Finite Group Databasebeta

← #13159698

Subgroups — #13159698

Order 1728.

764 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723501C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic, complemented
3#11722184C211yesmaximal, characteristic, complemented
4C3×#63648C211yesmaximal, characteristic
5C32×#818C211yesmaximal, characteristic, complemented
6#11722053C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic
8#10592590—31nomaximal
9#10590780—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×Dic3×C3⋊D3—21no—
12C2×C33×C8C2211yescharacteristic
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88807—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20#88803—21no—
21#64102—31no—
22#64099—31no—
23#63898—34no—
24#63872—34no—
25C32×#107—31no—
26#63648D311yescharacteristic
27#63641—31no—
28#63641—34no—
29#63641—64no—
30#63636—31no—
31#63635—31no—
32C3×#821—34no—
33C3×#818D314yescomplemented
34#63574—34no—
35#63562—34no—
36C22×C3×C3⋊D3—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C3×C4⋊(C3⋊D3)—21no—
42C3×C4⋊(C3⋊D3)Q812yes—
43C2×C32×Dic3—21no—
44C2×C32×Dic3—21no—
45Dic3×C3⋊D3—42no—
46C33×C8—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48#4891—94no—
49#4766—91no—
50C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
51C2×C4×C3⋊D3—61no—
52C2×C3×D12—34no—
53C2×D3×Dic3—64no—
54C2×C32×C8—24no—
55C2×C32×C8C2×D314yes—
56C2×C32×C8C2×D311yescharacteristic
57C32×C4⋊C4—31no—
58C32×C4⋊C4—31no—
59#3752—61no—
60C4⋊(C3⋊Dic3)—61no—
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—61no—
63C4×C3⋊Dic3—31no—
64C4×C3⋊Dic3—34no—
65C4×C3⋊Dic3—64no—
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×C4⋊Dic3—34no—
70C3×C4⋊Dic3C2×D314yes—
71C3×C4⋊Dic3C2×D314yes—
72Dic3⋊Dic3—64no—
73Dic3⋊Dic3—64no—
74(C2×D3)⋊Dic3—64no—
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—22no—
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3—41no—
79C33×C4C2×Q811yescharacteristic
80C33×C4#4111yescharacteristic
81C3⋊(C3⋊Dic3)—21no—
82C3⋊(C3⋊Dic3)—21no—
83C3×C3⋊Dic3—41no—
84C32×Dic3—41no—
85C32×Dic3—41no—
86D4×Dic3—91no—
87#893—94no—
88#890—94no—
89#890—91no—
90C3×#107—34no—
91C3×#102—91no—
92#832—91no—
93#821—34no—
94#818C3⋊D311yescharacteristic, complemented
95Dic3⋊C8—94no—
96Dic3⋊C8—91no—
97Dic3⋊C8—184no—
98#809—94no—
99#808—94no—
100#808—91no—
101C22×C3⋊D3—21no—
102C22×C3×D3—64no—
103C2×C32×C4—24no—
104C2×C32×C4—61no—
105C2×C32×C4—61no—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—34no—
112C2×C3⋊Dic3—34no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—64no—
115C4⋊(C3⋊D3)—21no—
116C4⋊(C3⋊D3)Dic612yes—
117C4×C3⋊D3—122no—
118C2×C3×Dic3—24no—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—64no—
121C3×D12—38no—
122C3×D12—64no—
123D3×Dic3—128no—
124C32×C8—21no—
125C32×C8—24no—
126C32×C8—44no—
127C3⋊(C3⋊C8)—61no—
128C3⋊(C3⋊C8)—64no—
129C3⋊(C3⋊C8)—124no—
130#478—271noSylow
131C2×C33C22⋊Q811yescharacteristic
132C2×C33#13511yescharacteristic
133C2×C33AGL(1,ℤ/8)11yescharacteristic
134C3×C3⋊D3—42no—
135C2×C3×D4—91no—
136C22×Dic3—181no—
137C2×D12—34no—
138C2×C4×D3—184no—
139C2×C3×C8—24no—
140C2×C3×C8D3214yes—
141C2×C3×C8C2×C3⋊D311yescharacteristic
142C3×C4⋊C4—34no—
143C3×C4⋊C4—34no—
144C3×C4⋊C4—91no—
145#217—181no—
146(C2×D3)⋊C4—184no—
147C4⋊Dic3—34no—
148C4⋊Dic3—181no—
149C4⋊Dic3—184no—
150C4⋊Dic3C2×C3⋊D311yescharacteristic
151C4⋊Dic3C2×C3⋊D311yescharacteristic
152Dic3⋊C4—181no—
153Dic3⋊C4—184no—
154C4×Dic3—94no—
155C4×Dic3—91no—
156C4×Dic3—184no—
157C2×C3⋊C8—94no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—184no—
160C22×C32—24no—
161C22×C32D3×D414yes—
162C22×C32D3×D411yescharacteristic
