Finite Group Databasebeta

← #30648044

Subgroups — #30648044

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×#16861229C211yesmaximal, characteristic, complemented
3#22550454C211yesmaximal, characteristic, complemented
4#22550037C212yesmaximal, complemented
5#22550030C212yesmaximal, complemented
6#22550020C212yesmaximal, complemented
7#22550013C212yesmaximal, complemented
8#22549878C211yesmaximal, characteristic, complemented
9C2×#16861133C211yesmaximal, characteristic, complemented
10#22549844C211yesmaximal, characteristic, complemented
11C2×#137×C127C211yesmaximal, characteristic, complemented
12#16861257—24no—
13#16861257—24no—
14C2×#14523249—21no—
15C2×#14523249C2212yes—
16C2×#14523248—22no—
17C2×#14523248C2211yescharacteristic
18C2×#14523248C2211yescharacteristic
19C2×Q8×D127—21no—
20C2×Q8×D127C2212yes—
21#16861229—24no—
22C2×#14523238C2211yescharacteristic
23C2×#14523236C2211yescharacteristic
24C2×#14523235C2211yescharacteristic
25C2×#14523233C2211yescharacteristic
26C22×C127⋊C8C2211yescharacteristic
27#16861188—21no—
28#16861185C2212yes—
29#16861183—21no—
30#16861181—22no—
31#16861180C2212yes—
32C2×C4⋊Dic127C2211yescharacteristic
33#16861134C2212yes—
34#16861133C2212yes—
35#16861133C2212yes—
36#16861131C2212yes—
37(C2×D127)⋊C8C2212yes—
38(C2×D127)⋊C8C2212yes—
39C2×#41×C127C2211yescharacteristic
40C22×Q8×C127C2211yescharacteristic
41#137×C127—24no—
42C2×Q16×C127C2212yes—
43C2×SD16×C127C2212yes—
44C2×M16×C127C2211yescharacteristic
45#14523249—28no—
46#14523249—24no—
47#14523249—24no—
48#14523248—216no—
49#14523248—22no—
50#14523248—22no—
51#14523248—22no—
52#14523248D414yescomplemented
53C2×Dic254—22no—
54C2×Dic254D411yescharacteristic
55C2×Dic254C2311yescharacteristic
56Q8×D127—28no—
57Q8×D127—24no—
58Q8×D127—24no—
59C2×D508—22no—
60C2×D508D411yescharacteristic
61C2×D508C2311yescharacteristic
62C2×C22⋊D127—22no—
63C2×C22⋊D127—21no—
64C2×C22⋊D127D412yes—
65C22×Dic127—21no—
66C2×C4×D127—22no—
67C2×C4×D127—22no—
68C2×C4×D127—21no—
69C2×C4×D127C2312yes—
70#14523238—22no—
71#14523236—22no—
72#14523235—22no—
73#14523233—22no—
74C2×C127⋊C8—22no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C4⋊Dic127—21no—
78C4⋊Dic127C2312yes—
79Dic127⋊C4—22no—
80C22⋊Dic127—22no—
81(C2×D127)⋊C4—22no—
82#41×C127—22no—
83#41×C127—22no—
84C2×Q8×C127—21no—
85C2×Q8×C127—22no—
86C2×Q8×C127D414yescomplemented
87C2×Q8×C127C2311yescharacteristic
88C2×Q8×C127C2312yes—
89C2×D4×C127—21no—
90C2×D4×C127C2311yescharacteristic
91C22×C4×C127—21no—
92C22×C4×C127—21no—
93C22×C4×C127C2311yescharacteristic
94Q16×C127—24no—
95SD16×C127—24no—
96M16×C127—22no—
97C2×C8×C127C2312yes—
98C22×D127—21no—
99C22×D127C2×D412yes—
100Dic254—28no—
101Dic254—22no—
102Dic254—21no—
103Dic254C2×D412yes—
104D508—28no—
105D508—22no—
106D508—21no—
107D508C2×D412yes—
108C22⋊D127—24no—
109C22⋊D127—24no—
110C22⋊D127—82no—
111C2×Dic127—21no—
112C2×Dic127—22no—
113C2×Dic127—81no—
114C2×Dic127C2×D412yes—
115C4×D127—28no—
116C4×D127—28no—
117C4×D127—24no—
118C4×D127—22no—
119C4×D127—22no—
120C127⋊C8—21no—
121C127⋊C8—21no—
122C23×C127—21no—
123C23×C127C2×D411yescharacteristic
124Q8×C127—24no—
125Q8×C127—22no—
126Q8×C127—21no—
127Q8×C127—21no—
128Q8×C127C2×D414yes—
129D4×C127—21no—
130D4×C127—21no—
131D4×C127—42no—
132C2×C4×C127—24no—
133C2×C4×C127—22no—
134C2×C4×C127—21no—
135C2×C4×C127—42no—
136C2×C4×C127C2×D411yescharacteristic
137C2×C4×C127C2×D412yes—
138C2×C4×C127C2×D412yes—
139C2×C4×C127C2411yescharacteristic
140C8×C127—22no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—41no—
144C22×C127—41no—
145C22×C127—41no—
146C22×C127#12012yes—
147C22×C127C22×D411yescharacteristic
148C2×D127—22no—
149C2×D127—22no—
150C2×D127—22no—
151C2×D127—22no—
152C2×D127—41no—
153C2×D127—41no—
154C4×C127—24no—
155C4×C127—41no—
156C4×C127#12012yes—
157C4×C127C22×D411yescharacteristic
158C4×C127C22×D411yescharacteristic
