Finite Group Databasebeta

← #13144794

Subgroups — #13144794

Order 1728.

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

normal subgroups only

This group has 836 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#11722675C211yesmaximal, characteristic
2H(3,𝔽3)×#137C211yesmaximal, characteristic, complemented
3#11721864C211yesmaximal, characteristic, complemented
4#11721859C211yesmaximal, characteristic, complemented
5#11720586C211yesmaximal, characteristic, complemented
6#11720562C211yesmaximal, characteristic, complemented
7#11720560C211yesmaximal, characteristic, complemented
8C3×#5202—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
11#88748—21no—
12D4×#272—22no—
13D4×#272—21no—
14#88740—21no—
15#88740C2211yescharacteristic
16C2×C4⋊#272—21no—
17C2×C4⋊#272C2211yescharacteristic
18Q16×H(3,𝔽3)—21no—
19SD16×H(3,𝔽3)—21no—
20M16×H(3,𝔽3)C2211yescharacteristic, complemented
21#88550—21no—
22#88549—21no—
23#88542—22no—
24C4×#1084C2211yescharacteristic, complemented
25#88538C2211yescharacteristic, complemented
26C3×#1009—34no—
27C32×#137D314yescomplemented
28C3×#947—34no—
29C3×#942—34no—
30C3×#837—34no—
31C3×#825—34no—
32C3×#823—34no—
33C22×#272—22no—
34C22×#272D412yes—
35Q8×H(3,𝔽3)—21no—
36Q8×H(3,𝔽3)—21no—
37Q8×H(3,𝔽3)D411yescharacteristic
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43#6235—22no—
44#6235—21no—
45C2×#1084—21no—
46C4⋊#272—22no—
47C4⋊#272—21no—
48C4⋊#272—21no—
49C4⋊#272D411yescharacteristic
50C4×#272—22no—
51C4×#272—21no—
52#6229D411yescharacteristic
53C8×H(3,𝔽3)—21no—
54#6180—21no—
55C3×#455—91no—
56#5202—34no—
57C32×#41C2×D314yes—
58C2×C32×Q8C2×D314yescomplemented
59C3×#239—64no—
60C3×D3×D4—68no—
61C3×D3×D4—64no—
62C3×#235—34no—
63C3×#235—64no—
64C2×C3×D12—34no—
65C2×C3×D12—64no—
66C32×Q16—24no—
67C32×SD16—24no—
68C32×M16C2×D314yescomplemented
69C3×#216—64no—
70C3×#215—64no—
71C3×(C2×D3)⋊C4—68no—
72C3×C4×Dic3—34no—
73C3×#208—34no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C2×#272—22no—
77C2×#272—21no—
78C2×#272—42no—
79C2×#272—82no—
80C4×H(3,𝔽3)—21no—
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)C2×D411yescharacteristic
83C4×H(3,𝔽3)C2×D411yescharacteristic
84#1084—21no—
85#1084—41no—
86C3×H(5,𝔽2)—91no—
87#1009—34no—
88C3×#137—34no—
89C3×#137—91no—
90C3×#137C3⋊D311yescharacteristic
91C3×#124—91no—
92#947—34no—
93#942—34no—
94C3×#104—91no—
95C3×#104—91no—
96C3×#100—91no—
97#837—34no—
98#825—34no—
99#823—34no—
100C22×C3×D3—38no—
101C22×C3×D3—68no—
102C32×Q8—24no—
103C32×Q8—24no—
104C32×Q8#6714yes—
105C32×D4—24no—
106C32×D4#6714yes—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4C22×D314yes—
110C3×#67—68no—
111C3×#67—64no—
112C2×C3×Dic3—64no—
113C3×D12—34no—
114C3×D12—68no—
115C3×D12—64no—
116C3×D12—64no—
117C3×C4×D3—68no—
118C3×C4×D3—64no—
119C3×Dic6—34no—
120C32×C8—24no—
121C3×C3⋊C8—64no—
122#455—91noSylow
123C2×H(3,𝔽3)—21no—
124C2×H(3,𝔽3)—41no—
125C2×H(3,𝔽3)#12011yescharacteristic
126#272—42no—
127#272—41no—
128C3×#41—34no—
129C3×#41—91no—
130C3×#41—181no—
131C3×#41—181no—
132C3×#41C2×C3⋊D311yescharacteristic
133C2×C3×Q8—34no—
134C2×C3×Q8C2×C3⋊D311yescharacteristic
135C2×C3×D4—91no—
136C2×C3×D4—181no—
137C2×C3×D4—182no—
138C2×C3×D4—181no—
139#239—64no—
140D3×D4—64no—
141D3×D4—68no—
142#235—34no—
143#235—64no—
144C2×D12—34no—
145C2×D12—64no—
146C3×Q16—21no—
147C3×Q16—64no—
148C3×Q16—181no—
149C3×SD16—21no—
150C3×SD16—64no—
151C3×SD16—181no—
152C3×M16—34no—
153C3×M16—91no—
154C3×M16C2×C3⋊D311yescharacteristic
155C3×C22⋊C4—182no—
156C3×C42—91no—
157#216—64no—
158#215—64no—
159(C2×D3)⋊C4—68no—
160C4×Dic3—34no—
161#208—34no—
162C22×C32—24no—
163C22×C32D3×D414yes—
164C2×C3×D3—64no—
165C2×C3×D3—68no—
