Finite Group Databasebeta

← #13138903

Subgroups — #13138903

Order 1728.

475 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 79 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×#272C211yesmaximal, characteristic
2C2×#3746⋊C3C211yesmaximal, characteristic, complemented
3#11721094C212yesmaximal
4#63645⋊C3C211yesmaximal, characteristic, complemented
5#11720313C212yesmaximal, complemented
6#10590852—31nomaximal
7#10590851—31nomaximal
8C2×C4×#272C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12C8×#272—22no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14#3746⋊C3—22no—
15#88451—22no—
16#88448—22no—
17C2×#3746D311yescharacteristic, complemented
18C2×C3×Dic12—31no—
19C2×C3×D3×C8—31no—
20C2×C3×D3×C8—31no—
21C2×C3×D3×C8—61no—
22#63857—32no—
23#63856—32no—
24#63645D311yescharacteristic, complemented
25C3×#818—31no—
26#63567—32no—
27#63566—32no—
28C22×#272D411yescharacteristic
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084D411yescharacteristic
31C4×#272—21no—
32C4×#272—21no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37C8×H(3,𝔽3)D412yes—
38#6180—21no—
39#5188—91no—
40#5116—91no—
41#4887—91no—
42C2×#634C2×D312yes—
43C2×C3×C4×D3—31no—
44C2×C3×C4×D3—31no—
45C2×C3×C4×D3—61no—
46C2×C3×Dic6—32no—
47C2×C32×C8—21no—
48C2×C32×C8C2×D311yescharacteristic
49C2×C32×C8C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)C2×D312yes—
51#3746—22no—
52C3×C4⋊Dic3—32no—
53C2×C3×C3⋊C8—31no—
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8—61no—
56C3×Dic12—62no—
57C3×D3×C8—64no—
58C3×D3×C8—62no—
59C3×D3×C8—62no—
60#3724—62no—
61Dic3⋊Dic3—62no—
62(C2×D3)⋊Dic3—62no—
63(C2×D3)⋊Dic3—62no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272—21no—
66C2×#272—21no—
67C2×#272Q1612yescomplemented
68C4×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)#4111yescharacteristic
70#1084—21no—
71(C3⋊Dic3)⋊C3—42no—
72(C3⋊Dic3)⋊C3—42no—
73C2×C3×Q16—91no—
74C22×C3×C8—91no—
75#946—92no—
76#924—92no—
77C2×Dic12—31no—
78C2×Dic12—91no—
79C2×D3×C8—31no—
80C2×D3×C8—31no—
81C2×D3×C8—61no—
82#897—92no—
83C3×#107—91no—
84#835—92no—
85#818—31no—
86#818—91no—
87#816—92no—
88#810—92no—
89C22×C3×D3—31no—
90C22×C3×D3—31no—
91C22×C3×D3—61no—
92C2×C32×C4—21no—
93C2×C32×C4C22×D311yescharacteristic
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—22no—
97#634—22no—
98#634—22no—
99C2×C3×Dic3—31no—
100C2×C3×Dic3—31no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C3×C4×D3—62no—
105C3×C4×D3—62no—
106C3×C4×D3—61no—
107C3×C4×D3—61no—
108C3×C4×D3—61no—
109C3×C4×D3—61no—
110C3×Dic6—62no—
111C3×Dic6—62no—
112C32×C8—22no—
113C32×C8D1212yes—
114C32×C8#6712yes—
115C3×C3⋊C8—62no—
116C3×C3⋊C8—61no—
117C3×C3⋊C8—61no—
118#467—271noSylow
119C2×H(3,𝔽3)#12111yescharacteristic
120C2×H(3,𝔽3)C2×Q1611yescharacteristic
121C2×H(3,𝔽3)#13511yescharacteristic
122#272—22no—
123C2×C3×Q8—92no—
124C22×C3×C4—91no—
125C2×C4×D3—31no—
126C2×C4×D3—31no—
127C2×C4×D3—61no—
128C2×Dic6—32no—
129C2×Dic6—92no—
130C3×Q16—182no—
131C2×C3×C8—31no—
132C2×C3×C8—31no—
133C2×C3×C8—61no—
134C2×C3×C8—91no—
135C2×C3×C8—182no—
136C2×C3×C8D3211yescharacteristic
137C3×C4⋊C4—92no—
138#217—182no—
139(C2×D3)⋊C4—182no—
140(C2×D3)⋊C4—182no—
141C4⋊Dic3—32no—
142C4⋊Dic3—92no—
143C4⋊Dic3—182no—
144Dic3⋊C4—182no—
145Dic3⋊C4—182no—
146C2×C3⋊C8—31no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—61no—
149Dic12—62no—
150Dic12—182no—
151D3×C8—62no—
152D3×C8—62no—
153D3×C8—64no—
154C22×C32—21no—
155C22×C32D3×D411yescharacteristic
156C22×C32D3×D411yescharacteristic
157C2×C3×D3—32no—
158C2×C3×D3—32no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—62no—
