Finite Group Databasebeta

← #21455188

Subgroups — #21455188

Order 7808.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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#16840598C211yesmaximal, characteristic, complemented
2C2×#14519803C211yesmaximal, characteristic, complemented
3#16840329C211yesmaximal, characteristic, complemented
4#16839933C214yesmaximal, complemented
5#16839928C214yesmaximal, complemented
6#16839779C211yesmaximal, characteristic, complemented
7#16839778C211yesmaximal, characteristic, complemented
8#16839744C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C61C211yesmaximal, characteristic, complemented
10#14519825—21no—
11#14519824—21no—
12#14519824C2212yescomplemented
13#14519823—21no—
14#14519823C2212yescomplemented
15#14519822—21no—
16#14519822—22no—
17C2×#11758222—22no—
18C2×#11758222C2211yescharacteristic
19C2×#11758222C2211yescharacteristic
20#41×D61—22no—
21#41×D61—22no—
22#14519803—22no—
23#14519803C2214yescomplemented
24C2×#11758200—21no—
25C2×#11758199—21no—
26C2×#11758194C2211yescharacteristic, complemented
27#14519773C2212yescomplemented
28#14519764C2212yescomplemented
29#14519748—22no—
30#14519746—22no—
31#14519736C2211yescharacteristic, complemented
32#14519704C2214yescomplemented
33#14519703C2214yescomplemented
34#14519695C2212yescomplemented
35#14519693C2212yescomplemented
36C2×#41×C61C2211yescharacteristic
37C22×D4×C61C2211yescharacteristic, complemented
38AGL(1,ℤ/8)×C61—22no—
39AGL(1,ℤ/8)×C61C2214yescomplemented
40C2×SD16×C61—21no—
41C2×D8×C61—21no—
42C2×M16×C61C2211yescharacteristic, complemented
43C24×C61—21no—
44#41×C61—21no—
45#41×C61—21no—
46#41×C61D412yescomplemented
47#41×C61C2312yes—
48C2×Q8×C61D411yescharacteristic, complemented
49C2×D4×C61—21no—
50C2×D4×C61—21no—
51C2×D4×C61—21no—
52C2×D4×C61—21no—
53C2×D4×C61—21no—
54C2×D4×C61—21no—
55C2×D4×C61D411yescharacteristic, complemented
56C2×D4×C61C2312yes—
57C22×C4×C61—21no—
58C22×C4×C61C2311yescharacteristic
59C2×C22⋊D61—24no—
60C2×C22⋊D61—21no—
61C2×C22⋊D61—24no—
62C2×C22⋊D61D412yes—
63#11758226—22no—
64#11758226—22no—
65Q8×D61—22no—
66Q8×D61—22no—
67#11758224—24no—
68#11758224—22no—
69#11758224—22no—
70#11758224—24no—
71D4×D61—24no—
72D4×D61—22no—
73D4×D61—24no—
74D4×D61—22no—
75#11758222—24no—
76#11758222—24no—
77#11758222—21no—
78#11758222—21no—
79#11758222—21no—
80#11758222—21no—
81#11758222—21no—
82#11758222—21no—
83#11758222—22no—
84#11758222—22no—
85#11758222D412yescomplemented
86#11758222C2312yes—
87C2×D244—22no—
88C2×D244D411yescharacteristic, complemented
89C2×D244C2311yescharacteristic
90C2×C4×D61—24no—
91C2×C4×D61—21no—
92C2×C4×D61D412yes—
93C2×Dic122—22no—
94C2×Dic122D411yescharacteristic, complemented
95C2×Dic122C2311yescharacteristic
96SD16×C61—24no—
97D8×C61—24no—
98M16×C61—21no—
99M16×C61C2312yes—
100C2×C8×C61—21no—
101C22⋊Dic61—21no—
102C22⋊Dic61—24no—
103#11758200—24no—
104#11758199—24no—
105C4⋊Dic61—21no—
106C4×Dic61C2312yes—
107#11758194—21no—
108#11758194C2312yes—
109C2×#92418—21no—
110C23×C61—21no—
111C23×C61—22no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—82no—
115C23×C61C2×D411yescharacteristic
116C22×D61—24no—
117C22×D61—21no—
118C22×D61C2×D412yes—
119Q8×C61—21no—
