Finite Group Databasebeta

← #30639031

Subgroups — #30639031

Order 16256.

376 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 85 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×M16×D127C211yesmaximal, characteristic, complemented
2#22550388C211yesmaximal, characteristic
3C2×#16861137C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic, complemented
5C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×C4×D127C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11C2×#14523231—21no—
12M16×D127—22no—
13M16×D127—22no—
14C2×C8×D127—21no—
15#16861193—21no—
16#16861189C411yescharacteristic, complemented
17#16861189C411yescharacteristic, complemented
18#16861184C411yescharacteristic, complemented
19#16861184C411yescharacteristic, complemented
20#16861179—21no—
21C4×D508—21no—
22C4×D508—21no—
23C2×(C2×D127)⋊C4—21no—
24C2×C4×Dic127C2211yescharacteristic
25#16861137—22no—
26#16861135—22no—
27C127⋊#95—21no—
28C2×M16×C127C2211yescharacteristic
29#117×C127C2211yescharacteristic
30C2×C42×C127C2211yescharacteristic
31C23×D127D411yescharacteristic
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35#14523248—21no—
36C2×Dic254C2×C411yescharacteristic
37C2×D508—21no—
38C2×D508—21no—
39C2×D508C2×C411yescharacteristic
40C2×C22⋊D127—21no—
41C22×Dic127—21no—
42C22×Dic127D411yescharacteristic
43C2×C4×D127—21no—
44C2×C4×D127—21no—
45C2×C4×D127—21no—
46C2×C4×D127—21no—
47C2×C4×D127—21no—
48C2×C4×D127C2×C411yescharacteristic
49C2×C4×D127C2×C411yescharacteristic
50C2×C4×D127D411yescharacteristic
51C2×C4×D127D411yescharacteristic
52#14523232—21no—
53#14523232D412yescomplemented
54#14523231—24no—
55C2×C127⋊C8—21no—
56C8×D127—24no—
57C4⋊Dic127—21no—
58C4⋊Dic127—21no—
59(C2×D127)⋊C4—24no—
60(C2×D127)⋊C4—21no—
61(C2×D127)⋊C4—21no—
62(C2×D127)⋊C4—21no—
63(C2×D127)⋊C4—21no—
64C4×Dic127—21no—
65C4×Dic127—21no—
66C22×C4×C127—21no—
67C22×C4×C127C2311yescharacteristic
68M16×C127—21no—
69M16×C127D412yescomplemented
70C2×C8×C127—21no—
71C4⋊C4×C127C2×C411yescharacteristic
72C4⋊C4×C127C2×C411yescharacteristic
73C22⋊C4×C127—21no—
74C42×C127—21no—
75C42×C127—21no—
76C42×C127—21no—
77C22×D127—21no—
78C22×D127—21no—
79C22×D127—22no—
80C22×D127—21no—
81C22×D127—41no—
82C22×D127C22⋊C411yescharacteristic
83C22×D127C22⋊C411yescharacteristic
84Dic254—21no—
85Dic254—21no—
86D508—21no—
87D508—21no—
88D508—42no—
89D508—42no—
90C22⋊D127—41no—
91C22⋊D127—41no—
92C2×Dic127—21no—
93C2×Dic127—21no—
94C2×Dic127—22no—
95C2×Dic127—21no—
96C2×Dic127—41no—
97C2×Dic127—41no—
98C2×Dic127C22⋊C411yescharacteristic
99C2×Dic127C22⋊C411yescharacteristic
100C4×D127—21no—
101C4×D127—21no—
102C4×D127—21no—
103C4×D127—21no—
104C4×D127—21no—
105C4×D127—21no—
106C4×D127—21no—
107C4×D127—21no—
108C4×D127—41no—
109C4×D127—41no—
110C127⋊C8—22no—
111C23×C127C2×D411yescharacteristic
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—41no—
116C2×C4×C127—41no—
117C2×C4×C127C22×C411yescharacteristic
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127#4111yescharacteristic
122C2×C4×C127#4111yescharacteristic
123C8×C127—22no—
124C22×C127—21no—
125C22×C127—21no—
126C22×C127C2×C22⋊C411yescharacteristic
127C22×C127#12111yescharacteristic
128C22×C127#12111yescharacteristic
129C2×D127—21no—
130C2×D127—21no—
131C2×D127—21no—
132C2×D127—21no—
133C2×D127—21no—
134C2×D127—42no—
135C2×D127—42no—
136C2×D127—41no—
137C2×D127—41no—
138C2×D127—41no—
139C2×D127#10412yescomplemented
140C4×C127—42no—
141C4×C127—41no—
142C4×C127—41no—
143C4×C127C4×D411yescharacteristic
144C4×C127C4×D411yescharacteristic
145C4×C127#12011yescharacteristic
146C4×C127#12311yescharacteristic
147Dic127—21no—
148Dic127—21no—
149Dic127—42no—
150Dic127—41no—
151C2×C127—21no—
152C2×C127—21no—
153C2×C127#38611yescharacteristic
154C2×C127C2×#10411yescharacteristic
155C2×C127#42111yescharacteristic
156D127—22no—
157D127—41no—
158D127—81no—
