Finite Group Databasebeta

← C32×#2186 #11769235

Subgroups — C32×#2186 #11769235

Order 1152.

690 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 204 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×C32×#135C211yesmaximal, characteristic, complemented
2C2×C32×SD32C211yesmaximal, characteristic, complemented
3C2×C32×D16C211yesmaximal, characteristic, complemented
4C32×#466C211yesmaximal, characteristic, complemented
5C32×#358C211yesmaximal, characteristic, complemented
6C32×#357C211yesmaximal, characteristic, complemented
7C32×C22⋊C16C211yesmaximal, characteristic, complemented
8C3×#2186C314yesmaximal, direct product factor
9C2×C32×#41—21no—
10C32×#135—22no—
11C32×#135—22no—
12C2×C32×Q16C2211yescharacteristic
13C2×C32×SD16—21no—
14C2×C32×D8C2211yescharacteristic
15C2×C32×D8C2211yescharacteristic
16C22×C32×C8C2211yescharacteristic
17C32×#121—21no—
18C32×SD32—22no—
19C32×D16—22no—
20C2×C32×C16C2211yescharacteristic
21C2×C32×C16C2211yescharacteristic
22C32×#107C2211yescharacteristic
23C32×#102—21no—
24C2×C3×#135C2×C314yescomplemented
25C2×C3×SD32C2×C314yescomplemented
26C2×C3×D16C2×C314yescomplemented
27C3×#466C2×C314yescomplemented
28C3×#358C2×C314yescomplemented
29C3×#357C2×C314yescomplemented
30C3×C22⋊C16C2×C314yescomplemented
31C32×#41—24no—
32C32×#41—42no—
33C2×C32×Q8—21no—
34C2×C32×D4—21no—
35C2×C32×D4—21no—
36C2×C32×D4—41no—
37C2×C32×D4—41no—
38C22×C32×C4—41no—
39C22×C32×C4D411yescharacteristic
40C32×Q16—21no—
41C32×Q16D412yescomplemented
42C32×SD16—24no—
43C32×D8—21no—
44C32×D8—21no—
45C32×D8—21no—
46C32×D8D412yescomplemented
47C2×C32×C8—21no—
48C2×C32×C8—21no—
49C2×C32×C8D411yescharacteristic
50C2×C32×C8C2311yescharacteristic
51C32×C4⋊C4—21no—
52C32×C22⋊C4—41no—
53C32×C16—21no—
54C32×C16—21no—
55#2186C3211yescharacteristic, direct product factor, Sylow
56C2×C3×#41—24no—
57C3×#135—28no—
58C3×#135—28no—
59C2×C3×Q16C22×C314yes—
60C2×C3×SD16—24no—
61C2×C3×D8C22×C314yes—
62C2×C3×D8C22×C314yes—
63C22×C3×C8C22×C314yes—
64C3×#121—24no—
65C3×SD32—28no—
66C3×D16—28no—
67C2×C3×C16C22×C314yes—
68C2×C3×C16C22×C314yes—
69C3×#107C22×C314yes—
70C3×#102—24no—
71C23×C32—41no—
72C23×C32—41no—
73C23×C32D811yescharacteristic
74C32×Q8—22no—
75C32×Q8—41no—
76C32×D4—22no—
77C32×D4—44no—
78C32×D4—41no—
79C32×D4—41no—
80C32×D4—41no—
81C32×D4—82no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—44no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C32×C4D811yescharacteristic
88C2×C32×C4C2×D411yescharacteristic
89C32×C8—21no—
90C32×C8C2×D411yescharacteristic
91C32×C8C2×D411yescharacteristic
92C2×#135C2×C3211yescharacteristic, complemented
93C2×SD32C2×C3211yescharacteristic, complemented
94C2×D16C2×C3211yescharacteristic, complemented
95#466C2×C3211yescharacteristic, complemented
96#358C2×C3211yescharacteristic, complemented
97#357C2×C3211yescharacteristic, complemented
98C22⋊C16C2×C3211yescharacteristic, complemented
99C3×#41—216no—
100C3×#41—48no—
101C2×C3×Q8—24no—
102C2×C3×D4—24no—
103C2×C3×D4—24no—
104C2×C3×D4—44no—
105C2×C3×D4—44no—
106C22×C3×C4—44no—
107C22×C3×C4C3×D414yes—
108C3×Q16—24no—
109C3×Q16C3×D418yescomplemented
110C3×SD16—216no—
111C3×D8—24no—
112C3×D8—24no—
113C3×D8—24no—
114C3×D8C3×D418yescomplemented
115C2×C3×C8—24no—
116C2×C3×C8—24no—
117C2×C3×C8C3×D414yes—
118C2×C3×C8C23×C314yes—
119C3×C4⋊C4—24no—
120C3×C22⋊C4—44no—
121C3×C16—24no—
122C3×C16—24no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—42no—
127C22×C32—81no—
128C22×C32—81no—
129C22×C32—81no—
130C22×C32—81no—
131C22×C32—81no—
132C22×C32C2×D811yescharacteristic
133C32×C4—21no—
134C32×C4—42no—
135C32×C4—81no—
136C32×C4#12011yescharacteristic
137C32×C4AGL(1,ℤ/8)11yescharacteristic
138C2×#41—21no—
139#135—22no—
140#135—22no—