166C2×C3×D3—128no—
167C2×C3×D3—248no—
168C32×C4—24no—
169C32×C4—44no—
170C32×C4D3×D414yes—
171C32×C4C2×#6714yes—
172C3×Dic3—64no—
173C3×Dic3—124no—
174H(5,𝔽2)—91no—
175#137—91no—
176#137#27211yescharacteristic, complemented
177#124—91no—
178#104—91no—
179#104—91no—
180#100—91no—
181H(3,𝔽3)#45511yescharacteristic, complemented, Sylow
182C23×C3—92no—
183C23×C3—182no—
184C22×D3—38no—
185C22×D3—68no—
186C3×Q8—21no—
187C3×Q8—21no—
188C3×Q8—34no—
189C3×Q8—64no—
190C3×Q8—64no—
191C3×Q8—91no—
192C3×Q8#63811yescharacteristic
193C3×D4—21no—
194C3×D4—34no—
195C3×D4—64no—
196C3×D4—91no—
197C3×D4—182no—
198C3×D4—182no—
199C3×D4—181no—
200C3×D4—181no—
201C3×D4—181no—
202C3×D4#63811yescharacteristic
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—34no—
206C2×C3×C4—64no—
207C2×C3×C4—64no—
208C2×C3×C4—182no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211C2×C3×C4C22×C3⋊D311yescharacteristic
212#67—64no—
213#67—68no—
214C2×Dic3—64no—
215D12—34no—
216D12—64no—
217D12—64no—
218D12—68no—
219C4×D3—64no—
220C4×D3—68no—
221Dic6—34no—
222C3×C8—21no—
223C3×C8—64no—
224C3×C8—181no—
225C3⋊C8—64no—
226C2×C32—24no—
227C2×C32—44no—
228C2×C32#93714yes—
229C3×D3—128no—
230C3×D3—124no—
231#41—91no—
232#41—181no—
233#41—181no—
234#41C2×#27211yescharacteristic
235C2×Q8C2×#27211yescharacteristic, complemented
236C2×D4—91no—
237C2×D4—181no—
238C2×D4—182no—
239C2×D4—181no—
240Q16—21no—
241Q16—181no—
242SD16—21no—
243SD16—181no—
244M16—91no—
245M16C2×#27211yescharacteristic, complemented
246C22⋊C4—182no—
247C42—91no—
248C22×C3—21no—
249C22×C3—34no—
250C22×C3—64no—
251C22×C3—181no—
252C22×C3—182no—
253C22×C3—362no—
254C22×C3—722no—
255C22×C3D4×C3⋊D311yescharacteristic
256C2×D3—64no—
257C2×D3—68no—
258C2×D3—128no—
259C2×D3—248no—
260C3×C4—21no—
261C3×C4—34no—
262C3×C4—34no—
263C3×C4—41no—
264C3×C4—64no—
265C3×C4—124no—
266C3×C4—181no—
267C3×C4—361no—
268C3×C4D4×C3⋊D311yescharacteristic
269C3×C4C2×#63811yescharacteristic
270Dic3—64no—
271Dic3—124no—
272C32#520214yescomplemented
273C23—92no—
274C23—182no—
275Q8—21no—
276Q8—21no—
277Q8—91no—
278Q8#623511yescharacteristic
279D4—21no—
280D4—91no—
281D4—182no—
282D4—182no—
283D4—181no—
284D4—181no—
285D4—181no—
286D4#623511yescharacteristic
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—182no—
290C2×C4—181no—
291C2×C4—181no—
292C2×C4C22×#27211yescharacteristic
293C8—21no—
294C8—181no—
295C2×C3—21no—
296C2×C3—34no—
297C2×C3—41no—
298C2×C3—64no—
299C2×C3—124no—
300C2×C3—362no—
301C2×C3—361no—
302C2×C3#6414611yescharacteristic
303D3—128no—
304D3—124no—
305C22—21no—
306C22—181no—
307C22—182no—
308C22—362no—
309C22—722no—
310C22D4×#27211yescharacteristic
311C4—21no—
312C4—41no—
313C4—181no—
314C4—361no—
315C4D4×#27211yescharacteristic
316C4C2×#623511yescharacteristic
317C3—34no—
318C3#1059290111yescharacteristic
319C2—21no—
320C2—41no—
321C2—362no—
322C2—361no—
323C2#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#11722675C2
hypercenterC3×#137C3⋊D3
hypocenterH(3,𝔽3)#455
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. #11721864C2semidirect
4. #11721859C2semidirect
5. #11720586C2semidirect
6. #11720562C2semidirect
7. #11720560C2semidirect
10. C2×Q8×H(3,𝔽3)C22semidirect
20. M16×H(3,𝔽3)C22semidirect
24. C4×#1084C22semidirect
25. #88538C22semidirect
27. C32×#137D3semidirect
58. C2×C32×Q8C2×D3semidirect
68. C32×M16C2×D3semidirect
176. #137#272semidirect
181. H(3,𝔽3)#455semidirect
235. C2×Q8C2×#272semidirect
245. M16C2×#272semidirect
272. C32#5202semidirect

Maximal subgroups

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

subgroupindex
#117226752
H(3,𝔽3)×#1372
#117218642
#117218592
#117205862
#117205622
#117205602
C3×#52023