Finite Group Databasebeta

← #30639625

Subgroups — #30639625

Order 16256.

284 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16861229C211yesmaximal, characteristic
2#22549912C211yesmaximal, characteristic, complemented
3#22549894C211yesmaximal, characteristic, complemented
4#22549893C211yesmaximal, characteristic, complemented
5#22549849C211yesmaximal, characteristic, complemented
6#22549832C211yesmaximal, characteristic, complemented
7#410×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861229—21no—
10#16861229—21no—
11#16861229—21no—
12#16861229—21no—
13C2×#14523238C411yescharacteristic, complemented
14C2×#14523236C411yescharacteristic, complemented
15C2×#14523235C411yescharacteristic, complemented
16C2×#14523233C411yescharacteristic, complemented
17C2×#14523232C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19#16861168C2211yescharacteristic
20#16861142—21no—
21#16861140—21no—
22#16861137—22no—
23#16861136—22no—
24#16861123—22no—
25#16861120—21no—
26C4×C127⋊C8—21no—
27C2×#41×C127C2211yescharacteristic
28#117×C127C2211yescharacteristic
29#117×C127C2211yescharacteristic
30#99×C127—22no—
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35C2×Dic254C2×C411yescharacteristic
36C2×D508C2×C411yescharacteristic
37C2×C22⋊D127—21no—
38C2×C4×D127—21no—
39#14523238—21no—
40#14523238—21no—
41#14523236—21no—
42#14523236—21no—
43#14523235—21no—
44#14523235—21no—
45#14523233—21no—
46#14523233—21no—
47#14523232—21no—
48#14523232D412yes—
49C2×C127⋊C8—21no—
50C2×C127⋊C8C2×C411yescharacteristic
51C2×C127⋊C8C2×C411yescharacteristic
52C2×C127⋊C8D411yescharacteristic
53C2×C127⋊C8D411yescharacteristic
54C2×C127⋊C8D411yescharacteristic
55C2×C127⋊C8D411yescharacteristic
56C4⋊Dic127—21no—
57C22⋊Dic127—21no—
58C4×Dic127—21no—
59#41×C127—21no—
60#41×C127—21no—
61#41×C127—21no—
62#41×C127—21no—
63C2×Q8×C127C2×C411yescharacteristic
64C2×D4×C127—21no—
65C2×D4×C127C2×C411yescharacteristic
66C22×C4×C127—21no—
67C22×C4×C127C2311yescharacteristic
68C4⋊C4×C127—21no—
69C4⋊C4×C127—21no—
70C22⋊C4×C127—21no—
71C22⋊C4×C127D412yes—
72C42×C127—21no—
73C42×C127—21no—
74C22×D127—21no—
75Dic254—21no—
76Dic254—21no—
77D508—21no—
78D508—21no—
79C22⋊D127—41no—
80C22⋊D127—41no—
81C2×Dic127—21no—
82C2×Dic127—41no—
83C4×D127—41no—
84C4×D127—41no—
85C127⋊C8—22no—
86C127⋊C8C22⋊C411yescharacteristic
87C127⋊C8C22⋊C411yescharacteristic
88C127⋊C8C22⋊C411yescharacteristic
89C127⋊C8C22⋊C411yescharacteristic
90C23×C127—21no—
91C23×C127C2×D411yescharacteristic
92Q8×C127—21no—
93Q8×C127—21no—
94D4×C127—21no—
95D4×C127—21no—
96D4×C127—41no—
97D4×C127—41no—
98C2×C4×C127—21no—
99C2×C4×C127—22no—
100C2×C4×C127—41no—
101C2×C4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127C22×C411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127#4111yescharacteristic
108C2×C4×C127#4111yescharacteristic
109C22×C127—41no—
110C22×C127—41no—
111C22×C127—81no—
112C22×C127C4×D411yescharacteristic
113C22×C127#12111yescharacteristic
114C22×C127#12711yescharacteristic
115C2×D127—41no—
116C2×D127—81no—
117C4×C127—42no—
118C4×C127—41no—
119C4×C127—81no—
120C4×C127C2×C22⋊C411yescharacteristic
121C4×C127C4×D411yescharacteristic
122C4×C127#12111yescharacteristic
123C4×C127#12411yescharacteristic
124Dic127—41no—
125Dic127—81no—
126C2×C127—21no—
127C2×C127—21no—
