Finite Group Databasebeta

← #13143954

Subgroups — #13143954

Order 1728.

836 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 113 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×#88744C211yesmaximal, characteristic
2C2×#88546C211yesmaximal, characteristic, complemented
3C2×#88530C211yesmaximal, characteristic, complemented
4#11721765C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#11720553C211yesmaximal, characteristic, complemented
7#11720532C211yesmaximal, characteristic, complemented
8C3×#4783—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C22×#1084—21no—
12#88744—22no—
13#88744—22no—
14C2×C4×#272C2211yescharacteristic
15C2×#6229C2211yescharacteristic
16C2×#6229C2211yescharacteristic
17C2×C8×H(3,𝔽3)C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19#88546—22no—
20#88542—21no—
21C4⋊#1084—21no—
22C2×#6180C2211yescharacteristic
23#88530—22no—
24C2×C3×#237—34no—
25C2×C3×#214—34no—
26C2×C3×Dic12—34no—
27C3×#897—34no—
28C32×#102D314yescomplemented
29C3×(C2×D3)⋊C8—34no—
30C3×#810—34no—
31C23×H(3,𝔽3)—21no—
32C22×#272D411yescharacteristic
33D4×H(3,𝔽3)—21no—
34D4×H(3,𝔽3)D412yes—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37#6235—24no—
38C2×#1084—24no—
39C2×#1084—21no—
40C2×#1084—21no—
41C2×#1084D411yescharacteristic
42C4×#272—21no—
43C4×#272—21no—
44#6229—21no—
45#6229—21no—
46#6229—21no—
47#6229D412yes—
48C8×H(3,𝔽3)—21no—
49#6180—21no—
50C3×#452—91no—
51#4783—34no—
52C2×C32×D4C2×D314yes—
53C2×C3×#67—64no—
54C22×C3×Dic3—64no—
55C3×#237—68no—
56C3×#237—68no—
57C2×C3×C4×D3—34no—
58C2×C3×Dic6—34no—
59C2×C3×Dic6—34no—
60C2×C32×C8C2×D314yes—
61C32×C4⋊C4C2×D314yes—
62C3×#214—68no—
63C3×(C2×D3)⋊C4—64no—
64C3×C4⋊Dic3—64no—
65C2×C3×C3⋊C8—34no—
66C3×Dic12—68no—
67C22×H(3,𝔽3)—22no—
68C22×H(3,𝔽3)—41no—
69C22×H(3,𝔽3)—41no—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C2×#272—21no—
72C2×#272—21no—
73C2×#272—21no—
74C4×H(3,𝔽3)—41no—
75C4×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)C2×D411yescharacteristic
77#1084—22no—
78#1084—21no—
79#1084—41no—
80C2×C3×#41—91no—
81C2×#237—34no—
82C2×C3×Q16—91no—
83C2×C3×SD16—91no—
84C3×C22⋊Q8—91no—
85C2×#214—34no—
86C2×Dic12—34no—
87#897—34no—
88C3×#103—91no—
89C3×#102—34no—
90C3×#102C3⋊D311yescharacteristic
91C3×C22⋊C8—91no—
92(C2×D3)⋊C8—34no—
93#810—34no—
94C23×C32—24no—
95C22×C3×D3—34no—
96C32×D4—24no—
97C32×D4D1218yes—
98C2×C32×C4—24no—
99C2×C32×C4C22×D314yes—
100C3×#67—616no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—616no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—64no—
105C3×C4×D3—64no—
106C3×C4×D3—64no—
107C3×Dic6—38no—
108C3×Dic6—64no—
109C3×Dic6—64no—
110C3×Dic6—64no—
111C32×C8—24no—
112C3×C3⋊C8—64no—
113#452—91noSylow
114C2×H(3,𝔽3)—42no—
115C2×H(3,𝔽3)#12011yescharacteristic
116C2×H(3,𝔽3)#13511yescharacteristic
117C2×H(3,𝔽3)#13711yescharacteristic
118#272—41no—
119C3×#41—182no—
120C3×#41—182no—
121C2×C3×Q8—91no—
122C2×C3×Q8—91no—
123C2×C3×D4—34no—
124C2×C3×D4—181no—
125C2×C3×D4C2×C3⋊D311yescharacteristic
126C22×C3×C4—91no—
127C22×C3×C4—181no—
128C2×#67—64no—
129C22×Dic3—64no—
130#237—68no—
131#237—68no—
132C2×C4×D3—34no—
133C2×Dic6—34no—
134C2×Dic6—34no—
135C3×Q16—182no—
136C3×SD16—182no—
137C2×C3×C8—34no—
138C2×C3×C8—91no—
139C2×C3×C8C2×C3⋊D311yescharacteristic
140C3×C4⋊C4—34no—
141C3×C4⋊C4—181no—
142C3×C4⋊C4C2×C3⋊D311yescharacteristic
143C3×C22⋊C4—181no—
144#214—68no—
145(C2×D3)⋊C4—64no—
146C4⋊Dic3—64no—
147C2×C3⋊C8—34no—
148Dic12—68no—
149C22×C32—28no—
150C22×C32—44no—
