Finite Group Databasebeta

← #13159700

Subgroups — #13159700

Order 1728.

764 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 154 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#11723499C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic, complemented
3#11722184C211yesmaximal, characteristic, complemented
4C3×#63644C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic
6#11722056C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic
8#10592462—31nomaximal
9#10590782—34nomaximal
10C2×C32×D12C2211yescharacteristic
11C2×D3×C3⋊Dic3—21no—
12C2×C33×C8C2211yescharacteristic
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88807—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21D4×C3⋊Dic3—31no—
22#64099—31no—
23#63898—34no—
24#63872—34no—
25C32×#102—31no—
26#63659—31no—
27#63644D311yescharacteristic, complemented
28#63641—31no—
29#63641—34no—
30#63641—64no—
31#63634—31no—
32C3×#821D314yes—
33C3×#817—34no—
34#63576—34no—
35#63562—34no—
36C22×C32×D3—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C32×Dic3—21no—
43C32×D12—21no—
44C32×D12Q812yes—
45D3×C3⋊Dic3—42no—
46C33×C8—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48#4871—91no—
49#4763—94no—
50C2×C32×D4—31no—
51C22×C3⋊Dic3—61no—
52C2×C3×D12C2×D314yes—
53C2×D3×Dic3—64no—
54C2×C32×C8—24no—
55C2×C32×C8C2×D314yes—
56C2×C32×C8C2×D311yescharacteristic
57C32×C4⋊C4—31no—
58#3757—61no—
59C4⋊(C3⋊Dic3)—61no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
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⋊Dic3—34no—
71C3×C4⋊Dic3C2×D314yes—
72Dic3⋊Dic3—64no—
73Dic3⋊Dic3—64no—
74(C2×D3)⋊Dic3—64no—
75C22×C33C2×D411yescharacteristic
76C2×C32×D3—22no—
77C2×C32×D3—41no—
78C2×C32×D3—41no—
79C33×C4C2×Q811yescharacteristic
80C33×C4#4111yescharacteristic
81C3⋊(C3⋊Dic3)—21no—
82C3⋊(C3⋊Dic3)—21no—
83C3×C3⋊Dic3—41no—
84C3×C3⋊Dic3—41no—
85C32×Dic3—41no—
86D4×Dic3—94no—
87#893—91no—
88#890—94no—
89#890—91no—
90C3×#106—91no—
91C3×#102—34no—
92#832—94no—
93#821C3⋊D311yescharacteristic
94#817—34no—
95Dic3⋊C8—94no—
96Dic3⋊C8—91no—
97Dic3⋊C8—184no—
98#809—91no—
99#807—94no—
100#807—91no—
101C23×C32—61no—
102C22×C3×D3—24no—
103C32×D4—32no—
104C32×D4—61no—
105C2×C32×C4—24no—
106C2×C32×C4—61no—
107C2×C32×C4C22×D314yes—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—34no—
114C2×C3⋊Dic3—34no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—122no—
118C2×C3×Dic3—24no—
119C2×C3×Dic3—64no—
120C2×C3×Dic3—64no—
121C3×D12—24no—
122C3×D12Dic618yes—
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
134C32×D3—42no—
135C2×C3×D4—34no—
136C22×Dic3—184no—
137C2×D12C2×C3⋊D311yescharacteristic
138C2×C4×D3—181no—
139C2×C3×C8—24no—
140C2×C3×C8D3214yes—
141C2×C3×C8C2×C3⋊D311yescharacteristic
142C3×C4⋊C4—34no—
143C3×C4⋊C4—91no—
144C3×C4⋊C4—91no—
145#217—184no—
146(C2×D3)⋊C4—181no—
147C4⋊Dic3—34no—
148C4⋊Dic3—34no—
149C4⋊Dic3—181no—
150C4⋊Dic3—184no—
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×C32—62no—
162C22×C32—121no—
163C22×C32—121no—
164C22×C32D3×D414yes—
165C22×C32D3×D411yescharacteristic
166C2×C3×D3—28no—
167C2×C3×D3—44no—
168C2×C3×D3—44no—
169C32×C4—24no—
170C32×C4—24no—
