Finite Group Databasebeta

← #13138859

Subgroups — #13138859

Order 1728.

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

normal subgroups only

This group has 435 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×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#3746⋊C3C211yesmaximal, characteristic, complemented
3#11721106C212yesmaximal
4#11720551C211yesmaximal, characteristic, complemented
5#11720309C212yesmaximal, complemented
6#10590857—31nomaximal
7#10590852—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
13C8×(C3⋊D3)⋊C3—22no—
14#3746⋊C3—22no—
15#88450—22no—
16#88449—22no—
17C2×#3746D311yescharacteristic, complemented
18C2×C8×C3⋊D3D311yescharacteristic
19C2×C3×Dic12—31no—
20C2×C3×D3×C8—31no—
21#63871—32no—
22#63857—32no—
23C3×#818—31no—
24C3×#818—31no—
25C3×#818—61no—
26#63567—32no—
27#63565—32no—
28C22×(C3⋊D3)⋊C3D411yescharacteristic
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—22no—
31C2×(C3⋊Dic3)⋊C3—22no—
32C2×(C3⋊Dic3)⋊C3D411yescharacteristic
33C4×(C3⋊D3)⋊C3—21no—
34C4×(C3⋊D3)⋊C3—21no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37C8×H(3,𝔽3)D412yes—
38(C3⋊(C3⋊C8))⋊C3—21no—
39#5188—91no—
40#5116—91no—
41#4887—91no—
42C2×C4×C3⋊D3C2×D311yescharacteristic
43C2×#634C2×D312yes—
44C2×C3×C4×D3—31no—
45C2×C3×Dic6—32no—
46C2×C32×C8—21no—
47C2×C32×C8C2×D311yescharacteristic
48C2×C32×C8C2×D311yescharacteristic
49C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
50#3746—22no—
51C8×C3⋊D3—22no—
52C3×C4⋊Dic3—32no—
53C3×C4⋊Dic3—32no—
54C3×C4⋊Dic3—62no—
55C2×C3×C3⋊C8—31no—
56C3×Dic12—62no—
57C3×D3×C8—62no—
58Dic3⋊Dic3—62no—
59Dic3⋊Dic3—62no—
60#3722—62no—
61(C2×D3)⋊Dic3—62no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×(C3⋊D3)⋊C3—21no—
64C2×(C3⋊D3)⋊C3—21no—
65C2×(C3⋊D3)⋊C3Q1612yescomplemented
66C4×H(3,𝔽3)C2×D411yescharacteristic
67C4×H(3,𝔽3)#4111yescharacteristic
68#1084—42no—
69(C3⋊Dic3)⋊C3—21no—
70(C3⋊Dic3)⋊C3—42no—
71C2×C3×Q16—91no—
72C22×C3×C8—91no—
73#946—92no—
74#924—92no—
75C2×Dic12—31no—
76C2×Dic12—91no—
77C2×D3×C8—31no—
78C2×D3×C8—91no—
79#897—92no—
80C3×#107—91no—
81#835—92no—
82#818—31no—
83#818—31no—
84#818—61no—
85#816—92no—
86#810—92no—
87C22×C3⋊D3D1211yescharacteristic
88C22×C3×D3—31no—
89C2×C32×C4—21no—
90C2×C32×C4C22×D311yescharacteristic
91C2×C32×C4C22×D311yescharacteristic
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3D1211yescharacteristic
94C4×C3⋊D3—21no—
95C4×C3⋊D3—21no—
96#634—22no—
97#634—22no—
98C2×C3×Dic3—31no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—122no—
103C3×C4×D3—61no—
104C3×C4×D3—61no—
105C3×Dic6—62no—
106C3×Dic6—62no—
107C32×C8—22no—
108C32×C8#6712yes—
109C32×C8#6712yes—
110C3⋊(C3⋊C8)—21no—
111C3×C3⋊C8—61no—
112#467—271noSylow
113C2×H(3,𝔽3)#12111yescharacteristic
114C2×H(3,𝔽3)C2×Q1611yescharacteristic
115C2×H(3,𝔽3)#13511yescharacteristic
116(C3⋊D3)⋊C3—22no—
117C2×C3×Q8—92no—
118C22×C3×C4—91no—
119C2×C4×D3—31no—
120C2×C4×D3—91no—
121C2×Dic6—32no—
122C2×Dic6—92no—
123C3×Q16—182no—
124C2×C3×C8—31no—
125C2×C3×C8—31no—
126C2×C3×C8—61no—
127C2×C3×C8—91no—
128C2×C3×C8—182no—
129C2×C3×C8D3211yescharacteristic
130C3×C4⋊C4—92no—
131#217—182no—
132(C2×D3)⋊C4—182no—
133(C2×D3)⋊C4—182no—
134C4⋊Dic3—32no—
135C4⋊Dic3—32no—
136C4⋊Dic3—62no—
137C4⋊Dic3—182no—
138Dic3⋊C4—182no—
139Dic3⋊C4—182no—
140C2×C3⋊C8—31no—
141C2×C3⋊C8—91no—
142Dic12—62no—
143Dic12—182no—
144D3×C8—62no—
145D3×C8—182no—
146C22×C32—21no—
147C22×C32C2×D1211yescharacteristic
148C22×C32D3×D411yescharacteristic
149C2×C3⋊D3—21no—
150C2×C3⋊D3—21no—
151C2×C3⋊D3Dic1212yescomplemented
152C2×C3×D3—32no—
153C2×C3×D3—61no—
