Finite Group Databasebeta

← C2×#28409201 #44409059

Subgroups — C2×#28409201 #44409059

Order 29056.

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×#20300334C211yesmaximal, characteristic, complemented
2C2×#20300316C211yesmaximal, characteristic, complemented
3C2×#20300313C211yesmaximal, characteristic, complemented
4C2×#20300285C211yesmaximal, characteristic, complemented
5#28409201C218yesmaximal, direct product factor
6C2×#20300216C211yesmaximal, characteristic, complemented
7C2×#20300204C211yesmaximal, characteristic, complemented
8C2×#104×C227C211yesmaximal, characteristic, complemented
9#20300334—22no—
10#20300334—22no—
11#20300334—22no—
12#20300334C2214yescomplemented
13C2×#16813885—21no—
14C2×#16813884—21no—
15C2×#16813884C2211yescharacteristic
16C22×D908—21no—
17C22×D908—22no—
18C22×D908C2211yescharacteristic
19C22×D908C2211yescharacteristic, complemented
20C2×D4×D227—22no—
21C2×D4×D227—21no—
22#20300316—22no—
23#20300316C2214yescomplemented
24#20300313—22no—
25#20300313C2214yescomplemented
26C2×#16813873—21no—
27C2×#16813871—21no—
28C2×D1816—21no—
29C2×#16813869—21no—
30C2×#16813868C2211yescharacteristic, complemented
31#20300285—22no—
32#20300285C2214yescomplemented
33#20300216C2214yescomplemented
34#20300204C2214yescomplemented
35C2×#41×C227C2211yescharacteristic
36C2×M16×C227C2211yescharacteristic, complemented
37C2×C42×C227C2211yescharacteristic, complemented
38#104×C227C2214yescomplemented
39C23×D227—22no—
40C23×D227—21no—
41C23×D227D412yes—
42#16813885—24no—
43#16813885—24no—
44#16813884—24no—
45#16813884—24no—
46#16813884—21no—
47#16813884—21no—
48#16813884D412yescomplemented
49#16813884C2312yes—
50C2×Dic454D411yescharacteristic, complemented
51C2×D908—22no—
52C2×D908—24no—
53C2×D908—24no—
54C2×D908—28no—
55C2×D908—21no—
56C2×D908—21no—
57C2×D908—21no—
58C2×D908—21no—
59C2×D908—21no—
60C2×D908—21no—
61C2×D908—21no—
62C2×D908—21no—
63C2×D908—21no—
64C2×D908—48no—
65C2×D908D412yes—
66C2×D908D411yescharacteristic, complemented
67C2×D908C2312yes—
68C2×D908C2312yescomplemented
69C2×C22⋊D227—22no—
70C2×C22⋊D227—21no—
71D4×D227—28no—
72D4×D227—24no—
73D4×D227—28no—
74D4×D227—24no—
75C2×C4×D227—22no—
76C2×C4×D227—21no—
77#16813873—24no—
78#16813871—24no—
79D1816—24no—
80#16813869—24no—
81#16813868—21no—
82#16813868C2312yescomplemented
83C2×C227⋊C8—21no—
84#41×C227—21no—
85#41×C227—21no—
86#41×C227D412yescomplemented
87#41×C227C2312yes—
88C2×Q8×C227D411yescharacteristic, complemented
89C2×D4×C227—21no—
90C2×D4×C227D411yescharacteristic, complemented
91C22×C4×C227—21no—
92C22×C4×C227—21no—
93C22×C4×C227C2311yescharacteristic
94M16×C227—21no—
95M16×C227C2312yescomplemented
96C2×C8×C227—21no—
97C42×C227—21no—
98C42×C227C2312yescomplemented
99C22×D227—22no—
100C22×D227—21no—
101C22×D227—28no—
102C22×D227—22no—
103C22×D227—22no—
104C22×D227—44no—
105C22×D227—44no—
106C22×D227—41no—
107C22×D227—41no—
108C22×D227—84no—
109C22×D227C2×D414yes—
110Dic454—21no—
111Dic454C2×D412yescomplemented
112D908—28no—
113D908—22no—
114D908—22no—
115D908—21no—
116D908—22no—
117D908—22no—
118D908—22no—
119D908—22no—
120D908—416no—
121D908—41no—
122D908—41no—
123D908C2×D412yescomplemented
124C22⋊D227—28no—
125C22⋊D227—22no—
126C22⋊D227—22no—
127C2×Dic227—21no—
128C4×D227—28no—
129C4×D227—22no—
