Finite Group Databasebeta

← #30639182

Subgroups — #30639182

Order 16256.

598 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16861179C211yesmaximal, characteristic
2C2×C22⋊C4×D127C211yesmaximal, characteristic, complemented
3C2×#16861123C211yesmaximal, characteristic, complemented
4C2×#16861123C211yesmaximal, characteristic, complemented
5#22549731C211yesmaximal, characteristic, complemented
6#22549643C211yesmaximal, characteristic, complemented
7#342×C127C211yesmaximal, characteristic, complemented
8C24×D127C2211yescharacteristic
9C22×D508C412yescomplemented
10C22×C22⋊D127C2211yescharacteristic
11C22×C4×D127—21no—
12#16861179—21no—
13#16861179—22no—
14#16861179—21no—
15C2×C22⋊Dic127C2211yescharacteristic
16C2×C22⋊Dic127C2211yescharacteristic
17C2×(C2×D127)⋊C4—21no—
18C2×(C2×D127)⋊C4—22no—
19C2×(C2×D127)⋊C4C412yescomplemented
20C22⋊C4×D127—22no—
21C22⋊C4×D127—22no—
22#16861123—22no—
23#16861123—22no—
24C2×C22⋊C4×C127C2211yescharacteristic
25C2×C22⋊C4×C127C2211yescharacteristic
26C2×C22⋊C4×C127C2211yescharacteristic
27C23×D127—21no—
28C23×D127—21no—
29C23×D127—21no—
30C23×D127—21no—
31C23×D127—44no—
32C23×D127C2×C412yes—
33C23×D127C2×C411yescharacteristic, complemented
34C23×D127D411yescharacteristic
35C23×D127D412yes—
36C23×D127D411yescharacteristic
37C2×D508—22no—
38C2×D508—22no—
39C2×D508—44no—
40C2×C22⋊D127—21no—
41C2×C22⋊D127—21no—
42C2×C22⋊D127—21no—
43C2×C22⋊D127—21no—
44C2×C22⋊D127—21no—
45C2×C22⋊D127—21no—
46C22×Dic127—21no—
47C22×Dic127—21no—
48C22×Dic127C2×C411yescharacteristic, complemented
49C2×C4×D127—24no—
50C2×C4×D127—44no—
51C22⋊Dic127—21no—
52C22⋊Dic127—21no—
53C22⋊Dic127—21no—
54C22⋊Dic127D412yescomplemented
55(C2×D127)⋊C4—24no—
56(C2×D127)⋊C4—22no—
57(C2×D127)⋊C4—22no—
58(C2×D127)⋊C4—44no—
59C24×C127C2311yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127C2×C412yescomplemented
63C22⋊C4×C127—21no—
64C22⋊C4×C127—21no—
65C22⋊C4×C127—21no—
66C22⋊C4×C127—21no—
67C22⋊C4×C127—21no—
68C22⋊C4×C127D412yescomplemented
69C22×D127—21no—
70C22×D127—21no—
71C22×D127—24no—
72C22×D127—22no—
73C22×D127—22no—
74C22×D127—21no—
75C22×D127—22no—
76C22×D127—22no—
77C22×D127—21no—
78C22×D127—24no—
79C22×D127—24no—
80C22×D127—21no—
81C22×D127—21no—
82C22×D127—21no—
83C22×D127—42no—
84C22×D127—42no—
85C22×D127—42no—
86C22×D127—44no—
87C22×D127—44no—
88C22×D127—44no—
89C22×D127—44no—
90C22×D127—42no—
91C22×D127C22⋊C412yes—
92C22×D127C22⋊C412yes—
93D508—44no—
94D508—44no—
95C22⋊D127—41no—
96C22⋊D127—41no—
97C22⋊D127—41no—
98C22⋊D127—41no—
99C2×Dic127—21no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127—22no—
103C2×Dic127—42no—
104C2×Dic127—42no—
105C2×Dic127—41no—
106C4×D127—48no—
107C23×C127—42no—
108C23×C127C22×C411yescharacteristic
109C23×C127C2×D411yescharacteristic
110C23×C127C2×D411yescharacteristic
111C23×C127C2×D411yescharacteristic
112C23×C127C2×D411yescharacteristic
113C23×C127#4111yescharacteristic
114C23×C127#4111yescharacteristic
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127—42no—
118C2×C4×C127—42no—
119C2×C4×C127—42no—
120C2×C4×C127—41no—
121C22×C127—22no—
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—41no—
125C22×C127—41no—
126C22×C127—41no—
127C22×C127—41no—
128C22×C127C2×C22⋊C411yescharacteristic
129C22×C127C4×D411yescharacteristic
130C22×C127C4×D411yescharacteristic
131C22×C127#12011yescharacteristic
132C22×C127#12111yescharacteristic
133C22×C127#12111yescharacteristic
134C22×C127#12311yescharacteristic
135C2×D127—24no—
136C2×D127—24no—
137C2×D127—22no—
138C2×D127—22no—
139C2×D127—22no—
140C2×D127—22no—
141C2×D127—22no—
142C2×D127—44no—
143C2×D127—44no—
144C2×D127—44no—
145C2×D127—44no—
146C2×D127—42no—
147C2×D127—42no—
148C2×D127—41no—
149C2×D127—42no—
150C2×D127—81no—
151C2×D127—81no—
152C2×D127#9914yescomplemented
153C4×C127—42no—
154C4×C127—42no—
155C4×C127—81no—
156Dic127—42no—
157Dic127—41no—
158Dic127—81no—
159C2×C127—21no—
160C2×C127—21no—
161C2×C127—21no—
162C2×C127—21no—
163C2×C127—21no—
164C2×C127—21no—
165C2×C127#38611yescharacteristic
166C2×C127C2×#9912yes—