162C2×C3×D3—61no—
163C2×C3×D3—61no—
164C2×C3×D3—61no—
165C2×C3×D3—61no—
166C32×C4—21no—
167C32×C4—21no—
168C32×C4C2×D1211yescharacteristic
169C32×C4#23711yescharacteristic
170C32×C4#23911yescharacteristic
171C32×C4C2×#6711yescharacteristic
172C3⋊Dic3—42no—
173C3⋊Dic3—42no—
174C3×Dic3—62no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C3×Dic3—122no—
178C3×Dic3—122no—
179C2×Q16—91no—
180C22×C8—91no—
181C22⋊Q8—272no—
182#107—91no—
183#103—272no—
184H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
185C23×C3—91no—
186C22×D3—31no—
187C22×D3—31no—
188C22×D3—61no—
189C3×Q8—182no—
190C3×Q8—182no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—61no—
194C2×C3×C4—91no—
195C2×C3×C4—181no—
196C2×C3×C4—181no—
197C2×C3×C4—182no—
198C2×C3×C4—182no—
199C2×C3×C4C2×D3211yescharacteristic
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—61no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—182no—
206C2×Dic3—182no—
207C4×D3—61no—
208C4×D3—61no—
209C4×D3—61no—
210C4×D3—61no—
211C4×D3—62no—
212C4×D3—62no—
213Dic6—62no—
214Dic6—62no—
215Dic6—182no—
216Dic6—182no—
217C3×C8—32no—
218C3×C8—32no—
219C3×C8—62no—
220C3×C8—181no—
221C3×C8#62612yes—
222C3⋊C8—61no—
223C3⋊C8—61no—
224C3⋊C8—62no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32#89511yescharacteristic
229C2×C32#91211yescharacteristic
230C2×C32D3×Q1611yescharacteristic
231C2×C32D3×Q1611yescharacteristic
232C2×C32#91911yescharacteristic
233C2×C32#93811yescharacteristic
234C3×D3—64no—
235C3×D3—62no—
236C3×D3—62no—
237C2×Q8—92no—
238C22×C4—91no—
239Q16—182no—
240C2×C8—91no—
241C2×C8—182no—
242C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C4⋊C4—92no—
244C4⋊C4—542no—
245C22⋊C4—542no—
246C22×C3—31no—
247C22×C3—31no—
248C22×C3—61no—
249C22×C3—92no—
250C22×C3—181no—
251C22×C3—181no—
252C22×C3#380211yescharacteristic
253C2×D3—32no—
254C2×D3—32no—
255C2×D3—62no—
256C2×D3—61no—
257C2×D3—61no—
258C2×D3—61no—
259C2×D3—61no—
260C2×D3—62no—
261C2×D3—62no—
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—61no—
267C3×C4—61no—
268C3×C4—181no—
269C3×C4—362no—
270C3×C4—362no—
271C3×C4#379611yescharacteristic
272C3×C4C2×#62611yescharacteristic
273Dic3—61no—
274Dic3—61no—
275Dic3—62no—
276Dic3—122no—
277Dic3—122no—
278Dic3—362no—
279Dic3—362no—
280C32—21no—
281C32#488711yescharacteristic, complemented
282C32#518811yescharacteristic, complemented
283C23—91no—
284Q8—182no—
285Q8—182no—
286C2×C4—91no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4—182no—
290C2×C4—182no—
291C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
292C8—181no—
293C8#620012yes—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—61no—
301C2×C3—61no—
302C2×C3—61no—
303C2×C3—182no—
304C2×C3#6380211yescharacteristic
305C2×C3#6381011yescharacteristic
306C2×C3#6391611yescharacteristic
307D3—62no—
308D3—62no—
309D3—64no—
310C22—92no—
311C22—181no—
312C22—181no—
313C22#8865511yescharacteristic
314C4—181no—
315C4—362no—
316C4—362no—
317C4#8864911yescharacteristic
318C4C2×#620011yescharacteristic
319C3—31no—
320C3—31no—
321C3—61no—
322C3#1059085711yescharacteristic
323C2—182no—
324C2#1172083311yescharacteristic
325C2#1172084411yescharacteristic
326C2#1172115911yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×#272D4
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. #63645⋊C3C2semidirect
5. #11720313C2semidirect
17. C2×#3746D3semidirect
24. #63645D3semidirect
67. C2×#272Q16semidirect
184. H(3,𝔽3)#467semidirect
281. C32#4887semidirect
282. C32#5188semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
C2×#3746⋊C32
#117210942
#63645⋊C32
#117203132
#105908523
#105908513