Finite Group Databasebeta

← #21455189

Subgroups — #21455189

Order 7808.

736 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#16840599C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D61C212yesmaximal, complemented
3#16840330C211yesmaximal, characteristic, complemented
4#16839937C211yesmaximal, characteristic, complemented
5#16839935C211yesmaximal, characteristic, complemented
6#16839934C211yesmaximal, characteristic, complemented
7#16839931C212yesmaximal, complemented
8#16839928C212yesmaximal, complemented
9#16839777C212yesmaximal, complemented
10#16839745C211yesmaximal, characteristic, complemented
11C61×#454C211yesmaximal, characteristic, complemented
12#14519825—22no—
13#14519825C2211yescharacteristic, complemented
14#14519823—22no—
15#14519823—22no—
16#14519823C2212yescomplemented
17C2×#11758226—22no—
18C2×#11758226C2211yescharacteristic
19#41×D61—22no—
20#41×D61—22no—
21#41×D61C2212yes—
22C2×D4×D61C2211yescharacteristic, complemented
23#14519805C2212yescomplemented
24#14519804—22no—
25#14519803C2212yescomplemented
26#14519802C2212yescomplemented
27#14519801—22no—
28SD16×D61—22no—
29D8×D61—22no—
30M16×D61C2212yescomplemented
31#14519774C2211yescharacteristic, complemented
32#14519773C2211yescharacteristic, complemented
33#14519766C2211yescharacteristic, complemented
34#14519750—22no—
35#14519747—22no—
36#14519734C2211yescharacteristic, complemented
37#14519705C2212yescomplemented
38#14519704C2212yescomplemented
39#14519703C2212yescomplemented
40#14519696C2211yescharacteristic, complemented
41#14519693C2212yescomplemented
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—22no—
55C2×C22⋊D61—21no—
56#11758226—22no—
57#11758226—24no—
58#11758226—22no—
59#11758226—22no—
60#11758226—22no—
61#11758226—22no—
62#11758226D412yescomplemented
63Q8×D61—22no—
64Q8×D61—22no—
65Q8×D61D412yescomplemented
66#11758224—22no—
67#11758224—22no—
68#11758224D412yescomplemented
69D4×D61—24no—
70D4×D61—22no—
71D4×D61—24no—
72D4×D61—22no—
73D4×D61—22no—
74D4×D61—22no—
75D4×D61—22no—
76D4×D61D412yescomplemented
77#11758222—24no—
78#11758222—24no—
79#11758222—22no—
80#11758222—22no—
81#11758222—22no—
82#11758222C2312yes—
83C2×D244—24no—
84C2×D244—22no—
85C2×D244—22no—
86C2×D244D412yes—
87C2×D244C2311yescharacteristic
88C2×D244C2311yescharacteristic
89C2×C4×D61—24no—
90C2×C4×D61—22no—
91C2×C4×D61D412yes—
92C2×C4×D61C2311yescharacteristic
93SD16×C61—22no—
94D8×C61—22no—
95M16×C61C2312yes—
96C22⋊C4×C61—22no—
97C42×C61C2311yescharacteristic
98#11758201—22no—
99#11758200—22no—
100#11758199—22no—
101#11758199—22no—
102#11758198—21no—
103#11758198—22no—
104#11758196—21no—
105C4×Dic61C2311yescharacteristic
106#11758194C2312yes—
107D488—22no—
108#11758190—22no—
109#11758189—22no—
110C8×D61—22no—
111C23×C61—21no—
112C22×D61—24no—
113C22×D61—22no—
114C22×D61—21no—
115C22×D61—42no—
116C22×D61—82no—
117C22×D61C2×D412yes—
118C22×D61C2×D411yescharacteristic
119Q8×C61—22no—
120Q8×C61—22no—
121Q8×C61C2×D412yes—
122D4×C61—22no—
