Finite Group Databasebeta

← #454×D61 #21455152

Subgroups — #454×D61 #21455152

Order 7808.

728 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
12−1+4×D61C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D61C212yesmaximal, complemented
3#124×D61C211yesmaximal, characteristic, complemented
4#16839937C211yesmaximal, characteristic, complemented
5#16839936C212yesmaximal, complemented
6#16839935C211yesmaximal, characteristic, complemented
7#16839934C211yesmaximal, characteristic, complemented
8#16839933C212yesmaximal, complemented
9#104×D61C212yesmaximal, complemented
10#101×D61C211yesmaximal, characteristic, complemented
11C61×#454C211yesmaximal, characteristic, complemented
12#14519825—22no—
13#14519825C2211yescharacteristic, complemented
14#14519824—22no—
15#14519824—22no—
16#14519824C2212yescomplemented
17#41×D61—22no—
18#41×D61—22no—
19#41×D61C2212yes—
20C2×Q8×D61—22no—
21C2×Q8×D61C2211yescharacteristic
22C2×D4×D61C2211yescharacteristic
23#14519805C2212yescomplemented
24#14519804—22no—
25#14519803C2212yescomplemented
26#14519802C2212yescomplemented
27#14519801—22no—
28SD16×D61—22no—
29D8×D61—22no—
30M16×D61C2212yes—
31#14519766C2211yescharacteristic, complemented
32#14519765C2211yescharacteristic, complemented
33#14519764C2211yescharacteristic, complemented
34#14519763—22no—
35C22⋊C4×D61—22no—
36C42×D61C2211yescharacteristic
37#14519705C2212yescomplemented
38#14519704C2212yescomplemented
39#14519703C2212yescomplemented
40#14519696C2211yescharacteristic, complemented
41#14519695C2212yescomplemented
422−1+4×C61C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C61C2212yescomplemented
44#124×C61C2211yescharacteristic, complemented
45#104×C61C2212yescomplemented
46#101×C61C2211yescharacteristic, complemented
47C23×D61—21no—
48#41×C61—22no—
49#41×C61—22no—
50#41×C61C2312yes—
51C2×Q8×C61—22no—
52C2×Q8×C61C2311yescharacteristic
53C2×D4×C61C2311yescharacteristic
54C2×C22⋊D61—21no—
55#11758226—22no—
56#11758226—22no—
57#11758226D412yes—
58Q8×D61—22no—
59Q8×D61—24no—
60Q8×D61—22no—
61Q8×D61—22no—
62Q8×D61—22no—
63Q8×D61—22no—
64Q8×D61D412yes—
65#11758224—22no—
66#11758224—24no—
67#11758224—22no—
68#11758224—24no—
69#11758224—22no—
70#11758224D412yes—
71D4×D61—22no—
72D4×D61—22no—
73D4×D61—22no—
74D4×D61—22no—
75D4×D61D412yes—
76#11758222—24no—
77#11758222—24no—
78#11758222—22no—
79#11758222—22no—
80#11758222—22no—
81#11758222C2312yes—
82C2×D244C2311yescharacteristic
83C2×C4×D61—24no—
84C2×C4×D61—22no—
85C2×C4×D61—21no—
86C2×C4×D61D412yes—
87C2×C4×D61C2311yescharacteristic
88C2×Dic122—24no—
89C2×Dic122—22no—
90C2×Dic122D412yes—
91C2×Dic122C2311yescharacteristic
92SD16×C61—22no—
93D8×C61—22no—
94M16×C61C2312yes—
95C22⋊C4×C61—22no—
96C42×C61C2311yescharacteristic
97C22⋊Dic61—22no—
98#11758201—22no—
99#11758200—22no—
100#11758199—22no—
101#11758199—22no—
102#11758198—22no—
103#11758198—22no—
104C4×Dic61—21no—
105C4×Dic61C2311yescharacteristic
106#11758194C2312yes—
107D488—22no—
108#11758190—22no—
109#11758189—22no—
110C8×D61—22no—
111C23×C61—21no—