130C4×D227—22no—
131C227⋊C8—22no—
132C23×C227—21no—
133C23×C227C2×D411yescharacteristic
134Q8×C227—21no—
135Q8×C227C2×D412yescomplemented
136D4×C227—22no—
137D4×C227—22no—
138D4×C227—21no—
139D4×C227C2×D412yescomplemented
140C2×C4×C227—24no—
141C2×C4×C227—21no—
142C2×C4×C227—22no—
143C2×C4×C227—22no—
144C2×C4×C227—22no—
145C2×C4×C227C2×D411yescharacteristic
146C2×C4×C227C2×D411yescharacteristic
147C2×C4×C227C2×D411yescharacteristic
148C2×C4×C227C2×D411yescharacteristic
149C2×C4×C227C2×D411yescharacteristic
150C2×C4×C227C2411yescharacteristic
151C8×C227—22no—
152C22×C227—22no—
153C22×C227—22no—
154C22×C227—42no—
155C22×C227#12011yescharacteristic
156C22×C227#12011yescharacteristic
157C22×C227C22×D411yescharacteristic
158C2×D227—24no—
159C2×D227—22no—
160C2×D227—44no—
161C2×D227—42no—
162C2×D227—44no—
163C2×D227—44no—
164C2×D227—42no—
165C2×D227—82no—
166C2×D227—88no—
167C2×D227—82no—
168C2×D227—82no—
169C4×C227—24no—
170C4×C227—22no—
171C4×C227#12011yescharacteristic
172C4×C227#12011yescharacteristic
173C4×C227C22×D411yescharacteristic
174C4×C227C22×D411yescharacteristic
175Dic227—22no—
176C2×C227—21no—
177C2×C227—21no—
178C2×C227—42no—
179C2×C227#45312yescomplemented
180C2×C227C2×#12011yescharacteristic
181D227—44no—
182D227—42no—
183D227—82no—
184C227C2×#45311yescharacteristic, complemented, Sylow
185C2×#453—2271nomaximal, Sylow
186C2×H(5,𝔽2)—2271no—
187C2×AGL(1,ℤ/8)—2271no—
188C2×AGL(1,ℤ/8)—2271no—
189C2×#127—2271no—
190#453—2278no—
191C2×#104—2271no—
192C2×#104D22711yescharacteristic, complemented
193C2×#100—2271no—
194H(5,𝔽2)—4542no—
195H(5,𝔽2)—4542no—
196H(5,𝔽2)—4542no—
197H(5,𝔽2)—2274no—
198C2×#41—4541no—
199C2×#41—4541no—
200C2×#41—2271no—
201C2×#41C2×D22711yescharacteristic
202C22×D4—4542no—
203C22×D4—4541no—
204C22×D4—4542no—
205C22×D4—4541no—
206C22×D4—2271no—
207C22×D4—2271no—
208AGL(1,ℤ/8)—4542no—
209AGL(1,ℤ/8)—4542no—
210AGL(1,ℤ/8)—2274no—
211AGL(1,ℤ/8)—2274no—
212C2×SD16—4541no—
213C2×SD16—4541no—
214C2×D8—4541no—
215C2×D8—4541no—
216C2×M16—2271no—
217C2×M16C2×D22711yescharacteristic, complemented
218#127—4542no—
219#127—2274no—
220C2×C42C2×D22711yescharacteristic, complemented
221#104—2274no—
222#104C2×D22714yescomplemented
223#100—2274no—
224C24—4542no—
225C24—4541no—
226C24—2272no—
227#41—21no—
228#41—21no—
229#41—4544no—
230#41—4544no—
231#41—4544no—
232#41—4544no—
233#41—4541no—
234#41—4541no—
235#41—2272no—
236#41—2272no—
237#41D90812yescomplemented
238#41C22×D22712yes—
239C2×Q8—2271no—
240C2×Q8D90811yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—9088no—
243C2×D4—4548no—
244C2×D4—4542no—
245C2×D4—4544no—
246C2×D4—4544no—
247C2×D4—4548no—
248C2×D4—4544no—
249C2×D4—4548no—
250C2×D4—4542no—
251C2×D4—4541no—
252C2×D4—4544no—
253C2×D4—4541no—
254C2×D4—4541no—
255C2×D4—4541no—
256C2×D4—4541no—
257C2×D4—4541no—
258C2×D4—4541no—
259C2×D4—4541no—
260C2×D4—4541no—
261C2×D4—4541no—
262C2×D4—2272no—
263C2×D4—2271no—
264C2×D4—2272no—
265C2×D4—2272no—
266C2×D4D90811yescharacteristic, complemented
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—4542no—
270C22×C4—4541no—
271C22×C4C22×D22711yescharacteristic
272SD16—4544no—
273SD16—4544no—
274D8—4544no—
275D8—4544no—
276M16—21no—
277M16—4541no—
278M16—2272no—
279M16C22×D22712yescomplemented
280C2×C8—21no—
281C2×C8—4541no—
