Finite Group Databasebeta

← #13144021

Subgroups — #13144021

Order 1728.

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

normal subgroups only

This group has 891 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×#88748C211yesmaximal, characteristic, complemented
2C2×#88549C211yesmaximal, characteristic, complemented
3C2×#88529C211yesmaximal, characteristic, complemented
4#11721763C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#11720553C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#4810—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10#88748—22no—
11#88748—22no—
12C2×C4⋊#272—21no—
13C2×C4⋊#272—21no—
14C2×C4⋊#272C2211yescharacteristic
15C2×C4⋊#272C2211yescharacteristic
16C2×C4×#272—21no—
17C2×C4×#272C2211yescharacteristic
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20#88549—22no—
21#88542—21no—
22C2×#6180C2211yescharacteristic
23#88529—22no—
24C2×C3×#239—34no—
25C2×C3×#215—34no—
26C2×C3×D24—34no—
27C3×#895—34no—
28C32×#103D314yescomplemented
29C3×(C2×D3)⋊C8—34no—
30C3×#809—34no—
31C22×#272—21no—
32C22×#272—21no—
33C22×#272D411yescharacteristic
34Q8×H(3,𝔽3)—21no—
35Q8×H(3,𝔽3)D412yescomplemented
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084D411yescharacteristic
40C4⋊#272—24no—
41C4⋊#272—21no—
42C4⋊#272—21no—
43C4⋊#272—21no—
44C4⋊#272—42no—
45C4⋊#272D412yescomplemented
46C4×#272—24no—
47C4×#272—21no—
48C4×#272—21no—
49C8×H(3,𝔽3)—21no—
50#6180—21no—
51C3×#449—91no—
52#4810—34no—
53C2×C32×Q8C2×D314yes—
54C3×#239—68no—
55C3×#239—68no—
56C2×C3×D12—34no—
57C2×C3×D12—34no—
58C2×C3×D12—64no—
59C2×C3×D12—64no—
60C2×C3×C4×D3—34no—
61C2×C3×C4×D3—64no—
62C2×C32×C8C2×D314yes—
63C32×C4⋊C4C2×D314yes—
64C3×#215—68no—
65C3×(C2×D3)⋊C4—64no—
66C2×C3×C3⋊C8—34no—
67C3×D24—68no—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C2×#272—21no—
70C2×#272—22no—
71C2×#272—21no—
72C2×#272—21no—
73C2×#272—41no—
74C2×#272—41no—
75C2×#272—41no—
76C2×#272—41no—
77C2×#272—41no—
78C4×H(3,𝔽3)—22no—
79C4×H(3,𝔽3)—41no—
80C4×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)C2×D411yescharacteristic
82#1084—21no—
83C2×C3×#41—91no—
84C2×#239—34no—
85C2×C3×SD16—91no—
86C2×C3×D8—91no—
87C3×#121—91no—
88C2×#215—34no—
89C2×D24—34no—
90#895—34no—
91C3×#103—34no—
92C3×#103C3⋊D311yescharacteristic
93C3×#102—91no—
94C3×C22⋊C8—91no—
95(C2×D3)⋊C8—34no—
96#809—34no—
97C22×C3×D3—34no—
98C22×C3×D3—64no—
99C22×C3×D3—64no—
100C32×Q8—24no—
101C32×Q8D1218yescomplemented
102C2×C32×C4—24no—
103C2×C32×C4—24no—
104C2×C32×C4C22×D314yes—
105C2×C3×Dic3—34no—
106C3×D12—38no—
107C3×D12—616no—
108C3×D12—64no—
109C3×D12—64no—
110C3×D12—64no—
111C3×D12—128no—
112C3×C4×D3—616no—
113C3×C4×D3—64no—
114C3×C4×D3—64no—
115C32×C8—24no—
116C3×C3⋊C8—64no—
117#449—91noSylow
118C2×H(3,𝔽3)#12011yescharacteristic
119C2×H(3,𝔽3)#13511yescharacteristic
120C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
121#272—41no—
122#272—42no—
123#272—81no—
124C3×#41—182no—
125C3×#41—182no—
126C2×C3×Q8—34no—
127C2×C3×Q8C2×C3⋊D311yescharacteristic
128C2×C3×D4—91no—
129C2×C3×D4—91no—
130C2×C3×D4—181no—
131C2×C3×D4—181no—
132C22×C3×C4—91no—
133C22×C3×C4—181no—
134#239—68no—
135#239—68no—
136C2×D12—34no—
137C2×D12—34no—
138C2×D12—64no—
139C2×D12—64no—
140C2×C4×D3—34no—
141C2×C4×D3—64no—
142C3×SD16—182no—
143C3×D8—182no—
144C2×C3×C8—34no—
145C2×C3×C8—91no—
146C2×C3×C8C2×C3⋊D311yescharacteristic
147C3×C4⋊C4—34no—
148C3×C4⋊C4C2×C3⋊D311yescharacteristic
149C3×C22⋊C4—181no—
150#215—68no—
151(C2×D3)⋊C4—64no—
152C2×C3⋊C8—34no—
153D24—68no—
154C22×C32D3×D414yes—
155C2×C3×D3—68no—
156C2×C3×D3—64no—
157C2×C3×D3—64no—
158C2×C3×D3—64no—
159C2×C3×D3—124no—
160C2×C3×D3—124no—
161C2×C3×D3—124no—
