Finite Group Databasebeta

← #13144787

Subgroups — #13144787

Order 1728.

803 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 803 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#11722676C211yesmaximal, characteristic, complemented
2H(3,𝔽3)×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#11721846C211yesmaximal, characteristic, complemented
4#11721842C211yesmaximal, characteristic, complemented
5#11720586C211yesmaximal, characteristic, complemented
6#11720562C211yesmaximal, characteristic, complemented
7#11720561C211yesmaximal, characteristic, complemented
8C3×#5201—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11Q8×#272—21no—
12#88744—21no—
13#88744—22no—
14#88740—21no—
15#88740C2211yescharacteristic
16C2×#6229—21no—
17C2×#6229C2211yescharacteristic
18SD16×H(3,𝔽3)—21no—
19D8×H(3,𝔽3)—21no—
20M16×H(3,𝔽3)C2211yescharacteristic
21#88552—22no—
22#88546—21no—
23#88545—21no—
24C4×#1084C2211yescharacteristic
25#88538C2211yescharacteristic
26C3×#1010—34no—
27C32×AGL(1,ℤ/8)D314yescomplemented
28C3×#936—34no—
29C3×#932—34no—
30C3×#837—34no—
31C3×#825—34no—
32C3×#824—34no—
33C23×H(3,𝔽3)—21no—
34Q8×H(3,𝔽3)D411yescharacteristic
35D4×H(3,𝔽3)—21no—
36D4×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41#6235—22no—
42#6235—21no—
43C2×#1084—22no—
44C2×#1084—21no—
45C2×#1084D412yes—
46C4⋊#272D411yescharacteristic
47C4×#272—22no—
48C4×#272—21no—
49#6229—22no—
50#6229—21no—
51#6229—21no—
52#6229D411yescharacteristic
53C8×H(3,𝔽3)—21no—
54#6180—21no—
55C3×#454—91no—
56#5201—34no—
57C32×#41C2×D314yes—
58C2×C32×D4C2×D314yes—
59C3×D3×Q8—64no—
60C3×#237—68no—
61C3×#237—64no—
62C3×#235—34no—
63C3×#235—64no—
64C2×C3×Dic6—34no—
65C2×C3×Dic6—64no—
66C32×SD16—24no—
67C32×D8—24no—
68C32×M16C2×D314yes—
69C3×#217—68no—
70C3×#214—64no—
71C3×#213—64no—
72C3×C4×Dic3—34no—
73C3×#208—34no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)—42no—
76C22×H(3,𝔽3)—41no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×#272—21no—
79C4×H(3,𝔽3)—21no—
80C4×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)C2×D411yescharacteristic
82#1084—22no—
83#1084—21no—
84#1084—41no—
85C3×2−1+4—91no—
86#1010—34no—
87C3×AGL(1,ℤ/8)—34no—
88C3×AGL(1,ℤ/8)—91no—
89C3×AGL(1,ℤ/8)C3⋊D311yescharacteristic
90C3×#124—91no—
91#936—34no—
92#932—34no—
93C3×#104—91no—
94C3×#104—91no—
95C3×#101—91no—
96#837—34no—
97#825—34no—
98#824—34no—
99C23×C32—24no—
100C32×Q8#6714yes—
101C32×D4—24no—
102C32×D4—24no—
103C32×D4—24no—
104C32×D4#6714yes—
105C2×C32×C4—24no—
106C2×C32×C4C22×D314yes—
107C3×#67—68no—
108C3×#67—64no—
109C2×C3×Dic3—38no—
110C2×C3×Dic3—68no—
111C2×C3×Dic3—64no—
112C3×D12—34no—
113C3×C4×D3—68no—
114C3×C4×D3—64no—
115C3×Dic6—34no—
116C3×Dic6—68no—
117C3×Dic6—64no—
118C3×Dic6—64no—
119C32×C8—24no—
120C3×C3⋊C8—64no—
121#454—91noSylow
122C2×H(3,𝔽3)—21no—
123C2×H(3,𝔽3)—41no—
124C2×H(3,𝔽3)—81no—
125C2×H(3,𝔽3)#12011yescharacteristic
126#272—41no—
127C3×#41—34no—
128C3×#41—91no—
129C3×#41—181no—
130C3×#41—182no—
131C3×#41—181no—
132C3×#41C2×C3⋊D311yescharacteristic
133C2×C3×Q8—91no—
134C2×C3×Q8—181no—
135C2×C3×Q8—181no—
136C2×C3×D4—34no—
137C2×C3×D4C2×C3⋊D311yescharacteristic
138D3×Q8—64no—
139#237—68no—
140#237—64no—
141#235—34no—
142#235—64no—
143C2×Dic6—34no—
144C2×Dic6—64no—
145C3×SD16—21no—
146C3×SD16—64no—
147C3×SD16—181no—
148C3×D8—21no—
149C3×D8—64no—
150C3×D8—181no—
151C3×M16—34no—
152C3×M16—91no—
153C3×M16C2×C3⋊D311yescharacteristic
154C3×C22⋊C4—182no—
155C3×C42—91no—
156#217—68no—
157#214—64no—
158#213—64no—
159C4×Dic3—34no—
160#208—34no—
161C22×C32—24no—