112C22×D61—22no—
113C22×D61—21no—
114C22×D61—44no—
115C22×D61—41no—
116C22×D61—41no—
117C22×D61C2×D411yescharacteristic
118Q8×C61—22no—
119Q8×C61—22no—
120Q8×C61C2×D412yes—
121D4×C61—22no—
122D4×C61—22no—
123D4×C61—22no—
124D4×C61C2×D412yes—
125C2×C4×C61—24no—
126C2×C4×C61—22no—
127C2×C4×C61—21no—
128C2×C4×C61C2×D412yes—
129C2×C4×C61C2411yescharacteristic
130C22⋊D61—24no—
131C22⋊D61—22no—
132C22⋊D61—22no—
133C22⋊D61—22no—
134C22⋊D61—42no—
135C2×Dic61—24no—
136C2×Dic61—22no—
137C2×Dic61—21no—
138C2×Dic61C2×D412yes—
139C2×Dic61C2×D411yescharacteristic
140D244—22no—
141D244—22no—
142D244—22no—
143D244C2×D412yes—
144C4×D61—24no—
145C4×D61—24no—
146C4×D61—22no—
147C4×D61—22no—
148C4×D61—22no—
149C4×D61—22no—
150C4×D61—22no—
151C4×D61—42no—
152C4×D61C2×D412yes—
153C4×D61C2×D412yes—
154Dic122—24no—
155Dic122—22no—
156Dic122—22no—
157Dic122—22no—
158Dic122—24no—
159Dic122C2×D412yes—
160C8×C61—22no—
161#92418—22no—
162C22×C61—22no—
163C22×C61—42no—
164C22×C61—41no—
165C22×C61C22×D411yescharacteristic
166C2×D61—22no—
167C2×D61—22no—
168C2×D61—44no—
169C2×D61—42no—
170C2×D61—41no—
171C2×D61—82no—
172C2×D61—82no—
173C2×D61#12011yescharacteristic
174C2×D61#12011yescharacteristic
175C4×C61—22no—
176C4×C61—22no—
177C4×C61—41no—
178C4×C61C22×D412yes—
179Dic61—22no—
180Dic61—22no—
181Dic61—41no—
182Dic61#12012yes—
183C2×#454—611nomaximal, Sylow
184C2×C61—21no—
185C2×C61—42no—
186C2×C61—81no—
187C2×C61C2×#12011yescharacteristic
188D61—21no—
189D61—42no—
190D61—81no—
191D61#45412yesdirect product factor
192C2×2−1+4—611no—
193C2×AGL(1,ℤ/8)—612no—
194C2×#124—611no—
195#454—611no—
196#454—612no—
197#454—611no—
198#454—612no—
199#454—611no—
200#454D6111yescharacteristic, direct product factor
201C2×#104—612no—
202C2×#101—611no—
203C61C2×#45411yescharacteristic, complemented, Sylow
2042−1+4—612no—
2052−1+4—611no—
2062−1+4—1222no—
2072−1+4—1222no—
2082−1+4—1222no—
2092−1+4C2×D6111yescharacteristic, complemented
210C2×#41—612no—
211C2×#41—1222no—
212C2×#41—1222no—
213C22×Q8—611no—
214C22×Q8—1222no—
215C22×D4—611no—
216AGL(1,ℤ/8)—612no—
217AGL(1,ℤ/8)—612no—
218AGL(1,ℤ/8)—612no—
219AGL(1,ℤ/8)—1222no—
220AGL(1,ℤ/8)—1222no—
221AGL(1,ℤ/8)C2×D6112yescomplemented
222C2×SD16—1222no—
223C2×D8—1222no—
224C2×M16—612no—
225#124—611no—
226#124—611no—
227#124—611no—
228#124—1222no—
229#124C2×D6111yescharacteristic, complemented
230C2×C22⋊C4—1222no—
231C2×C42—611no—
232#104—612no—
233#104—612no—
234#104—612no—
235#104C2×D6112yescomplemented
236#101—611no—
237#101—612no—
238#101C2×D6111yescharacteristic, complemented
239C24—1221no—
240#41—22no—
241#41—22no—
242#41—612no—
243#41—612no—
244#41—612no—
245#41—1222no—
246#41—1222no—
247#41—1224no—
248#41—1224no—
249#41—1222no—
250#41—1224no—
251#41—1222no—
252#41—1224no—
253#41—1222no—
254#41—1222no—
255#41—1222no—
256#41—1222no—
257#41C22×D6112yes—
258C2×Q8—22no—
259C2×Q8—612no—
260C2×Q8—611no—
261C2×Q8—612no—
262C2×Q8—1222no—
263C2×Q8—1224no—
264C2×Q8—1222no—
265C2×Q8—1222no—
266C2×Q8—1222no—
267C2×Q8—1222no—
268C2×Q8—1224no—
269C2×Q8—1222no—
