Finite Group Databasebeta

← #30639727

Subgroups — #30639727

Order 16256.

496 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C22×#14523233C211yesmaximal, characteristic, complemented
2C2×#16861184C211yesmaximal, characteristic, complemented
3C2×#16861180C211yesmaximal, characteristic, complemented
4C2×#16861131C211yesmaximal, characteristic, complemented
5C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic, complemented
8C22×D508—21no—
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—21no—
11C22×C4×D127C2211yescharacteristic
12C2×#14523233—22no—
13C2×#14523233—22no—
14C2×#14523233—22no—
15C22×C127⋊C8C2211yescharacteristic
16#16861184—22no—
17#16861184—22no—
18#16861180—22no—
19#16861180—22no—
20C2×C4⋊Dic127C2211yescharacteristic
21C2×C22⋊Dic127—21no—
22C2×(C2×D127)⋊C4—21no—
23#16861131—22no—
24(C2×D127)⋊C8—22no—
25C22×D4×C127C2211yescharacteristic
26C22×C8×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2211yescharacteristic
28#102×C127—22no—
29C23×D127—21no—
30C23×D127D411yescharacteristic
31C2×D508—24no—
32C2×D508—21no—
33C2×D508—21no—
34C2×D508—21no—
35C2×D508—21no—
36C2×D508—21no—
37C2×D508—42no—
38C2×D508—42no—
39C2×D508D412yescomplemented
40C2×C22⋊D127—24no—
41C2×C22⋊D127—42no—
42C2×C22⋊D127—42no—
43C22×Dic127—21no—
44C22×Dic127D411yescharacteristic
45C2×C4×D127—21no—
46C2×C4×D127—21no—
47C2×C4×D127—21no—
48C2×C4×D127—21no—
49C2×C4×D127—21no—
50C2×C4×D127—21no—
51#14523233—28no—
52C2×C127⋊C8—21no—
53C2×C127⋊C8—21no—
54C2×C127⋊C8D412yescomplemented
55C4⋊Dic127—21no—
56C4⋊Dic127D412yescomplemented
57C22⋊Dic127—24no—
58(C2×D127)⋊C4—24no—
59C24×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127D412yescomplemented
66C22×C4×C127—21no—
67C22×C4×C127C2311yescharacteristic
68C2×C8×C127—21no—
69C2×C8×C127—21no—
70C2×C8×C127—21no—
71C4⋊C4×C127—21no—
72C4⋊C4×C127D412yescomplemented
73C22×D127—21no—
74C22×D127—21no—
75C22×D127—21no—
76C22×D127—21no—
77C22×D127—21no—
78C22×D127—21no—
79C22×D127—22no—
80C22×D127—41no—
81C22×D127—41no—
82C22×D127—41no—
83C22×D127—41no—
84C22×D127—41no—
85C22×D127—41no—
86C22×D127D812yescomplemented
87D508—22no—
88D508—22no—
89D508—48no—
90D508—41no—
91D508—41no—
92C22⋊D127—48no—
93C2×Dic127—21no—
94C2×Dic127—21no—
95C2×Dic127—22no—
96C2×Dic127—21no—
97C2×Dic127—41no—
98C2×Dic127—41no—
99C4×D127—41no—
100C4×D127—41no—
101C4×D127—41no—
102C4×D127—41no—
103C127⋊C8—22no—
104C23×C127—22no—
105C23×C127—41no—
106C23×C127—41no—
107C23×C127—41no—
108C23×C127—41no—
109C23×C127—41no—
110C23×C127—41no—
111C23×C127C2×D411yescharacteristic
112D4×C127—22no—
113D4×C127—22no—
114D4×C127—41no—
115D4×C127—41no—
116C2×C4×C127—22no—
117C2×C4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D411yescharacteristic
122C2×C4×C127C2×D411yescharacteristic
123C2×C4×C127C2×D411yescharacteristic
124C2×C4×C127#4111yescharacteristic
125C8×C127—41no—
126C22×C127—42no—
127C22×C127—42no—
128C22×C127—42no—
129C22×C127—81no—
130C22×C127—81no—
131C22×C127—81no—
132C22×C127—81no—
133C22×C127#12011yescharacteristic
134C22×C127#12011yescharacteristic
135C22×C127#12111yescharacteristic
136C22×C127C2×D811yescharacteristic
137C22×C127#13511yescharacteristic
138C22×C127AGL(1,ℤ/8)11yescharacteristic
139C22×C127AGL(1,ℤ/8)11yescharacteristic
140C2×D127—22no—
141C2×D127—22no—
142C2×D127—22no—
143C2×D127—41no—
144C2×D127—41no—
145C2×D127—41no—
146C2×D127—41no—
147C2×D127—42no—
148C2×D127—42no—
149C2×D127—42no—
150C2×D127—81no—
151C2×D127—81no—
152C2×D127—81no—
153C2×D127—81no—
154C4×C127—42no—
155C4×C127#12011yescharacteristic
156C4×C127#12111yescharacteristic
157C4×C127#12111yescharacteristic
158C4×C127#12711yescharacteristic
159Dic127—41no—
160Dic127—42no—
161C2×C127—82no—
162C2×C127#39211yescharacteristic
163C2×C127#44711yescharacteristic
164C2×C127#44711yescharacteristic
165C2×C127#44911yescharacteristic
166C2×C127#44911yescharacteristic
167C2×C127#46611yescharacteristic
168C2×C127#46911yescharacteristic
169D127—42no—
170D127—82no—
171#1999—1271nomaximal, Sylow
