Finite Group Databasebeta

← #13138856

Subgroups — #13138856

Order 1728.

505 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721141C212yesmaximal, complemented
4#11720551C211yesmaximal, characteristic, complemented
5#11720308C212yesmaximal, complemented
6#10590844—31nomaximal
7#10590832—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×#6198—22no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084C2212yes—
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14#3745⋊C3—22no—
15C8×(C3⋊D3)⋊C3—22no—
16#88450—22no—
17C2×#3745D311yescharacteristic, complemented
18C2×C8×C3⋊D3D311yescharacteristic, complemented
19C2×C3×D24—31no—
20C2×C3×D3×C8—31no—
21#63916—32no—
22#63909—32no—
23C3×#818—31no—
24C3×#818—31no—
25C3×#818—61no—
26#63564—32no—
27#63562—32no—
28C22×(C3⋊D3)⋊C3—22no—
29C22×(C3⋊D3)⋊C3D411yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084—22no—
32C2×(C3⋊Dic3)⋊C3D411yescharacteristic
33C4⋊((C3⋊D3)⋊C3)—22no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35C4×(C3⋊D3)⋊C3—21no—
36C4×(C3⋊D3)⋊C3—21no—
37#6198—44no—
38C8×H(3,𝔽3)D412yescomplemented
39(C3⋊(C3⋊C8))⋊C3—21no—
40#5157—91no—
41#5115—91no—
42#4883—91no—
43C2×C4⋊(C3⋊D3)C2×D312yes—
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×C3×D12—32no—
46C2×C3×C4×D3—31no—
47C2×#626—62no—
48C2×#626—62no—
49C2×C32×C8—21no—
50C2×C32×C8C2×D311yescharacteristic
51C2×C32×C8C2×D311yescharacteristic
52C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
53#3745—22no—
54C8×C3⋊D3—22no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—62no—
58C2×C3×C3⋊C8—31no—
59C3×D24—62no—
60C3×D3×C8—62no—
61#3722—62no—
62(C2×D3)⋊Dic3—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)C2×D411yescharacteristic
71C4×H(3,𝔽3)#4111yescharacteristic
72#1084—42no—
73(C3⋊Dic3)⋊C3—21no—
74C2×C3×D8—91no—
75C22×C3×C8—91no—
76#938—92no—
77#930—92no—
78C2×D24—31no—
79C2×D24—91no—
80C2×D3×C8—31no—
81C2×D3×C8—91no—
82#895—92no—
83C3×#107—91no—
84#832—92no—
85#821—92no—
86#818—31no—
87#818—31no—
88#818—61no—
89#809—92no—
90C22×C3⋊D3—22no—
91C22×C3⋊D3D1211yescharacteristic
92C22×C3×D3—31no—
93C22×C3×D3—62no—
94C2×C32×C4—21no—
95C2×C32×C4C22×D311yescharacteristic
96C2×C32×C4C22×D311yescharacteristic
97C2×C3⋊Dic3D1211yescharacteristic
98C4⋊(C3⋊D3)—22no—
99C4⋊(C3⋊D3)—22no—
100C4×C3⋊D3—21no—
101C4×C3⋊D3—21no—
102C2×C3×Dic3—31no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—122no—
106C3×D12—62no—
107C3×D12—62no—
108C3×C4×D3—61no—
109C3×C4×D3—61no—
110#626—124no—
111#626—124no—
112C32×C8—22no—
113C32×C8#6712yescomplemented
114C32×C8#6712yescomplemented
115C3⋊(C3⋊C8)—21no—
116C3×C3⋊C8—61no—
117#466—271noSylow
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—
124C22×C3×C4—91no—
125C2×#67—182no—
126C2×#67—182no—
127C2×D12—32no—
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×C8D3211yescharacteristic
139C3×C4⋊C4—92no—
140#217—182no—
141(C2×D3)⋊C4—182no—
142(C2×D3)⋊C4—182no—
143C4⋊Dic3—32no—
144C4⋊Dic3—32no—
145C4⋊Dic3—62no—
146C2×C3⋊C8—31no—
147C2×C3⋊C8—91no—
148D24—62no—
149D24—182no—
150D3×C8—62no—
151D3×C8—182no—
152C22×C32—21no—
153C22×C32C2×D1211yescharacteristic
154C22×C32D3×D411yescharacteristic
155C2×C3⋊D3—21no—
156C2×C3⋊D3—21no—
157C2×C3⋊D3—42no—
158C2×C3⋊D3—42no—
159C2×C3⋊D3—42no—
