Finite Group Databasebeta

← #10592582⋊C3 #13142224

Subgroups — #10592582⋊C3 #13142224

Order 1728.

505 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 100 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×#3745⋊C3C211yesmaximal, characteristic, complemented
2C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#64104⋊C3C212yesmaximal, complemented
4H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
5#63636⋊C3C212yesmaximal, complemented
6#10592582C311yesmaximal, characteristic, complemented
7C3×#4883—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)—22no—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2212yes—
13#3752⋊C3—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15#3745⋊C3—22no—
16C8×(C3⋊D3)⋊C3—22no—
17C2×#3745C2×C311yescharacteristic, complemented
18C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
19#64104C2×C312yescomplemented
20C2×C3×D24—31no—
21C2×C3×D3×C8—31no—
22C3×#895—32no—
23C32×#107—21no—
24C32×#107D311yescharacteristic, complemented
25C32×#107C2×C311yescharacteristic, complemented
26#63636C2×C312yescomplemented
27C3×#809—32no—
28C22×(C3⋊D3)⋊C3—22no—
29C22×(C3⋊D3)⋊C3D411yescharacteristic
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3D411yescharacteristic
33C4⋊((C3⋊D3)⋊C3)—22no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35C4⋊((C3⋊D3)⋊C3)—44no—
36C4×(C3⋊D3)⋊C3—21no—
37C4×(C3⋊D3)⋊C3—21no—
38C8×H(3,𝔽3)D412yescomplemented
39(C3⋊(C3⋊C8))⋊C3—21no—
40C3×#466—91no—
41#4883—31no—
42#4883—91no—
43C2×C4⋊(C3⋊D3)—22no—
44C2×C4⋊(C3⋊D3)C22×C312yes—
45C2×C4×C3⋊D3C22×C311yescharacteristic
46C2×C3×D12—32no—
47C2×C3×D12—62no—
48C2×C3×C4×D3—31no—
49C2×C32×C8—21no—
50C2×C32×C8C2×D311yescharacteristic
51C2×C32×C8C22×C311yescharacteristic
52C32×C4⋊C4—22no—
53C32×C4⋊C4C2×D312yes—
54C32×C4⋊C4C22×C312yes—
55#3752—22no—
56C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
57#3745—22no—
58C8×C3⋊D3—22no—
59C3×(C2×D3)⋊C4—62no—
60C2×C3×C3⋊C8—31no—
61C3×D24—62no—
62C3×D3×C8—62no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×(C3⋊D3)⋊C3—21no—
65C2×(C3⋊D3)⋊C3—21no—
66C2×(C3⋊D3)⋊C3—42no—
67C2×(C3⋊D3)⋊C3—42no—
68C2×(C3⋊D3)⋊C3—42no—
69C2×(C3⋊D3)⋊C3D812yescomplemented
70C4×H(3,𝔽3)—42no—
71C4×H(3,𝔽3)C2×D411yescharacteristic
72C4×H(3,𝔽3)#4111yescharacteristic
73(C3⋊Dic3)⋊C3—21no—
74C2×C3×D8—91no—
75C22×C3×C8—91no—
76C3×#121—92no—
77C2×D24—31no—
78C2×D24—91no—
79C2×D3×C8—31no—
80C2×D3×C8—91no—
81#895—32no—
82#895—92no—
83C3×#107—31no—
84C3×#107—31no—
85C3×#107—61no—
86C3×#107C3×D311yescharacteristic
87C3×#102—92no—
88#809—32no—
89#809—92no—
90C22×C3⋊D3—22no—
91C22×C3⋊D3C3×D411yescharacteristic
92C22×C3×D3—31no—
93C22×C3×D3—62no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—21no—
97C2×C32×C4—42no—
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C23×C311yescharacteristic