128C2×C127—21no—
129C2×C127—81no—
130C2×C127#39011yescharacteristic
131D127—81no—
132#1976—1271nomaximal, Sylow
133C127#197611yescharacteristic, complemented, Sylow
134C2×#135—1271no—
135#427—1271no—
136#421—1271no—
137#421—1271no—
138#410—1271no—
139#410D12711yescharacteristic, complemented
140#405—1271no—
141C2×#41—1271no—
142C2×#41C2×D12711yescharacteristic
143#135—2541no—
144#135—2541no—
145#135—2541no—
146#135—2541no—
147C2×Q16—1271no—
148C2×SD16—1271no—
149C2×SD16—1271no—
150C2×D8—1271no—
151C2×M16—1271no—
152C22×C8—1271no—
153#117—1271no—
154#117C2×D12711yescharacteristic
155#117C2×D12711yescharacteristic
156#107—2541no—
157#106—2541no—
158#104—2542no—
159#104—2542no—
160#99—22no—
161#99—2542no—
162#97—2541no—
163C4×C8—2541no—
164#41—21no—
165#41—21no—
166#41—21no—
167#41—21no—
168#41—2541no—
169#41—2541no—
170#41—2541no—
171#41—2541no—
172C2×Q8—1271no—
173C2×Q8C4×D12711yescharacteristic
174C2×D4—21no—
175C2×D4—1271no—
176C2×D4—2541no—
177C2×D4C4×D12711yescharacteristic
178C22×C4—21no—
179C22×C4—2541no—
180C22×C4C22×D12711yescharacteristic
181Q16—2541no—
182Q16—2541no—
183SD16—2541no—
184SD16—2541no—
185SD16—2541no—
186SD16—2541no—
187D8—2541no—
188D8—2541no—
189M16—1272no—
190M16—2541no—
191C2×C8—1271no—
192C2×C8—1271no—
193C2×C8—1271no—
194C2×C8—1271no—
195C2×C8—1271no—
196C2×C8—1271no—
197C2×C8—2541no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—2541no—
201C22⋊C4—21no—
202C22⋊C4—2541no—
203C22⋊C4C22⋊D12712yes—
204C42—21no—
205C42—21no—
206C42—2541no—
207C23—21no—
208C23—2541no—
209C23C2×C22⋊D12711yescharacteristic
210Q8—21no—
211Q8—21no—
212Q8—2541no—
213Q8—2541no—
214D4—21no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—5081no—
219D4—5081no—
220D4—2541no—
221D4—2541no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—5081no—
228C2×C4—5081no—
229C2×C4—5081no—
230C2×C4—2541no—
231C2×C4C2×C4×D12711yescharacteristic
232C2×C4D4×D12711yescharacteristic
233C2×C4D4×D12711yescharacteristic
234C2×C4D4×D12711yescharacteristic
235C2×C4#1452324711yescharacteristic
236C2×C4#1452324811yescharacteristic
237C8—1271no—
238C8—1271no—
239C8—1271no—
240C8—1271no—
241C8—2542no—
242C22—41no—
243C22—41no—
244C22—81no—
245C22—10161no—
246C22—5081no—
247C22C4×C22⋊D12711yescharacteristic
248C22#1686118111yescharacteristic
249C22#1686120511yescharacteristic
250C4—42no—
251C4—41no—
252C4—81no—
253C4—10161no—
254C4—5081no—
255C4C22⋊C4×D12711yescharacteristic
256C4(C22⋊D127)⋊C411yescharacteristic
257C4#1686118211yescharacteristic
258C4#1686119511yescharacteristic
259C2—21no—
260C2—21no—
261C2—21no—
262C2—81no—
263C2—10161no—
264C2#2254977611yescharacteristic
subgroupG/H
centerC4#16861182
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127#390
prime radicalC2×#16861229C2
cosocleC22×C4×C127C23
hypercenter#410D127
hypocenterC127#1976
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22549912C2semidirect
3. #22549894C2semidirect
4. #22549893C2semidirect
5. #22549849C2semidirect
6. #22549832C2semidirect
7. #410×C127C2semidirect
13. C2×#14523238C4semidirect
14. C2×#14523236C4semidirect
15. C2×#14523235C4semidirect
16. C2×#14523233C4semidirect
133. C127#1976semidirect
139. #410D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
#225499122
#225498942
#225498932
#225498492
#225498322
#410×C1272
#1976127