Finite Group Databasebeta

← #13139011

Subgroups — #13139011

Order 1728.

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

normal subgroups only

This group has 459 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×#88530C211yesmaximal, characteristic, complemented
2C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3#11721107C212yesmaximal
4#63645⋊C3C211yesmaximal, characteristic, complemented
5#11720326C212yesmaximal, complemented
6#10590857—31nomaximal
7#10590851—31nomaximal
8C2×#6229C2212yes—
9C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88530—22no—
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14C8×(C3⋊D3)⋊C3—22no—
15#88450—22no—
16#88448—22no—
17C2×C8×C3⋊D3D311yescharacteristic
18C2×C3×Dic12—31no—
19C2×C3×Dic12—31no—
20C2×C3×Dic12—61no—
21C2×C3×D3×C8—31no—
22#63871—32no—
23#63856—32no—
24#63645D311yescharacteristic, complemented
25C3×#818—31no—
26#63566—32no—
27#63565—32no—
28C22×(C3⋊D3)⋊C3D411yescharacteristic
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—22no—
31#6229—22no—
32#6229—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3D411yescharacteristic
35C4×(C3⋊D3)⋊C3—21no—
36C4×(C3⋊D3)⋊C3—21no—
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×C3×C4×D3—31no—
44C2×C3×Dic6—32no—
45C2×C3×Dic6—32no—
46C2×C3×Dic6—62no—
47C2×C32×C8—21no—
48C2×C32×C8C2×D311yescharacteristic
49C2×C32×C8C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)C2×D312yes—
51C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
52C8×C3⋊D3—22no—
53C3×C4⋊Dic3—32no—
54C2×C3×C3⋊C8—31no—
55C3×Dic12—64no—
56C3×Dic12—62no—
57C3×Dic12—62no—
58C3×D3×C8—62no—
59#3724—62no—
60Dic3⋊Dic3—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)⋊C3Q1612yescomplemented
67C4×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)#4111yescharacteristic
69#1084—42no—
70(C3⋊Dic3)⋊C3—21no—
71(C3⋊Dic3)⋊C3—42no—
72C2×C3×Q16—91no—
73C22×C3×C8—91no—
74#946—92no—
75#924—92no—
76C2×Dic12—31no—
77C2×Dic12—31no—
78C2×Dic12—61no—
79C2×D3×C8—31no—
80C2×D3×C8—91no—
81#897—92no—
82C3×#107—91no—
83#835—92no—
84#818—31no—
85#818—91no—
86#816—92no—
87#810—92no—
88C22×C3⋊D3D1211yescharacteristic
89C22×C3×D3—31no—
90C2×C32×C4—21no—
91C2×C32×C4C22×D311yescharacteristic
92C2×C32×C4C22×D311yescharacteristic
93C2×C3⋊Dic3—22no—
94C2×C3⋊Dic3D1211yescharacteristic
95C4×C3⋊D3—21no—
96C4×C3⋊D3—21no—
97C2×C3×Dic3—31no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—122no—
102C3×C4×D3—61no—
103C3×C4×D3—61no—
104C3×Dic6—64no—
105C3×Dic6—62no—
106C3×Dic6—62no—
107C3×Dic6—62no—
108C3×Dic6—62no—
109C3×Dic6—122no—
110C32×C8—22no—
111C32×C8D1212yes—
112C32×C8#6712yes—
113C3⋊(C3⋊C8)—21no—
114C3×C3⋊C8—61no—
115#467—271noSylow
116C2×H(3,𝔽3)#12111yescharacteristic
117C2×H(3,𝔽3)C2×Q1611yescharacteristic
118C2×H(3,𝔽3)#13511yescharacteristic
119(C3⋊D3)⋊C3—22no—
120C2×C3×Q8—92no—
121C22×C3×C4—91no—
122C2×C4×D3—31no—
123C2×C4×D3—91no—
124C2×Dic6—32no—
125C2×Dic6—32no—
126C2×Dic6—62no—
127C3×Q16—182no—
128C2×C3×C8—31no—
129C2×C3×C8—31no—
130C2×C3×C8—61no—
131C2×C3×C8—91no—
132C2×C3×C8—182no—
133C2×C3×C8D3211yescharacteristic
134C3×C4⋊C4—92no—
135#217—182no—
136(C2×D3)⋊C4—182no—
137(C2×D3)⋊C4—182no—
138C4⋊Dic3—32no—
139C4⋊Dic3—92no—
140C4⋊Dic3—182no—
141Dic3⋊C4—182no—
142Dic3⋊C4—182no—
143C2×C3⋊C8—31no—
144C2×C3⋊C8—91no—
145Dic12—62no—
146Dic12—62no—
147Dic12—64no—
148D3×C8—62no—
149D3×C8—182no—
150C22×C32—21no—
151C22×C32C2×D1211yescharacteristic
152C22×C32D3×D411yescharacteristic
153C2×C3⋊D3—21no—
154C2×C3⋊D3—21no—
155C2×C3⋊D3Dic1212yescomplemented
156C2×C3×D3—32no—
