Finite Group Databasebeta

← #13136465

Subgroups — #13136465

Order 1728.

584 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 155 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×#88403C212yesmaximal, complemented
2C27×C4⋊Q8C211yesmaximal, characteristic, complemented
3#11720195C212yesmaximal, complemented
4#11720144C212yesmaximal, complemented
5#11720143C214yesmaximal, complemented
6C4⋊C4×D27C212yesmaximal, complemented
7C4×D108C211yesmaximal, characteristic, complemented
8C4×D108C211yesmaximal, characteristic, complemented
9#10589828—31nomaximal
10C2×Q8×C27C2212yes—
11#88403—28no—
12C2×D108C2211yescharacteristic
13C2×D108C2214yes—
14C2×D108C2211yescharacteristic
15C2×C4×D27C2212yes—
16C2×C4×D27C2212yes—
17C2×C4×D27C2214yes—
18C4⋊C4×C27C2212yes—
19C4⋊C4×C27C2212yes—
20C42×C27C2211yescharacteristic
21(C2×D27)⋊C4C2214yes—
22(C2×D27)⋊C4C2212yes—
23(C2×D27)⋊C4C2212yes—
24C4⋊Dic27C2211yescharacteristic
25C4⋊Dic27C2211yescharacteristic
26Dic27⋊C4C2214yes—
27C2×#3701—32no—
28C9×C4⋊Q8D311yescharacteristic
29#63508—32no—
30#63457—32no—
31#63456—34no—
32C4⋊C4×D9—32no—
33C4×D36—31no—
34C4×D36—31no—
35C22×D27C2312yes—
36C22×D27C2312yes—
37Q8×C27—24no—
38C2×C4×C27C2314yes—
39C2×C4×C27C2311yescharacteristic
40C2×C4×C27C2311yescharacteristic
41C2×C4×C27C2311yescharacteristic
42C2×Dic27C2312yes—
43C2×Dic27C2312yes—
44D108—28no—
45D108—22no—
46D108D414yescomplemented
47C4×D27—28no—
48C4×D27—24no—
49C4×D27—24no—
50#5726—91no—
51C2×Q8×C9C2×D312yes—
52#3701—68no—
53C2×D36—31no—
54C2×D36—31no—
55C2×D36—34no—
56C2×C4×D9—32no—
57C2×C4×D9—32no—
58C2×C4×D9—34no—
59C9×C4⋊C4C2×D312yes—
60C9×C4⋊C4C2×D312yes—
61C42×C9C2×D311yescharacteristic
62(C2×D9)⋊C4—34no—
63(C2×D9)⋊C4—32no—
64(C2×D9)⋊C4—32no—
65C4⋊Dic9—31no—
66C4⋊Dic9—31no—
67Dic9⋊C4—34no—
68C22×C27C2411yescharacteristic
69C2×D27—22no—
70C2×D27—22no—
71C2×D27—22no—
72C2×D27—22no—
73C2×D27—22no—
74C2×D27C2×D414yes—
75C4×C27—24no—
76C4×C27—21no—
77C4×C27C2×D412yes—
78C4×C27#4112yescomplemented
79Dic27—22no—
80Dic27—22no—
81C2×#239—92no—
82C3×C4⋊Q8D911yescharacteristic
83#946—92no—
84#895—92no—
85#894—94no—
86D3×C4⋊C4—92no—
87C4×D12—91no—
88C4×D12—91no—
89C22×D9—32no—
90C22×D9—32no—
91Q8×C9—24no—
92C2×C4×C9C22×D314yes—
93C2×C4×C9C22×D311yescharacteristic
94C2×C4×C9C22×D311yescharacteristic
95C2×C4×C9C22×D311yescharacteristic
96C2×Dic9—32no—
97C2×Dic9—32no—
98D36—34no—
99D36—68no—
100D36—62no—
101C4×D9—68no—
102C4×D9—64no—
103C4×D9—64no—
104#547—271noSylow
105C2×C27C22×D411yescharacteristic
106C2×C27C2×#4111yescharacteristic
107C2×C272−1+411yescharacteristic
108D27—24no—
109D27—42no—
110C2×C3×Q8C2×D912yes—
111#239—188no—
112C2×D12—91no—
113C2×D12—91no—
114C2×D12—94no—
115C2×C4×D3—92no—
116C2×C4×D3—92no—
117C2×C4×D3—94no—
118C3×C4⋊C4C2×D912yes—
119C3×C4⋊C4C2×D912yes—
120C3×C42C2×D911yescharacteristic
121(C2×D3)⋊C4—92no—
122(C2×D3)⋊C4—92no—
123(C2×D3)⋊C4—94no—
124C4⋊Dic3—91no—
125C4⋊Dic3—91no—
126Dic3⋊C4—94no—
