Finite Group Databasebeta

← #13136463

Subgroups — #13136463

Order 1728.

540 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 147 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×#88403C211yesmaximal, characteristic
2C2×Q8×D27C211yesmaximal, characteristic, complemented
3C27×C4⋊Q8C211yesmaximal, characteristic, complemented
4#11720196C211yesmaximal, characteristic, complemented
5#11720193C211yesmaximal, characteristic, complemented
6#11720145C214yesmaximal, complemented
7#11720143C214yesmaximal, complemented
8#11720124C211yesmaximal, characteristic, complemented
9#11720121C211yesmaximal, characteristic, complemented
10#10589826—31nomaximal
11C2×Q8×C27C2211yescharacteristic
12C2×Q8×C27C2211yescharacteristic
13#88403—24no—
14Q8×D27—24no—
15C2×D108C2211yescharacteristic
16C2×D108C2212yes—
17C2×C4×D27C2212yes—
18C2×C4×D27C2212yes—
19C2×C4×D27C2211yescharacteristic
20C2×Dic54C2211yescharacteristic
21C2×Dic54C2212yes—
22C4⋊C4×C27C2214yes—
23C42×C27C2211yescharacteristic
24(C2×D27)⋊C4C2214yes—
25(C2×D27)⋊C4C2212yes—
26(C2×D27)⋊C4C2214yes—
27Dic27⋊C4C2214yes—
28Dic27⋊C4C2212yes—
29C4×Dic27C2211yescharacteristic
30C2×#3701—31no—
31C2×Q8×D9—31no—
32C9×C4⋊Q8D311yescharacteristic
33#63509—31no—
34#63506—31no—
35#63458—34no—
36#63456—34no—
37#63437—31no—
38#63434—31no—
39C22×D27C2312yes—
40C22×D27C2311yescharacteristic
41Q8×C27—22no—
42Q8×C27—22no—
43C2×C4×C27C2312yes—
44C2×C4×C27C2312yes—
45C2×C4×C27C2312yes—
46C2×C4×C27C2311yescharacteristic
47C2×Dic27C2312yes—
48C2×Dic27C2312yes—
49C2×Dic27C2311yescharacteristic
50D108—24no—
51D108—22no—
52C4×D27—24no—
53C4×D27—24no—
54C4×D27D414yes—
55Dic54—22no—
56Dic54—24no—
57#5724—91no—
58C2×Q8×C9C2×D311yescharacteristic
59C2×Q8×C9C2×D311yescharacteristic
60#3701—64no—
61Q8×D9—64no—
62C2×D36—31no—
63C2×D36—32no—
64C2×C4×D9—31no—
65C2×C4×D9—32no—
66C2×C4×D9—32no—
67C2×Dic18—31no—
68C2×Dic18—32no—
69C9×C4⋊C4C2×D314yes—
70C42×C9C2×D311yescharacteristic
71(C2×D9)⋊C4—34no—
72(C2×D9)⋊C4—32no—
73(C2×D9)⋊C4—34no—
74Dic9⋊C4—34no—
75Dic9⋊C4—32no—
76C4×Dic9—31no—
77C22×C27C2411yescharacteristic
78C2×D27—22no—
79C2×D27—22no—
80C2×D27—24no—
81C2×D27C2×D412yes—
82C4×C27—22no—
83C4×C27—22no—
84C4×C27—22no—
85C4×C27C2×D412yes—
86Dic27—22no—
87Dic27—22no—
88Dic27C2×D412yes—
89C2×#239—91no—
90C2×D3×Q8—91no—
91C3×C4⋊Q8D911yescharacteristic
92#947—91no—
93#944—91no—
94#896—94no—
95#894—94no—
96#875—91no—
97#872—91no—
98C22×D9—31no—
99C22×D9—32no—
100Q8×C9—22no—
101Q8×C9—22no—
102C2×C4×C9C22×D312yes—
103C2×C4×C9C22×D312yes—
104C2×C4×C9C22×D312yes—
105C2×C4×C9C22×D311yescharacteristic
106C2×Dic9—31no—
107C2×Dic9—32no—
108C2×Dic9—32no—
109D36—64no—
110D36—62no—
111C4×D9—34no—
112C4×D9—64no—
113C4×D9—64no—
114Dic18—64no—
115Dic18—62no—
116#536—271noSylow
117C2×C27C22×D411yescharacteristic
118C2×C272−1+412yes—
119D27—22no—
120D27—42no—
121C2×C3×Q8C2×D911yescharacteristic
122C2×C3×Q8C2×D911yescharacteristic
123#239—184no—
124D3×Q8—184no—
125C2×D12—91no—
126C2×D12—92no—
127C2×C4×D3—91no—
128C2×C4×D3—92no—