282C42—21no—
283C42C22×D22712yescomplemented
284C23—21no—
285C23—18164no—
286C23—9084no—
287C23—9084no—
288C23—9081no—
289C23—9081no—
290C23—4548no—
291C23—4542no—
292C23—4542no—
293C23—4542no—
294C23—4541no—
295C23—2274no—
296C23D4×D22711yescharacteristic
297Q8—21no—
298Q8—4541no—
299Q8—2272no—
300Q8C2×D90812yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—90816no—
305D4—9081no—
306D4—9081no—
307D4—4548no—
308D4—4548no—
309D4—4542no—
310D4—4542no—
311D4—4542no—
312D4—4542no—
313D4—4542no—
314D4—4542no—
315D4—4542no—
316D4—4541no—
317D4—4542no—
318D4—2272no—
319D4C2×D90812yescomplemented
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—4548no—
326C2×C4—4541no—
327C2×C4—4542no—
328C2×C4—4542no—
329C2×C4D4×D22711yescharacteristic
330C2×C4D4×D22711yescharacteristic
331C2×C4D4×D22711yescharacteristic
332C2×C4C2×D90811yescharacteristic
333C2×C4C2×D90811yescharacteristic
334C2×C4C23×D22711yescharacteristic
335C8—22no—
336C8—4542no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—18162no—
341C22—18162no—
342C22—18168no—
343C22—18162no—
344C22—9084no—
345C22—9084no—
346C22—9082no—
347C22—9082no—
348C22—9084no—
349C22—4544no—
350C22—4542no—
351C22#2030025811yescharacteristic
352C22#2030025811yescharacteristic
353C22C2×D4×D22711yescharacteristic
354C4—24no—
355C4—22no—
356C4—4542no—
357C4#2030025811yescharacteristic
358C4#2030025811yescharacteristic
359C4C2×D4×D22711yescharacteristic
360C4C22×D90811yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—18162no—
365C2—9082no—
366C2—9084no—
367C2#2840920112yesdirect product factor
368C2C2×#2030025811yescharacteristic
subgroupG/H
centerC22#20300258
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
FittingC2×#104×C227C2
socleC22×C227#120
hypercenterC2×#104D227
hypocenterC227C2×#453
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#20300334C2semidirect
2. C2×#20300316C2semidirect
3. C2×#20300313C2semidirect
4. C2×#20300285C2semidirect
5. #28409201C2direct (N × H)
6. C2×#20300216C2semidirect
7. C2×#20300204C2semidirect
8. C2×#104×C227C2semidirect
12. #20300334C22semidirect
19. C22×D908C22semidirect
23. #20300316C22semidirect
25. #20300313C22semidirect
30. C2×#16813868C22semidirect
32. #20300285C22semidirect
33. #20300216C22semidirect
34. #20300204C22semidirect
36. C2×M16×C227C22semidirect
37. C2×C42×C227C22semidirect
38. #104×C227C22semidirect
48. #16813884D4semidirect
50. C2×Dic454D4semidirect
66. C2×D908D4semidirect
68. C2×D908C23semidirect
82. #16813868C23semidirect
86. #41×C227D4semidirect
88. C2×Q8×C227D4semidirect
90. C2×D4×C227D4semidirect
95. M16×C227C23semidirect
98. C42×C227C23semidirect
111. Dic454C2×D4semidirect
123. D908C2×D4semidirect
135. Q8×C227C2×D4semidirect
139. D4×C227C2×D4semidirect
179. C2×C227#453semidirect
184. C227C2×#453semidirect
192. C2×#104D227semidirect
217. C2×M16C2×D227semidirect
220. C2×C42C2×D227semidirect
222. #104C2×D227semidirect
237. #41D908semidirect
240. C2×Q8D908semidirect
266. C2×D4D908semidirect
279. M16C22×D227semidirect
283. C42C22×D227semidirect
300. Q8C2×D908semidirect
319. D4C2×D908semidirect
367. C2#28409201direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#203003342
C2×#203003162
C2×#203003132
C2×#203002852
#284092012
C2×#203002162
C2×#203002042
C2×#104×C2272
C2×#453227