Finite Group Databasebeta

← #13144795

Subgroups — #13144795

Order 1728.

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

normal subgroups only

This group has 781 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
2H(3,𝔽3)×#137C211yesmaximal, characteristic, complemented
3#11721861C211yesmaximal, characteristic, complemented
4#11721859C211yesmaximal, characteristic, complemented
5#11720586C211yesmaximal, characteristic, complemented
6#11720562C211yesmaximal, characteristic, complemented
7#11720561C211yesmaximal, characteristic, complemented
8C3×#5205—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11Q8×#272—21no—
12#88744—21no—
13#88744—22no—
14#88740—21no—
15#88740C2211yescharacteristic
16C2×#6229—21no—
17C2×#6229C2211yescharacteristic
18Q16×H(3,𝔽3)—21no—
19SD16×H(3,𝔽3)—21no—
20M16×H(3,𝔽3)C2211yescharacteristic
21#88550—21no—
22#88549—21no—
23#88540—22no—
24C4×#1084C2211yescharacteristic
25#88538C2211yescharacteristic
26C3×#1010—34no—
27C32×#137D314yescomplemented
28C3×#944—34no—
29C3×#942—34no—
30C3×#837—34no—
31C3×#825—34no—
32C3×#824—34no—
33Q8×H(3,𝔽3)—21no—
34Q8×H(3,𝔽3)—21no—
35Q8×H(3,𝔽3)D411yescharacteristic
36D4×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
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×#456—91no—
56#5205—34no—
57C32×#41C2×D314yes—
58C2×C32×Q8C2×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×Q16—24no—
67C32×SD16—24no—
68C32×M16C2×D314yes—
69C3×#216—64no—
70C3×#215—64no—
71C3×Dic3⋊C4—68no—
72C3×C4×Dic3—34no—
73C3×#208—34no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C2×#272—21no—
77C4×H(3,𝔽3)—21no—
78C4×H(3,𝔽3)—41no—
79C4×H(3,𝔽3)C2×D411yescharacteristic
80C4×H(3,𝔽3)C2×D411yescharacteristic
81#1084—22no—
82#1084—22no—
83#1084—21no—
84C3×2−1+4—91no—
85#1010—34no—
86C3×#137—34no—
87C3×#137—91no—
88C3×#137C3⋊D311yescharacteristic
89C3×C4⋊Q8—91no—
90#944—34no—
91#942—34no—
92C3×#104—91no—
93C3×#104—91no—
94C3×#101—91no—
95#837—34no—
96#825—34no—
97#824—34no—
98C32×Q8—24no—
99C32×Q8—24no—
100C32×Q8#6714yes—
101C32×D4—24no—
102C32×D4#6714yes—
103C2×C32×C4—24no—
104C2×C32×C4—24no—
105C2×C32×C4C22×D314yes—
106C3×#67—68no—
107C3×#67—64no—
108C2×C3×Dic3—38no—
109C2×C3×Dic3—68no—
110C2×C3×Dic3—64no—
111C3×D12—34no—
112C3×C4×D3—68no—
113C3×C4×D3—64no—
114C3×Dic6—34no—
115C3×Dic6—68no—
116C3×Dic6—64no—
117C3×Dic6—64no—
118C32×C8—24no—
119C3×C3⋊C8—64no—
120#456—91noSylow
121C2×H(3,𝔽3)—21no—
122C2×H(3,𝔽3)—41no—
123C2×H(3,𝔽3)#12011yescharacteristic
124#272—41no—
125C3×#41—34no—
126C3×#41—91no—
127C3×#41—181no—
128C3×#41—182no—
129C3×#41—181no—
130C3×#41C2×C3⋊D311yescharacteristic
131C2×C3×Q8—34no—
132C2×C3×Q8—91no—
133C2×C3×Q8—181no—
134C2×C3×Q8—181no—
135C2×C3×Q8C2×C3⋊D311yescharacteristic
136D3×Q8—64no—
137#237—68no—
138#237—64no—
139#235—34no—
140#235—64no—
141C2×Dic6—34no—
142C2×Dic6—64no—
143C3×Q16—21no—
144C3×Q16—64no—
145C3×Q16—181no—
146C3×SD16—21no—
147C3×SD16—64no—
148C3×SD16—181no—
149C3×M16—34no—
150C3×M16—91no—
151C3×M16C2×C3⋊D311yescharacteristic
152C3×C4⋊C4—182no—
153C3×C42—91no—
154#216—64no—
155#215—64no—
156Dic3⋊C4—68no—
157C4×Dic3—34no—
158#208—34no—
159C22×C32—24no—
160C22×C32D3×D414yes—
161C2×C3×D3—64no—
162C32×C4—24no—
163C32×C4—44no—
164C32×C4D3×D414yes—
165C32×C4C2×#6714yes—
166C3×Dic3—68no—
167C3×Dic3—64no—
168C3×Dic3—68no—
