Finite Group Databasebeta

← C127×#12316 #44575989

Subgroups — C127×#12316 #44575989

Order 32512.

458 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 90 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
1D4×D8×C127C211yesmaximal, characteristic, complemented
2C127×#2246C211yesmaximal, characteristic, complemented
3C127×#2206C211yesmaximal, characteristic, complemented
4C127×#1667C211yesmaximal, characteristic, complemented
5C127×#1361C211yesmaximal, characteristic, complemented
6C127×#1331C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic, complemented
8C22×D8×C127—21no—
9D42×C127—21no—
10C2×D16×C127—21no—
11C2×D16×C127—22no—
12#493×C127C2211yescharacteristic
13#493×C127C2211yescharacteristic, complemented
14#466×C127—21no—
15#459×C127—21no—
16#459×C127—21no—
17#447×C127—22no—
18C4×D8×C127C2211yescharacteristic
19C8×D4×C127C2211yescharacteristic
20C4⋊C16×C127C2211yescharacteristic, complemented
21#357×C127—21no—
22C4×C16×C127C2211yescharacteristic, complemented
23#335×C127C2211yescharacteristic, complemented
24#331×C127—21no—
25C22×D4×C127—22no—
26C22×D4×C127—41no—
27C2×D8×C127—22no—
28C2×D8×C127—21no—
29C2×D8×C127—21no—
30C2×D8×C127—21no—
31C2×D8×C127—42no—
32C2×D8×C127—42no—
33C2×D8×C127—41no—
34C2×D8×C127D411yescharacteristic
35C2×D8×C127D412yes—
36C22×C8×C127—21no—
37#127×C127—21no—
38#127×C127—21no—
39#121×C127—22no—
40#121×C127—41no—
41#120×C127—42no—
42C4×D4×C127—21no—
43C4×D4×C127D411yescharacteristic
44D16×C127—44no—
45D16×C127—42no—
46C2×C16×C127—21no—
47C2×C16×C127—21no—
48#107×C127D411yescharacteristic
49C4⋊C8×C127—21no—
50C4⋊C8×C127D411yescharacteristic
51#102×C127—22no—
52#102×C127—21no—
53#102×C127—41no—
54C22⋊C8×C127—21no—
55C4×C8×C127C2311yescharacteristic
56C24×C127—42no—
57C2×D4×C127—21no—
58C2×D4×C127—22no—
59C2×D4×C127—42no—
60C2×D4×C127—44no—
61C2×D4×C127—42no—
62C2×D4×C127—41no—
63C2×D4×C127—41no—
64C2×D4×C127—41no—
65C2×D4×C127—41no—
66C2×D4×C127—41no—
67C2×D4×C127—82no—
68C2×D4×C127—82no—
69C2×D4×C127—82no—
70C2×D4×C127D811yescharacteristic
71C22×C4×C127—21no—
72C22×C4×C127—41no—
73D8×C127—21no—
74D8×C127—42no—
75D8×C127—42no—
76D8×C127—44no—
77D8×C127—41no—
78D8×C127—41no—
79D8×C127—41no—
80D8×C127—41no—
81D8×C127—82no—
82D8×C127D812yescomplemented
83C2×C8×C127—21no—
84C2×C8×C127—41no—
85C2×C8×C127—41no—
86C2×C8×C127—41no—
87C2×C8×C127C2×D411yescharacteristic
88C2×C8×C127C2×D411yescharacteristic
89C4⋊C4×C127—21no—
90C4⋊C4×C127D811yescharacteristic
91C22⋊C4×C127—21no—
92C22⋊C4×C127—42no—
93C22⋊C4×C127—41no—
94C42×C127C2×D411yescharacteristic
95C16×C127—22no—
96C16×C127—41no—
97C23×C127—21no—
98C23×C127—41no—
99C23×C127—42no—
100C23×C127—84no—
101C23×C127—84no—
102C23×C127—81no—
103C23×C127—162no—
104D4×C127—22no—
105D4×C127—21no—
106D4×C127—42no—
107D4×C127—41no—
108D4×C127—42no—
109D4×C127—84no—
110D4×C127—84no—
111D4×C127—81no—
112D4×C127—81no—
113D4×C127—81no—
114D4×C127—81no—
115D4×C127—81no—
116D4×C127—81no—
117D4×C127—161no—
118D4×C127—161no—
119D4×C127D1612yescomplemented
120C2×C4×C127—21no—
121C2×C4×C127—41no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—81no—
125C2×C4×C127—81no—
126C2×C4×C127#12011yescharacteristic
127C2×C4×C127C2×D811yescharacteristic
128C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
129C8×C127—21no—
130C8×C127—41no—
131C8×C127—81no—
132C8×C127C2×D811yescharacteristic
133C8×C127AGL(1,ℤ/8)11yescharacteristic
134C22×C127—22no—
135C22×C127—41no—
136C22×C127—41no—
137C22×C127—42no—
138C22×C127—81no—
139C22×C127—81no—
140C22×C127—82no—
141C22×C127—82no—
142C22×C127—82no—
143C22×C127—168no—
144C22×C127—161no—
145C22×C127—161no—
146C22×C127—161no—
147C22×C127#44711yescharacteristic
148C4×C127—21no—
149C4×C127—41no—
150C4×C127—81no—
151C4×C127#44711yescharacteristic
152C4×C127#45311yescharacteristic
153C4×C127C2×D1611yescharacteristic
154C4×C127#50911yescharacteristic
155#12316C12711yesmaximal, characteristic, direct product factor, Sylow
156C2×C127—42no—
157C2×C127—82no—
158C2×C127—162no—
159C2×C127—321no—
160C2×C127#161911yescharacteristic
161C2×C127#218411yescharacteristic
162C2×C127#219211yescharacteristic
163D4×D8C2×C12711yescharacteristic, complemented
164#2246C2×C12711yescharacteristic, complemented
