Finite Group Databasebeta

← #13143852

Subgroups — #13143852

Order 1728.

869 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 869 conjugacy classes of subgroups — too many to draw. Only its 125 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×#88740C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic, complemented
3C2×C8×#272C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
5#11720573C211yesmaximal, characteristic, complemented
6#11720553C211yesmaximal, characteristic, complemented
7#11720553C211yesmaximal, characteristic, complemented
8C3×#4727—34nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235C411yescharacteristic
11C2×#6235C411yescharacteristic
12#88740—22no—
13#88740—22no—
14C2×C4⋊#272C411yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×#6229C411yescharacteristic
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19C2×C8×H(3,𝔽3)C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22#88527—22no—
23C8×#272—22no—
24C2×C3×#235—34no—
25C2×C3×ΓL(1,𝔽25)—34no—
26C2×C3×D3×C8—34no—
27C32×C22⋊C8D314yescomplemented
28C3×#830—34no—
29C3×(C2×D3)⋊C8—34no—
30C3×(C2×D3)⋊C8—34no—
31C23×H(3,𝔽3)C2×C411yescharacteristic
32C22×#272C2×C411yescharacteristic
33C22×#272C2×C411yescharacteristic
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38#6235—22no—
39#6235—22no—
40C2×#1084C2×C411yescharacteristic
41C2×#1084C2×C411yescharacteristic
42C4⋊#272—22no—
43C4×#272—21no—
44C4×#272—21no—
45C4×#272D412yes—
46C4×#272D412yes—
47#6229—22no—
48C8×H(3,𝔽3)—21no—
49C8×H(3,𝔽3)—21no—
50#6180—21no—
51#6180—21no—
52C3×#408—91no—
53#4727—34no—
54C22×C32×C4C2×D314yes—
55C2×C3×#67—34no—
56C2×C3×#67—34no—
57C3×#235—68no—
58C3×#235—68no—
59C2×C3×D12—34no—
60C2×C3×C4×D3—34no—
61C2×C3×C4×D3—34no—
62C2×C3×Dic6—34no—
63C2×C32×C8C2×D314yes—
64C2×C32×C8C2×D314yes—
65C2×C3×C3⋊C8—34no—
66C2×C3×C3⋊C8—34no—
67C3×ΓL(1,𝔽25)—68no—
68C3×D3×C8—68no—
69C22×H(3,𝔽3)—21no—
70C22×H(3,𝔽3)—21no—
71C22×H(3,𝔽3)—21no—
72C22×H(3,𝔽3)C22×C411yescharacteristic
73C2×#272—21no—
74C2×#272—21no—
75C2×#272—21no—
76C2×#272—21no—
77C2×#272—21no—
78C2×#272C22⋊C412yes—
79C4×H(3,𝔽3)—21no—
80C4×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)C2×D411yescharacteristic
82#1084—21no—
83#1084C22⋊C412yes—
84C2×C3×#41—91no—
85C2×#235—34no—
86C2×C3×M16—91no—
87C22×C3×C8—91no—
88C2×ΓL(1,𝔽25)—34no—
89C2×D3×C8—34no—
90C3×C22⋊C8—34no—
91C3×C22⋊C8—91no—
92C3×C22⋊C8—91no—
93C3×C22⋊C8—91no—
94C3×C22⋊C8C3⋊D311yescharacteristic
95#830—34no—
96(C2×D3)⋊C8—34no—
97(C2×D3)⋊C8—34no—
98C23×C32C4×D314yes—
99C22×C3×D3—34no—
100C22×C3×D3—34no—
101C2×C32×C4—24no—
102C2×C32×C4—24no—
103C2×C32×C4C4×D314yes—
104C2×C32×C4C22×D314yes—
105C3×#67—68no—
106C3×#67—68no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—34no—
109C3×D12—68no—
110C3×C4×D3—38no—
111C3×C4×D3—38no—
112C3×C4×D3—64no—
113C3×C4×D3—64no—
114C3×Dic6—68no—
115C32×C8—24no—
116C32×C8—24no—
117C3×C3⋊C8—64no—
118C3×C3⋊C8—64no—
119#408—91noSylow
120C2×H(3,𝔽3)—41no—
121C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
122C2×H(3,𝔽3)#13111yescharacteristic
123C2×H(3,𝔽3)#13111yescharacteristic
124#272—22no—
125#272—41no—
126C3×#41—182no—
127C3×#41—182no—
128C2×C3×Q8—91no—
129C2×C3×D4—91no—
130C2×C3×D4—91no—
131C2×C3×D4—91no—
132C22×C3×C4—34no—
133C22×C3×C4—91no—
134C22×C3×C4—91no—
135C22×C3×C4C2×C3⋊D311yescharacteristic
136C2×#67—34no—
137C2×#67—34no—
138#235—68no—
139#235—68no—
140C2×D12—34no—
141C2×C4×D3—34no—
142C2×C4×D3—34no—
143C2×Dic6—34no—
144C3×M16—182no—
145C2×C3×C8—34no—
146C2×C3×C8—34no—
147C2×C3×C8—91no—
148C2×C3×C8—91no—
149C2×C3×C8—182no—
150C2×C3×C8C2×C3⋊D311yescharacteristic
151C2×C3×C8C2×C3⋊D311yescharacteristic
152C2×C3⋊C8—34no—
153C2×C3⋊C8—34no—
154ΓL(1,𝔽25)—68no—
155D3×C8—68no—
156C22×C32—24no—
157C22×C32—24no—
158C22×C32—24no—
159C22×C32C2×C4×D314yes—
160C2×C3×D3—38no—
