Finite Group Databasebeta

← C2×#30631856 #44559690

Subgroups — C2×#30631856 #44559690

Order 32128.

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×#22548761C211yesmaximal, characteristic, complemented
2C2×#22548743C211yesmaximal, characteristic, complemented
3C2×#22548740C211yesmaximal, characteristic, complemented
4C2×#22548712C211yesmaximal, characteristic, complemented
5#30631856C218yesmaximal, direct product factor
6C2×#22548643C211yesmaximal, characteristic, complemented
7C2×#22548631C211yesmaximal, characteristic, complemented
8C2×#104×C251C211yesmaximal, characteristic, complemented
9#22548761—22no—
10#22548761—22no—
11#22548761—22no—
12#22548761C2214yescomplemented
13C2×#16860811—21no—
14C2×#16860810—21no—
15C2×#16860810C2211yescharacteristic
16C22×D1004—21no—
17C22×D1004—22no—
18C22×D1004C2211yescharacteristic
19C22×D1004C2211yescharacteristic, complemented
20C2×D4×D251—22no—
21C2×D4×D251—21no—
22#22548743—22no—
23#22548743C2214yescomplemented
24#22548740—22no—
25#22548740C2214yescomplemented
26C2×#16860799—21no—
27C2×#16860797—21no—
28C2×D2008—21no—
29C2×#16860795—21no—
30C2×#16860794C2211yescharacteristic, complemented
31#22548712—22no—
32#22548712C2214yescomplemented
33#22548643C2214yescomplemented
34#22548631C2214yescomplemented
35C2×#41×C251C2211yescharacteristic
36C2×M16×C251C2211yescharacteristic, complemented
37C2×C42×C251C2211yescharacteristic, complemented
38#104×C251C2214yescomplemented
39C23×D251—22no—
40C23×D251—21no—
41C23×D251D412yes—
42#16860811—24no—
43#16860811—24no—
44#16860810—24no—
45#16860810—24no—
46#16860810—21no—
47#16860810—21no—
48#16860810D412yescomplemented
49#16860810C2312yes—
50C2×Dic502D411yescharacteristic, complemented
51C2×D1004—22no—
52C2×D1004—24no—
53C2×D1004—24no—
54C2×D1004—28no—
55C2×D1004—21no—
56C2×D1004—21no—
57C2×D1004—21no—
58C2×D1004—21no—
59C2×D1004—21no—
60C2×D1004—21no—
61C2×D1004—21no—
62C2×D1004—21no—
63C2×D1004—21no—
64C2×D1004—48no—
65C2×D1004D412yes—
66C2×D1004D411yescharacteristic, complemented
67C2×D1004C2312yes—
68C2×D1004C2312yescomplemented
69C2×C22⋊D251—22no—
70C2×C22⋊D251—21no—
71D4×D251—28no—
72D4×D251—24no—
73D4×D251—28no—
74D4×D251—24no—
75C2×C4×D251—22no—
76C2×C4×D251—21no—
77#16860799—24no—
78#16860797—24no—
79D2008—24no—
80#16860795—24no—
81#16860794—21no—
82#16860794C2312yescomplemented
83C2×C251⋊C8—21no—
84#41×C251—21no—
85#41×C251—21no—
86#41×C251D412yescomplemented
87#41×C251C2312yes—
88C2×Q8×C251D411yescharacteristic, complemented
89C2×D4×C251—21no—
90C2×D4×C251D411yescharacteristic, complemented
91C22×C4×C251—21no—
92C22×C4×C251—21no—
93C22×C4×C251C2311yescharacteristic
94M16×C251—21no—
95M16×C251C2312yescomplemented
96C2×C8×C251—21no—
97C42×C251—21no—
98C42×C251C2312yescomplemented
99C22×D251—22no—
100C22×D251—21no—
101C22×D251—28no—
102C22×D251—22no—
103C22×D251—22no—
104C22×D251—44no—
105C22×D251—44no—
106C22×D251—41no—
107C22×D251—41no—
108C22×D251—84no—
109C22×D251C2×D414yes—
110Dic502—21no—
111Dic502C2×D412yescomplemented
112D1004—28no—
113D1004—22no—
114D1004—22no—
115D1004—21no—
116D1004—22no—
117D1004—22no—
118D1004—22no—
119D1004—22no—
120D1004—416no—
121D1004—41no—
122D1004—41no—
123D1004C2×D412yescomplemented
124C22⋊D251—28no—
125C22⋊D251—22no—