165#2206C2×C12711yescharacteristic, complemented
166#1667C2×C12711yescharacteristic, complemented
167#1361C2×C12711yescharacteristic, complemented
168#1331C2×C12711yescharacteristic, complemented
169D4⋊C16C2×C12711yescharacteristic, complemented
170C127#1231611yescharacteristic, direct product factor, Sylow
171C22×D8—21no—
172D42—21no—
173C2×D16—21no—
174C2×D16—22no—
175#493C22×C12711yescharacteristic
176#493C22×C12711yescharacteristic, complemented
177#466—21no—
178#459—21no—
179#459—21no—
180#447—22no—
181C4×D8C22×C12711yescharacteristic
182C8×D4C22×C12711yescharacteristic
183C4⋊C16C22×C12711yescharacteristic, complemented
184#357—21no—
185C4×C16C22×C12711yescharacteristic, complemented
186#335C22×C12711yescharacteristic, complemented
187#331—21no—
188C22×D4—22no—
189C22×D4—41no—
190C2×D8—22no—
191C2×D8—21no—
192C2×D8—21no—
193C2×D8—21no—
194C2×D8—42no—
195C2×D8—42no—
196C2×D8—41no—
197C2×D8D4×C12711yescharacteristic
198C2×D8D4×C12712yes—
199C22×C8—21no—
200#127—21no—
201#127—21no—
202#121—22no—
203#121—41no—
204#120—42no—
205C4×D4—21no—
206C4×D4D4×C12711yescharacteristic
207D16—44no—
208D16—42no—
209C2×C16—21no—
210C2×C16—21no—
211#107D4×C12711yescharacteristic
212C4⋊C8—21no—
213C4⋊C8D4×C12711yescharacteristic
214#102—22no—
215#102—21no—
216#102—41no—
217C22⋊C8—21no—
218C4×C8C23×C12711yescharacteristic
219C24—42no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—42no—
223C2×D4—44no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—82no—
231C2×D4—82no—
232C2×D4—82no—
233C2×D4D8×C12711yescharacteristic
234C22×C4—21no—
235C22×C4—41no—
236D8—21no—
237D8—42no—
238D8—42no—
239D8—44no—
240D8—41no—
241D8—41no—
242D8—41no—
243D8—41no—
244D8—82no—
245D8D8×C12712yescomplemented
246C2×C8—21no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8C2×D4×C12711yescharacteristic
251C2×C8C2×D4×C12711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4D8×C12711yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—42no—
256C22⋊C4—41no—
257C42C2×D4×C12711yescharacteristic
258C16—22no—
259C16—41no—
260C23—21no—
261C23—41no—
262C23—42no—
263C23—84no—
264C23—84no—
265C23—81no—
266C23—162no—
267D4—21no—
268D4—22no—
269D4—42no—
270D4—41no—
271D4—42no—
272D4—84no—
273D4—84no—
274D4—81no—
275D4—81no—
276D4—81no—
277D4—81no—
278D4—81no—
279D4—81no—
280D4—161no—
281D4—161no—
282D4D16×C12712yescomplemented
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4#120×C12711yescharacteristic
290C2×C4C2×D8×C12711yescharacteristic
291C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
292C8—21no—
293C8—41no—
294C8—81no—
295C8C2×D8×C12711yescharacteristic
296C8AGL(1,ℤ/8)×C12711yescharacteristic
297C22—22no—
298C22—41no—
299C22—41no—
300C22—42no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—168no—
307C22—161no—
308C22—161no—
309C22—161no—
310C22#447×C12711yescharacteristic
311C4—21no—
312C4—41no—
313C4—81no—
314C4#447×C12711yescharacteristic
315C4#453×C12711yescharacteristic
316C4C2×D16×C12711yescharacteristic
317C4#509×C12711yescharacteristic
318C2—42no—
319C2—82no—
320C2—162no—
321C2—321no—
322C2C127×#161911yescharacteristic
323C2C127×#218411yescharacteristic
324C2C127×#219211yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. D4×D8×C127C2semidirect
2. C127×#2246C2semidirect
3. C127×#2206C2semidirect
4. C127×#1667C2semidirect
5. C127×#1361C2semidirect
6. C127×#1331C2semidirect
7. C127×D4⋊C16C2semidirect
13. #493×C127C22semidirect
20. C4⋊C16×C127C22semidirect
22. C4×C16×C127C22semidirect
23. #335×C127C22semidirect
82. D8×C127D8semidirect
119. D4×C127D16semidirect
155. #12316C127direct (N × H)
163. D4×D8C2×C127semidirect
164. #2246C2×C127semidirect
165. #2206C2×C127semidirect
166. #1667C2×C127semidirect
167. #1361C2×C127semidirect
168. #1331C2×C127semidirect
169. D4⋊C16C2×C127semidirect
170. C127#12316direct (N × H)
176. #493C22×C127semidirect
183. C4⋊C16C22×C127semidirect
185. C4×C16C22×C127semidirect
186. #335C22×C127semidirect
245. D8D8×C127semidirect
282. D4D16×C127semidirect

Maximal subgroups

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

subgroupindex
D4×D8×C1272
C127×#22462
C127×#22062
C127×#16672
C127×#13612
C127×#13312
C127×D4⋊C162
#12316127