Finite Group Databasebeta

← #21455187

Subgroups — #21455187

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#16840597C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D61C212yesmaximal, complemented
3#16840325C211yesmaximal, characteristic, complemented
4#16839936C212yesmaximal, complemented
5#16839933C212yesmaximal, complemented
6#16839932C211yesmaximal, characteristic, complemented
7#16839930C211yesmaximal, characteristic, complemented
8#16839929C211yesmaximal, characteristic, complemented
9#16839777C212yesmaximal, complemented
10#16839743C211yesmaximal, characteristic, complemented
11C61×#453C211yesmaximal, characteristic, complemented
12#14519824—22no—
13#14519824C2212yescomplemented
14#14519822—22no—
15#14519822—22no—
16#14519822C2211yescharacteristic, complemented
17C2×#11758224—22no—
18C2×#11758224—22no—
19C2×#11758224C2211yescharacteristic
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#14519773C2211yescharacteristic, complemented
32#14519765C2211yescharacteristic, complemented
33#14519764C2211yescharacteristic, complemented
34#14519749—22no—
35#14519745—22no—
36#14519734C2211yescharacteristic, complemented
37#14519705C2212yescomplemented
38#14519704C2212yescomplemented
39#14519703C2212yescomplemented
40#14519695C2212yescomplemented
41#14519694C2211yescharacteristic, complemented
42H(5,𝔽2)×C61C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C61C2212yescomplemented
44#127×C61C2211yescharacteristic, complemented
45#104×C61C2212yescomplemented
46#100×C61C2211yescharacteristic, complemented
47C23×D61—21no—
48#41×C61—22no—
49#41×C61C2312yes—
50C2×D4×C61—22no—
51C2×D4×C61—22no—
52C2×D4×C61—22no—
53C2×D4×C61C2311yescharacteristic
54C2×D4×C61C2311yescharacteristic
55C2×C22⋊D61—24no—
56C2×C22⋊D61—22no—
57C2×C22⋊D61—22no—
58C2×C22⋊D61—22no—
59C2×C22⋊D61—21no—
60C2×C22⋊D61D412yes—
61C22×Dic61—22no—
62C22×Dic61D412yes—
63#11758226D412yescomplemented
64Q8×D61—22no—
65Q8×D61D412yescomplemented
66#11758224—24no—
67#11758224—24no—
68#11758224—24no—
69#11758224—22no—
70#11758224—22no—
71#11758224—24no—
72#11758224—22no—
73#11758224—22no—
74#11758224—22no—
75#11758224D412yescomplemented
76D4×D61—24no—
77D4×D61—22no—
78D4×D61—22no—
79D4×D61—22no—
80D4×D61—22no—
81D4×D61D412yescomplemented
82#11758222—24no—
83#11758222—22no—
84#11758222—22no—
85#11758222C2312yes—
86C2×D244C2311yescharacteristic
87C2×C4×D61—22no—
88C2×C4×D61—22no—
89C2×C4×D61C2311yescharacteristic
90C2×Dic122—22no—
91C2×Dic122—22no—
92C2×Dic122C2311yescharacteristic
93SD16×C61—22no—
94D8×C61—22no—
95M16×C61C2312yes—
96C42×C61C2311yescharacteristic
97C22⋊Dic61—22no—
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—22no—
112C23×C61—21no—
113C23×C61C2×D412yes—
114C22×D61—22no—
115C22×D61—22no—
116C22×D61—21no—
117C22×D61—42no—
118C22×D61—82no—
119C22×D61C2×D411yescharacteristic
120Q8×C61C2×D412yes—
121D4×C61—24no—
122D4×C61—22no—
123D4×C61—22no—
124D4×C61—22no—
125D4×C61—82no—
126D4×C61C2×D412yes—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—21no—