167D127—24no—
168D127—42no—
169D127—81no—
170#1898—1271nomaximal, Sylow
171C127#189811yescharacteristic, complemented, Sylow
172C2×#120—1271no—
173C22×C22⋊C4—1271no—
174C2×#99—1271no—
175C2×#99—1271no—
176#379—1271no—
177#342—1271no—
178#342D12711yescharacteristic, complemented
179C25—1271no—
180C22×D4—1271no—
181C22×D4—1272no—
182C23×C4—2541no—
183#120—2541no—
184#120—2542no—
185#120—2541no—
186C2×C22⋊C4—1271no—
187C2×C22⋊C4—1272no—
188C2×C22⋊C4—1271no—
189C2×C22⋊C4—2542no—
190C2×C22⋊C4—2542no—
191C2×C22⋊C4—2541no—
192C2×C22⋊C4—2542no—
193C2×C22⋊C4C2×D12711yescharacteristic
194C2×C22⋊C4C2×D12711yescharacteristic
195C2×C22⋊C4C2×D12711yescharacteristic
196#99—2542no—
197#99—2542no—
198C24—1271no—
199C24—1272no—
200C24—1272no—
201C24—1271no—
202C24—1271no—
203C24—5084no—
204C24—2541no—
205C24—2541no—
206C24—2541no—
207C24—2541no—
208C24C22×D12711yescharacteristic
209C2×D4—5084no—
210C2×D4—2542no—
211C2×D4—2542no—
212C2×D4—2541no—
213C2×D4—2541no—
214C2×D4—2541no—
215C2×D4—2541no—
216C2×D4—2541no—
217C2×D4—2541no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—1271no—
221C22×C4—5084no—
222C22×C4—2544no—
223C22×C4—2541no—
224C22×C4—2541no—
225C22×C4C4×D12712yescomplemented
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—21no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—1272no—
232C22⋊C4—5084no—
233C22⋊C4—2542no—
234C22⋊C4—2542no—
235C22⋊C4—2544no—
236C22⋊C4—2541no—
237C22⋊C4—2541no—
238C22⋊C4—2541no—
239C22⋊C4D50812yescomplemented
240C23—42no—
241C23—1272no—
242C23—1272no—
243C23—5082no—
244C23—5082no—
245C23—5082no—
246C23—5082no—
247C23—5084no—
248C23—5084no—
249C23—5084no—
250C23—5084no—
251C23—2544no—
252C23—2544no—
253C23—2544no—
254C23—2541no—
255C23—2541no—
256C23—2541no—
257C23—2542no—
258C23—2541no—
259C23—2542no—
260C23—2542no—
261C23—2541no—
262C23—2542no—
263C23—2541no—
264C23—2541no—
265C23C2×C4×D12711yescharacteristic
266C23D4×D12711yescharacteristic
267C23D4×D12711yescharacteristic
268C23D4×D12711yescharacteristic
269C23C2×D50811yescharacteristic
270C23#1452324811yescharacteristic
271C23#1452324911yescharacteristic
272D4—5084no—
273D4—5084no—
274D4—5081no—
275D4—5081no—
276D4—5081no—
277D4—5081no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—5088no—
285C2×C4—5082no—
286C2×C4—5081no—
287C2×C4—5082no—
288C2×C4—2541no—
289C2×C4—2541no—
290C2×C4—2541no—
291C2×C4—2542no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—1274no—
300C22—10161no—
301C22—10161no—
302C22—5082no—
303C22—5082no—
304C22—5084no—
305C22—5084no—
306C22—5084no—
307C22—5084no—
308C22—5082no—
309C22—5081no—
310C22—2544no—
311C22—2544no—
312C22—2542no—
313C22—2542no—
314C22—2542no—
315C22—2542no—
316C22—2542no—
317C22C22⋊C4×D12711yescharacteristic
318C22#1686117111yescharacteristic
319C22C4×D50811yescharacteristic
320C22#1686117911yescharacteristic
321C22#1686118211yescharacteristic
322C22#1686118411yescharacteristic
323C22#1686119411yescharacteristic
324C4—42no—
325C4—42no—
326C4—81no—
327C4—10161no—
328C4—5082no—
329C4—5081no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2—21no—
335C2—21no—
336C2—10161no—
337C2—5082no—
338C2—2544no—
339C2#2254975411yescharacteristic
340C2#99×D12712yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#342×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#16861179C2
cosocleC24×C127C23
hypercenter#342D127
hypocenterC127#1898
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×D127C2semidirect
3. C2×#16861123C2semidirect
4. C2×#16861123C2semidirect
5. #22549731C2semidirect
6. #22549643C2semidirect
7. #342×C127C2semidirect
9. C22×D508C4semidirect
19. C2×(C2×D127)⋊C4C4semidirect
33. C23×D127C2×C4semidirect
48. C22×Dic127C2×C4semidirect
54. C22⋊Dic127D4semidirect
62. C22×C4×C127C2×C4semidirect
68. C22⋊C4×C127D4semidirect
152. C2×D127#99semidirect
171. C127#1898semidirect
178. #342D127semidirect
225. C22×C4C4×D127semidirect
239. C22⋊C4D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168611792
C2×C22⋊C4×D1272
C2×#168611232
#225497312
#225496432
#342×C1272
#1898127