123D4×C61—22no—
124D4×C61—22no—
125D4×C61C2×D412yes—
126C2×C4×C61—24no—
127C2×C4×C61—22no—
128C2×C4×C61—21no—
129C2×C4×C61C2×D412yes—
130C2×C4×C61C2411yescharacteristic
131C22⋊D61—24no—
132C22⋊D61—22no—
133C22⋊D61—22no—
134C22⋊D61—22no—
135C22⋊D61—41no—
136C22⋊D61—41no—
137C22⋊D61—82no—
138C2×Dic61—22no—
139C2×Dic61—21no—
140C2×Dic61C2×D411yescharacteristic
141D244—24no—
142D244—22no—
143D244—22no—
144D244—24no—
145D244—22no—
146D244—22no—
147D244—82no—
148D244C2×D412yes—
149C4×D61—24no—
150C4×D61—24no—
151C4×D61—22no—
152C4×D61—22no—
153C4×D61—22no—
154C4×D61—22no—
155C4×D61—22no—
156C4×D61C2×D412yes—
157C4×D61C2×D412yes—
158Dic122—22no—
159Dic122—22no—
160Dic122C2×D412yes—
161C8×C61—22no—
162#92418—22no—
163C22×C61—22no—
164C22×C61—42no—
165C22×C61—41no—
166C22×C61C22×D411yescharacteristic
167C2×D61—22no—
168C2×D61—22no—
169C2×D61—41no—
170C2×D61—42no—
171C2×D61—42no—
172C2×D61—41no—
173C2×D61—82no—
174C2×D61—84no—
175C2×D61—84no—
176C2×D61#12012yes—
177C4×C61—22no—
178C4×C61—22no—
179C4×C61—41no—
180C4×C61C22×D412yes—
181Dic61—22no—
182Dic61—41no—
183Dic61#12011yescharacteristic
184Dic61#12011yescharacteristic
185#3021—611nomaximal, Sylow
186C2×C61—21no—
187C2×C61—42no—
188C2×C61—81no—
189C2×C61C2×#12011yescharacteristic
190D61—22no—
191D61—42no—
192D61—42no—
193D61—81no—
194#585—611no—
195C2×AGL(1,ℤ/8)—612no—
196#535—611no—
197#454—611no—
198#454—611no—
199#454—611no—
200#454D6111yescharacteristic, complemented
201#453—612no—
202#453—612no—
203#421—612no—
204#413—611no—
205C61#302111yescharacteristic, complemented, Sylow
2062−1+4—611no—
2072−1+4—1222no—
2082−1+4C2×D6111yescharacteristic, complemented
209H(5,𝔽2)—612no—
210H(5,𝔽2)—1222no—
211H(5,𝔽2)—1222no—
212C2×#41—611no—
213C2×#41—612no—
214C2×#41—1222no—
215C2×#41—1222no—
216C2×#41—1222no—
217C22×D4—611no—
218AGL(1,ℤ/8)—612no—
219AGL(1,ℤ/8)—612no—
220AGL(1,ℤ/8)—612no—
221AGL(1,ℤ/8)—1222no—
222AGL(1,ℤ/8)—1222no—
223AGL(1,ℤ/8)C2×D6112yescomplemented
224C2×SD16—1222no—
225C2×D8—1222no—
226C2×M16—612no—
227#127—611no—
228#127—611no—
229#124—611no—
230#124C2×D6111yescharacteristic, complemented
231#121—1222no—
232#120—1222no—
233#117—611no—
234#104—612no—
235#104—612no—
236#104—612no—
237#104C2×D6112yescomplemented
238#101—611no—
239#101C2×D6111yescharacteristic, complemented
240#100—612no—
241C24—1221no—
242#41—22no—
243#41—22no—
244#41—612no—
245#41—612no—
246#41—612no—
247#41—1222no—
248#41—1222no—
249#41—1224no—
250#41—1224no—
251#41—1222no—
252#41—1222no—
253#41—1222no—
254#41—1222no—
255#41—1222no—
256#41—1222no—
257#41—1224no—
258#41—1222no—
259#41—1222no—
260#41C22×D6112yes—
261C2×Q8—22no—
262C2×Q8—612no—
263C2×Q8—1222no—
264C2×Q8—1222no—
265C2×Q8C22×D6111yescharacteristic
266C2×D4—612no—
267C2×D4—612no—
268C2×D4—611no—
269C2×D4—611no—
270C2×D4—1222no—
271C2×D4—1224no—
272C2×D4—1222no—
273C2×D4—1222no—
274C2×D4—1222no—
275C2×D4—1221no—
276C2×D4—1222no—
277C2×D4—1224no—
278C2×D4—1222no—
279C2×D4—1224no—
280C2×D4—1222no—
281C2×D4—1222no—
282C2×D4C22×D6111yescharacteristic