162C2×C3×D3—124no—
163C2×C3×D3—124no—
164C32×C4—28no—
165C32×C4—44no—
166C32×C4C2×D1214yes—
167C32×C4D3×D414yes—
168C3×Dic3—64no—
169C2×#41—91no—
170C2×SD16—91no—
171C2×D8—91no—
172#121—91no—
173#103#27211yescharacteristic, complemented
174#102—91no—
175C22⋊C8—91no—
176H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
177C23×C3—91no—
178C23×C3—181no—
179C23×C3—181no—
180C22×D3—34no—
181C22×D3—64no—
182C22×D3—64no—
183C3×Q8—21no—
184C3×Q8—38no—
185C3×Q8—64no—
186C3×Q8C4⋊(C3⋊D3)12yes—
187C3×D4—92no—
188C3×D4—184no—
189C3×D4—181no—
190C3×D4—181no—
191C3×D4—181no—
192C3×D4—362no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—34no—
196C2×C3×C4—64no—
197C2×C3×C4—64no—
198C2×C3×C4—91no—
199C2×C3×C4—184no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202C2×C3×C4C22×C3⋊D311yescharacteristic
203C2×Dic3—34no—
204D12—38no—
205D12—64no—
206D12—64no—
207D12—64no—
208D12—616no—
209D12—128no—
210C4×D3—64no—
211C4×D3—64no—
212C4×D3—616no—
213C3×C8—21no—
214C3×C8—64no—
215C3×C8—181no—
216C3⋊C8—64no—
217C2×C32#88214yes—
218C2×C32#91414yes—
219C2×C32#91914yes—
220C3×D3—124no—
221C3×D3—128no—
222C3×D3—244no—
223#41—182no—
224#41—182no—
225C2×Q8C2×#27211yescharacteristic
226C2×D4—91no—
227C2×D4—91no—
228C2×D4—181no—
229C2×D4—181no—
230C22×C4—91no—
231C22×C4—181no—
232SD16—182no—
233D8—182no—
234C2×C8—91no—
235C2×C8C2×#27211yescharacteristic
236C4⋊C4C2×#27211yescharacteristic
237C22⋊C4—181no—
238C22×C3—34no—
239C22×C3—182no—
240C22×C3—181no—
241C22×C3—181no—
242C22×C3—181no—
243C22×C3—361no—
244C22×C3—361no—
245C22×C3—361no—
246C22×C3—361no—
247C22×C3—361no—
248C22×C3D4×C3⋊D311yescharacteristic
249C2×D3—68no—
250C2×D3—64no—
251C2×D3—64no—
252C2×D3—64no—
253C2×D3—124no—
254C2×D3—124no—
255C2×D3—124no—
256C2×D3—124no—
257C2×D3—124no—
258C3×C4—22no—
259C3×C4—34no—
260C3×C4—34no—
261C3×C4—41no—
262C3×C4—68no—
263C3×C4—124no—
264C3×C4—181no—
265C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
266C3×C4D4×C3⋊D311yescharacteristic
267Dic3—64no—
268C32#481014yescomplemented
269C23—91no—
270C23—181no—
271C23—181no—
272Q8—21no—
273Q8C4⋊#27212yescomplemented
274D4—92no—
275D4—184no—
276D4—181no—
277D4—181no—
278D4—181no—
279D4—362no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—91no—
283C2×C4—184no—
284C2×C4—181no—
285C2×C4—181no—
286C2×C4C22×#27211yescharacteristic
287C8—21no—
288C8—181no—
289C2×C3—34no—
290C2×C3—34no—
291C2×C3—34no—
292C2×C3—361no—
293C2×C3—362no—
294C2×C3—721no—
295C2×C3#6409111yescharacteristic
296C2×C3#6412311yescharacteristic
297C2×C3#6412811yescharacteristic
298D3—124no—
299D3—128no—
300D3—244no—
301C22—182no—
302C22—181no—
303C22—181no—
304C22—181no—
305C22—361no—
306C22—361no—
307C22—361no—
308C22—361no—
309C22—361no—
310C22D4×#27211yescharacteristic
311C4—22no—
312C4—41no—
313C4—181no—
314C4C2×C4⋊#27211yescharacteristic
315C4D4×#27211yescharacteristic
316C3—34no—
317C3#1059250911yescharacteristic
318C2—361no—
319C2—362no—
320C2—721no—
321C2#1172173811yescharacteristic
322C2#1172180011yescharacteristic
323C2#1172180811yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#449
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
1. C2×#88748C2semidirect
2. C2×#88549C2semidirect
3. C2×#88529C2semidirect
4. #11721763C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #11720553C2semidirect
7. #11720531C2semidirect
28. C32×#103D3semidirect
35. Q8×H(3,𝔽3)D4semidirect
45. C4⋊#272D4semidirect
101. C32×Q8D12semidirect
173. #103#272semidirect
176. H(3,𝔽3)#449semidirect
268. C32#4810semidirect
273. Q8C4⋊#272semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
C2×#885492
C2×#885292
#117217632
H(3,𝔽3)×#1032
#117205532
#117205312
C3×#48103