161C2×C3×D3—64no—
162C2×C3×D3—64no—
163C2×C3×D3—64no—
164C2×C3×D3—64no—
165C2×C3×D3—64no—
166C32×C4—24no—
167C32×C4D3×D414yes—
168C32×C4D3×D414yes—
169C3×Dic3—38no—
170C3×Dic3—64no—
171C2×#41—91no—
172C2×M16—91no—
173C22×C8—91no—
174C22⋊C8—91no—
175C22⋊C8—91no—
176C22⋊C8—91no—
177C22⋊C8#27211yescharacteristic, complemented
178H(3,𝔽3)#40811yescharacteristic, complemented, Sylow
179C23×C3—34no—
180C23×C3—91no—
181C23×C3—91no—
182C23×C3C4×C3⋊D311yescharacteristic
183C22×D3—34no—
184C22×D3—34no—
185C3×Q8—182no—
186C3×D4—182no—
187C3×D4—182no—
188C3×D4—182no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—34no—
192C2×C3×C4—34no—
193C2×C3×C4—64no—
194C2×C3×C4—64no—
195C2×C3×C4—91no—
196C2×C3×C4—92no—
197C2×C3×C4—92no—
198C2×C3×C4—91no—
199C2×C3×C4—181no—
200C2×C3×C4—181no—
201C2×C3×C4C4×C3⋊D311yescharacteristic
202C2×C3×C4C22×C3⋊D311yescharacteristic
203#67—68no—
204#67—68no—
205C2×Dic3—34no—
206C2×Dic3—34no—
207D12—68no—
208C4×D3—38no—
209C4×D3—38no—
210C4×D3—64no—
211C4×D3—64no—
212Dic6—68no—
213C3×C8—21no—
214C3×C8—21no—
215C3×C8—64no—
216C3×C8—64no—
217C3×C8—181no—
218C3×C8—181no—
219C3⋊C8—64no—
220C3⋊C8—64no—
221C2×C32—44no—
222C2×C32D3×C22⋊C414yes—
223C2×C32#90114yes—
224C2×C32#90714yes—
225C3×D3—68no—
226C3×D3—124no—
227#41—182no—
228#41—182no—
229C2×Q8—91no—
230C2×D4—91no—
231C2×D4—91no—
232C2×D4—91no—
233C22×C4—91no—
234C22×C4—91no—
235C22×C4C2×#27211yescharacteristic
236M16—182no—
237C2×C8—91no—
238C2×C8—91no—
239C2×C8—182no—
240C2×C8C2×#27211yescharacteristic
241C2×C8C2×#27211yescharacteristic
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—34no—
246C22×C3—64no—
247C22×C3—64no—
248C22×C3—64no—
249C22×C3—92no—
250C22×C3—181no—
251C22×C3—181no—
252C22×C3—181no—
253C22×C3—181no—
254C22×C3—181no—
255C22×C3C2×C4×C3⋊D311yescharacteristic
256C2×D3—38no—
257C2×D3—64no—
258C2×D3—64no—
259C2×D3—64no—
260C2×D3—64no—
261C2×D3—64no—
262C3×C4—21no—
263C3×C4—34no—
264C3×C4—34no—
265C3×C4—64no—
266C3×C4—92no—
267C3×C4—181no—
268C3×C4D4×C3⋊D311yescharacteristic
269C3×C4D4×C3⋊D311yescharacteristic
270Dic3—38no—
271Dic3—64no—
272C32#472714yescomplemented
273C23—91no—
274C23—91no—
275C23C4×#27211yescharacteristic
276Q8—182no—
277D4—182no—
278D4—182no—
279D4—182no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—91no—
283C2×C4—92no—
284C2×C4—92no—
285C2×C4—91no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4C4×#27211yescharacteristic
289C2×C4C22×#27211yescharacteristic
290C8—21no—
291C8—21no—
292C8—181no—
293C8—181no—
294C2×C3—34no—
295C2×C3—34no—
296C2×C3—34no—
297C2×C3—41no—
298C2×C3—124no—
299C2×C3—182no—
300C2×C3—361no—
301C2×C3C22⋊C4×C3⋊D311yescharacteristic
302C2×C3#6411011yescharacteristic
303C2×C3#6411611yescharacteristic
304D3—68no—
305D3—124no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—92no—
310C22—181no—
311C22—181no—
312C22—181no—
313C22—181no—
314C22—181no—
315C22C2×C4×#27211yescharacteristic
316C4—21no—
317C4—92no—
318C4—181no—
319C4D4×#27211yescharacteristic
320C4D4×#27211yescharacteristic
321C3—34no—
322C3#1059242611yescharacteristic
323C2—41no—
324C2—182no—
325C2—361no—
326C2C22⋊C4×#27211yescharacteristic
327C2#1172177611yescharacteristic
328C2#1172178611yescharacteristic
subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C4×C3⋊D3
prime radicalC2×#88740C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×C22⋊C8C3⋊D3
hypocenterH(3,𝔽3)#408
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#88527C2semidirect
3. C2×C8×#272C2semidirect
4. H(3,𝔽3)×C22⋊C8C2semidirect
5. #11720573C2semidirect
6. #11720553C2semidirect
7. #11720553C2semidirect
27. C32×C22⋊C8D3semidirect
177. C22⋊C8#272semidirect
178. H(3,𝔽3)#408semidirect
272. C32#4727semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
C2×#885272
C2×C8×#2722
H(3,𝔽3)×C22⋊C82
#117205732
#117205532
C3×#47273