Finite Group Databasebeta

← #13137908

Subgroups — #13137908

Order 1728.

337 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#64099⋊C3C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic
3#11720530C211yesmaximal, characteristic
4#63634⋊C3C211yesmaximal, characteristic
5#11720314C211yesmaximal, characteristic
6#11720310C211yesmaximal, characteristic
7#11720306C211yesmaximal, characteristic
8#10590417—31nomaximal
9#10590414—31nomaximal
10C4⋊C4×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)—21no—
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88449—21no—
20#88448—21no—
21#64099D311yescharacteristic
22C3×#890—31no—
23#63872—31no—
24#63869—31no—
25#63634D311yescharacteristic
26C3×#808—31no—
27C3×#808—31no—
28C3×#808—61no—
29C3×#807—31no—
30#63576—31no—
31#63575—31no—
32#63573—31no—
33#63573—31no—
34#63553—31no—
35Dic3×C3⋊C8—31no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43#6180Q812yes—
44(C3⋊(C3⋊C8))⋊C3Q812yes—
45#5065—91no—
46#4878—91no—
47#4858—91no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4C2×D311yescharacteristic
50C32×C4⋊C4C2×D311yescharacteristic
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)C2×D311yescharacteristic
54#3750—21no—
55C4×C3⋊Dic3C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3—61no—
62C3×C4⋊Dic3—61no—
63C3×Dic3⋊C4—61no—
64C3×C4×Dic3—31no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—61no—
69#3724—61no—
70Dic3⋊Dic3—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73C22×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)—41no—
75C4×H(3,𝔽3)C2×Q811yescharacteristic
76C4×H(3,𝔽3)C2×Q811yescharacteristic
77#1084—41no—
78(C3⋊Dic3)⋊C3—21no—
79(C3⋊Dic3)⋊C3—21no—
80(C3⋊Dic3)⋊C3—41no—
81(C3⋊Dic3)⋊C3—41no—
82C3×#125—91no—
83#890—31no—
84#890—91no—
85#890—91no—
86#890—91no—
87#870—91no—
88C3×#107—91no—
89C3×#106—91no—
90#818—91no—
91#817—91no—
92C8×Dic3—91no—
93C8×Dic3—91no—
94#808—31no—
95#808—31no—
96#808—61no—
97#808—91no—
98#808—91no—
99#807—31no—
100#807—91no—
101#807—91no—
102#807—91no—
103C4×C3⋊C8—91no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—41no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3#6711yescharacteristic
113C2×C3⋊Dic3#6711yescharacteristic
114C2×C3×Dic3—31no—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—121no—
121C3⋊(C3⋊C8)Dic612yes—
122C3×C3⋊C8—32no—
123C3×C3⋊C8—32no—
124C3×C3⋊C8—32no—
125C3×C3⋊C8—62no—
126#499—271noSylow
127C2×H(3,𝔽3)C4⋊Q811yescharacteristic
128C2×H(3,𝔽3)#13511yescharacteristic
129C2×H(3,𝔽3)#13511yescharacteristic
130C2×C3×C8—91no—
131C2×C3×C8—91no—
132C3×C4⋊C4—31no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—91no—
137C3×C4⋊C4—91no—
138C3×C4⋊C4—181no—
139C3×C4⋊C4—181no—
140C3×C4⋊C4D3211yescharacteristic
141C3×C42—91no—
142C4⋊Dic3—31no—
143C4⋊Dic3—31no—
144C4⋊Dic3—31no—
145C4⋊Dic3—31no—
146C4⋊Dic3—61no—
147C4⋊Dic3—61no—
148C4⋊Dic3—91no—
149C4⋊Dic3—91no—
150C4⋊Dic3—181no—
151C4⋊Dic3—181no—
152C4⋊Dic3—181no—
153Dic3⋊C4—61no—
154Dic3⋊C4—181no—
155Dic3⋊C4—181no—
156Dic3⋊C4—181no—
157Dic3⋊C4—181no—
158Dic3⋊C4—181no—
159C4×Dic3—31no—
160C4×Dic3—91no—
161C2×C3⋊C8—31no—
162C2×C3⋊C8—31no—
163C2×C3⋊C8—31no—
164C2×C3⋊C8—61no—
165C2×C3⋊C8—91no—
166C22×C32—21no—
167C22×C32D3×D411yescharacteristic
