C2×Dic122C22×C412yes—
114#11758217—44no—
115C2×#92420—22no—
116C2×#92420—22no—
117C2×#92420—44no—
118C2×#92420C2×D411yescharacteristic
119C2×#92420C2×D411yescharacteristic
120C2×#92420C2×D411yescharacteristic
121C2×#92420C2×D411yescharacteristic
122C4×#6923—24no—
123C4×#6923—22no—
124C4×#6923—22no—
125C4×#6923—42no—
126#11758213—24no—
127#11758213—24no—
128#11758213—44no—
129D61⋊C8—22no—
130D61⋊C8—22no—
131D61⋊C8—44no—
132SD16×C61—44no—
133C2×C8×C61C22×C412yes—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C42×C61C22×C411yescharacteristic
137C22⋊Dic61—22no—
138C22⋊Dic61—22no—
139#11758202—44no—
140#11758199—44no—
141#11758198—22no—
142#11758198—22no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4×Dic61C22×C411yescharacteristic
146C4×Dic61C2×D411yescharacteristic
147C4×Dic61C2×D411yescharacteristic
148C2×#92418C22×C412yes—
149#11758190—44no—
150C8×D61—22no—
151C8×D61—22no—
152C23×C61—22no—
153C22×D61—22no—
154C22×D61C2×#4111yescharacteristic
155C2×#6923—24no—
156C2×#6923—42no—
157C2×#6923—42no—
158Q8×C61—24no—
159D4×C61—24no—
160D4×C61—42no—
161D4×C61—42no—
162C2×C4×C61—22no—
163C2×C4×C61C4×D411yescharacteristic
164C2×C4×C61C4×D411yescharacteristic
165C2×C4×C61C23×C411yescharacteristic
166C22⋊D61—44no—
167C2×Dic61—22no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61C4×D411yescharacteristic
171C2×Dic61C4×D411yescharacteristic
172C2×Dic61C22×D411yescharacteristic
173D244—24no—
174D244—44no—
175D244—42no—
176D244—42no—
177C4×D61—22no—
178C4×D61—22no—
179C4×D61—42no—
180C4×D61—42no—
181C4×D61H(5,𝔽2)12yes—
182C4×D61H(5,𝔽2)12yes—
183Dic122—24no—
184#92420—21no—
185#92420—21no—
186#92420—21no—
187#92420—21no—
188#92420—42no—
189#92420—42no—
190C8×C61—22no—
191#92418—22no—
192#22280—611nomaximal, Sylow
193C22×C61—42no—
194C22×C61—42no—
195C22×C61—42no—
196C22×C61C2×C4×D411yescharacteristic
197C2×D61—21no—
198C2×D61—21no—
199C2×D61—42no—
200C2×D61—42no—
201C2×D61—42no—
202C2×D61#54012yes—
203#6923—44no—
204C4×C61—42no—
205C4×C61#44312yes—
206C4×C61#44412yes—
207C4×C61#51812yes—
208Dic61—21no—
209Dic61—21no—
210Dic61—42no—
211Dic61#53511yescharacteristic
212Dic61#53511yescharacteristic
213#3147—611no—
214#2931—611no—
215#2931—611no—
216#1962—611no—
217#1962—611no—
218#1876—614no—
219#1876—614no—
220#1837—611no—
221#1744—611no—
222C2×C61—82no—
223C2×C61#232811yescharacteristic
224C2×C61C2×#44311yescharacteristic
225C2×C61C2×#44411yescharacteristic
226D61—22no—
227D61—82no—
228C2×#135—614no—
229C2×#131—612no—
230C2×#131—612no—
231#532—611no—
232#532—611no—
233#495—614no—
234#494—611no—
235#493—611no—
236#492—611no—
237#492#692311yescharacteristic, complemented
238#436—1221no—
239#436—1221no—
240#436—1221no—
241#436—1221no—
242C8×D4—1221no—
243C8×D4—1221no—
244C8×D4—1221no—
245C8×D4—1221no—
246#423—611no—
247#423—611no—
248C2×#104—614no—
249C2×#104—614no—
250#408—612no—
251#408—612no—
252C2×C4×C8—611no—
253C2×C4×C8—611no—
254C43—611no—
255#341—614no—
256#336—612no—
257#336—612no—
258C61#2228011yescharacteristic, complemented, Sylow
259C2×#41—612no—
260C2×#41—612no—
261#135—2444no—
262#135—2444no—
263C2×Q16—614no—
264C2×SD16—614no—
265C2×SD16C2×#692314yes—
266C2×D8—614no—
267#131—2444no—
268#131—2444no—
269C2×M16—612no—
270C2×M16—612no—
271C2×M16—1222no—
272C2×M16—1222no—
273C22×C8—611no—
274C22×C8—611no—
275C22×C8—612no—
276C22×C8—1222no—
277C22×C8—1222no—
278C4⋊Q8—611no—
279C4⋊Q8C2×#692311yescharacteristic
280#127—611no—
281#127C2×#692311yescharacteristic
