Finite Group Databasebeta

← #30650532

Subgroups — #30650532

Order 16256.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#22550691C212yesmaximal, complemented
2#22550690C211yesmaximal, characteristic, complemented
3#22550254C211yesmaximal, characteristic, complemented
4#22550151C211yesmaximal, characteristic, complemented
5#22550149C212yesmaximal, complemented
6#22550148C211yesmaximal, characteristic, complemented
7#22550059C212yesmaximal, complemented
8#22550057C212yesmaximal, complemented
9#22549993C211yesmaximal, characteristic, complemented
10#22549991C211yesmaximal, characteristic, complemented
11#578×C127C211yesmaximal, characteristic, complemented
12#16861256C2212yes—
13#16861240—22no—
14#16861240C2212yes—
15#16861239—22no—
16#16861239C2212yes—
17#16861239C2212yes—
18#16861238—22no—
19#16861229—22no—
20#16861229C2211yescharacteristic
21#16861228—21no—
22C2×Dic508C2211yescharacteristic
23C2×#14523238—22no—
24C2×#14523238C2212yes—
25Q16×D127—21no—
26#16861216C2212yes—
27#16861215C2211yescharacteristic
28#16861207C2212yes—
29#16861145C2211yescharacteristic
30#16861144C2211yescharacteristic
31#16861136C2212yes—
32#16861136C2211yescharacteristic
33#16861135C2211yescharacteristic
34#16861128C2212yes—
35#16861127C2212yes—
36C8×Dic127C2211yescharacteristic
372−1+4×C127C2212yes—
38#137×C127—22no—
39#137×C127C2212yes—
40#135×C127—21no—
41#135×C127C2211yescharacteristic
42C2×Q16×C127—21no—
43C2×Q16×C127C2211yescharacteristic
44#131×C127C2211yescharacteristic
45#14523248—22no—
46#14523248C2311yescharacteristic
47#14523247—22no—
48#14523247—42no—
49C2×Dic254—22no—
50C2×Dic254C2312yes—
51Q8×D127—22no—
52Dic508—21no—
53Dic508—21no—
54#14523238—22no—
55#14523238—22no—
56#14523238—22no—
57#14523238—22no—
58#14523238—22no—
59#14523238D411yescharacteristic
60#14523236—22no—
61#14523236—22no—
62#14523236D411yescharacteristic
63#14523235—22no—
64#14523235D411yescharacteristic
65#14523233D411yescharacteristic
66#14523232—22no—
67#14523232C2311yescharacteristic
68#14523232C2312yes—
69#14523231—21no—
70C2×C127⋊C8—22no—
71C2×C127⋊C8C2311yescharacteristic
72C8×D127—21no—
73Dic127⋊C4—42no—
74C4×Dic127C2311yescharacteristic
75#41×C127—22no—
76#41×C127—22no—
77#41×C127—42no—
78#41×C127C2312yes—
79#41×C127C2311yescharacteristic
80C2×Q8×C127—22no—
81C2×Q8×C127—22no—
82C2×Q8×C127C2312yes—
83Q16×C127—22no—
84Q16×C127—21no—
85Q16×C127—21no—
86Q16×C127D411yescharacteristic
87SD16×C127—22no—
88SD16×C127D412yes—
89D8×C127D411yescharacteristic
90M16×C127—21no—
91M16×C127C2311yescharacteristic
92C2×C8×C127—21no—
93C2×C8×C127C2311yescharacteristic
94Dic254—22no—
95Dic254—22no—
96Dic254—42no—
97Dic254C2×D411yescharacteristic
98D508C2×D411yescharacteristic
99C22⋊D127—21no—
100C22⋊D127—42no—
101C2×Dic127—21no—
102C2×Dic127—22no—
103C2×Dic127—42no—
104C4×D127—21no—
105C4×D127—42no—
106C127⋊C8—21no—
107C127⋊C8—22no—
108C127⋊C8C2×D411yescharacteristic
109C127⋊C8C2×D411yescharacteristic
110Q8×C127—22no—
111Q8×C127—22no—
112Q8×C127—42no—
113Q8×C127C2×D411yescharacteristic
114Q8×C127C2×D412yes—
115D4×C127—21no—
116D4×C127—22no—
117D4×C127—42no—
118D4×C127C2×D411yescharacteristic
119D4×C127C2×D412yes—
120C2×C4×C127—21no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—42no—
124C2×C4×C127—42no—
125C2×C4×C127C2411yescharacteristic
126C8×C127—21no—
127C8×C127C2×D411yescharacteristic
128C8×C127C2×D411yescharacteristic
129C22×C127—21no—
130C22×C127—22no—
131C22×C127H(5,𝔽2)11yescharacteristic
132C2×D127—21no—
133C4×C127—21no—
134C4×C127—22no—
135C4×C127—42no—
136C4×C127C22×D411yescharacteristic
