Finite Group Databasebeta

← #13154290

Subgroups — #13154290

Order 1728.

1540 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1540 conjugacy classes of subgroups — too many to draw. Only its 235 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×Q8×#272C212yesmaximal, complemented
2H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
3#11721863C212yesmaximal, complemented
4#11721764C214yesmaximal, complemented
5#11721761C212yesmaximal
6#11721756C212yesmaximal, complemented
7C4×C4⋊#272C211yesmaximal, characteristic
8C4×#6229C211yesmaximal, characteristic, complemented
9C3×#5727—34nomaximal
10C2×Q8×H(3,𝔽3)C2212yes—
11Q8×#272—28no—
12C2×C4⋊#272C2211yescharacteristic
13C2×C4×#272C2214yes—
14C2×C4×#272C2212yes—
15C2×#6229C2214yes—
16C2×#6229C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2212yes—
18C4⋊C4×H(3,𝔽3)C2212yes—
19C42×H(3,𝔽3)C2211yescharacteristic
20#88542C2214yes—
21#88542C2212yes—
22C4⋊#1084C2211yescharacteristic
23C4⋊#1084C2211yescharacteristic
24#88540C2214yes—
25#88540C2212yes—
26C4×#1084C2212yes—
27C2×C3×D3×Q8—38no—
28C32×C4⋊Q8D314yescomplemented
29C3×#946—38no—
30C3×#896—316no—
31C3×#893—38no—
32C3×#888—38no—
33C3×C4×D12—34no—
34C3×C4×Dic6—34no—
35C22×#272C2312yes—
36Q8×H(3,𝔽3)—24no—
37C2×C4×H(3,𝔽3)C2314yes—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2312yes—
42C2×#1084C2312yes—
43C2×#1084C2312yes—
44C4⋊#272Q814yescomplemented
45C4×#272—24no—
46C4×#272—28no—
47#6229—28no—
48#6229D414yescomplemented
49C3×D4×Q8—91no—
50#5727—34no—
51C2×C32×Q8C2×D318yes—
52C3×D3×Q8—632no—
53C2×C3×D12—34no—
54C2×C3×C4×D3—316no—
55C2×C3×C4×D3—38no—
56C2×C3×Dic6—34no—
57C2×C3×Dic6—316no—
58C32×C4⋊C4C2×D318yes—
59C32×C4⋊C4C2×D318yes—
60C32×C42C2×D314yes—
61C3×(C2×D3)⋊C4—316no—
62C3×(C2×D3)⋊C4—38no—
63C3×C4⋊Dic3—34no—
64C3×C4⋊Dic3—34no—
65C3×Dic3⋊C4—316no—
66C3×Dic3⋊C4—38no—
67C3×C4×Dic3—38no—
68C22×H(3,𝔽3)C2411yescharacteristic
69C2×#272—22no—
70C2×#272—22no—
71C2×#272C2×Q814yescomplemented
72C4×H(3,𝔽3)—24no—
73C4×H(3,𝔽3)—21no—
74C4×H(3,𝔽3)C2×D412yes—
75C4×H(3,𝔽3)C2×Q812yescomplemented
76#1084—22no—
77#1084—22no—
78#1084C2×D414yes—
79C22×C3×Q8—92no—
80C2×D3×Q8—38no—
81C3×C4⋊Q8—34no—
82C3×C4⋊Q8—92no—
83C3×C4⋊Q8C3⋊D311yescharacteristic
84C3×C22⋊Q8—92no—
85C3×C22⋊Q8—94no—
86C3×C4×Q8—91no—
87C3×C4×D4—92no—
88C3×C4×D4—91no—
89#946—38no—
90#896—316no—
91#893—38no—
92#888—38no—
93C4×D12—34no—
94C4×Dic6—34no—
95C22×C3×D3—38no—
96C32×Q8—216no—
97C2×C32×C4C22×D314yes—
98C2×C32×C4C22×D3116yes—
99C2×C32×C4C22×D314yes—
100C2×C32×C4C22×D314yes—
101C2×C3×Dic3—38no—
102C2×C3×Dic3—38no—
103C2×C3×Dic3—38no—
104C3×D12—316no—
105C3×C4×D3—632no—
106C3×C4×D3—616no—
107C3×Dic6—316no—
108C3×Dic6—632no—
109D4×Q8—91noSylow
110C2×H(3,𝔽3)C22×D411yescharacteristic
111C2×H(3,𝔽3)C22×Q811yescharacteristic
112C2×H(3,𝔽3)2−1+411yescharacteristic
113#272—24no—
114C2×C3×Q8—38no—
115C2×C3×Q8—91no—
116C2×C3×Q8—94no—
117C2×C3×Q8—188no—
118C2×C3×Q8C2×C3⋊D312yes—
119C2×C3×D4—91no—
120C22×C3×C4—94no—
121C22×C3×C4—92no—
122D3×Q8—632no—
123C2×D12—34no—
124C2×C4×D3—38no—
125C2×C4×D3—316no—
126C2×Dic6—34no—
127C2×Dic6—316no—
128C3×C4⋊C4—38no—
129C3×C4⋊C4—38no—
130C3×C4⋊C4—92no—
131C3×C4⋊C4—94no—
132C3×C4⋊C4—91no—
133C3×C4⋊C4—91no—
134C3×C4⋊C4C2×C3⋊D312yes—
135C3×C4⋊C4C2×C3⋊D312yes—
136C3×C22⋊C4—94no—
137C3×C22⋊C4—92no—
138C3×C42—34no—
139C3×C42—92no—
140C3×C42C2×C3⋊D311yescharacteristic
141(C2×D3)⋊C4—38no—
142(C2×D3)⋊C4—316no—