168C22×C32C2×#6711yescharacteristic
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—41no—
172C32×C4—41no—
173C32×C4—81no—
174C32×C4C2×Dic611yescharacteristic
175C32×C4D3×Q811yescharacteristic
176C32×C4D3×Q811yescharacteristic
177C32×C4D3×Q811yescharacteristic
178C3⋊Dic3—21no—
179C3⋊Dic3—21no—
180C3⋊Dic3—41no—
181C3⋊Dic3—41no—
182C3×Dic3—61no—
183C3×Dic3—61no—
184C3×Dic3—121no—
185C3×Dic3—121no—
186C3×Dic3—121no—
187C3×Dic3—121no—
188C3×Dic3—241no—
189#125—91no—
190#125—271no—
191#107—91no—
192#107—271no—
193#106—91no—
194#106—271no—
195C4×C8—271no—
196H(3,𝔽3)#49911yescharacteristic, complemented, Sylow
197C2×C3×C4—21no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—121no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208C2×C3×C4—181no—
209C2×C3×C4C2×D3211yescharacteristic
210C2×Dic3—31no—
211C2×Dic3—31no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—91no—
217C2×Dic3—91no—
218C2×Dic3—121no—
219C2×Dic3—181no—
220C2×Dic3—181no—
221C3×C8—92no—
222C3×C8—92no—
223C3⋊C8—32no—
224C3⋊C8—32no—
225C3⋊C8—32no—
226C3⋊C8—62no—
227C3⋊C8—92no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32#88811yescharacteristic
232C2×C32#91211yescharacteristic
233C2×C32#91911yescharacteristic
234C2×C32#94411yescharacteristic
235C2×C32#95011yescharacteristic
236C2×C32#95011yescharacteristic
237C2×C8—91no—
238C2×C8—91no—
239C4⋊C4—91no—
240C4⋊C4—91no—
241C4⋊C4—91no—
242C4⋊C4—181no—
243C4⋊C4—181no—
244C4⋊C4—541no—
245C4⋊C4—541no—
246C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
247C42—91no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—61no—
251C22×C3D3×#6711yescharacteristic
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—41no—
257C3×C4—61no—
258C3×C4—61no—
259C3×C4—121no—
260C3×C4—121no—
261C3×C4—181no—
262C3×C4—181no—
263C3×C4—241no—
264C3×C4—361no—
265C3×C4—361no—
266C3×C4—361no—
267C3×C4D3×Dic611yescharacteristic
268C3×C4#379711yescharacteristic
269Dic3—61no—
270Dic3—61no—
271Dic3—121no—
272Dic3—121no—
273Dic3—121no—
274Dic3—121no—
275Dic3—181no—
276Dic3—181no—
277Dic3—241no—
278Dic3—361no—
279Dic3—361no—
280C32—21no—
281C32#487811yescharacteristic, complemented
282C32#506511yescharacteristic, complemented
283C2×C4—21no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C8—92no—
291C8—92no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—61no—
299C2×C3—61no—
300C2×C3—61no—
301C2×C3#6383911yescharacteristic
302C2×C3#6393311yescharacteristic
303C2×C3#6394911yescharacteristic
304C22#8867611yescharacteristic
305C4—41no—
306C4—181no—
307C4—181no—
308C4—361no—
309C4—361no—
310C4—361no—
311C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
312C4#8865111yescharacteristic
313C3—31no—
314C3—31no—
315C3—61no—
316C3#1059041311yescharacteristic
317C2#1172107911yescharacteristic
318C2#1172120111yescharacteristic
319C2#1172123511yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#499
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
196. H(3,𝔽3)#499semidirect
281. C32#4878semidirect
282. C32#5065semidirect

Maximal subgroups

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

subgroupindex
#64099⋊C32
#117210932
#117205302
#63634⋊C32
#117203142
#117203102
#117203062
#105904173
#105904143