120Q8×C61C2×D412yes—
121D4×C61—22no—
122D4×C61—22no—
123D4×C61—21no—
124D4×C61—22no—
125D4×C61—22no—
126D4×C61—41no—
127D4×C61—41no—
128D4×C61C2×D412yes—
129C2×C4×C61—21no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61C2×D411yescharacteristic
133C2×C4×C61C2×D411yescharacteristic
134C2×C4×C61C2×D411yescharacteristic
135C2×C4×C61C2×D411yescharacteristic
136C2×C4×C61C2×D411yescharacteristic
137C2×C4×C61C2411yescharacteristic
138C22⋊D61—24no—
139C22⋊D61—24no—
140C22⋊D61—22no—
141C22⋊D61—22no—
142C22⋊D61—22no—
143C22⋊D61—22no—
144C22⋊D61—84no—
145C2×Dic61—24no—
146C2×Dic61—21no—
147C2×Dic61—22no—
148C2×Dic61C2×D412yes—
149D244—24no—
150D244—21no—
151D244—21no—
152D244—21no—
153D244C2×D412yes—
154C4×D61—24no—
155C4×D61—24no—
156C4×D61—22no—
157C4×D61—22no—
158C4×D61—22no—
159C4×D61—22no—
160Dic122—24no—
161Dic122—21no—
162Dic122—21no—
163Dic122—21no—
164Dic122C2×D412yes—
165C8×C61—22no—
166#92418—22no—
167C22×C61—22no—
168C22×C61—41no—
169C22×C61—44no—
170C22×C61—42no—
171C22×C61—81no—
172C22×C61—82no—
173C22×C61—82no—
174C22×C61#12011yescharacteristic
175C22×C61#12011yescharacteristic
176C22×C61C22×D411yescharacteristic
177C2×D61—22no—
178C2×D61—22no—
179C2×D61—42no—
180C2×D61—81no—
181C2×D61—84no—
182C4×C61—22no—
183C4×C61#12011yescharacteristic
184C4×C61#12011yescharacteristic
185C4×C61C22×D411yescharacteristic
186C4×C61C22×D411yescharacteristic
187Dic61—22no—
188Dic61—22no—
189Dic61—42no—
190#3021—611nomaximal, Sylow
191C2×C61—21no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—42no—
195C2×C61—82no—
196C2×C61C2×#12011yescharacteristic
197D61—42no—
198D61—42no—
199#585—611no—
200C2×AGL(1,ℤ/8)—611no—
201C2×AGL(1,ℤ/8)D6111yescharacteristic, complemented
202#535—611no—
203#454—614no—
204#453—614no—
205#421—611no—
206#421—611no—
207#413—611no—
208C61#302111yescharacteristic, complemented, Sylow
2092−1+4—612no—
2102−1+4—1221no—
2112−1+4—1221no—
212H(5,𝔽2)—612no—
213H(5,𝔽2)—1222no—
214H(5,𝔽2)—1221no—
215H(5,𝔽2)—1221no—
216C2×#41—611no—
217C2×#41—611no—
218C2×#41—1222no—
219C2×#41—1222no—
220C2×#41—1222no—
221C2×#41C2×D6111yescharacteristic
222C22×D4C2×D6111yescharacteristic, complemented
223AGL(1,ℤ/8)—22no—
224AGL(1,ℤ/8)—614no—
225AGL(1,ℤ/8)—1222no—
226AGL(1,ℤ/8)C2×D6114yescomplemented
227C2×SD16—21no—
228C2×SD16—1221no—
229C2×D8—21no—
230C2×D8—1221no—
231C2×M16—611no—
232C2×M16C2×D6111yescharacteristic, complemented
233#127—612no—
234#124—612no—
235#121—1222no—
236#120—1222no—
237#117—611no—
238#104—614no—
239#104—614no—
240#101—612no—
241#100—612no—
242C24—21no—
243#41—21no—
244#41—21no—
245#41—612no—
246#41—612no—
247#41—1222no—
248#41—1222no—
249#41—1224no—
250#41—1224no—
251#41—1224no—
252#41—1224no—
253#41—1222no—
254#41—1222no—
255#41—1222no—
256#41—1222no—
257#41—1221no—
258#41—1221no—
259#41—1221no—
260#41—1221no—
261#41—1221no—
262#41—1221no—
263#41C22⋊D6112yescomplemented
264#41C22×D6112yes—
265C2×Q8—611no—
266C2×Q8—611no—
267C2×Q8—1222no—
268C2×Q8—1222no—
269C2×Q8—1222no—
270C2×Q8C22⋊D6111yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—612no—
278C2×D4—611no—
279C2×D4—611no—
280C2×D4—1222no—
281C2×D4—1224no—
282C2×D4—1224no—
283C2×D4—1222no—
284C2×D4—1224no—
285C2×D4—1221no—
286C2×D4—1222no—
287C2×D4—1224no—