154C2×C3×D3—61no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4#23511yescharacteristic
158C32×C4#23711yescharacteristic
159C32×C4C2×#6711yescharacteristic
160C32×C4C2×#6711yescharacteristic
161C3⋊Dic3—21no—
162C3⋊Dic3—42no—
163C3×Dic3—61no—
164C3×Dic3—122no—
165C3×Dic3—122no—
166C3×Dic3—122no—
167C3×Dic3—242no—
168C2×Q16—91no—
169C22×C8—91no—
170C22⋊Q8—272no—
171#107—91no—
172#103—272no—
173H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
174C23×C3—91no—
175C22×D3—31no—
176C22×D3—91no—
177C3×Q8—182no—
178C3×Q8—182no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—61no—
182C2×C3×C4—91no—
183C2×C3×C4—181no—
184C2×C3×C4—181no—
185C2×C3×C4—182no—
186C2×C3×C4—182no—
187C2×C3×C4C2×D3211yescharacteristic
188C2×Dic3—31no—
189C2×Dic3—62no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192C2×Dic3—91no—
193C2×Dic3—122no—
194C2×Dic3—182no—
195C4×D3—61no—
196C4×D3—61no—
197C4×D3—181no—
198C4×D3—181no—
199Dic6—62no—
200Dic6—62no—
201Dic6—182no—
202Dic6—182no—
203C3×C8—32no—
204C3×C8—32no—
205C3×C8—62no—
206C3×C8—181no—
207C3×C8#62512yes—
208C3⋊C8—61no—
209C3⋊C8—181no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32#90411yescharacteristic
214C2×C32C2×Dic1211yescharacteristic
215C2×C32D3×Q1611yescharacteristic
216C2×C32#91911yescharacteristic
217C2×C32#93011yescharacteristic
218C2×C32#93811yescharacteristic
219C3⋊D3—22no—
220C3×D3—62no—
221C2×Q8—92no—
222C22×C4—91no—
223Q16—182no—
224C2×C8—91no—
225C2×C8—182no—
226C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
227C4⋊C4—92no—
228C4⋊C4—542no—
229C22⋊C4—542no—
230C22×C3—31no—
231C22×C3—31no—
232C22×C3—61no—
233C22×C3—92no—
234C22×C3—181no—
235C22×C3—181no—
236C22×C3D3×D1211yescharacteristic
237C2×D3—32no—
238C2×D3—61no—
239C2×D3—61no—
240C2×D3—92no—
241C2×D3—181no—
242C2×D3—181no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4—181no—
250C3×C4—362no—
251C3×C4—362no—
252C3×C4#379511yescharacteristic
253C3×C4C2×#62511yescharacteristic
254Dic3—61no—
255Dic3—122no—
256Dic3—122no—
257Dic3—122no—
258Dic3—181no—
259Dic3—242no—
260Dic3—362no—
261C32—21no—
262C32#511611yescharacteristic, complemented
263C32#518811yescharacteristic, complemented
264C23—91no—
265Q8—182no—
266Q8—182no—
267C2×C4—91no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4—182no—
271C2×C4—182no—
272C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
273C8—181no—
274C8#619812yes—
275C2×C3—31no—
276C2×C3—31no—
277C2×C3—31no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—61no—
282C2×C3—61no—
283C2×C3—61no—
284C2×C3—182no—
285C2×C3D3×Dic1211yescharacteristic
286C2×C3#6380811yescharacteristic
287C2×C3#6390811yescharacteristic
288D3—62no—
289D3—182no—
290C22—92no—
291C22—181no—
292C22—181no—
293C22#8865411yescharacteristic
294C4—181no—
295C4—362no—
296C4—362no—
297C4#8864811yescharacteristic
298C4C2×#619811yescharacteristic
299C3—31no—
300C3—31no—
301C3—61no—
302C3#1059085111yescharacteristic
303C2—182no—
304C2#1172085011yescharacteristic
305C2#1172085111yescharacteristic
306C2#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
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#467
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
2. C2×#3746⋊C3C2semidirect
4. #11720551C2semidirect
5. #11720309C2semidirect
17. C2×#3746D3semidirect
65. C2×(C3⋊D3)⋊C3Q16semidirect
151. C2×C3⋊D3Dic12semidirect
173. H(3,𝔽3)#467semidirect
262. C32#5116semidirect
263. C32#5188semidirect

Maximal subgroups

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

subgroupindex
C2×C8×(C3⋊D3)⋊C32
C2×#3746⋊C32
#117211062
#117205512
#117203092
#105908573
#105908523