Finite Group Databasebeta

← #30648050

Subgroups — #30648050

Order 16256.

712 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 712 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×#16861257C211yesmaximal, characteristic, complemented
2C2×#16861240C211yesmaximal, characteristic, complemented
3C2×#16861231C211yesmaximal, characteristic, complemented
4#22550457C211yesmaximal, characteristic, complemented
5#22550041C212yesmaximal, complemented
6#22550033C212yesmaximal, complemented
7#22550024C212yesmaximal, complemented
8#22550016C212yesmaximal, complemented
9#22549875C211yesmaximal, characteristic, complemented
10#22549842C211yesmaximal, characteristic, complemented
11C2×#103×C127C211yesmaximal, characteristic, complemented
12#16861257—24no—
13#16861257—24no—
14C2×#14523249—21no—
15C2×#14523249C2212yes—
16C2×#14523248—22no—
17C2×#14523248C2211yescharacteristic
18C2×#14523248C2211yescharacteristic
19C2×#14523248C2211yescharacteristic
20C2×Q8×D127—21no—
21C2×Q8×D127C2212yes—
22#16861240—24no—
23#16861231—24no—
24C2×Dic508C2211yescharacteristic
25C2×#14523238C2212yes—
26C2×#14523237C2211yescharacteristic
27C2×#14523237C2211yescharacteristic
28C2×#14523235C2212yes—
29C2×D1016C2211yescharacteristic
30C2×#14523232C2211yescharacteristic
31#16861189—21no—
32#16861189C2212yes—
33#16861184—21no—
34#16861184C2212yes—
35#16861183—22no—
36#16861132C2212yes—
37#16861129C2212yes—
38(C2×D127)⋊C8C2212yes—
39(C2×D127)⋊C8C2212yes—
40C22×Q8×C127C2211yescharacteristic
41C22×C8×C127C2211yescharacteristic
42C2×C4⋊C4×C127C2211yescharacteristic
43#103×C127C2212yes—
44#103×C127C2212yes—
45#14523249—28no—
46#14523249—24no—
47#14523249—24no—
48#14523248—216no—
49#14523248—22no—
50#14523248—22no—
51#14523248—22no—
52#14523248—22no—
53#14523248—22no—
54#14523248D414yescomplemented
55C2×Dic254—22no—
56C2×Dic254D411yescharacteristic
57C2×Dic254C2311yescharacteristic
58C2×Dic254C2311yescharacteristic
59Q8×D127—28no—
60Q8×D127—24no—
61Q8×D127—24no—
62C2×D508—22no—
63C2×D508—22no—
64C2×D508D411yescharacteristic
65C2×D508C2311yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127D412yes—
70C2×C4×D127—22no—
71C2×C4×D127—22no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127C2312yes—
75Dic508—22no—
76#14523238—24no—
77#14523237—22no—
78#14523237—22no—
79#14523235—24no—
80D1016—22no—
81#14523232—22no—
82C2×C127⋊C8C2312yes—
83C4⋊Dic127—22no—
84(C2×D127)⋊C4—22no—
85(C2×D127)⋊C4—22no—
86C2×Q8×C127—21no—
87C2×Q8×C127—22no—
88C2×Q8×C127D414yescomplemented
89C2×Q8×C127C2312yes—
90C22×C4×C127—21no—
91C22×C4×C127—21no—
92C22×C4×C127C2311yescharacteristic
93C2×C8×C127—22no—
94C2×C8×C127C2311yescharacteristic
95C2×C8×C127C2311yescharacteristic
96C4⋊C4×C127—21no—
97C4⋊C4×C127C2312yes—
98C22×D127—21no—
99C22×D127—21no—
100C22×D127C2×D412yes—
101Dic254—28no—
102Dic254—22no—
103Dic254—21no—
104Dic254—21no—
105Dic254—21no—
106Dic254C2×D412yes—
107D508—28no—
108D508—22no—
109D508—21no—
110D508—21no—
111D508—21no—
112D508—82no—
113D508C2×D412yes—
114C22⋊D127—24no—
115C22⋊D127—24no—
116C22⋊D127—42no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127C2×D412yes—
120C4×D127—28no—
121C4×D127—28no—
122C4×D127—24no—
123C4×D127—22no—
124C4×D127—22no—
125C4×D127—42no—
126C127⋊C8—22no—
127C23×C127C2×D411yescharacteristic
128Q8×C127—24no—
129Q8×C127—22no—
130Q8×C127C2×D414yes—
131C2×C4×C127—24no—
132C2×C4×C127—22no—
133C2×C4×C127—22no—
134C2×C4×C127—81no—
135C2×C4×C127C2×D411yescharacteristic
136C2×C4×C127C2×D412yes—
137C2×C4×C127C2×D412yes—
138C2×C4×C127C2411yescharacteristic
139C8×C127—21no—
140C8×C127—21no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127#12012yes—
144C22×C127C22×D411yescharacteristic
145C2×D127—22no—
146C2×D127—22no—
147C2×D127—22no—
148C2×D127—22no—
149C2×D127—41no—
150C2×D127—41no—
151C2×D127—41no—
152C2×D127—41no—
153C2×D127—41no—
154C4×C127—24no—
155C4×C127—42no—
156C4×C127#12012yes—
157C4×C127C22×D411yescharacteristic
158C4×C127C22×D411yescharacteristic
159Dic127—22no—
160Dic127—22no—
161Dic127—41no—
162C2×C127—22no—
163C2×C127C2×#12011yescharacteristic