137C4×C127C22×D411yescharacteristic
138Dic127—22no—
139Dic127—21no—
140Dic127—42no—
141C2×C127—21no—
142C2×C127—41no—
143C2×C127—42no—
144C2×C127D4211yescharacteristic
145D127—41no—
146#3293—1271nomaximal, Sylow
147C127#329311yescharacteristic, complemented, Sylow
148#578—1272no—
149#578—1271no—
150#578D12711yescharacteristic, complemented
151#494—1271no—
152#473—1272no—
153#473—1271no—
154#473—1271no—
155#456—1272no—
156#456—1272no—
157#443—1271no—
158#443—1271no—
1592−1+4—1272no—
1602−1+4C2×D12712yes—
161#137—22no—
162#137—1272no—
163#137—1272no—
164#137—1272no—
165#137—2542no—
166#137—2542no—
167#137—2542no—
168#137C2×D12712yes—
169#135—21no—
170#135—1271no—
171#135—2542no—
172#135—2541no—
173#135C2×D12711yescharacteristic
174C2×Q16—21no—
175C2×Q16—1272no—
176C2×Q16—1271no—
177C2×Q16—2542no—
178C2×Q16—2541no—
179C2×Q16C2×D12711yescharacteristic
180#131—1272no—
181#131—1271no—
182#131C2×D12711yescharacteristic
183C4⋊Q8—1272no—
184#108—1271no—
185#108—1271no—
186#104—1271no—
187#104—1272no—
188#104—1271no—
189#101—1272no—
190#101—1272no—
191C4×C8—1271no—
192#41—22no—
193#41—22no—
194#41—42no—
195#41—1271no—
196#41—5082no—
197#41—2542no—
198#41—2542no—
199#41C22×D12711yescharacteristic
200#41C22×D12712yes—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—1272no—
204C2×Q8—2542no—
205C2×Q8—2542no—
206C2×Q8C22×D12712yes—
207Q16—22no—
208Q16—21no—
209Q16—21no—
210Q16—1271no—
211Q16—2542no—
212Q16—2542no—
213Q16—2542no—
214Q16—2542no—
215Q16—2542no—
216Q16—2541no—
217Q16—2541no—
218Q16C22⋊D12711yescharacteristic
219SD16—22no—
220SD16—1271no—
221SD16—1271no—
222SD16—2542no—
223SD16—2542no—
224SD16—2542no—
225SD16C22⋊D12712yes—
226D8—1271no—
227D8C22⋊D12711yescharacteristic
228M16—21no—
229M16—1272no—
230M16—1271no—
231M16—2542no—
232M16—2541no—
233M16C22×D12711yescharacteristic
234C2×C8—21no—
235C2×C8—1271no—
236C2×C8—2542no—
237C2×C8—2541no—
238C2×C8C22×D12711yescharacteristic
239C4⋊C4—5082no—
240C42—1271no—
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—1271no—
245Q8—5082no—
246Q8—2542no—
247Q8—2542no—
248Q8D4×D12711yescharacteristic
249Q8C2×C22⋊D12712yes—
250D4—21no—
251D4—22no—
252D4—42no—
253D4—1271no—
254D4—5082no—
255D4—2541no—
256D4D4×D12711yescharacteristic
257D4C2×C22⋊D12712yes—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—5082no—
264C2×C4—5082no—
265C2×C4—2542no—
266C2×C4—2541no—
267C2×C4—2541no—
268C2×C4C23×D12711yescharacteristic
269C8—21no—
270C8—1271no—
271C8—1271no—
272C8—2542no—
273C8—2541no—
274C8C2×C22⋊D12711yescharacteristic
275C8C2×C22⋊D12711yescharacteristic
276C22—21no—
277C22—22no—
278C22—2541no—
279C22#1686125411yescharacteristic
280C4—21no—
281C4—22no—
282C4—42no—
283C4—5082no—
284C4—2542no—
285C4—2541no—
286C4C2×D4×D12711yescharacteristic
287C4C22×C22⋊D12711yescharacteristic
288C2—21no—
289C2—41no—
290C2—42no—
291C2—5081no—
292C2D4×C22⋊D12711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#578×C127C2
socleC2×C127D42
hypercenter#578D127
hypocenterC127#3293
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #22550691C2semidirect
2. #22550690C2semidirect
3. #22550254C2semidirect
4. #22550151C2semidirect
5. #22550149C2semidirect
6. #22550148C2semidirect
7. #22550059C2semidirect
8. #22550057C2semidirect
9. #22549993C2semidirect
10. #22549991C2semidirect
11. #578×C127C2semidirect
147. C127#3293semidirect
150. #578D127semidirect

Maximal subgroups

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

subgroupindex
#225506912
#225506902
#225502542
#225501512
#225501492
#225501482
#225500592
#225500572
#225499932
#225499912
#578×C1272
#3293127