162C22×C32—48no—
163C22×C32—44no—
164C22×C32D3×D414yes—
165C2×C3×D3—64no—
166C32×C4—24no—
167C32×C4D3×D414yes—
168C32×C4C2×#6714yes—
169C3×Dic3—64no—
170C3×Dic3—68no—
171C3×Dic3—124no—
1722−1+4—91no—
173AGL(1,ℤ/8)—91no—
174AGL(1,ℤ/8)#27211yescharacteristic, complemented
175#124—91no—
176#104—91no—
177#104—91no—
178#101—91no—
179H(3,𝔽3)#45411yescharacteristic, complemented, Sylow
180C23×C3—21no—
181C23×C3—64no—
182C3×Q8—34no—
183C3×Q8—91no—
184C3×Q8—182no—
185C3×Q8—181no—
186C3×Q8—181no—
187C3×Q8#63811yescharacteristic
188C3×D4—21no—
189C3×D4—21no—
190C3×D4—21no—
191C3×D4—34no—
192C3×D4—64no—
193C3×D4—64no—
194C3×D4—64no—
195C3×D4—91no—
196C3×D4—182no—
197C3×D4—181no—
198C3×D4#63811yescharacteristic
199C2×C3×C4—21no—
200C2×C3×C4—34no—
201C2×C3×C4—64no—
202C2×C3×C4—92no—
203C2×C3×C4—182no—
204C2×C3×C4—182no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208#67—64no—
209#67—68no—
210C2×Dic3—38no—
211C2×Dic3—68no—
212C2×Dic3—64no—
213D12—34no—
214C4×D3—64no—
215C4×D3—68no—
216Dic6—34no—
217Dic6—64no—
218Dic6—64no—
219Dic6—68no—
220C3×C8—21no—
221C3×C8—64no—
222C3×C8—181no—
223C3⋊C8—64no—
224C2×C32—24no—
225C2×C32—44no—
226C2×C32—84no—
227C2×C32#93714yes—
228C3×D3—124no—
229#41—91no—
230#41—181no—
231#41—182no—
232#41—181no—
233#41C2×#27211yescharacteristic
234C2×Q8—91no—
235C2×Q8—181no—
236C2×Q8—181no—
237C2×D4C2×#27211yescharacteristic
238SD16—21no—
239SD16—181no—
240D8—21no—
241D8—181no—
242M16—91no—
243M16C2×#27211yescharacteristic
244C22⋊C4—182no—
245C42—91no—
246C22×C3—21no—
247C22×C3—34no—
248C22×C3—42no—
249C22×C3—41no—
250C22×C3—64no—
251C22×C3—124no—
252C22×C3—128no—
253C22×C3—181no—
254C22×C3D4×C3⋊D311yescharacteristic
255C2×D3—64no—
256C3×C4—21no—
257C3×C4—34no—
258C3×C4—34no—
259C3×C4—64no—
260C3×C4—182no—
261C3×C4—181no—
262C3×C4—361no—
263C3×C4D4×C3⋊D311yescharacteristic
264C3×C4C2×#63811yescharacteristic
265Dic3—64no—
266Dic3—68no—
267Dic3—124no—
268C32#520114yescomplemented
269C23—21no—
270Q8—91no—
271Q8—182no—
272Q8—181no—
273Q8—181no—
274Q8#623511yescharacteristic
275D4—21no—
276D4—21no—
277D4—21no—
278D4—91no—
279D4—182no—
280D4—181no—
281D4#623511yescharacteristic
282C2×C4—21no—
283C2×C4—92no—
284C2×C4—182no—
285C2×C4—182no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4C22×#27211yescharacteristic
289C8—21no—
290C8—181no—
291C2×C3—21no—
292C2×C3—34no—
293C2×C3—41no—
294C2×C3—64no—
295C2×C3—81no—
296C2×C3—124no—
297C2×C3—244no—
298C2×C3—361no—
299C2×C3#6414611yescharacteristic
300D3—124no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—181no—
305C22D4×#27211yescharacteristic
306C4—21no—
307C4—182no—
308C4—181no—
309C4—361no—
310C4D4×#27211yescharacteristic
311C4C2×#623511yescharacteristic
312C3—34no—
313C3#1059290011yescharacteristic
314C2—21no—
315C2—41no—
316C2—81no—
317C2—361no—
318C2#1172184711yescharacteristic
subgroupG/H
center, socleC2×C3#64146
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×AGL(1,ℤ/8)C2
hypercenterC3×AGL(1,ℤ/8)C3⋊D3
hypocenterH(3,𝔽3)#454
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #11722676C2semidirect
2. H(3,𝔽3)×AGL(1,ℤ/8)C2semidirect
3. #11721846C2semidirect
4. #11721842C2semidirect
5. #11720586C2semidirect
6. #11720562C2semidirect
7. #11720561C2semidirect
27. C32×AGL(1,ℤ/8)D3semidirect
174. AGL(1,ℤ/8)#272semidirect
179. H(3,𝔽3)#454semidirect
268. C32#5201semidirect

Maximal subgroups

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

subgroupindex
#117226762
H(3,𝔽3)×AGL(1,ℤ/8)2
#117218462
#117218422
#117205862
#117205622
#117205612
C3×#52013