126C22⋊D251—22no—
127C2×Dic251—21no—
128C4×D251—28no—
129C4×D251—22no—
130C4×D251—22no—
131C251⋊C8—22no—
132C23×C251—21no—
133C23×C251C2×D411yescharacteristic
134Q8×C251—21no—
135Q8×C251C2×D412yescomplemented
136D4×C251—22no—
137D4×C251—22no—
138D4×C251—21no—
139D4×C251C2×D412yescomplemented
140C2×C4×C251—24no—
141C2×C4×C251—21no—
142C2×C4×C251—22no—
143C2×C4×C251—22no—
144C2×C4×C251—22no—
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2×D411yescharacteristic
150C2×C4×C251C2411yescharacteristic
151C8×C251—22no—
152C22×C251—22no—
153C22×C251—22no—
154C22×C251—42no—
155C22×C251#12011yescharacteristic
156C22×C251#12011yescharacteristic
157C22×C251C22×D411yescharacteristic
158C2×D251—24no—
159C2×D251—22no—
160C2×D251—44no—
161C2×D251—42no—
162C2×D251—44no—
163C2×D251—44no—
164C2×D251—42no—
165C2×D251—82no—
166C2×D251—88no—
167C2×D251—82no—
168C2×D251—82no—
169C4×C251—24no—
170C4×C251—22no—
171C4×C251#12011yescharacteristic
172C4×C251#12011yescharacteristic
173C4×C251C22×D411yescharacteristic
174C4×C251C22×D411yescharacteristic
175Dic251—22no—
176C2×C251—21no—
177C2×C251—21no—
178C2×C251—42no—
179C2×C251#45312yescomplemented
180C2×C251C2×#12011yescharacteristic
181D251—44no—
182D251—42no—
183D251—82no—
184C251C2×#45311yescharacteristic, complemented, Sylow
185C2×#453—2511nomaximal, Sylow
186C2×H(5,𝔽2)—2511no—
187C2×AGL(1,ℤ/8)—2511no—
188C2×AGL(1,ℤ/8)—2511no—
189C2×#127—2511no—
190#453—2518no—
191C2×#104—2511no—
192C2×#104D25111yescharacteristic, complemented
193C2×#100—2511no—
194H(5,𝔽2)—5022no—
195H(5,𝔽2)—5022no—
196H(5,𝔽2)—5022no—
197H(5,𝔽2)—2514no—
198C2×#41—5021no—
199C2×#41—5021no—
200C2×#41—2511no—
201C2×#41C2×D25111yescharacteristic
202C22×D4—5022no—
203C22×D4—5021no—
204C22×D4—5022no—
205C22×D4—5021no—
206C22×D4—2511no—
207C22×D4—2511no—
208AGL(1,ℤ/8)—5022no—
209AGL(1,ℤ/8)—5022no—
210AGL(1,ℤ/8)—2514no—
211AGL(1,ℤ/8)—2514no—
212C2×SD16—5021no—
213C2×SD16—5021no—
214C2×D8—5021no—
215C2×D8—5021no—
216C2×M16—2511no—
217C2×M16C2×D25111yescharacteristic, complemented
218#127—5022no—
219#127—2514no—
220C2×C42C2×D25111yescharacteristic, complemented
221#104—2514no—
222#104C2×D25114yescomplemented
223#100—2514no—
224C24—5022no—
225C24—5021no—
226C24—2512no—
227#41—21no—
228#41—21no—
229#41—5024no—
230#41—5024no—
231#41—5024no—
232#41—5024no—
233#41—5021no—
234#41—5021no—
235#41—2512no—
236#41—2512no—
237#41D100412yescomplemented
238#41C22×D25112yes—
239C2×Q8—2511no—
240C2×Q8D100411yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—10048no—
243C2×D4—5028no—
244C2×D4—5022no—
245C2×D4—5024no—
246C2×D4—5024no—
247C2×D4—5028no—
248C2×D4—5024no—
249C2×D4—5028no—
250C2×D4—5022no—
251C2×D4—5021no—
252C2×D4—5024no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—5021no—
256C2×D4—5021no—
257C2×D4—5021no—
258C2×D4—5021no—
259C2×D4—5021no—
260C2×D4—5021no—
261C2×D4—5021no—
262C2×D4—2512no—
263C2×D4—2511no—
264C2×D4—2512no—
265C2×D4—2512no—
266C2×D4D100411yescharacteristic, complemented
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—5022no—
270C22×C4—5021no—
271C22×C4C22×D25111yescharacteristic
272SD16—5024no—
273SD16—5024no—
274D8—5024no—
275D8—5024no—
276M16—21no—
277M16—5021no—
278M16—2512no—
279M16C22×D25112yescomplemented
280C2×C8—21no—
281C2×C8—5021no—
