Finite Group Databasebeta

← C2×#20298764 #28399896

Subgroups — C2×#20298764 #28399896

Order 14464.

844 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#16813668C211yesmaximal, characteristic, complemented
2C2×#16813648C211yesmaximal, characteristic, complemented
3C2×#16813618C211yesmaximal, characteristic, complemented
4#20298764C218yesmaximal, direct product factor
5C2×#16813549C211yesmaximal, characteristic, complemented
6C2×#16813548C211yesmaximal, characteristic, complemented
7C2×#16813538C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
9#16813668—22no—
10#16813668—22no—
11#16813668—22no—
12#16813668C2214yescomplemented
13C2×#14268648—21no—
14C2×#14268647—21no—
15C2×#14268647C2211yescharacteristic
16C22×D452—21no—
17C22×D452C2211yescharacteristic
18C22×C22⋊D113—22no—
19C2×D4×D113—22no—
20C2×D4×D113—21no—
21#16813648—22no—
22#16813648C2214yescomplemented
23C2×#14268635—21no—
24C2×#14268632—21no—
25C2×#14268631C2211yescharacteristic, complemented
26#16813618—22no—
27#16813618C2214yescomplemented
28C2×C4×Dic113C2211yescharacteristic, complemented
29#16813549C2214yescomplemented
30#16813548C2214yescomplemented
31#16813538C2214yescomplemented
32C2×#41×C113C2211yescharacteristic
33C22×D4×C113C2211yescharacteristic, complemented
34AGL(1,ℤ/8)×C113—22no—
35AGL(1,ℤ/8)×C113C2214yescomplemented
36C2×SD16×C113—21no—
37C2×D8×C113—21no—
38C2×M16×C113C2211yescharacteristic, complemented
39C23×D113—22no—
40C23×D113D412yes—
41#14268648—24no—
42#14268648—24no—
43#14268647—24no—
44#14268647—21no—
45#14268647—21no—
46#14268647—24no—
47#14268647D412yescomplemented
48#14268647C2312yes—
49C2×Dic226D411yescharacteristic, complemented
50C2×D452—22no—
51C2×D452—24no—
52C2×D452—21no—
53C2×D452—21no—
54C2×D452—21no—
55C2×D452—21no—
56C2×D452—24no—
57C2×D452D412yes—
58C2×D452D411yescharacteristic, complemented
59C2×D452C2312yes—
60C2×C22⋊D113—22no—
61C2×C22⋊D113—21no—
62C2×C22⋊D113—28no—
63C2×C22⋊D113—48no—
64D4×D113—28no—
65D4×D113—24no—
66D4×D113—28no—
67D4×D113—24no—
68C22×Dic113—21no—
69C2×C4×D113—22no—
70C2×C4×D113—21no—
71#14268635—24no—
72#14268632—24no—
73#14268631—21no—
74#14268631C2312yescomplemented
75C2×#11744751—21no—
76C4×Dic113—21no—
77C4×Dic113C2312yescomplemented
78C24×C113—21no—
79#41×C113—21no—
80#41×C113—21no—
81#41×C113D412yescomplemented
82#41×C113C2312yes—
83C2×Q8×C113D411yescharacteristic, complemented
84C2×D4×C113—21no—
85C2×D4×C113—21no—
86C2×D4×C113—21no—
87C2×D4×C113—21no—
88C2×D4×C113—21no—
89C2×D4×C113—21no—
90C2×D4×C113D411yescharacteristic, complemented
91C2×D4×C113C2312yescomplemented
92C22×C4×C113—21no—
93C22×C4×C113C2311yescharacteristic
94SD16×C113—24no—
95D8×C113—24no—
96M16×C113—21no—
97M16×C113C2312yescomplemented
98C2×C8×C113—21no—
99C23×C113—21no—
100C23×C113—22no—
101C23×C113—44no—
102C23×C113—41no—
103C23×C113—41no—
104C23×C113C2×D411yescharacteristic
105C22×D113—22no—
106C22×D113—21no—
107C22×D113—28no—
108C22×D113—22no—
109C22×D113—44no—
110C22×D113—84no—
111C22×D113C2×D414yes—
112Q8×C113—21no—
113Q8×C113C2×D412yescomplemented
114D4×C113—22no—
115D4×C113—22no—
116D4×C113—21no—
117D4×C113—22no—
118D4×C113—22no—
119D4×C113—41no—
120D4×C113—41no—
121D4×C113C2×D412yescomplemented
122C2×C4×C113—21no—
123C2×C4×C113—22no—
124C2×C4×C113—22no—
125C2×C4×C113C2×D411yescharacteristic