163C2×C3⋊D3—22no—
164C2×C3⋊D3—41no—
165C2×C3⋊D3—41no—
166C2×C3×D3—68no—
167C2×C3×D3—124no—
168C2×C3×D3—124no—
169C32×C4—24no—
170C32×C4—24no—
171C32×C4—121no—
172C32×C4—121no—
173C32×C4C2×Dic611yescharacteristic
174C32×C4#23514yes—
175C32×C4#23711yescharacteristic
176C32×C4D3×Q814yes—
177C3⋊Dic3—41no—
178C3⋊Dic3—68no—
179C3⋊Dic3—61no—
180C3⋊Dic3—61no—
181C3⋊Dic3—64no—
182C3⋊Dic3—64no—
183C3⋊Dic3—124no—
184C3×Dic3—44no—
185C3×Dic3—44no—
186C3×Dic3—124no—
187#125—271no—
188C4×D4—271no—
189#107—31no—
190#106—271no—
191C4⋊C8—271no—
192#102—91no—
193#102—271no—
194C33#47811yescharacteristic, complemented, Sylow
195C23×C3—181no—
196C22×D3—64no—
197C3×D4—92no—
198C3×D4—181no—
199C2×C3×C4—24no—
200C2×C3×C4—64no—
201C2×C3×C4—64no—
202C2×C3×C4—181no—
203C2×C3×C4C2×D3214yes—
204C2×C3×C4C22×C3⋊D311yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—64no—
208C2×Dic3—94no—
209C2×Dic3—94no—
210C2×Dic3—91no—
211C2×Dic3—91no—
212C2×Dic3—184no—
213C2×Dic3—184no—
214C2×Dic3—362no—
215D12—38no—
216D12—64no—
217C4×D3—368no—
218C3×C8—24no—
219C3×C8—21no—
220C3×C8—44no—
221C3⋊C8—184no—
222C3⋊C8—181no—
223C3⋊C8—364no—
224C2×C32—24no—
225C2×C32—24no—
226C2×C32—24no—
227C2×C32#87811yescharacteristic
228C2×C32#89614yes—
229C2×C32#91114yes—
230C2×C32#91211yescharacteristic
231C2×C32#91411yescharacteristic
232C2×C32#91914yes—
233C3⋊D3—42no—
234C3×D3—128no—
235C2×D4—91no—
236C22×C4—541no—
237C2×C8—271no—
238C2×C8D3×C3⋊D311yescharacteristic
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—91no—
242C4⋊C4—541no—
243C4⋊C4—541no—
244C22⋊C4—541no—
245C42—271no—
246C22×C3—24no—
247C22×C3—182no—
248C22×C3—361no—
249C22×C3—361no—
250C22×C3#380214yes—
251C22×C3D4×C3⋊D311yescharacteristic
252C2×D3—68no—
253C2×D3—124no—
254C2×D3—124no—
255C3×C4—24no—
256C3×C4—24no—
257C3×C4—124no—
258C3×C4—124no—
259C3×C4—361no—
260C3×C4D3×Dic614yes—
261C3×C4#380414yes—
262C3×C4#382811yescharacteristic
263C3×C4Q8×C3⋊D311yescharacteristic
264Dic3—41no—
265Dic3—41no—
266Dic3—124no—
267Dic3—188no—
268Dic3—184no—
269Dic3—184no—
270Dic3—181no—
271Dic3—181no—
272Dic3—364no—
273C32—24no—
274C32#476611yescharacteristic, complemented
275C32#489114yescomplemented
276C23—181no—
277D4—92no—
278D4—181no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—181no—
282C2×C4—271no—
283C2×C4—271no—
284C2×C4—1082no—
285C2×C4C2×D3×C3⋊D311yescharacteristic
286C8—21no—
287C8—541no—
288C2×C3—24no—
289C2×C3—24no—
290C2×C3—24no—
291C2×C3—362no—
292C2×C3#6379814yes—
293C2×C3#6381014yes—
294C2×C3#6389914yes—
295C2×C3#6410511yescharacteristic
296C2×C3#6412011yescharacteristic
297C2×C3#6412811yescharacteristic
298D3—128no—
299C22—182no—
300C22—361no—
301C22—361no—
302C22#8902611yescharacteristic
303C4—121no—
304C4—121no—
305C4—361no—
306C4—541no—
307C4—541no—
308C4C3⋊D3×Dic611yescharacteristic
309C4#8903111yescharacteristic
310C3—24no—
311C3#1059078014yescomplemented
312C3#1059259011yescharacteristic, complemented
313C2—362no—
314C2#1172332411yescharacteristic
315C2#1172334511yescharacteristic
316C2#1172350211yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#478
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11723450C2semidirect
3. #11722184C2semidirect
5. C32×#818C2semidirect
6. #11722053C2semidirect
33. C3×#818D3semidirect
94. #818C3⋊D3semidirect
194. C33#478semidirect
274. C32#4766semidirect
275. C32#4891semidirect
311. C3#10590780semidirect
312. C3#10592590semidirect

Maximal subgroups

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

subgroupindex
#117235012
#117234502
#117221842
C3×#636482
C32×#8182
#117220532
#117220332
#105925903
#105907803