130C2×C4×C61C2411yescharacteristic
131C22⋊D61—24no—
132C22⋊D61—24no—
133C22⋊D61—22no—
134C22⋊D61—24no—
135C22⋊D61—22no—
136C22⋊D61—22no—
137C22⋊D61—41no—
138C22⋊D61—41no—
139C22⋊D61—82no—
140C2×Dic61—24no—
141C2×Dic61—22no—
142C2×Dic61—22no—
143C2×Dic61—22no—
144C2×Dic61—22no—
145C2×Dic61—21no—
146C2×Dic61C2×D412yes—
147C2×Dic61C2×D411yescharacteristic
148D244—22no—
149D244—22no—
150D244C2×D412yes—
151C4×D61—24no—
152C4×D61—22no—
153C4×D61—22no—
154C4×D61—22no—
155C4×D61C2×D412yes—
156C4×D61C2×D412yes—
157Dic122—24no—
158Dic122—22no—
159Dic122—22no—
160Dic122C2×D412yes—
161C8×C61—22no—
162#92418—22no—
163C22×C61—22no—
164C22×C61—22no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—41no—
168C22×C61—82no—
169C22×C61C22×D411yescharacteristic
170C2×D61—22no—
171C2×D61—41no—
172C2×D61—42no—
173C2×D61—41no—
174C2×D61—82no—
175C2×D61—84no—
176C2×D61—82no—
177C2×D61#12012yes—
178C4×C61—22no—
179C4×C61—41no—
180C4×C61C22×D412yes—
181Dic61—22no—
182Dic61—22no—
183Dic61—41no—
184Dic61#12011yescharacteristic
185Dic61#12011yescharacteristic
186#3021—611nomaximal, Sylow
187C2×C61—21no—
188C2×C61—42no—
189C2×C61—42no—
190C2×C61—81no—
191C2×C61C2×#12011yescharacteristic
192D61—22no—
193D61—42no—
194D61—81no—
195#585—611no—
196C2×AGL(1,ℤ/8)—612no—
197#535—611no—
198#454—612no—
199#454—612no—
200#453—611no—
201#453—611no—
202#453—611no—
203#453D6111yescharacteristic, complemented
204#421—612no—
205#413—611no—
206C61#302111yescharacteristic, complemented, Sylow
2072−1+4—612no—
2082−1+4—1222no—
209H(5,𝔽2)—611no—
210H(5,𝔽2)—1222no—
211H(5,𝔽2)—1222no—
212H(5,𝔽2)C2×D6111yescharacteristic, complemented
213C2×#41—611no—
214C2×#41—612no—
215C2×#41—1222no—
216C2×#41—1222no—
217C2×#41—1222no—
218C22×D4—611no—
219AGL(1,ℤ/8)—612no—
220AGL(1,ℤ/8)—612no—
221AGL(1,ℤ/8)—612no—
222AGL(1,ℤ/8)—1222no—
223AGL(1,ℤ/8)—1222no—
224AGL(1,ℤ/8)C2×D6112yescomplemented
225C2×SD16—1222no—
226C2×D8—1222no—
227C2×M16—612no—
228#127—611no—
229#127C2×D6111yescharacteristic, complemented
230#124—611no—
231#124—611no—
232#121—1222no—
233#120—1222no—
234#117—611no—
235#104—612no—
236#104—612no—
237#104—612no—
238#104C2×D6112yescomplemented
239#101—612no—
240#100—611no—
241#100C2×D6111yescharacteristic, complemented
242C24—1221no—
243#41—22no—
244#41—612no—
245#41—612no—
246#41—612no—
247#41—1222no—
248#41—1222no—
249#41—1224no—
250#41—1222no—
251#41—1222no—
252#41—1222no—
253#41—1224no—
254#41—1222no—
255#41—1224no—
256#41—1224no—
257#41—1224no—
258#41—1222no—
259#41C22×D6112yes—
260C2×Q8—611no—
261C2×Q8—612no—
262C2×Q8—1222no—
263C2×Q8—1222no—
264C2×Q8—1222no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—612no—
269C2×D4—612no—
270C2×D4—611no—
271C2×D4—1222no—
272C2×D4—1222no—
273C2×D4—1221no—
274C2×D4—1222no—
275C2×D4—1222no—
276C2×D4—1224no—
277C2×D4—1222no—
278C2×D4—1224no—
279C2×D4—1222no—
280C2×D4—1222no—
281C2×D4C22×D6111yescharacteristic
282C2×D4C22×D6111yescharacteristic
283C22×C4—611no—
284C22×C4—612no—