1692−1+4—91no—
170#137—91no—
171#137#27211yescharacteristic, complemented
172C4⋊Q8—91no—
173#104—91no—
174#104—91no—
175#101—91no—
176H(3,𝔽3)#45611yescharacteristic, complemented, Sylow
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—34no—
180C3×Q8—64no—
181C3×Q8—64no—
182C3×Q8—91no—
183C3×Q8—182no—
184C3×Q8—181no—
185C3×Q8—181no—
186C3×Q8#63811yescharacteristic
187C3×D4—21no—
188C3×D4—34no—
189C3×D4—64no—
190C3×D4—91no—
191C3×D4—182no—
192C3×D4—181no—
193C3×D4#63811yescharacteristic
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—34no—
197C2×C3×C4—64no—
198C2×C3×C4—64no—
199C2×C3×C4—92no—
200C2×C3×C4—182no—
201C2×C3×C4—182no—
202C2×C3×C4—181no—
203C2×C3×C4—181no—
204C2×C3×C4C22×C3⋊D311yescharacteristic
205#67—64no—
206#67—68no—
207C2×Dic3—38no—
208C2×Dic3—64no—
209C2×Dic3—68no—
210D12—34no—
211C4×D3—64no—
212C4×D3—68no—
213Dic6—34no—
214Dic6—64no—
215Dic6—64no—
216Dic6—68no—
217C3×C8—21no—
218C3×C8—64no—
219C3×C8—181no—
220C3⋊C8—64no—
221C2×C32—24no—
222C2×C32—44no—
223C2×C32#93714yes—
224C3×D3—124no—
225#41—91no—
226#41—181no—
227#41—182no—
228#41—181no—
229#41C2×#27211yescharacteristic
230C2×Q8—91no—
231C2×Q8—181no—
232C2×Q8—181no—
233C2×Q8C2×#27211yescharacteristic
234Q16—21no—
235Q16—181no—
236SD16—21no—
237SD16—181no—
238M16—91no—
239M16C2×#27211yescharacteristic
240C4⋊C4—182no—
241C42—91no—
242C22×C3—21no—
243C22×C3—34no—
244C22×C3—64no—
245C22×C3—181no—
246C22×C3D4×C3⋊D311yescharacteristic
247C2×D3—64no—
248C3×C4—21no—
249C3×C4—34no—
250C3×C4—34no—
251C3×C4—41no—
252C3×C4—64no—
253C3×C4—124no—
254C3×C4—182no—
255C3×C4—182no—
256C3×C4—181no—
257C3×C4D4×C3⋊D311yescharacteristic
258C3×C4C2×#63811yescharacteristic
259Dic3—64no—
260Dic3—68no—
261Dic3—68no—
262C32#520514yescomplemented
263Q8—21no—
264Q8—21no—
265Q8—91no—
266Q8—182no—
267Q8—181no—
268Q8—181no—
269Q8#623511yescharacteristic
270D4—21no—
271D4—91no—
272D4—182no—
273D4—181no—
274D4#623511yescharacteristic
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—92no—
278C2×C4—182no—
279C2×C4—182no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4C22×#27211yescharacteristic
283C8—21no—
284C8—181no—
285C2×C3—21no—
286C2×C3—34no—
287C2×C3—41no—
288C2×C3—64no—
289C2×C3—124no—
290C2×C3—361no—
291C2×C3#6414611yescharacteristic
292D3—124no—
293C22—21no—
294C22—181no—
295C22D4×#27211yescharacteristic
296C4—21no—
297C4—41no—
298C4—182no—
299C4—182no—
300C4—181no—
301C4D4×#27211yescharacteristic
302C4C2×#623511yescharacteristic
303C3—34no—
304C3#1059290411yescharacteristic
305C2—21no—
306C2—41no—
307C2—361no—
308C2#1172184711yescharacteristic
subgroupG/H
center, socleC2×C3#64146
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#137C2
prime radical#11722676C2
hypercenterC3×#137C3⋊D3
hypocenterH(3,𝔽3)#456
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#137

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#137C2semidirect
3. #11721861C2semidirect
4. #11721859C2semidirect
5. #11720586C2semidirect
6. #11720562C2semidirect
7. #11720561C2semidirect
27. C32×#137D3semidirect
171. #137#272semidirect
176. H(3,𝔽3)#456semidirect
262. C32#5205semidirect

Maximal subgroups

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

subgroupindex
#117226762
H(3,𝔽3)×#1372
#117218612
#117218592
#117205862
#117205622
#117205612
C3×#52053