129C2×C4×D3—92no—
130C2×Dic6—91no—
131C2×Dic6—92no—
132C3×C4⋊C4C2×D914yes—
133C3×C42C2×D911yescharacteristic
134(C2×D3)⋊C4—92no—
135(C2×D3)⋊C4—94no—
136(C2×D3)⋊C4—94no—
137Dic3⋊C4—92no—
138Dic3⋊C4—94no—
139C4×Dic3—91no—
140C22×C9C23×D311yescharacteristic
141C2×D9—32no—
142C2×D9—62no—
143C2×D9—62no—
144C2×D9—64no—
145C4×C9—22no—
146C4×C9—22no—
147C4×C9—22no—
148C4×C9D3×D412yes—
149Dic9—32no—
150Dic9—62no—
151Dic9—62no—
152C2×#41—271no—
153C22×Q8—271no—
154C4⋊Q8—271no—
155C4⋊Q8D2711yescharacteristic, complemented
156#124—271no—
157#124—271no—
158#123—274no—
159C22⋊Q8—274no—
160#117—271no—
161C27#53611yescharacteristic, complemented, Sylow
162C22×D3—91no—
163C22×D3—92no—
164C3×Q8—22no—
165C3×Q8—22no—
166C2×C3×C4C22×D912yes—
167C2×C3×C4C22×D912yes—
168C2×C3×C4C22×D912yes—
169C2×C3×C4C22×D911yescharacteristic
170C2×Dic3—91no—
171C2×Dic3—92no—
172C2×Dic3—92no—
173D12—184no—
174D12—182no—
175C4×D3—94no—
176C4×D3—184no—
177C4×D3—184no—
178Dic6—184no—
179Dic6—182no—
180C2×C9C2×D3×D411yescharacteristic
181C2×C9#100712yes—
182D9—62no—
183D9—122no—
184#41—544no—
185C2×Q8—272no—
186C2×Q8—271no—
187C2×Q8—544no—
188C2×Q8C2×D2711yescharacteristic
189C2×Q8C2×D2711yescharacteristic
190C2×D4—272no—
191C2×D4—271no—
192C22×C4—271no—
193C22×C4—272no—
194C22×C4—272no—
195C4⋊C4—272no—
196C4⋊C4—274no—
197C4⋊C4C2×D2714yes—
198C22⋊C4—272no—
199C22⋊C4—274no—
200C22⋊C4—274no—
201C42—271no—
202C42C2×D2711yescharacteristic
203C22×C3C23×D911yescharacteristic
204C2×D3—92no—
205C2×D3—182no—
206C2×D3—182no—
207C2×D3—184no—
208C3×C4—22no—
209C3×C4—22no—
210C3×C4—22no—
211C3×C4D4×D912yes—
212Dic3—92no—
213Dic3—182no—
214Dic3—182no—
215C9#572411yescharacteristic
216C23—271no—
217C23—272no—
218Q8—22no—
219Q8—22no—
220Q8—544no—
221Q8—542no—
222D4—544no—
223D4—542no—
224C2×C4—271no—
225C2×C4—274no—
226C2×C4—272no—
227C2×C4—272no—
228C2×C4—544no—
229C2×C4—544no—
230C2×C4C22×D2712yes—
231C2×C4C22×D2712yes—
232C2×C4C22×D2712yes—
233C2×C4C22×D2711yescharacteristic
234C2×C3C2×D4×D911yescharacteristic
235C2×C3#6371312yes—
236D3—182no—
237D3—362no—
238C22—272no—
239C22—542no—
240C22—542no—
241C22—544no—
242C22C23×D2711yescharacteristic
243C4—22no—
244C4—22no—
245C4—22no—
246C4—272no—
247C4—542no—
248C4—542no—
249C4D4×D2712yes—
250C3#1058982611yescharacteristic
251C2—542no—
252C2—1082no—
253C2C2×D4×D2711yescharacteristic
254C2#1172067312yes—
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×C4⋊Q8C2
socleC22×C3C23×D9
prime radicalC2×#88403C2
hypercenterC4⋊Q8D27
hypocenterC27#536
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×Q8×D27C2semidirect
3. C27×C4⋊Q8C2semidirect
4. #11720196C2semidirect
5. #11720193C2semidirect
6. #11720145C2semidirect
7. #11720143C2semidirect
8. #11720124C2semidirect
9. #11720121C2semidirect
155. C4⋊Q8D27semidirect
161. C27#536semidirect

Maximal subgroups

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

subgroupindex
C2×#884032
C2×Q8×D272
C27×C4⋊Q82
#117201962
#117201932
#117201452
#117201432
#117201242
#117201212
#105898263