160C2×C3⋊D3D2412yescomplemented
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#23511yescharacteristic
170C32×C4#23711yescharacteristic
171C32×C4C2×#6711yescharacteristic
172C32×C4C2×#6711yescharacteristic
173C3⋊Dic3—21no—
174C3×Dic3—61no—
175C3×Dic3—122no—
176C3×Dic3—122no—
177C3×Dic3—242no—
178C2×D8—91no—
179C22×C8—91no—
180#121—272no—
181#107—91no—
182#102—272no—
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—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—61no—
195C2×C3×C4—91no—
196C2×C3×C4—181no—
197C2×C3×C4—181no—
198C2×C3×C4—182no—
199C2×C3×C4C2×D3211yescharacteristic
200#67—364no—
201#67—364no—
202C2×Dic3—31no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—91no—
206C2×Dic3—122no—
207D12—62no—
208D12—62no—
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×C8#62512yes—
221C3⋊C8—61no—
222C3⋊C8—181no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32C2×D2411yescharacteristic
227C2×C32#90411yescharacteristic
228C2×C32D3×D811yescharacteristic
229C2×C32#91211yescharacteristic
230C2×C32#93011yescharacteristic
231C2×C32#93811yescharacteristic
232C3⋊D3—22no—
233C3⋊D3—82no—
234C3×D3—62no—
235C3×D3—242no—
236C2×D4—92no—
237C2×D4—542no—
238C22×C4—91no—
239D8—182no—
240C2×C8—91no—
241C2×C8—182no—
242C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C4⋊C4—92no—
244C22⋊C4—542no—
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×C3D3×D1211yescharacteristic
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—61no—
272C3×C4—61no—
273C3×C4—181no—
274C3×C4—362no—
275C3×C4#379511yescharacteristic
276C3×C4C2×#62511yescharacteristic
277Dic3—61no—
278Dic3—122no—
279Dic3—122no—
280Dic3—181no—
281Dic3—242no—
282C32—21no—
283C32#511511yescharacteristic, complemented
284C32#515711yescharacteristic, complemented
285C23—91no—
286C23—182no—
287D4—182no—
288D4—182no—
289D4—1084no—
290C2×C4—91no—
291C2×C4—181no—
292C2×C4—181no—
293C2×C4—182no—
294C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
295C8—181no—
296C8#619812yescomplemented
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×C3D3×D2411yescharacteristic
309C2×C3#6380711yescharacteristic
310C2×C3#6390811yescharacteristic
311D3—62no—
312D3—182no—
313D3—242no—
314D3—722no—
315C22—92no—
316C22—181no—
317C22—181no—
318C22—362no—
319C22—362no—
320C22—362no—
321C22#8865411yescharacteristic
322C4—181no—
323C4—362no—
324C4#8864811yescharacteristic
325C4C2×#619811yescharacteristic
326C3—31no—
327C3—31no—
328C3—61no—
329C3#1059083111yescharacteristic
330C2—182no—
331C2—722no—
332C2#1172084711yescharacteristic
333C2#1172085511yescharacteristic
334C2#1172114111yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#466
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

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. #11721141C2semidirect
4. #11720551C2semidirect
5. #11720308C2semidirect
17. C2×#3745D3semidirect
18. C2×C8×C3⋊D3D3semidirect
38. C8×H(3,𝔽3)D4semidirect
69. C2×(C3⋊D3)⋊C3D8semidirect
113. C32×C8#67semidirect
114. C32×C8#67semidirect
160. C2×C3⋊D3D24semidirect
183. H(3,𝔽3)#466semidirect
283. C32#5115semidirect
284. C32#5157semidirect
296. C8#6198semidirect

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
#117211412
#117205512
#117203082
#105908443
#105908323