172C127#199911yescharacteristic, complemented, Sylow
173C22×D8—1271no—
174C2×#121—1271no—
175C2×#121—1271no—
176C2×#102—1271no—
177C2×#102D12711yescharacteristic, complemented
178C2×C22⋊C8—1271no—
179#340—1271no—
180C22×D4—1271no—
181C22×D4—2541no—
182C22×D4—2541no—
183C22×D4C2×D12711yescharacteristic
184C23×C4—1271no—
185C2×D8—2542no—
186C2×D8—2542no—
187C2×D8—2542no—
188C22×C8—1271no—
189C22×C8C2×D12711yescharacteristic
190#121—2542no—
191#121—2542no—
192#121—2542no—
193#121—2542no—
194C2×C4⋊C4—1271no—
195C2×C4⋊C4C2×D12711yescharacteristic
196C2×C22⋊C4—2541no—
197C2×C22⋊C4—2541no—
198#102—22no—
199#102—2542no—
200C22⋊C8—2542no—
201C24—21no—
202C24—1271no—
203C24—2541no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—1272no—
210C2×D4—5082no—
211C2×D4—5082no—
212C2×D4—5082no—
213C2×D4—5082no—
214C2×D4—2544no—
215C2×D4—2544no—
216C2×D4—2541no—
217C2×D4—2541no—
218C2×D4—2541no—
219C2×D4—2541no—
220C2×D4—2541no—
221C2×D4D50812yescomplemented
222C22×C4—21no—
223C22×C4—1271no—
224C22×C4—2541no—
225C22×C4—2541no—
226C22×C4—2541no—
227C22×C4—2541no—
228C22×C4—2541no—
229C22×C4—2541no—
230C22×C4—2541no—
231C22×C4C22×D12711yescharacteristic
232D8—2548no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—1272no—
237C2×C8—2541no—
238C2×C8—2541no—
239C4⋊C4—21no—
240C4⋊C4—1272no—
241C4⋊C4—2541no—
242C4⋊C4C22⋊D12712yescomplemented
243C22⋊C4—2544no—
244C22⋊C4—2544no—
245C23—22no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—1272no—
253C23—5081no—
254C23—5081no—
255C23—5081no—
256C23—5081no—
257C23—5081no—
258C23—5081no—
259C23—2542no—
260C23—2541no—
261C23—2541no—
262C23—2541no—
263C23—2541no—
264C23—2541no—
265C23—2541no—
266C23D4×D12711yescharacteristic
267D4—22no—
268D4—22no—
269D4—41no—
270D4—41no—
271D4—5088no—
272D4—5088no—
273D4—5081no—
274D4—5081no—
275D4—2542no—
276D4—2542no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—5081no—
281C2×C4—5081no—
282C2×C4—5081no—
283C2×C4—5081no—
284C2×C4—5081no—
285C2×C4—5081no—
286C2×C4—2541no—
287C2×C4—2541no—
288C2×C4—2542no—
289C2×C4—2541no—
290C2×C4D4×D12711yescharacteristic
291C2×C4D4×D12711yescharacteristic
292C2×C4D4×D12711yescharacteristic
293C2×C4C2×C22⋊D12711yescharacteristic
294C2×C4C2×D50811yescharacteristic
295C2×C4#1452324811yescharacteristic
296C8—41no—
297C8—2542no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—10161no—
306C22—10161no—
307C22—10161no—
308C22—10161no—
309C22—5082no—
310C22—5082no—
311C22—5082no—
312C22—5081no—
313C22—5081no—
314C22—5081no—
315C22—5081no—
316C22—2542no—
317C22—2542no—
318C22—2542no—
319C22#1686117711yescharacteristic
320C22#1686117911yescharacteristic
321C22#1686118211yescharacteristic
322C22D8×D12711yescharacteristic
323C22#1686123011yescharacteristic
324C22#1686123311yescharacteristic
325C22#1686123611yescharacteristic
326C4—42no—
327C4—5081no—
328C4—5082no—
329C4#1686117911yescharacteristic
330C4#1686118211yescharacteristic
331C4#1686118311yescharacteristic
332C4#1686120511yescharacteristic
333C2—82no—
334C2—10162no—
335C2—5082no—
336C2#2254978511yescharacteristic
337C2#2255000811yescharacteristic
338C2#2255001111yescharacteristic
339C2#2255002011yescharacteristic
340C2#2255002211yescharacteristic
341C2#2255011111yescharacteristic
342C2#2255012811yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
hypercenterC2×#102D127
hypocenterC127#1999
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#14523233C2semidirect
2. C2×#16861184C2semidirect
3. C2×#16861180C2semidirect
4. C2×#16861131C2semidirect
5. C2×(C2×D127)⋊C8C2semidirect
6. #22549638C2semidirect
7. C2×#102×C127C2semidirect
39. C2×D508D4semidirect
54. C2×C127⋊C8D4semidirect
56. C4⋊Dic127D4semidirect
65. C2×D4×C127D4semidirect
72. C4⋊C4×C127D4semidirect
86. C22×D127D8semidirect
172. C127#1999semidirect
177. C2×#102D127semidirect
221. C2×D4D508semidirect
242. C4⋊C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
C2×#168611842
C2×#168611802
C2×#168611312
C2×(C2×D127)⋊C82
#225496382
C2×#102×C1272
#1999127