126C2×C4×C113C2×D411yescharacteristic
127C2×C4×C113C2×D411yescharacteristic
128C2×C4×C113C2×D411yescharacteristic
129C2×C4×C113C2×D411yescharacteristic
130C2×C4×C113C2411yescharacteristic
131C22⋊D113—28no—
132C22⋊D113—22no—
133C22⋊D113—22no—
134C22⋊D113—416no—
135C2×Dic113—24no—
136C2×Dic113—21no—
137C2×Dic113—22no—
138D452—28no—
139D452—22no—
140D452—22no—
141D452—21no—
142D452—22no—
143D452—22no—
144D452C2×D412yescomplemented
145C4×D113—28no—
146C4×D113—22no—
147C4×D113—22no—
148Dic226—21no—
149Dic226C2×D412yescomplemented
150C8×C113—22no—
151#11744751—22no—
152C22×C113—22no—
153C22×C113—22no—
154C22×C113—44no—
155C22×C113—42no—
156C22×C113—42no—
157C22×C113—82no—
158C22×C113—82no—
159C22×C113#12011yescharacteristic
160C22×C113#12011yescharacteristic
161C22×C113C22×D411yescharacteristic
162C2×D113—24no—
163C2×D113—22no—
164C2×D113—44no—
165C2×D113—42no—
166C2×D113—44no—
167C2×D113—82no—
168C2×D113—88no—
169C4×C113—22no—
170C4×C113#12011yescharacteristic
171C4×C113#12011yescharacteristic
172C4×C113C22×D411yescharacteristic
173C4×C113C22×D411yescharacteristic
174Dic113—24no—
175Dic113—22no—
176C2×C113—21no—
177C2×C113—21no—
178C2×C113—42no—
179C2×C113—82no—
180C2×C113#45312yescomplemented
181C2×C113C2×#12011yescharacteristic
182D113—44no—
183D113—42no—
184C2×#453—1131nomaximal, Sylow
185C113C2×#45311yescharacteristic, complemented, Sylow
186C2×H(5,𝔽2)—1131no—
187C2×AGL(1,ℤ/8)—1131no—
188C2×AGL(1,ℤ/8)D11311yescharacteristic, complemented
189C2×#127—1131no—
190#453—1138no—
191C2×#104—1131no—
192C2×#104—1131no—
193C2×#100—1131no—
194H(5,𝔽2)—1134no—
195H(5,𝔽2)—2262no—
196H(5,𝔽2)—2262no—
197H(5,𝔽2)—2262no—
198C2×#41—1131no—
199C2×#41—2261no—
200C2×#41—2261no—
201C2×#41C2×D11311yescharacteristic
202C22×D4—1131no—
203C22×D4—2261no—
204C22×D4—2262no—
205C22×D4—2262no—
206C22×D4—2261no—
207C22×D4C2×D11311yescharacteristic, complemented
208AGL(1,ℤ/8)—22no—
209AGL(1,ℤ/8)—1134no—
210AGL(1,ℤ/8)—2262no—
211AGL(1,ℤ/8)C2×D11314yescomplemented
212C2×SD16—21no—
213C2×SD16—2261no—
214C2×D8—21no—
215C2×D8—2261no—
216C2×M16—1131no—
217C2×M16C2×D11311yescharacteristic, complemented
218#127—1134no—
219#127—2262no—
220C2×C42—1131no—
221#104—1134no—
222#104—1134no—
223#100—1134no—
224C24—21no—
225C24—1132no—
226C24—2262no—
227#41—21no—
228#41—21no—
229#41—1132no—
230#41—1132no—
231#41—2264no—
232#41—2264no—
233#41—2264no—
234#41—2264no—
235#41—2261no—
236#41—2261no—
237#41C22⋊D11312yescomplemented
238#41C22×D11312yes—
239C2×Q8—1131no—
240C2×Q8C22⋊D11311yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—1131no—
248C2×D4—1132no—
249C2×D4—1132no—
250C2×D4—4528no—
251C2×D4—2264no—
252C2×D4—2262no—
253C2×D4—2264no—
254C2×D4—2268no—
255C2×D4—2268no—
256C2×D4—2264no—
257C2×D4—2268no—
258C2×D4—2262no—
259C2×D4—2261no—
260C2×D4—2264no—
261C2×D4—2261no—
262C2×D4—2261no—
263C2×D4—2261no—
264C2×D4—2261no—
265C2×D4C22⋊D11311yescharacteristic, complemented
266C2×D4C22×D11312yescomplemented
267C22×C4—21no—
268C22×C4—2262no—
269C22×C4—2261no—
270C22×C4—2261no—
271C22×C4C22×D11311yescharacteristic
272SD16—24no—
273SD16—2264no—
274D8—24no—
275D8—2264no—
276M16—21no—
277M16—1132no—
278M16—2261no—
279M16C22×D11312yescomplemented
280C2×C8—21no—
281C2×C8—2261no—
282C42—1132no—
283C42—2261no—
284C23—21no—
285C23—22no—