282#124—612no—
283#124—612no—
284#121—1221no—
285#121—1221no—
286#121—1221no—
287#121—1221no—
288C4×D4—1221no—
289C4×D4—1221no—
290C4×D4—1221no—
291C4×D4—1221no—
292C2×C42—611no—
293C2×C42—612no—
294C2×C42—1221no—
295C2×C42—1221no—
296C4⋊C8—1221no—
297C4⋊C8—1221no—
298C4⋊C8—1221no—
299C4⋊C8—1221no—
300#104—2444no—
301#104—2444no—
302C22⋊C8—1222no—
303C22⋊C8—1222no—
304C22⋊C8—1222no—
305C22⋊C8—1222no—
306C4×C8—611no—
307C4×C8—611no—
308C4×C8—611no—
309C4×C8—611no—
310C4×C8—612no—
311C4×C8—611no—
312C4×C8C2×#692311yescharacteristic
313#41—1224no—
314#41—1224no—
315#41—1224no—
316#41—1224no—
317C2×Q8—612no—
318C2×Q8C22×#692312yes—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—612no—
322C2×D4—1222no—
323C2×D4—1222no—
324C2×D4—1221no—
325C2×D4—1221no—
326C2×D4C22×#692312yes—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—611no—
330C22×C4—1222no—
331C22×C4—1222no—
332C22×C4—1222no—
333Q16—2444no—
334SD16—44no—
335SD16—2444no—
336D8—2444no—
337M16—1224no—
338M16—1224no—
339M16—2444no—
340M16—2444no—
341C2×C8—611no—
342C2×C8—611no—
343C2×C8—611no—
344C2×C8—611no—
345C2×C8—612no—
346C2×C8—1222no—
347C2×C8—1222no—
348C2×C8—1222no—
349C2×C8—1222no—
350C2×C8—1222no—
351C2×C8—1222no—
352C2×C8—2444no—
353C2×C8—2444no—
354C2×C8C22×#692312yes—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—1221no—
358C4⋊C4—1221no—
359C22⋊C4—1222no—
360C22⋊C4—1222no—
361C22⋊C4—1222no—
362C22⋊C4—1222no—
363C42—611no—
364C42—611no—
365C42—611no—
366C42—1222no—
367C42—1222no—
368C42—1224no—
369C42—2442no—
370C42C22×#692311yescharacteristic
371C23—22no—
372C23—611no—
373C23—1222no—
374Q8—24no—
375Q8—1224no—
376D4—24no—
377D4—42no—
378D4—42no—
379D4—1224no—
380D4—2444no—
381D4—2444no—
382D4—2442no—
383D4—2442no—
384C2×C4—22no—
385C2×C4—611no—
386C2×C4—612no—
387C2×C4—612no—
388C2×C4—611no—
389C2×C4—611no—
390C2×C4—1222no—
391C2×C4—1222no—
392C2×C4—1224no—
393C2×C4—1222no—
394C2×C4—2442no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4D4×#692311yescharacteristic
401C2×C4D4×#692311yescharacteristic
402C2×C4C23×#692311yescharacteristic
403C8—22no—
404C8—1221no—
405C8—1221no—
406C8—1221no—
407C8—1221no—
408C8—1222no—
409C8—2442no—
410C8—2442no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—612no—
415C22—1221no—
416C22—1221no—
417C22—2442no—
418C22—2442no—
419C22—2442no—
420C22C2×D4×#692311yescharacteristic
421C4—42no—
422C4—611no—
423C4—611no—
424C4—1221no—
425C4—1221no—
426C4—2444no—
427C4—2442no—
428C4#1684080512yes—
429C4#1684080712yes—
430C4#1684082912yes—
431C2—82no—
432C2—1222no—
433C2—4882no—
434C2#2146231111yescharacteristic
435C2C2×#1684080511yescharacteristic
436C2C2×#1684080711yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#492C4
socleC22×C61C2×C4×D4
prime radical#21456242C2
cosocleC2×C4×D61C24
hypercenter#492#6923
hypocenterC61#22280
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #21462544C2semidirect
2. #21462541C2semidirect
3. #21461953C2semidirect
4. #21461952C2semidirect
5. #21461811C2semidirect
6. #21461808C2semidirect
7. #21461743C2semidirect
8. #21461532C2semidirect
34. #16840143C4semidirect
35. #16840141C4semidirect
36. #16840138C4semidirect
37. #16840135C4semidirect
39. C61×#492C4semidirect
237. #492#6923semidirect
258. C61#22280semidirect

Maximal subgroups

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

subgroupindex
#214625442
#214625412
#214619532
#214619522
#214618112
#214618082
#214617432
#214615322
#214562422
#2228061