Finite Group Databasebeta

← C2×#30631861 #44559702

Subgroups — C2×#30631861 #44559702

Order 32128.

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

normal subgroups only

This group has 728 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×#22548762C211yesmaximal, characteristic, complemented
2C2×#22548743C211yesmaximal, characteristic, complemented
3C2×#22548740C211yesmaximal, characteristic, complemented
4C2×#22548703C211yesmaximal, characteristic, complemented
5#30631861C218yesmaximal, direct product factor
6C2×#22548643C211yesmaximal, characteristic, complemented
7C2×#22548633C211yesmaximal, characteristic, complemented
8C2×#104×C251C211yesmaximal, characteristic, complemented
9#22548762—22no—
10#22548762—22no—
11#22548762—22no—
12#22548762C2214yescomplemented
13C2×#16860810—21no—
14C2×#16860810C2211yescharacteristic
15C2×#16860809—22no—
16C2×#16860809—21no—
17C22×Dic502—21no—
18C22×Dic502C2211yescharacteristic
19C2×Q8×D251—21no—
20C22×D1004C2211yescharacteristic
21#22548743—22no—
22#22548743C2214yescomplemented
23#22548740—22no—
24#22548740C2214yescomplemented
25C2×#16860799—21no—
26C2×#16860797—21no—
27C2×D2008—21no—
28C2×#16860795—21no—
29C2×#16860794C2211yescharacteristic
30#22548703—22no—
31#22548703C2214yescomplemented
32C2×(C2×D251)⋊C4—22no—
33#22548643C2214yescomplemented
34#22548633C2214yescomplemented
35C2×#41×C251C2211yescharacteristic
36C2×M16×C251C2211yescharacteristic
37C2×C42×C251C2211yescharacteristic
38#104×C251C2214yescomplemented
39C23×D251—21no—
40#16860810—24no—
41#16860810—24no—
42#16860810—21no—
43#16860810—21no—
44#16860810D412yes—
45#16860810C2312yes—
46#16860809—28no—
47#16860809—24no—
48#16860809—28no—
49#16860809—24no—
50C2×Dic502—22no—
51C2×Dic502—24no—
52C2×Dic502—24no—
53C2×Dic502—21no—
54C2×Dic502—21no—
55C2×Dic502—21no—
56C2×Dic502—21no—
57C2×Dic502D411yescharacteristic
58C2×Dic502D412yes—
59C2×Dic502C2312yes—
60Q8×D251—24no—
61Q8×D251—24no—
62C2×D1004—21no—
63C2×D1004—21no—
64C2×D1004—21no—
65C2×D1004—21no—
66C2×D1004—21no—
67C2×D1004D411yescharacteristic
68C2×D1004C2312yes—
69C2×C22⋊D251—22no—
70C2×C22⋊D251—21no—
71C22×Dic251—22no—
72C22×Dic251D412yes—
73C2×C4×D251—22no—
74C2×C4×D251—21no—
75#16860799—24no—
76#16860797—24no—
77D2008—24no—
78#16860795—24no—
79#16860794—21no—
80#16860794C2312yes—
81C2×C251⋊C8—21no—
82(C2×D251)⋊C4—28no—
83#41×C251—21no—
84#41×C251—21no—
85#41×C251D412yes—
86#41×C251C2312yes—
87C2×Q8×C251D411yescharacteristic
88C2×D4×C251—21no—
89C2×D4×C251D411yescharacteristic
90C22×C4×C251—21no—
91C22×C4×C251—21no—
92C22×C4×C251C2311yescharacteristic
93M16×C251—21no—
94M16×C251C2312yes—
95C2×C8×C251—21no—
96C42×C251—21no—
97C42×C251C2312yes—
98C22×D251—21no—
99C22×D251—22no—
100C22×D251—44no—
101C22×D251—41no—
102C22×D251—41no—
103Dic502—28no—
104Dic502—22no—
105Dic502—21no—
106Dic502—22no—
107Dic502—22no—