171C32×C4—121no—
172C32×C4C2×Dic614yes—
173C32×C4#23511yescharacteristic
174C32×C4#23714yes—
175C32×C4D3×Q811yescharacteristic
176C3⋊Dic3—41no—
177C3⋊Dic3—41no—
178C3⋊Dic3—68no—
179C3⋊Dic3—61no—
180C3⋊Dic3—64no—
181C3⋊Dic3—61no—
182C3⋊Dic3—64no—
183C3⋊Dic3—124no—
184C3×Dic3—44no—
185C3×Dic3—124no—
186C3×Dic3—124no—
187#125—271no—
188C4×D4—271no—
189#107—271no—
190#106—91no—
191C4⋊C8—271no—
192#102—31no—
193#102—271no—
194C33#47811yescharacteristic, complemented, Sylow
195C23×C3—64no—
196C22×D3—21no—
197C3×D4—38no—
198C3×D4—64no—
199C2×C3×C4—24no—
200C2×C3×C4—64no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4C2×D3214yes—
204C2×C3×C4C22×C3⋊D311yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—64no—
207C2×Dic3—64no—
208C2×Dic3—94no—
209C2×Dic3—94no—
210C2×Dic3—91no—
211C2×Dic3—91no—
212C2×Dic3—184no—
213C2×Dic3—184no—
214C2×Dic3—368no—
215D12—21no—
216D12#63412yes—
217C4×D3—362no—
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—122no—
228C2×C32#87814yes—
229C2×C32#89611yescharacteristic
230C2×C32#91114yes—
231C2×C32#91411yescharacteristic
232C2×C32#91614yes—
233C2×C32#91611yescharacteristic
234C3×D3—48no—
235C2×D4—31no—
236C22×C4—541no—
237C2×C8—271no—
238C2×C8D3×C3⋊D311yescharacteristic
239C4⋊C4—31no—
240C4⋊C4—91no—
241C4⋊C4—91no—
242C4⋊C4—541no—
243C4⋊C4—541no—
244C22⋊C4—541no—
245C42—271no—
246C22×C3—24no—
247C22×C3—68no—
248C22×C3—124no—
249C22×C3—124no—
250C22×C3#380214yes—
251C22×C3D4×C3⋊D311yescharacteristic
252C2×D3—22no—
253C2×D3—41no—
254C2×D3—41no—
255C3×C4—24no—
256C3×C4—24no—
257C3×C4—124no—
258C3×C4—361no—
259C3×C4—361no—
260C3×C4D3×Dic614yes—
261C3×C4#380414yes—
262C3×C4C2×#63411yescharacteristic
263C3×C4#383011yescharacteristic
264Dic3—41no—
265Dic3—124no—
266Dic3—124no—
267Dic3—188no—
268Dic3—184no—
269Dic3—184no—
270Dic3—181no—
271Dic3—181no—
272Dic3—364no—
273C32—24no—
274C32#476314yescomplemented
275C32#487111yescharacteristic, complemented
276C23—61no—
277D4—32no—
278D4—61no—
279C2×C4—61no—
280C2×C4—181no—
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—128no—
292C2×C3#6379814yes—
293C2×C3#6380914yes—
294C2×C3#6389914yes—
295C2×C3#6408711yescharacteristic
296C2×C3#6412011yescharacteristic
297C2×C3#6412511yescharacteristic
298D3—42no—
299C22—62no—
300C22—121no—
301C22—121no—
302C22#8902611yescharacteristic
303C4—121no—
304C4—361no—
305C4—361no—
306C4—541no—
307C4—541no—
308C4D3×#63411yescharacteristic
309C4#8902811yescharacteristic
310C3—24no—
311C3#1059078214yescomplemented
312C3#1059246211yescharacteristic, complemented
313C2—122no—
314C2#1172332411yescharacteristic
315C2#1172334211yescharacteristic
316C2#1172350011yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
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
4. C3×#63644C2semidirect
6. #11722056C2semidirect
27. #63644D3semidirect
194. C33#478semidirect
274. C32#4763semidirect
275. C32#4871semidirect
311. C3#10590782semidirect
312. C3#10592462semidirect

Maximal subgroups

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

subgroupindex
#117234992
#117234502
#117221842
C3×#636442
C32×#8212
#117220562
#117220312
#105924623
#105907823