286C23—44no—
287C23—41no—
288C23—41no—
289C23—1134no—
290C23—9044no—
291C23—4524no—
292C23—2268no—
293C23—2262no—
294C23—2261no—
295C23—2262no—
296C23D4×D11311yescharacteristic
297Q8—21no—
298Q8—1132no—
299Q8—2261no—
300Q8C2×C22⋊D11312yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—22no—
305D4—22no—
306D4—41no—
307D4—41no—
308D4—1132no—
309D4—45216no—
310D4—2268no—
311D4—2268no—
312D4—2262no—
313D4—2262no—
314D4—2261no—
315D4—2262no—
316D4—2262no—
317D4—2262no—
318D4—2262no—
319D4C2×C22⋊D11312yescomplemented
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—2268no—
324C2×C4—2264no—
325C2×C4—2261no—
326C2×C4—2262no—
327C2×C4—2262no—
328C2×C4—2262no—
329C2×C4D4×D11311yescharacteristic
330C2×C4D4×D11311yescharacteristic
331C2×C4D4×D11311yescharacteristic
332C2×C4C2×C22⋊D11311yescharacteristic
333C2×C4C2×C22⋊D11311yescharacteristic
334C2×C4C23×D11311yescharacteristic
335C8—22no—
336C8—2262no—
337C22—22no—
338C22—22no—
339C22—44no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22—9042no—
345C22—9048no—
346C22—4524no—
347C22—4522no—
348C22—4524no—
349C22—2262no—
350C22—2264no—
351C22#1681359011yescharacteristic
352C22#1681359011yescharacteristic
353C22C2×D4×D11311yescharacteristic
354C4—22no—
355C4—2264no—
356C4—2262no—
357C4#1681359011yescharacteristic
358C4#1681359011yescharacteristic
359C4C2×D4×D11311yescharacteristic
360C4C22×C22⋊D11311yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—82no—
365C2—4522no—
366C2—4524no—
367C2#2029876412yesdirect product factor
368C2C2×#1681359011yescharacteristic
subgroupG/H
centerC22#16813590
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×AGL(1,ℤ/8)×C113C2
socleC22×C113#120
hypercenterC2×AGL(1,ℤ/8)D113
hypocenterC113C2×#453
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16813668C2semidirect
2. C2×#16813648C2semidirect
3. C2×#16813618C2semidirect
4. #20298764C2direct (N × H)
5. C2×#16813549C2semidirect
6. C2×#16813548C2semidirect
7. C2×#16813538C2semidirect
8. C2×AGL(1,ℤ/8)×C113C2semidirect
12. #16813668C22semidirect
22. #16813648C22semidirect
25. C2×#14268631C22semidirect
27. #16813618C22semidirect
28. C2×C4×Dic113C22semidirect
29. #16813549C22semidirect
30. #16813548C22semidirect
31. #16813538C22semidirect
33. C22×D4×C113C22semidirect
35. AGL(1,ℤ/8)×C113C22semidirect
38. C2×M16×C113C22semidirect
47. #14268647D4semidirect
49. C2×Dic226D4semidirect
58. C2×D452D4semidirect
74. #14268631C23semidirect
77. C4×Dic113C23semidirect
81. #41×C113D4semidirect
83. C2×Q8×C113D4semidirect
90. C2×D4×C113D4semidirect
91. C2×D4×C113C23semidirect
97. M16×C113C23semidirect
113. Q8×C113C2×D4semidirect
121. D4×C113C2×D4semidirect
144. D452C2×D4semidirect
149. Dic226C2×D4semidirect
180. C2×C113#453semidirect
185. C113C2×#453semidirect
188. C2×AGL(1,ℤ/8)D113semidirect
207. C22×D4C2×D113semidirect
211. AGL(1,ℤ/8)C2×D113semidirect
217. C2×M16C2×D113semidirect
237. #41C22⋊D113semidirect
240. C2×Q8C22⋊D113semidirect
265. C2×D4C22⋊D113semidirect
266. C2×D4C22×D113semidirect
279. M16C22×D113semidirect
300. Q8C2×C22⋊D113semidirect
319. D4C2×C22⋊D113semidirect
367. C2#20298764direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168136682
C2×#168136482
C2×#168136182
#202987642
C2×#168135492
C2×#168135482
C2×#168135382
C2×AGL(1,ℤ/8)×C1132
C2×#453113