151C22×C32—44no—
152C22×C32D3×D414yes—
153C2×C3×D3—64no—
154C2×C3×D3—64no—
155C2×C3×D3—64no—
156C32×C4—44no—
157C32×C4C2×D1214yes—
158C32×C4D3×D414yes—
159C3×Dic3—68no—
160C3×Dic3—64no—
161C3×Dic3—124no—
162C2×#41—91no—
163C2×Q16—91no—
164C2×SD16—91no—
165C22⋊Q8—91no—
166#103—91no—
167#102#27211yescharacteristic, complemented
168C22⋊C8—91no—
169H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
170C23×C3—21no—
171C23×C3—64no—
172C23×C3—91no—
173C22×D3—34no—
174C3×Q8—92no—
175C3×Q8—181no—
176C3×Q8—181no—
177C3×Q8—181no—
178C3×D4—21no—
179C3×D4—38no—
180C3×D4—64no—
181C3×D4—184no—
182C3×D4C4⋊(C3⋊D3)12yes—
183C2×C3×C4—21no—
184C2×C3×C4—34no—
185C2×C3×C4—64no—
186C2×C3×C4—91no—
187C2×C3×C4—184no—
188C2×C3×C4—181no—
189C2×C3×C4—181no—
190C2×C3×C4—181no—
191C2×C3×C4—181no—
192C2×C3×C4C22×C3⋊D311yescharacteristic
193#67—616no—
194C2×Dic3—34no—
195C2×Dic3—64no—
196C2×Dic3—616no—
197C2×Dic3—64no—
198C4×D3—64no—
199C4×D3—64no—
200Dic6—38no—
201Dic6—64no—
202Dic6—64no—
203Dic6—64no—
204C3×C8—21no—
205C3×C8—64no—
206C3×C8—181no—
207C3⋊C8—64no—
208C2×C32—48no—
209C2×C32#88214yes—
210C2×C32#91214yes—
211C2×C32#91514yes—
212C3×D3—124no—
213#41—182no—
214#41—182no—
215C2×Q8—91no—
216C2×Q8—91no—
217C2×D4—181no—
218C2×D4C2×#27211yescharacteristic
219C22×C4—91no—
220C22×C4—181no—
221Q16—182no—
222SD16—182no—
223C2×C8—91no—
224C2×C8C2×#27211yescharacteristic
225C4⋊C4—181no—
226C4⋊C4C2×#27211yescharacteristic
227C22⋊C4—181no—
228C22×C3—22no—
229C22×C3—34no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—68no—
233C22×C3—124no—
234C22×C3—124no—
235C22×C3—181no—
236C22×C3—181no—
237C22×C3—181no—
238C22×C3D4×C3⋊D311yescharacteristic
239C2×D3—64no—
240C2×D3—64no—
241C2×D3—64no—
242C3×C4—34no—
243C3×C4—34no—
244C3×C4—41no—
245C3×C4—124no—
246C3×C4—181no—
247C3×C4—182no—
248C3×C4—361no—
249C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
250C3×C4D4×C3⋊D311yescharacteristic
251Dic3—68no—
252Dic3—64no—
253Dic3—124no—
254C32#478314yescomplemented
255C23—21no—
256C23—91no—
257Q8—92no—
258Q8—181no—
259Q8—181no—
260Q8—181no—
261D4—21no—
262D4—184no—
263D4C4⋊#27212yes—
264C2×C4—21no—
265C2×C4—91no—
266C2×C4—184no—
267C2×C4—181no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4C22×#27211yescharacteristic
272C8—21no—
273C8—181no—
274C2×C3—34no—
275C2×C3—34no—
276C2×C3—34no—
277C2×C3—42no—
278C2×C3—128no—
279C2×C3—361no—
280C2×C3#6409111yescharacteristic
281C2×C3#6412111yescharacteristic
282C2×C3#6412411yescharacteristic
283D3—124no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—181no—
288C22—181no—
289C22—181no—
290C22D4×#27211yescharacteristic
291C4—41no—
292C4—181no—
293C4—182no—
294C4—361no—
295C4C2×C4⋊#27211yescharacteristic
296C4D4×#27211yescharacteristic
297C3—34no—
298C3#1059248211yescharacteristic
299C2—42no—
300C2—361no—
301C2#1172173811yescharacteristic
302C2#1172179411yescharacteristic
303C2#1172180111yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radicalC2×#88744C2
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#452
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C2×#88546C2semidirect
3. C2×#88530C2semidirect
4. #11721765C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #11720553C2semidirect
7. #11720532C2semidirect
28. C32×#102D3semidirect
167. #102#272semidirect
169. H(3,𝔽3)#452semidirect
254. C32#4783semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
C2×#885462
C2×#885302
#117217652
H(3,𝔽3)×#1022
#117205532
#117205322
C3×#47833