100C2×C3⋊Dic3C3×D411yescharacteristic
101C4⋊(C3⋊D3)—22no—
102C4⋊(C3⋊D3)—22no—
103C4⋊(C3⋊D3)—44no—
104C4×C3⋊D3—21no—
105C4×C3⋊D3—21no—
106C2×C3×Dic3—31no—
107C3×D12—62no—
108C3×D12—62no—
109C3×D12—124no—
110C3×C4×D3—61no—
111C3×C4×D3—61no—
112C32×C8—22no—
113C32×C8D1212yescomplemented
114C32×C8C3×D412yescomplemented
115C3⋊(C3⋊C8)—21no—
116C3×C3⋊C8—61no—
117#466—91noSylow
118C2×H(3,𝔽3)#12111yescharacteristic
119C2×H(3,𝔽3)C2×D811yescharacteristic
120C2×H(3,𝔽3)#13511yescharacteristic
121(C3⋊D3)⋊C3—22no—
122(C3⋊D3)⋊C3—82no—
123C2×C3×D4—92no—
124C2×C3×D4—182no—
125C22×C3×C4—91no—
126C2×D12—32no—
127C2×D12—62no—
128C2×D12—92no—
129C2×D12—182no—
130C2×C4×D3—31no—
131C2×C4×D3—91no—
132C3×D8—182no—
133C2×C3×C8—31no—
134C2×C3×C8—31no—
135C2×C3×C8—61no—
136C2×C3×C8—91no—
137C2×C3×C8—182no—
138C2×C3×C8C2×C3×D311yescharacteristic
139C3×C4⋊C4—32no—
140C3×C4⋊C4—32no—
141C3×C4⋊C4—62no—
142C3×C4⋊C4C2×C3×D312yes—
143C3×C22⋊C4—182no—
144(C2×D3)⋊C4—62no—
145(C2×D3)⋊C4—182no—
146C2×C3⋊C8—31no—
147C2×C3⋊C8—91no—
148D24—62no—
149D24—182no—
150D3×C8—62no—
151D3×C8—182no—
152C22×C32—21no—
153C22×C32D3×D411yescharacteristic
154C22×C32C2×C3×D411yescharacteristic
155C2×C3⋊D3—21no—
156C2×C3⋊D3—21no—
157C2×C3⋊D3—42no—
158C2×C3⋊D3—42no—
159C2×C3⋊D3—42no—
160C2×C3⋊D3C3×D812yescomplemented
161C2×C3×D3—32no—
162C2×C3×D3—61no—
163C2×C3×D3—61no—
164C2×C3×D3—122no—
165C2×C3×D3—122no—
166C2×C3×D3—122no—
167C32×C4—21no—
168C32×C4—21no—
169C32×C4—42no—
170C32×C4—42no—
171C32×C4—82no—
172C32×C4C2×D1211yescharacteristic
173C32×C4#23911yescharacteristic
174C32×C4C2×C3×D411yescharacteristic
175C32×C4C3×#4111yescharacteristic
176C3⋊Dic3—21no—
177C3×Dic3—61no—
178C2×D8—91no—
179C22×C8—91no—
180#121—92no—
181#107(C3⋊D3)⋊C311yescharacteristic, complemented
182#102—92no—
183H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
184C23×C3—91no—
185C23×C3—182no—
186C22×D3—31no—
187C22×D3—62no—
188C22×D3—91no—
189C22×D3—182no—
190C3×D4—182no—
191C3×D4—182no—
192C3×D4—364no—
193C2×C3×C4—22no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—62no—
197C2×C3×C4—62no—
198C2×C3×C4—61no—
199C2×C3×C4—91no—
200C2×C3×C4—122no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4C22×C3×D311yescharacteristic
204C2×Dic3—31no—
205C2×Dic3—91no—
206D12—62no—
207D12—62no—
208D12—124no—
209D12—182no—
210D12—182no—
211D12—364no—
212C4×D3—61no—
213C4×D3—61no—
214C4×D3—181no—
215C4×D3—181no—
216C3×C8—32no—
217C3×C8—32no—
218C3×C8—62no—
219C3×C8—181no—
220C3×C8C3×D1212yes—
221C3⋊C8—61no—
222C3⋊C8—181no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32#89511yescharacteristic
227C2×C32D3×D811yescharacteristic
228C2×C32#91911yescharacteristic
229C2×C32C3×#12111yescharacteristic
230C2×C32C2×C3×D811yescharacteristic
231C2×C32C3×#13511yescharacteristic