159Dic127—22no—
160Dic127—22no—
161Dic127—42no—
162C2×C127—22no—
163C2×C127—81no—
164C2×C127C2×#12011yescharacteristic
165C2×C127#57711yescharacteristic
166C2×C127#57811yescharacteristic
167D127—42no—
168D127—42no—
169#3013—1271nomaximal, Sylow
170C127#301311yescharacteristic, complemented, Sylow
171C2×2−1+4—1271no—
172C2×#137D12711yescharacteristic, complemented
173C2×#135—1271no—
174#537—1271no—
175#452—1272no—
176#451—1272no—
177#449—1272no—
178#448—1272no—
179#417—1271no—
180C2×#103—1271no—
181#408—1271no—
1822−1+4—2544no—
1832−1+4—2544no—
184C2×#41—1271no—
185C2×#41—1272no—
186C2×#41—1271no—
187C2×#41—2542no—
188C2×#41—2541no—
189C2×#41C2×D12711yescharacteristic
190C22×Q8—1272no—
191C22×Q8—2541no—
192C22×Q8C2×D12711yescharacteristic
193#137—24no—
194#135—2544no—
195C2×Q16—1271no—
196C2×Q16C2×D12712yes—
197C2×SD16—1271no—
198C2×SD16—1271no—
199C2×SD16C2×D12712yes—
200C2×D8—1271no—
201C2×M16C2×D12711yescharacteristic
202C22×C8—1271no—
203C22⋊Q8—1272no—
204C22⋊Q8—2541no—
205#121—1272no—
206#121—2541no—
207#121—2542no—
208C2×C4⋊C4—1271no—
209#103—1272no—
210#103—1272no—
211#103—1272no—
212#102—1272no—
213C22⋊C8—1272no—
214C22⋊C8—1272no—
215#41—22no—
216#41—22no—
217#41—1274no—
218#41—25416no—
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—2542no—
230C2×Q8—2544no—
231C2×Q8—2548no—
232C2×Q8—2544no—
233C2×Q8C22⋊D12714yescomplemented
234C2×Q8C22×D12711yescharacteristic
235C2×Q8C22×D12712yes—
236C2×D4—21no—
237C2×D4—1272no—
238C2×D4—1271no—
239C2×D4—1271no—
240C2×D4—2542no—
241C2×D4—2542no—
242C2×D4—2541no—
243C2×D4C22×D12711yescharacteristic
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—24no—
253Q16—2542no—
254SD16—24no—
255SD16—2542no—
256SD16—2542no—
257D8—2542no—
258M16—22no—
259C2×C8—1271no—
260C2×C8—1271no—
261C2×C8—2542no—
262C2×C8C22×D12712yes—
263C4⋊C4—1272no—
264C4⋊C4—2542no—
265C4⋊C4—2541no—
266C22⋊C4—2542no—
267C22⋊C4—2542no—
268C23—21no—
269C23—1272no—
270C23—2541no—
271C23D4×D12711yescharacteristic
272Q8—22no—
273Q8—24no—
274Q8—21no—
275Q8—21no—
276Q8—1272no—
277Q8—2548no—
278Q8—2541no—
279Q8—2542no—
280Q8C2×C22⋊D12714yes—
281D4—21no—
282D4—21no—
283D4—42no—
284D4—1272no—
285D4—10162no—
286D4—2548no—
287D4—2544no—
288D4—2544no—
289D4—2541no—
290D4—2542no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—1272no—
296C2×C4—10161no—
297C2×C4—2548no—
298C2×C4—2548no—
299C2×C4—2542no—
300C2×C4—2541no—
301C2×C4—2542no—
302C2×C4—2542no—
303C2×C4—2544no—
304C2×C4D4×D12712yes—
305C2×C4D4×D12711yescharacteristic
306C2×C4C2×C22⋊D12712yes—
307C2×C4C23×D12711yescharacteristic
308C8—22no—
309C8—2541no—
310C8—2541no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—5081no—
317C22—5081no—
318C22—2542no—
319C22—2542no—
320C22—2542no—
321C22—2542no—
322C22#1686117712yes—
323C22C2×D4×D12711yescharacteristic
324C4—24no—
325C4—41no—
326C4—5082no—
327C4—2542no—
328C4—2542no—
329C4#1686117712yes—
330C4C2×D4×D12711yescharacteristic
331C4C22×C22⋊D12711yescharacteristic
332C2—22no—
333C2—81no—
334C2—5082no—
335C2—5082no—
336C2C2×#1686117711yescharacteristic
337C2#2255068511yescharacteristic
338C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
hypercenterC2×#137D127
hypocenterC127#3013
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16861257C2semidirect
2. C2×#16861229C2semidirect
3. #22550454C2semidirect
4. #22550037C2semidirect
5. #22550030C2semidirect
6. #22550020C2semidirect
7. #22550013C2semidirect
8. #22549878C2semidirect
9. C2×#16861133C2semidirect
10. #22549844C2semidirect
11. C2×#137×C127C2semidirect
52. #14523248D4semidirect
86. C2×Q8×C127D4semidirect
170. C127#3013semidirect
172. C2×#137D127semidirect
233. C2×Q8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612572
C2×#168612292
#225504542
#225500372
#225500302
#225500202
#225500132
#225498782
C2×#168611332
#225498442
C2×#137×C1272
#3013127