159#1871—1271nomaximal, Sylow
160C127#187111yescharacteristic, complemented, Sylow
161C22×M16—1271no—
162#525—1271no—
163C2×#104—1271no—
164C2×#104—1271no—
165C4×C22⋊C4—1271no—
166#339—1271no—
167#339D12711yescharacteristic, complemented
168C2×#41—1271no—
169C23×C4—1271no—
170C2×M16—1271no—
171C2×M16—2541no—
172C2×M16—2542no—
173C2×M16—2542no—
174C2×M16C2×D12711yescharacteristic
175C22×C8—2541no—
176#123—2541no—
177C22⋊Q8—1271no—
178C22⋊Q8—1271no—
179#121—1271no—
180#121—1271no—
181#120—2541no—
182C4×D4—2541no—
183C4×D4—2541no—
184#117C2×D12711yescharacteristic
185C2×C22⋊C4—2541no—
186C2×C42—1271no—
187C2×C42C2×D12711yescharacteristic
188#104—2542no—
189#104—2542no—
190#95—2541no—
191C24—1271no—
192#41—2541no—
193#41—2541no—
194#41—2541no—
195#41—2541no—
196C2×Q8—1271no—
197C2×D4—1271no—
198C2×D4—2541no—
199C2×D4—2541no—
200C2×D4—2541no—
201C22×C4—21no—
202C22×C4—1271no—
203C22×C4—1271no—
204C22×C4—1271no—
205C22×C4—1271no—
206C22×C4—1271no—
207C22×C4—2541no—
208C22×C4—2541no—
209C22×C4—2541no—
210C22×C4—2541no—
211C22×C4—2541no—
212C22×C4—2541no—
213C22×C4C22×D12711yescharacteristic
214M16—21no—
215M16—1272no—
216M16—2544no—
217M16—2541no—
218M16D50812yescomplemented
219C2×C8—21no—
220C2×C8—2544no—
221C2×C8—2541no—
222C4⋊C4—2541no—
223C4⋊C4—2541no—
224C4⋊C4C4×D12711yescharacteristic
225C4⋊C4C4×D12711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—2541no—
228C22⋊C4—2541no—
229C22⋊C4—2541no—
230C22⋊C4—2541no—
231C22⋊C4—2544no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—2541no—
236C42—2541no—
237C23—1271no—
238C23—1271no—
239C23—5081no—
240C23—2542no—
241C23—2541no—
242C23—2541no—
243C23—2541no—
244C23D4×D12711yescharacteristic
245Q8—2541no—
246Q8—2541no—
247D4—5082no—
248D4—5082no—
249D4—5081no—
250D4—5081no—
251D4—2541no—
252D4—2541no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—1271no—
259C2×C4—1271no—
260C2×C4—5081no—
261C2×C4—5081no—
262C2×C4—5081no—
263C2×C4—5081no—
264C2×C4—2542no—
265C2×C4—2541no—
266C2×C4—2541no—
267C2×C4—2541no—
268C2×C4—2541no—
269C2×C4—2541no—
270C2×C4—2541no—
271C2×C4—2541no—
272C2×C4—2541no—
273C2×C4—2541no—
274C2×C4—2541no—
275C2×C4—2541no—
276C2×C4C2×C4×D12711yescharacteristic
277C2×C4D4×D12711yescharacteristic
278C2×C4D4×D12711yescharacteristic
279C2×C4C2×D50811yescharacteristic
280C2×C4#1452324811yescharacteristic
281C2×C4#1452324911yescharacteristic
282C8—22no—
283C8—2542no—
284C22—21no—
285C22—21no—
286C22—1272no—
287C22—5082no—
288C22—5082no—
289C22—5081no—
290C22—5081no—
291C22—5081no—
292C22—2541no—
293C22—2541no—
294C22—2541no—
295C22—2541no—
296C22—2541no—
297C22C22⋊C4×D12711yescharacteristic
298C22#1686118211yescharacteristic
299C22#1686118411yescharacteristic
300C4—42no—
301C4—41no—
302C4—41no—
303C4—5082no—
304C4—5081no—
305C4—2541no—
306C4—2541no—
307C4#1686117111yescharacteristic
308C4C4×D50811yescharacteristic
309C4#1686117911yescharacteristic
310C4#1686119411yescharacteristic
311C2—21no—
312C2—21no—
313C2—10161no—
314C2—5081no—
315C2—2542no—
316C2#2254975411yescharacteristic
317C2#104×D12711yescharacteristic
318C2#2254989111yescharacteristic
subgroupG/H
centerC2×C4D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550388C2
cosocleC22×C4×C127C23
hypercenter#339D127
hypocenterC127#1871
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×M16×D127C2semidirect
3. C2×#16861137C2semidirect
4. C2×#16861135C2semidirect
5. C4×(C2×D127)⋊C4C2semidirect
6. #22549635C2semidirect
7. #339×C127C2semidirect
16. #16861189C4semidirect
17. #16861189C4semidirect
18. #16861184C4semidirect
19. #16861184C4semidirect
53. #14523232D4semidirect
69. M16×C127D4semidirect
139. C2×D127#104semidirect
160. C127#1871semidirect
167. #339D127semidirect
218. M16D508semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1272
#225503882
C2×#168611372
C2×#168611352
C4×(C2×D127)⋊C42
#225496352
#339×C1272
#1871127