232C3⋊D3—22no—
233C3⋊D3—82no—
234C3×D3—62no—
235C3×D3—242no—
236C2×D4—92no—
237C2×D4—182no—
238C22×C4—91no—
239D8—182no—
240C2×C8—91no—
241C2×C8—182no—
242C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
243C4⋊C4C2×(C3⋊D3)⋊C312yes—
244C22⋊C4—182no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—61no—
248C22×C3—92no—
249C22×C3—181no—
250C22×C3—181no—
251C22×C3—362no—
252C22×C3—362no—
253C22×C3—362no—
254C22×C3C3×D3×D411yescharacteristic
255C2×D3—32no—
256C2×D3—61no—
257C2×D3—61no—
258C2×D3—92no—
259C2×D3—122no—
260C2×D3—122no—
261C2×D3—122no—
262C2×D3—181no—
263C2×D3—181no—
264C2×D3—362no—
265C2×D3—362no—
266C2×D3—362no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—42no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—122no—
275C3×C4—122no—
276C3×C4—181no—
277C3×C4—242no—
278C3×C4C2×C3×D1211yescharacteristic
279C3×C4C3×#23911yescharacteristic
280Dic3—61no—
281Dic3—181no—
282C32—21no—
283C32#488311yescharacteristic, complemented
284C32C3×#46611yescharacteristic, complemented
285C23—91no—
286C23—182no—
287D4—182no—
288D4—182no—
289D4—364no—
290C2×C4—22no—
291C2×C4—91no—
292C2×C4—181no—
293C2×C4—181no—
294C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
295C8—181no—
296C8C4⋊((C3⋊D3)⋊C3)12yescomplemented
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—61no—
304C2×C3—61no—
305C2×C3—61no—
306C2×C3—182no—
307C2×C3—722no—
308C2×C3C3×#89511yescharacteristic
309C2×C3C3×D3×D811yescharacteristic
310C2×C3C3×#91911yescharacteristic
311D3—62no—
312D3—182no—
313D3—242no—
314D3—722no—
315C22—92no—
316C22—181no—
317C22—181no—
318C22—362no—
319C22—362no—
320C22—362no—
321C22D4×(C3⋊D3)⋊C311yescharacteristic
322C4—42no—
323C4—181no—
324C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
325C4#8872211yescharacteristic
326C3—31no—
327C3—31no—
328C3—61no—
329C3C3×#488311yescharacteristic
330C2—182no—
331C2—722no—
332C2#64104⋊C311yescharacteristic
333C2D8×(C3⋊D3)⋊C311yescharacteristic
334C2#64128⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#466
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
1. C2×#3745⋊C3C2semidirect
2. C2×C8×(C3⋊D3)⋊C3C2semidirect
3. #64104⋊C3C2semidirect
4. H(3,𝔽3)×#107C2semidirect
5. #63636⋊C3C2semidirect
6. #10592582C3semidirect
17. C2×#3745C2×C3semidirect
18. C2×C8×C3⋊D3C2×C3semidirect
19. #64104C2×C3semidirect
24. C32×#107D3semidirect
25. C32×#107C2×C3semidirect
26. #63636C2×C3semidirect
38. C8×H(3,𝔽3)D4semidirect
69. C2×(C3⋊D3)⋊C3D8semidirect
113. C32×C8D12semidirect
114. C32×C8C3×D4semidirect
160. C2×C3⋊D3C3×D8semidirect
181. #107(C3⋊D3)⋊C3semidirect
183. H(3,𝔽3)#466semidirect
283. C32#4883semidirect
284. C32C3×#466semidirect
296. C8C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
C2×C8×(C3⋊D3)⋊C32
#64104⋊C32
H(3,𝔽3)×#1072
#63636⋊C32
#105925823
C3×#48833