108Dic502—22no—
109Dic502C2×D412yes—
110D1004—21no—
111D1004—22no—
112D1004—22no—
113D1004—41no—
114D1004—41no—
115D1004C2×D412yes—
116C22⋊D251—28no—
117C22⋊D251—22no—
118C22⋊D251—22no—
119C2×Dic251—28no—
120C2×Dic251—21no—
121C2×Dic251—22no—
122C2×Dic251—22no—
123C2×Dic251C2×D414yes—
124C4×D251—28no—
125C4×D251—22no—
126C4×D251—22no—
127C251⋊C8—22no—
128C23×C251—21no—
129C23×C251C2×D411yescharacteristic
130Q8×C251—21no—
131Q8×C251C2×D412yes—
132D4×C251—22no—
133D4×C251—22no—
134D4×C251—21no—
135D4×C251C2×D412yes—
136C2×C4×C251—21no—
137C2×C4×C251—22no—
138C2×C4×C251—22no—
139C2×C4×C251—22no—
140C2×C4×C251—42no—
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2×D411yescharacteristic
143C2×C4×C251C2×D411yescharacteristic
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2411yescharacteristic
147C8×C251—22no—
148C22×C251—22no—
149C22×C251—22no—
150C22×C251—42no—
151C22×C251#12011yescharacteristic
152C22×C251#12011yescharacteristic
153C22×C251C22×D411yescharacteristic
154C2×D251—22no—
155C2×D251—42no—
156C2×D251—44no—
157C2×D251—42no—
158C2×D251—82no—
159C2×D251—82no—
160C4×C251—22no—
161C4×C251—42no—
162C4×C251#12011yescharacteristic
163C4×C251#12011yescharacteristic
164C4×C251C22×D411yescharacteristic
165C4×C251C22×D411yescharacteristic
166Dic251—22no—
167Dic251—24no—
168C2×C251—21no—
169C2×C251—21no—
170C2×C251—42no—
171C2×C251#45412yescomplemented
172C2×C251C2×#12011yescharacteristic
173D251—42no—
174D251—82no—
175C251C2×#45411yescharacteristic, complemented, Sylow
176C2×#454—2511nomaximal, Sylow
177C2×2−1+4—2511no—
178C2×AGL(1,ℤ/8)—2511no—
179C2×AGL(1,ℤ/8)—2511no—
180C2×#124—2511no—
181#454—2518no—
182C2×#104—2511no—
183C2×#104D25111yescharacteristic, complemented
184C2×#101—2511no—
1852−1+4—5022no—
1862−1+4—5022no—
1872−1+4—5022no—
1882−1+4—2514no—
189C2×#41—5021no—
190C2×#41—5021no—
191C2×#41—5022no—
192C2×#41—2511no—
193C2×#41C2×D25111yescharacteristic
194C22×Q8—5021no—
195C22×Q8—5021no—
196C22×Q8—2511no—
197C22×D4—2511no—
198AGL(1,ℤ/8)—5022no—
199AGL(1,ℤ/8)—5022no—
200AGL(1,ℤ/8)—2514no—
201AGL(1,ℤ/8)—2514no—
202C2×SD16—5021no—
203C2×SD16—5021no—
204C2×D8—5021no—
205C2×D8—5021no—
206C2×M16—2511no—
207C2×M16C2×D25111yescharacteristic
208#124—5022no—
209#124—2514no—
210C2×C22⋊C4—5022no—
211C2×C42C2×D25111yescharacteristic
212#104—2514no—
213#104C2×D25114yescomplemented
214#101—2514no—
215C24—5021no—
216#41—21no—
217#41—21no—
218#41—5024no—
219#41—5024no—
220#41—5028no—
221#41—5024no—
222#41—5024no—
223#41—5028no—
224#41—5021no—
225#41—5021no—
226#41—2512no—
227#41—2512no—
228#41D100412yes—
229#41C22×D25112yes—
230C2×Q8—5022no—
231C2×Q8—5024no—
232C2×Q8—5024no—
233C2×Q8—5024no—
234C2×Q8—5024no—
235C2×Q8—5021no—
236C2×Q8—5021no—
237C2×Q8—5021no—
238C2×Q8—5021no—
239C2×Q8—2511no—
240C2×Q8—2512no—
241C2×Q8—2512no—
242C2×Q8D100411yescharacteristic
243C2×D4—21no—