127C22×C9C23×D311yescharacteristic
128C2×D9—34no—
129C2×D9—62no—
130C2×D9—62no—
131C2×D9—62no—
132C2×D9—62no—
133C2×D9—62no—
134C4×C9—24no—
135C4×C9—21no—
136C4×C9D3×D412yes—
137C4×C9#23912yes—
138Dic9—62no—
139Dic9—62no—
140C2×#41—272no—
141C4⋊Q8D2711yescharacteristic, complemented
142#123—274no—
143C22⋊Q8—272no—
144#121—272no—
145C4×D4—271no—
146C4×D4—271no—
147C2×C4⋊C4—272no—
148C27#54711yescharacteristic, complemented, Sylow
149C22×D3—92no—
150C22×D3—92no—
151C3×Q8—24no—
152C2×C3×C4C22×D914yes—
153C2×C3×C4C22×D911yescharacteristic
154C2×C3×C4C22×D911yescharacteristic
155C2×C3×C4C22×D911yescharacteristic
156C2×Dic3—92no—
157C2×Dic3—92no—
158D12—94no—
159D12—188no—
160D12—182no—
161C4×D3—188no—
162C4×D3—184no—
163C4×D3—184no—
164C2×C9C2×D3×D411yescharacteristic
165C2×C9C2×#23911yescharacteristic
166C2×C9#100711yescharacteristic
167D9—64no—
168D9—122no—
169#41—548no—
170C2×Q8C2×D2712yes—
171C2×D4—271no—
172C2×D4—271no—
173C2×D4—274no—
174C22×C4—272no—
175C22×C4—274no—
176C22×C4—272no—
177C4⋊C4—271no—
178C4⋊C4—274no—
179C4⋊C4—271no—
180C4⋊C4C2×D2712yes—
181C4⋊C4C2×D2712yes—
182C22⋊C4—272no—
183C22⋊C4—272no—
184C22⋊C4—274no—
185C42C2×D2711yescharacteristic
186C22×C3C23×D911yescharacteristic
187C2×D3—94no—
188C2×D3—182no—
189C2×D3—182no—
190C2×D3—182no—
191C2×D3—182no—
192C2×D3—182no—
193C3×C4—24no—
194C3×C4—21no—
195C3×C4D4×D912yes—
196C3×C4#370112yes—
197Dic3—182no—
198Dic3—182no—
199C9#572611yescharacteristic
200C23—272no—
201C23—272no—
202Q8—24no—
203D4—274no—
204D4—548no—
205D4—542no—
206C2×C4—272no—
207C2×C4—272no—
208C2×C4—548no—
209C2×C4—544no—
210C2×C4—544no—
211C2×C4C22×D2711yescharacteristic
212C2×C4C22×D2714yes—
213C2×C4C22×D2711yescharacteristic
214C2×C4C22×D2711yescharacteristic
215C2×C3C2×D4×D911yescharacteristic
216C2×C3C2×#370111yescharacteristic
217C2×C3#6371311yescharacteristic
218D3—184no—
219D3—362no—
220C22—274no—
221C22—542no—
222C22—542no—
223C22—542no—
224C22—542no—
225C22—542no—
226C22C23×D2711yescharacteristic
227C4—24no—
228C4—21no—
229C4—542no—
230C4—542no—
231C4D4×D2712yes—
232C4#8840312yescomplemented
233C3#1058982811yescharacteristic
234C2—544no—
235C2—1082no—
236C2C2×D4×D2711yescharacteristic
237C2C2×#8840311yescharacteristic
238C2#1172067311yescharacteristic
subgroupG/H
centerC22C23×D27
commutator, cosocleC22×C27C24
FrattiniC22×C9C23×D3
FittingC27×C4⋊Q8C2
socleC22×C3C23×D9
hypercenterC4⋊Q8D27
hypocenterC27#547
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88403C2semidirect
2. C27×C4⋊Q8C2semidirect
3. #11720195C2semidirect
4. #11720144C2semidirect
5. #11720143C2semidirect
6. C4⋊C4×D27C2semidirect
7. C4×D108C2semidirect
8. C4×D108C2semidirect
46. D108D4semidirect
78. C4×C27#41semidirect
141. C4⋊Q8D27semidirect
148. C27#547semidirect
232. C4#88403semidirect

Maximal subgroups

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

subgroupindex
C2×#884032
C27×C4⋊Q82
#117201952
#117201442
#117201432
C4⋊C4×D272
C4×D1082
#105898283