141C2×Q16C22×C3211yescharacteristic
142C2×SD16—21no—
143C2×D8C22×C3211yescharacteristic
144C2×D8C22×C3211yescharacteristic
145C22×C8C22×C3211yescharacteristic
146#121—21no—
147SD32—22no—
148D16—22no—
149C2×C16C22×C3211yescharacteristic
150C2×C16C22×C3211yescharacteristic
151#107C22×C3211yescharacteristic
152#102—21no—
153C23×C3—44no—
154C23×C3—44no—
155C23×C3C3×D814yes—
156C3×Q8—28no—
157C3×Q8—44no—
158C3×D4—28no—
159C3×D4—416no—
160C3×D4—44no—
161C3×D4—44no—
162C3×D4—44no—
163C3×D4—88no—
164C2×C3×C4—24no—
165C2×C3×C4—24no—
166C2×C3×C4—416no—
167C2×C3×C4—44no—
168C2×C3×C4—44no—
169C2×C3×C4C3×D814yes—
170C2×C3×C4C2×C3×D414yes—
171C3×C8—24no—
172C3×C8C2×C3×D414yes—
173C3×C8C2×C3×D414yes—
174C2×C32—41no—
175C2×C32—82no—
176C2×C32—161no—
177C2×C32#44711yescharacteristic
178C2×C32#50811yescharacteristic
179C2×C32#50911yescharacteristic
180#41—24no—
181#41—42no—
182C2×Q8—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—41no—
186C2×D4—41no—
187C22×C4—41no—
188C22×C4C32×D411yescharacteristic
189Q16—21no—
190Q16C32×D412yescomplemented
191SD16—24no—
192D8—21no—
193D8—21no—
194D8—21no—
195D8C32×D412yescomplemented
196C2×C8—21no—
197C2×C8—21no—
198C2×C8C32×D411yescharacteristic
199C2×C8C23×C3211yescharacteristic
200C4⋊C4—21no—
201C22⋊C4—41no—
202C16—21no—
203C16—21no—
204C22×C3—24no—
205C22×C3—24no—
206C22×C3—24no—
207C22×C3—48no—
208C22×C3—84no—
209C22×C3—84no—
210C22×C3—84no—
211C22×C3—84no—
212C22×C3—84no—
213C22×C3C2×C3×D814yes—
214C3×C4—24no—
215C3×C4—48no—
216C3×C4—84no—
217C3×C4C3×#12014yes—
218C3×C4C3×AGL(1,ℤ/8)14yes—
219C32#218611yescharacteristic, direct product factor, Sylow
220C23—41no—
221C23—41no—
222C23C32×D811yescharacteristic
223Q8—22no—
224Q8—41no—
225D4—22no—
226D4—44no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—82no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—44no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4C32×D811yescharacteristic
237C2×C4C2×C32×D411yescharacteristic
238C8—21no—
239C8C2×C32×D411yescharacteristic
240C8C2×C32×D411yescharacteristic
241C2×C3—44no—
242C2×C3—88no—
243C2×C3—164no—
244C2×C3C3×#44714yes—
245C2×C3C3×#50814yes—
246C2×C3C3×#50914yes—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—42no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22C2×C32×D811yescharacteristic
257C4—21no—
258C4—42no—
259C4—81no—
260C4C32×#12011yescharacteristic
261C4C32×AGL(1,ℤ/8)11yescharacteristic
262C3C3×#218614yesdirect product factor
263C2—41no—
264C2—82no—
265C2—161no—
266C2C32×#44711yescharacteristic
267C2C32×#50811yescharacteristic
268C2C32×#50911yescharacteristic
subgroupG/H
center, socleC22×C32C2×D8
commutator, Frattini, cosocleC2×C8C23×C32
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C32 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C32×#135C2semidirect
2. C2×C32×SD32C2semidirect
3. C2×C32×D16C2semidirect
4. C32×#466C2semidirect
5. C32×#358C2semidirect
6. C32×#357C2semidirect
7. C32×C22⋊C16C2semidirect
8. C3×#2186C3direct (N × H)
24. C2×C3×#135C2×C3semidirect
25. C2×C3×SD32C2×C3semidirect
26. C2×C3×D16C2×C3semidirect
27. C3×#466C2×C3semidirect
28. C3×#358C2×C3semidirect
29. C3×#357C2×C3semidirect
30. C3×C22⋊C16C2×C3semidirect
41. C32×Q16D4semidirect
46. C32×D8D4semidirect
55. #2186C32direct (N × H)
92. C2×#135C2×C32semidirect
93. C2×SD32C2×C32semidirect
94. C2×D16C2×C32semidirect
95. #466C2×C32semidirect
96. #358C2×C32semidirect
97. #357C2×C32semidirect
98. C22⋊C16C2×C32semidirect
109. C3×Q16C3×D4semidirect
114. C3×D8C3×D4semidirect
190. Q16C32×D4semidirect
195. D8C32×D4semidirect
219. C32#2186direct (N × H)
262. C3C3×#2186direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1352
C2×C32×SD322
C2×C32×D162
C32×#4662
C32×#3582
C32×#3572
C32×C22⋊C162
C3×#21863