288C2×D4C22⋊D6111yescharacteristic, complemented
289C2×D4C22×D6112yes—
290C22×C4—21no—
291C22×C4—612no—
292C22×C4—1224no—
293C22×C4—1221no—
294C22×C4C22×D6111yescharacteristic
295SD16—24no—
296SD16—1224no—
297D8—24no—
298D8—1224no—
299M16—21no—
300M16—612no—
301M16—1221no—
302M16C22×D6112yes—
303C2×C8—21no—
304C2×C8—1221no—
305C4⋊C4—1221no—
306C22⋊C4—1224no—
307C22⋊C4—1221no—
308C42—612no—
309C23—21no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—82no—
314C23—612no—
315C23—1224no—
316C23—1221no—
317C23D4×D6111yescharacteristic
318Q8—21no—
319Q8—612no—
320Q8—1224no—
321Q8—1221no—
322Q8—1221no—
323Q8—1221no—
324Q8C2×C22⋊D6112yes—
325D4—22no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4—21no—
330D4—41no—
331D4—41no—
332D4—612no—
333D4—1224no—
334D4—1224no—
335D4—1224no—
336D4—1222no—
337D4—1222no—
338D4—1222no—
339D4—1222no—
340D4—1221no—
341D4—1221no—
342D4—1221no—
343D4—4884no—
344D4C2×C22⋊D6112yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—612no—
349C2×C4—1224no—
350C2×C4—1224no—
351C2×C4—1224no—
352C2×C4—1221no—
353C2×C4—1222no—
354C2×C4—1222no—
355C2×C4—1222no—
356C2×C4—1222no—
357C2×C4—1222no—
358C2×C4D4×D6111yescharacteristic
359C2×C4D4×D6111yescharacteristic
360C2×C4D4×D6111yescharacteristic
361C2×C4C2×C22⋊D6111yescharacteristic
362C2×C4C2×C22⋊D6111yescharacteristic
363C2×C4C23×D6111yescharacteristic
364C8—22no—
365C8—1222no—
366C22—22no—
367C22—41no—
368C22—44no—
369C22—42no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—1222no—
374C22—1222no—
375C22—4881no—
376C22—4884no—
377C22—2442no—
378C22#1451974511yescharacteristic
379C22#1451974511yescharacteristic
380C22C2×D4×D6111yescharacteristic
381C4—22no—
382C4—1222no—
383C4—1222no—
384C4—2442no—
385C4#1451974511yescharacteristic
386C4#1451974511yescharacteristic
387C4C2×D4×D6111yescharacteristic
388C4C22×C22⋊D6111yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—82no—
394C2—2442no—
395C2—2442no—
396C2C2×#1451974511yescharacteristic
subgroupG/H
centerC2C2×#14519745
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC2×AGL(1,ℤ/8)×C61C2
socleC2×C61C2×#120
hypercenterC2×AGL(1,ℤ/8)D61
hypocenterC61#3021
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #16840598C2semidirect
2. C2×#14519803C2semidirect
3. #16840329C2semidirect
4. #16839933C2semidirect
5. #16839928C2semidirect
6. #16839779C2semidirect
7. #16839778C2semidirect
8. #16839744C2semidirect
9. C2×AGL(1,ℤ/8)×C61C2semidirect
12. #14519824C22semidirect
14. #14519823C22semidirect
23. #14519803C22semidirect
26. C2×#11758194C22semidirect
27. #14519773C22semidirect
28. #14519764C22semidirect
31. #14519736C22semidirect
32. #14519704C22semidirect
33. #14519703C22semidirect
34. #14519695C22semidirect
35. #14519693C22semidirect
37. C22×D4×C61C22semidirect
39. AGL(1,ℤ/8)×C61C22semidirect
42. C2×M16×C61C22semidirect
46. #41×C61D4semidirect
48. C2×Q8×C61D4semidirect
55. C2×D4×C61D4semidirect
85. #11758222D4semidirect
88. C2×D244D4semidirect
94. C2×Dic122D4semidirect
201. C2×AGL(1,ℤ/8)D61semidirect
208. C61#3021semidirect
222. C22×D4C2×D61semidirect
226. AGL(1,ℤ/8)C2×D61semidirect
232. C2×M16C2×D61semidirect
263. #41C22⋊D61semidirect
270. C2×Q8C22⋊D61semidirect
288. C2×D4C22⋊D61semidirect