282C42—21no—
283C42C22×D25112yescomplemented
284C23—21no—
285C23—20084no—
286C23—10044no—
287C23—10044no—
288C23—10041no—
289C23—10041no—
290C23—5028no—
291C23—5022no—
292C23—5022no—
293C23—5022no—
294C23—5021no—
295C23—2514no—
296C23D4×D25111yescharacteristic
297Q8—21no—
298Q8—5021no—
299Q8—2512no—
300Q8C2×D100412yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—100416no—
305D4—10041no—
306D4—10041no—
307D4—5028no—
308D4—5028no—
309D4—5022no—
310D4—5022no—
311D4—5022no—
312D4—5022no—
313D4—5022no—
314D4—5022no—
315D4—5022no—
316D4—5021no—
317D4—5022no—
318D4—2512no—
319D4C2×D100412yescomplemented
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—5028no—
326C2×C4—5021no—
327C2×C4—5022no—
328C2×C4—5022no—
329C2×C4D4×D25111yescharacteristic
330C2×C4D4×D25111yescharacteristic
331C2×C4D4×D25111yescharacteristic
332C2×C4C2×D100411yescharacteristic
333C2×C4C2×D100411yescharacteristic
334C2×C4C23×D25111yescharacteristic
335C8—22no—
336C8—5022no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—20082no—
341C22—20082no—
342C22—20088no—
343C22—20082no—
344C22—10044no—
345C22—10044no—
346C22—10042no—
347C22—10042no—
348C22—10044no—
349C22—5024no—
350C22—5022no—
351C22#2254868511yescharacteristic
352C22#2254868511yescharacteristic
353C22C2×D4×D25111yescharacteristic
354C4—24no—
355C4—22no—
356C4—5022no—
357C4#2254868511yescharacteristic
358C4#2254868511yescharacteristic
359C4C2×D4×D25111yescharacteristic
360C4C22×D100411yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—20082no—
365C2—10042no—
366C2—10044no—
367C2#3063185612yesdirect product factor
368C2C2×#2254868511yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#104×C251C2
socleC22×C251#120
hypercenterC2×#104D251
hypocenterC251C2×#453
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#22548761C2semidirect
2. C2×#22548743C2semidirect
3. C2×#22548740C2semidirect
4. C2×#22548712C2semidirect
5. #30631856C2direct (N × H)
6. C2×#22548643C2semidirect
7. C2×#22548631C2semidirect
8. C2×#104×C251C2semidirect
12. #22548761C22semidirect
19. C22×D1004C22semidirect
23. #22548743C22semidirect
25. #22548740C22semidirect
30. C2×#16860794C22semidirect
32. #22548712C22semidirect
33. #22548643C22semidirect
34. #22548631C22semidirect
36. C2×M16×C251C22semidirect
37. C2×C42×C251C22semidirect
38. #104×C251C22semidirect
48. #16860810D4semidirect
50. C2×Dic502D4semidirect
66. C2×D1004D4semidirect
68. C2×D1004C23semidirect
82. #16860794C23semidirect
86. #41×C251D4semidirect
88. C2×Q8×C251D4semidirect
90. C2×D4×C251D4semidirect
95. M16×C251C23semidirect
98. C42×C251C23semidirect
111. Dic502C2×D4semidirect
123. D1004C2×D4semidirect
135. Q8×C251C2×D4semidirect
139. D4×C251C2×D4semidirect
179. C2×C251#453semidirect
184. C251C2×#453semidirect
192. C2×#104D251semidirect
217. C2×M16C2×D251semidirect
220. C2×C42C2×D251semidirect
222. #104C2×D251semidirect
237. #41D1004semidirect
240. C2×Q8D1004semidirect
266. C2×D4D1004semidirect
279. M16C22×D251semidirect
283. C42C22×D251semidirect
300. Q8C2×D1004semidirect
319. D4C2×D1004semidirect
367. C2#30631856direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225487612
C2×#225487432
C2×#225487402
C2×#225487122
#306318562
C2×#225486432
C2×#225486312
C2×#104×C2512
C2×#453251