285C22×C4—1222no—
286C22×C4—1222no—
287C22×C4—1222no—
288SD16—22no—
289SD16—1222no—
290SD16—1222no—
291SD16—1222no—
292D8—22no—
293D8—1222no—
294D8—1222no—
295D8—1222no—
296M16—612no—
297M16—1222no—
298M16C22×D6112yes—
299C2×C8—1222no—
300C4⋊C4—1221no—
301C22⋊C4—1222no—
302C22⋊C4—1221no—
303C22⋊C4—1222no—
304C42—611no—
305C42C22×D6111yescharacteristic
306C23—22no—
307C23—21no—
308C23—611no—
309C23—1222no—
310C23—1222no—
311C23—1221no—
312C23—4882no—
313C23—2442no—
314C23D4×D6112yes—
315Q8—612no—
316Q8—1222no—
317Q8—1224no—
318Q8—1222no—
319Q8D4×D6112yes—
320D4—24no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—82no—
325D4—612no—
326D4—1222no—
327D4—1222no—
328D4—1224no—
329D4—1222no—
330D4—1224no—
331D4—1224no—
332D4—1222no—
333D4—1222no—
334D4—4882no—
335D4—2441no—
336D4—2441no—
337D4D4×D6112yes—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—611no—
342C2×C4—612no—
343C2×C4—612no—
344C2×C4—612no—
345C2×C4—1222no—
346C2×C4—1222no—
347C2×C4—1224no—
348C2×C4—1222no—
349C2×C4—1222no—
350C2×C4—1222no—
351C2×C4—1224no—
352C2×C4—1222no—
353C2×C4—1222no—
354C2×C4—1221no—
355C2×C4C23×D6111yescharacteristic
356C8—22no—
357C8—1222no—
358C22—22no—
359C22—22no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—82no—
364C22—612no—
365C22—1222no—
366C22—4882no—
367C22—4882no—
368C22—4884no—
369C22—2441no—
370C22—2442no—
371C22—2441no—
372C22C2×D4×D6111yescharacteristic
373C4—22no—
374C4—41no—
375C4—611no—
376C4—611no—
377C4—1222no—
378C4—1222no—
379C4—2441no—
380C4C2×D4×D6112yes—
381C2—21no—
382C2—42no—
383C2—42no—
384C2—81no—
385C2—1222no—
386C2—4881no—
387C2—2442no—
388C2#120×D6111yescharacteristic
subgroupG/H
centerC2#120×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#453C2
socleC2×C61C2×#120
hypercenter#453D61
hypocenterC61#3021
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #16840597C2semidirect
2. AGL(1,ℤ/8)×D61C2semidirect
3. #16840325C2semidirect
4. #16839936C2semidirect
5. #16839933C2semidirect
6. #16839932C2semidirect
7. #16839930C2semidirect
8. #16839929C2semidirect
9. #16839777C2semidirect
10. #16839743C2semidirect
11. C61×#453C2semidirect
13. #14519824C22semidirect
16. #14519822C22semidirect
22. C2×D4×D61C22semidirect
23. #14519805C22semidirect
25. #14519803C22semidirect
26. #14519802C22semidirect
30. M16×D61C22semidirect
31. #14519773C22semidirect
32. #14519765C22semidirect
33. #14519764C22semidirect
36. #14519734C22semidirect
37. #14519705C22semidirect
38. #14519704C22semidirect
39. #14519703C22semidirect
40. #14519695C22semidirect
41. #14519694C22semidirect
42. H(5,𝔽2)×C61C22semidirect
43. AGL(1,ℤ/8)×C61C22semidirect
44. #127×C61C22semidirect
45. #104×C61C22semidirect
46. #100×C61C22semidirect
63. #11758226D4semidirect
65. Q8×D61D4semidirect
75. #11758224D4semidirect
81. D4×D61D4semidirect
203. #453D61semidirect
206. C61#3021semidirect
212. H(5,𝔽2)C2×D61semidirect
224. AGL(1,ℤ/8)C2×D61semidirect
229. #127C2×D61semidirect
238. #104C2×D61semidirect
241. #100C2×D61semidirect