270C2×Q8C22×D6111yescharacteristic
271C2×D4—612no—
272C2×D4—611no—
273C2×D4—1222no—
274C2×D4—1222no—
275C2×D4—1221no—
276C2×D4—1222no—
277C2×D4—1222no—
278C2×D4C22×D6111yescharacteristic
279C22×C4—611no—
280C22×C4—612no—
281C22×C4—1222no—
282C22×C4—1224no—
283C22×C4—1221no—
284SD16—22no—
285SD16—1222no—
286SD16—1222no—
287SD16—1222no—
288D8—22no—
289D8—1222no—
290D8—1222no—
291D8—1222no—
292M16—612no—
293M16—1222no—
294M16C22×D6112yes—
295C2×C8—1222no—
296C22⋊C4—22no—
297C22⋊C4—1222no—
298C22⋊C4—1222no—
299C22⋊C4—1222no—
300C42—611no—
301C42—1221no—
302C42C22×D6111yescharacteristic
303C23—21no—
304C23—611no—
305C23—1222no—
306C23—1221no—
307C23—2441no—
308C23—2441no—
309C23—2444no—
310Q8—22no—
311Q8—22no—
312Q8—612no—
313Q8—1222no—
314Q8—1224no—
315Q8—1224no—
316Q8—1222no—
317Q8—1222no—
318Q8D4×D6112yes—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—612no—
323D4—1222no—
324D4—1222no—
325D4—1222no—
326D4—1222no—
327D4—1224no—
328D4—1222no—
329D4—1222no—
330D4—2442no—
331D4D4×D6112yes—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—611no—
336C2×C4—612no—
337C2×C4—612no—
338C2×C4—612no—
339C2×C4—1222no—
340C2×C4—1222no—
341C2×C4—1222no—
342C2×C4—1222no—
343C2×C4—1224no—
344C2×C4—1224no—
345C2×C4—1222no—
346C2×C4—1224no—
347C2×C4—1222no—
348C2×C4—1221no—
349C2×C4—2442no—
350C2×C4D4×D6112yes—
351C2×C4C23×D6111yescharacteristic
352C8—22no—
353C8—1222no—
354C22—22no—
355C22—42no—
356C22—41no—
357C22—611no—
358C22—611no—
359C22—1222no—
360C22—1222no—
361C22—4882no—
362C22—4882no—
363C22—2442no—
364C22—2444no—
365C22—2441no—
366C22C2×D4×D6111yescharacteristic
367C4—22no—
368C4—22no—
369C4—41no—
370C4—612no—
371C4—1222no—
372C4—1222no—
373C4—2441no—
374C4C2×D4×D6112yes—
375C2—21no—
376C2—42no—
377C2—81no—
378C2—612no—
379C2—1221no—
380C2—4881no—
381C2—2442no—
382C2#120×D6111yescharacteristic
subgroupG/H
centerC2#120×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#454C2
socleC2×C61C2×#120
hypercenter#454D61
hypocenterC61C2×#454
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. 2−1+4×D61C2semidirect
2. AGL(1,ℤ/8)×D61C2semidirect
3. #124×D61C2semidirect
4. #16839937C2semidirect
5. #16839936C2semidirect
6. #16839935C2semidirect
7. #16839934C2semidirect
8. #16839933C2semidirect
9. #104×D61C2semidirect
10. #101×D61C2semidirect
11. C61×#454C2semidirect
13. #14519825C22semidirect
16. #14519824C22semidirect
23. #14519805C22semidirect
25. #14519803C22semidirect
26. #14519802C22semidirect
31. #14519766C22semidirect
32. #14519765C22semidirect
33. #14519764C22semidirect
37. #14519705C22semidirect
38. #14519704C22semidirect
39. #14519703C22semidirect
40. #14519696C22semidirect
41. #14519695C22semidirect
42. 2−1+4×C61C22semidirect
43. AGL(1,ℤ/8)×C61C22semidirect
44. #124×C61C22semidirect
45. #104×C61C22semidirect
46. #101×C61C22semidirect
191. D61#454direct (N × H)
200. #454D61direct (N × H)
203. C61C2×#454semidirect
209. 2−1+4C2×D61semidirect
221. AGL(1,ℤ/8)C2×D61semidirect
229. #124C2×D61semidirect
235. #104C2×D61semidirect
238. #101C2×D61semidirect