283C22×C4—611no—
284C22×C4—612no—
285C22×C4—1222no—
286C22×C4—1224no—
287SD16—22no—
288SD16—1222no—
289SD16—1222no—
290SD16—1222no—
291D8—22no—
292D8—1222no—
293D8—1222no—
294D8—1222no—
295M16—612no—
296M16—1222no—
297M16C22×D6112yes—
298C2×C8—1222no—
299C4⋊C4—1221no—
300C22⋊C4—22no—
301C22⋊C4—1221no—
302C22⋊C4—1222no—
303C42—611no—
304C42C22×D6111yescharacteristic
305C23—21no—
306C23—611no—
307C23—612no—
308C23—1222no—
309C23—1224no—
310C23—1221no—
311C23—4882no—
312C23—2442no—
313Q8—22no—
314Q8—22no—
315Q8—612no—
316Q8—1222no—
317Q8—1222no—
318Q8D4×D6112yes—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—612no—
323D4—1224no—
324D4—1222no—
325D4—1224no—
326D4—1222no—
327D4—1222no—
328D4—1222no—
329D4—1224no—
330D4—1222no—
331D4—1222no—
332D4—1222no—
333D4—4882no—
334D4—4882no—
335D4—2441no—
336D4—2441no—
337D4D4×D6112yes—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—611no—
342C2×C4—612no—
343C2×C4—612no—
344C2×C4—1222no—
345C2×C4—1222no—
346C2×C4—1222no—
347C2×C4—1222no—
348C2×C4—1224no—
349C2×C4—1224no—
350C2×C4—1222no—
351C2×C4—1222no—
352C2×C4—1221no—
353C2×C4D4×D6112yes—
354C2×C4C23×D6111yescharacteristic
355C8—22no—
356C8—1222no—
357C22—22no—
358C22—42no—
359C22—41no—
360C22—612no—
361C22—1222no—
362C22—1222no—
363C22—4882no—
364C22—4884no—
365C22—4884no—
366C22—2441no—
367C22—2442no—
368C22—2442no—
369C22—2441no—
370C22C2×D4×D6111yescharacteristic
371C4—22no—
372C4—22no—
373C4—41no—
374C4—611no—
375C4—611no—
376C4—1222no—
377C4—2441no—
378C4C2×D4×D6112yes—
379C2—21no—
380C2—42no—
381C2—81no—
382C2—1222no—
383C2—4881no—
384C2—2442no—
385C2—2442no—
386C2#120×D6111yescharacteristic
subgroupG/H
centerC2#120×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#454C2
socleC2×C61C2×#120
hypercenter#454D61
hypocenterC61#3021
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. #16840599C2semidirect
2. AGL(1,ℤ/8)×D61C2semidirect
3. #16840330C2semidirect
4. #16839937C2semidirect
5. #16839935C2semidirect
6. #16839934C2semidirect
7. #16839931C2semidirect
8. #16839928C2semidirect
9. #16839777C2semidirect
10. #16839745C2semidirect
11. C61×#454C2semidirect
13. #14519825C22semidirect
16. #14519823C22semidirect
22. C2×D4×D61C22semidirect
23. #14519805C22semidirect
25. #14519803C22semidirect
26. #14519802C22semidirect
30. M16×D61C22semidirect
31. #14519774C22semidirect
32. #14519773C22semidirect
33. #14519766C22semidirect
36. #14519734C22semidirect
37. #14519705C22semidirect
38. #14519704C22semidirect
39. #14519703C22semidirect
40. #14519696C22semidirect
41. #14519693C22semidirect
42. 2−1+4×C61C22semidirect
43. AGL(1,ℤ/8)×C61C22semidirect
44. #124×C61C22semidirect
45. #104×C61C22semidirect
46. #101×C61C22semidirect
62. #11758226D4semidirect
65. Q8×D61D4semidirect
68. #11758224D4semidirect
76. D4×D61D4semidirect
200. #454D61semidirect
205. C61#3021semidirect
208. 2−1+4C2×D61semidirect
223. AGL(1,ℤ/8)C2×D61semidirect
230. #124C2×D61semidirect
237. #104C2×D61semidirect
239. #101C2×D61semidirect