164C2×C127#57711yescharacteristic
165C2×C127#57811yescharacteristic
166D127—42no—
167D127—42no—
168D127—81no—
169#3013—1271nomaximal, Sylow
170C127#301311yescharacteristic, complemented, Sylow
171C2×2−1+4—1271no—
172C2×#137—1271no—
173C2×#135—1271no—
174#537—1271no—
175#452—1272no—
176#451—1272no—
177#449—1272no—
178#448—1272no—
179#417—1271no—
180C2×#103D12711yescharacteristic, complemented
181#408—1271no—
1822−1+4—2544no—
1832−1+4—2544no—
184C2×#41—1271no—
185C2×#41—1272no—
186C2×#41—1271no—
187C2×#41—1271no—
188C2×#41—2542no—
189C2×#41—2541no—
190C22×Q8—1272no—
191C22×Q8—2541no—
192C22×Q8C2×D12711yescharacteristic
193#137—2544no—
194#135—2544no—
195C2×Q16—1272no—
196C2×Q16—1271no—
197C2×SD16—1272no—
198C2×SD16—1271no—
199C2×SD16—1271no—
200C2×D8—1271no—
201C2×M16—1271no—
202C22×C8C2×D12711yescharacteristic
203C22⋊Q8—1272no—
204C22⋊Q8—2541no—
205#121—1272no—
206#121—2541no—
207#121—2542no—
208C2×C4⋊C4C2×D12711yescharacteristic
209#103—1272no—
210#103C2×D12712yes—
211#103C2×D12712yes—
212#102—1272no—
213C22⋊C8—1272no—
214C22⋊C8—1272no—
215#41—1274no—
216#41—25416no—
217#41—2542no—
218#41—2542no—
219#41—2544no—
220#41—2548no—
221#41—2544no—
222#41—2542no—
223#41—2542no—
224#41—2542no—
225C2×Q8—21no—
226C2×Q8—22no—
227C2×Q8—1271no—
228C2×Q8—1271no—
229C2×Q8—1271no—
230C2×Q8—2542no—
231C2×Q8—2544no—
232C2×Q8—2548no—
233C2×Q8—2544no—
234C2×Q8D50814yescomplemented
235C2×Q8C22×D12712yes—
236C2×D4—1272no—
237C2×D4—1271no—
238C2×D4—1271no—
239C2×D4—1271no—
240C2×D4—2542no—
241C2×D4—2542no—
242C2×D4—2541no—
243C2×D4—2541no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—1272no—
247C22×C4—2542no—
248C22×C4—2542no—
249C22×C4—2541no—
250C22×C4—2541no—
251C22×C4C22×D12711yescharacteristic
252Q16—2542no—
253Q16—2544no—
254SD16—2542no—
255SD16—2542no—
256SD16—2544no—
257D8—2542no—
258M16—2542no—
259C2×C8—22no—
260C2×C8—1272no—
261C2×C8C22×D12711yescharacteristic
262C2×C8C22×D12711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—2542no—
265C4⋊C4C22×D12712yes—
266C22⋊C4—2542no—
267C22⋊C4—2542no—
268C23—1272no—
269C23—2541no—
270C23—2541no—
271C23D4×D12711yescharacteristic
272Q8—22no—
273Q8—24no—
274Q8—1272no—
275Q8—2548no—
276Q8—2541no—
277Q8—2541no—
278Q8—2541no—
279Q8—2542no—
280Q8C2×D50814yes—
281D4—1272no—
282D4—10162no—
283D4—5082no—
284D4—2548no—
285D4—2544no—
286D4—2544no—
287D4—2541no—
288D4—2541no—
289D4—2541no—
290D4—2542no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—81no—
295C2×C4—1272no—
296C2×C4—5082no—
297C2×C4—2548no—
298C2×C4—2548no—
299C2×C4—2541no—
300C2×C4—2541no—
301C2×C4—2542no—
302C2×C4—2542no—
303C2×C4—2544no—
304C2×C4D4×D12712yes—
305C2×C4D4×D12711yescharacteristic
306C2×C4C2×D50812yes—
307C2×C4C23×D12711yescharacteristic
308C8—21no—
309C8—21no—
310C8—2542no—
311C22—21no—
312C22—21no—
313C22—5081no—
314C22—5081no—
315C22—5081no—
316C22—5081no—
317C22—5081no—
318C22—2542no—
319C22—2542no—
320C22—2542no—
321C22—2542no—
322C22#1686117912yes—
323C22C2×D4×D12711yescharacteristic
324C4—24no—
325C4—42no—
326C4—5081no—
327C4—2542no—
328C4—2542no—
329C4#1686117912yes—
330C4C2×D4×D12711yescharacteristic
331C4C22×D50811yescharacteristic
332C2—22no—
333C2—10161no—
334C2—5082no—
335C2—5082no—
336C2C2×#1686117911yescharacteristic
337C2#2255068211yescharacteristic
338C2#2255068911yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#103×C127C2
socleC22×C127C22×D4
hypercenterC2×#103D127
hypocenterC127#3013
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C2×#16861257C2semidirect
2. C2×#16861240C2semidirect
3. C2×#16861231C2semidirect
4. #22550457C2semidirect
5. #22550041C2semidirect
6. #22550033C2semidirect
7. #22550024C2semidirect
8. #22550016C2semidirect
9. #22549875C2semidirect
10. #22549842C2semidirect
11. C2×#103×C127C2semidirect
54. #14523248D4semidirect
88. C2×Q8×C127D4semidirect
170. C127#3013semidirect
180. C2×#103D127semidirect
234. C2×Q8D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612572
C2×#168612402
C2×#168612312
#225504572
#225500412
#225500332
#225500242
#225500162
#225498752
#225498422
C2×#103×C1272
#3013127