143C4⋊Dic3—34no—
144C4⋊Dic3—34no—
145Dic3⋊C4—38no—
146Dic3⋊C4—316no—
147C4×Dic3—38no—
148C22×C32C23×D314yes—
149C2×C3×D3—316no—
150C2×C3×D3—68no—
151C2×C3×D3—68no—
152C32×C4—216no—
153C32×C4—24no—
154C32×C4D3×D418yes—
155C32×C4D3×Q818yescomplemented
156C3×Dic3—316no—
157C3×Dic3—68no—
158C3×Dic3—68no—
159C22×Q8—92no—
160C4⋊Q8—92no—
161C4⋊Q8#27211yescharacteristic, complemented
162C22⋊Q8—92no—
163C22⋊Q8—94no—
164C4×Q8—91no—
165C4×D4—92no—
166C4×D4—91no—
167H(3,𝔽3)D4×Q811yescharacteristic, complemented, Sylow
168C23×C3—92no—
169C22×D3—38no—
170C3×Q8—24no—
171C3×Q8—616no—
172C3×Q8—94no—
173C3×Q8—188no—
174C3×D4—94no—
175C2×C3×C4—34no—
176C2×C3×C4—34no—
177C2×C3×C4—34no—
178C2×C3×C4—316no—
179C2×C3×C4—92no—
180C2×C3×C4—92no—
181C2×C3×C4—92no—
182C2×C3×C4—188no—
183C2×C3×C4—184no—
184C2×C3×C4C22×C3⋊D311yescharacteristic
185C2×C3×C4C22×C3⋊D314yes—
186C2×C3×C4C22×C3⋊D311yescharacteristic
187C2×C3×C4C22×C3⋊D311yescharacteristic
188C2×Dic3—38no—
189C2×Dic3—38no—
190C2×Dic3—38no—
191D12—316no—
192C4×D3—632no—
193C4×D3—616no—
194Dic6—316no—
195Dic6—632no—
196C2×C32C2×D3×D414yes—
197C2×C32C2×D3×Q814yes—
198C2×C32#100714yes—
199C3×D3—616no—
200C2×Q8—91no—
201C2×Q8—94no—
202C2×Q8—188no—
203C2×Q8C2×#27212yes—
204C2×D4—91no—
205C22×C4—94no—
206C22×C4—92no—
207C4⋊C4—91no—
208C4⋊C4—92no—
209C4⋊C4—91no—
210C4⋊C4—94no—
211C4⋊C4C2×#27212yes—
212C4⋊C4C2×#27212yes—
213C22⋊C4—94no—
214C22⋊C4—92no—
215C42—92no—
216C42C2×#27211yescharacteristic
217C22×C3—34no—
218C22×C3—94no—
219C22×C3—182no—
220C22×C3—182no—
221C22×C3C23×C3⋊D311yescharacteristic
222C2×D3—316no—
223C2×D3—68no—
224C2×D3—68no—
225C3×C4—24no—
226C3×C4—21no—
227C3×C4—38no—
228C3×C4—38no—
229C3×C4—616no—
230C3×C4—64no—
231C3×C4—94no—
232C3×C4—182no—
233C3×C4—182no—
234C3×C4D4×C3⋊D312yes—
235C3×C4Q8×C3⋊D312yes—
236Dic3—316no—
237Dic3—68no—
238Dic3—68no—
239C32#572714yescomplemented
240C23—92no—
241Q8—24no—
242Q8—94no—
243Q8—188no—
244D4—94no—
245C2×C4—92no—
246C2×C4—92no—
247C2×C4—92no—
248C2×C4—188no—
249C2×C4—184no—
250C2×C4C22×#27211yescharacteristic
251C2×C4C22×#27214yes—
252C2×C4C22×#27211yescharacteristic
253C2×C4C22×#27211yescharacteristic
254C2×C3—34no—
255C2×C3—34no—
256C2×C3—34no—
257C2×C3—184no—
258C2×C3C2×D4×C3⋊D311yescharacteristic
259C2×C3C2×Q8×C3⋊D311yescharacteristic
260C2×C3#6436411yescharacteristic
261D3—616no—
262C22—94no—
263C22—182no—
264C22—182no—
265C22C23×#27211yescharacteristic
266C4—24no—
267C4—21no—
268C4—94no—
269C4—182no—
270C4—182no—
271C4D4×#27212yes—
272C4Q8×#27212yescomplemented
273C3—34no—
274C3#1059619811yescharacteristic
275C2—184no—
276C2C2×D4×#27211yescharacteristic
277C2C2×Q8×#27211yescharacteristic
278C2#1172267011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC2×C4⋊#272C22
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterH(3,𝔽3)D4×Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×#272C2semidirect
2. H(3,𝔽3)×C4⋊Q8C2semidirect
3. #11721863C2semidirect
4. #11721764C2semidirect
6. #11721756C2semidirect
8. C4×#6229C2semidirect
28. C32×C4⋊Q8D3semidirect
44. C4⋊#272Q8semidirect
48. #6229D4semidirect
71. C2×#272C2×Q8semidirect
75. C4×H(3,𝔽3)C2×Q8semidirect
155. C32×C4D3×Q8semidirect
161. C4⋊Q8#272semidirect
167. H(3,𝔽3)D4×Q8semidirect
239. C32#5727semidirect
272. C4Q8×#272semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
H(3,𝔽3)×C4⋊Q82
#117218632
#117217642
#117217612
#117217562
C4×C4⋊#2722
C4×#62292
C3×#57273