157C2×C3×D3—61no—
158C2×C3×D3—61no—
159C32×C4—21no—
160C32×C4—21no—
161C32×C4C2×D1211yescharacteristic
162C32×C4#23511yescharacteristic
163C32×C4#23911yescharacteristic
164C32×C4C2×#6711yescharacteristic
165C3⋊Dic3—21no—
166C3⋊Dic3—42no—
167C3×Dic3—61no—
168C3×Dic3—124no—
169C3×Dic3—122no—
170C3×Dic3—122no—
171C3×Dic3—122no—
172C2×Q16—91no—
173C22×C8—91no—
174C22⋊Q8—272no—
175#107—91no—
176#103—272no—
177H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
178C23×C3—91no—
179C22×D3—31no—
180C22×D3—91no—
181C3×Q8—182no—
182C3×Q8—182no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—61no—
186C2×C3×C4—91no—
187C2×C3×C4—181no—
188C2×C3×C4—181no—
189C2×C3×C4—182no—
190C2×C3×C4—182no—
191C2×C3×C4C2×D3211yescharacteristic
192C2×Dic3—31no—
193C2×Dic3—62no—
194C2×Dic3—62no—
195C2×Dic3—62no—
196C2×Dic3—91no—
197C2×Dic3—122no—
198C2×Dic3—182no—
199C4×D3—61no—
200C4×D3—61no—
201C4×D3—181no—
202C4×D3—181no—
203Dic6—62no—
204Dic6—62no—
205Dic6—62no—
206Dic6—62no—
207Dic6—64no—
208Dic6—122no—
209C3×C8—32no—
210C3×C8—32no—
211C3×C8—62no—
212C3×C8—181no—
213C3×C8#62612yes—
214C3⋊C8—61no—
215C3⋊C8—181no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32#89511yescharacteristic
220C2×C32#90411yescharacteristic
221C2×C32C2×Dic1211yescharacteristic
222C2×C32#91211yescharacteristic
223C2×C32D3×Q1611yescharacteristic
224C2×C32#93011yescharacteristic
225C3⋊D3—22no—
226C3×D3—62no—
227C2×Q8—92no—
228C22×C4—91no—
229Q16—182no—
230C2×C8—91no—
231C2×C8—182no—
232C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
233C4⋊C4—92no—
234C4⋊C4—542no—
235C22⋊C4—542no—
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—61no—
239C22×C3—92no—
240C22×C3—181no—
241C22×C3—181no—
242C22×C3D3×D1211yescharacteristic
243C2×D3—32no—
244C2×D3—61no—
245C2×D3—61no—
246C2×D3—92no—
247C2×D3—181no—
248C2×D3—181no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—31no—
252C3×C4—31no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—181no—
256C3×C4—362no—
257C3×C4—362no—
258C3×C4#379911yescharacteristic
259C3×C4C2×#62611yescharacteristic
260Dic3—61no—
261Dic3—122no—
262Dic3—122no—
263Dic3—124no—
264Dic3—122no—
265Dic3—181no—
266Dic3—362no—
267C32—21no—
268C32#488711yescharacteristic, complemented
269C32#511611yescharacteristic, complemented
270C23—91no—
271Q8—182no—
272Q8—182no—
273C2×C4—91no—
274C2×C4—181no—
275C2×C4—181no—
276C2×C4—182no—
277C2×C4—182no—
278C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
279C8—181no—
280C8#620012yes—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—31no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—61no—
288C2×C3—61no—
289C2×C3—61no—
290C2×C3—182no—
291C2×C3D3×Dic1211yescharacteristic
292C2×C3#6380711yescharacteristic
293C2×C3#6390911yescharacteristic
294D3—62no—
295D3—182no—
296C22—92no—
297C22—181no—
298C22—181no—
299C22#8865411yescharacteristic
300C4—181no—
301C4—362no—
302C4—362no—
303C4#8865211yescharacteristic
304C4C2×#620011yescharacteristic
305C3—31no—
306C3—31no—
307C3—61no—
308C3#1059085211yescharacteristic
309C2—182no—
310C2#1172085111yescharacteristic
311C2#1172085511yescharacteristic
312C2#1172115611yescharacteristic
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
1. C2×#88530C2semidirect
4. #63645⋊C3C2semidirect
5. #11720326C2semidirect
24. #63645D3semidirect
66. C2×(C3⋊D3)⋊C3Q16semidirect
155. C2×C3⋊D3Dic12semidirect
177. H(3,𝔽3)#467semidirect
268. C32#4887semidirect
269. C32#5116semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
C2×C8×(C3⋊D3)⋊C32
#117211072
#63645⋊C32
#117203262
#105908573
#105908513