244C2×D4—5022no—
245C2×D4—5021no—
246C2×D4—5021no—
247C2×D4—5021no—
248C2×D4—5021no—
249C2×D4—5021no—
250C2×D4—5021no—
251C2×D4—2511no—
252C2×D4—2512no—
253C2×D4D100411yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—5022no—
257C22×C4—5022no—
258C22×C4—5021no—
259C22×C4—2512no—
260C22×C4C22×D25111yescharacteristic
261SD16—5024no—
262SD16—5024no—
263D8—5024no—
264D8—5024no—
265M16—21no—
266M16—5021no—
267M16—2512no—
268M16C22×D25112yes—
269C2×C8—21no—
270C2×C8—5021no—
271C22⋊C4—5028no—
272C42—21no—
273C42C22×D25112yes—
274C23—21no—
275C23—10044no—
276C23—10041no—
277C23—10041no—
278C23—5022no—
279C23—5021no—
280C23D4×D25111yescharacteristic
281Q8—21no—
282Q8—5028no—
283Q8—5022no—
284Q8—5022no—
285Q8—5021no—
286Q8—5022no—
287Q8—5022no—
288Q8—2512no—
289Q8C2×D100412yes—
290D4—22no—
291D4—22no—
292D4—21no—
293D4—10041no—
294D4—10041no—
295D4—5028no—
296D4—5022no—
297D4—5022no—
298D4—5022no—
299D4—5022no—
300D4—5021no—
301D4—2512no—
302D4C2×D100412yes—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—5028no—
309C2×C4—5028no—
310C2×C4—5022no—
311C2×C4—5021no—
312C2×C4—5022no—
313C2×C4—5022no—
314C2×C4—5022no—
315C2×C4—2514no—
316C2×C4D4×D25111yescharacteristic
317C2×C4D4×D25111yescharacteristic
318C2×C4D4×D25111yescharacteristic
319C2×C4C2×D100411yescharacteristic
320C2×C4C2×D100411yescharacteristic
321C2×C4C23×D25111yescharacteristic
322C8—22no—
323C8—5022no—
324C22—22no—
325C22—22no—
326C22—42no—
327C22—20082no—
328C22—20082no—
329C22—10044no—
330C22—10042no—
331C22—10042no—
332C22—5022no—
333C22#2254868511yescharacteristic
334C22#2254868511yescharacteristic
335C22C2×D4×D25111yescharacteristic
336C4—22no—
337C4—42no—
338C4—5022no—
339C4—5024no—
340C4#2254868511yescharacteristic
341C4#2254868511yescharacteristic
342C4C2×D4×D25111yescharacteristic
343C4C22×D100411yescharacteristic
344C2—21no—
345C2—21no—
346C2—42no—
347C2—20082no—
348C2—10042no—
349C2#3063186112yesdirect product factor
350C2C2×#2254868511yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#104×C251C2
socleC22×C251#120
hypercenterC2×#104D251
hypocenterC251C2×#454
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×#22548762C2semidirect
2. C2×#22548743C2semidirect
3. C2×#22548740C2semidirect
4. C2×#22548703C2semidirect
5. #30631861C2direct (N × H)
6. C2×#22548643C2semidirect
7. C2×#22548633C2semidirect
8. C2×#104×C251C2semidirect
12. #22548762C22semidirect
22. #22548743C22semidirect
24. #22548740C22semidirect
31. #22548703C22semidirect
33. #22548643C22semidirect
34. #22548633C22semidirect
38. #104×C251C22semidirect
171. C2×C251#454semidirect
175. C251C2×#454semidirect
183. C2×#104D251semidirect
213. #104C2×D251semidirect
349. C2#30631861direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225487622
C2×#225487432
C2×#225487402
C2×#225487032
#306318612
C2×#225486432
C2×#225486332
C2×#104×C2512
C2×#454251