Finite Group Databasebeta

← #13159696

Subgroups — #13159696

Order 1728.

800 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 156 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#11723536C211yesmaximal, characteristic, complemented
2#11723499C211yesmaximal, characteristic
3#11722184C211yesmaximal, characteristic, complemented
4C3×#63644C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic
6#11722053C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic
8#10592460—31nomaximal
9#10590776—34nomaximal
10C2×C32×D12C2211yescharacteristic
11C2×#6293—21no—
12C2×D3×C3⋊Dic3—21no—
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88806—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21D4×C3⋊Dic3—31no—
22#64097—31no—
23#63919—34no—
24#63898—34no—
25C32×#102—31no—
26#63659—31no—
27#63644D311yescharacteristic, complemented
28#63641—31no—
29#63641—34no—
30#63641—64no—
31#63635—31no—
32C3×#821D314yes—
33C3×#817—34no—
34#63574—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×D12Q812yescomplemented
45#6293—42no—
46D3×C3⋊Dic3—42no—
47C33×C8—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49#4870—91no—
50#4761—94no—
51C2×C32×D4—31no—
52C22×C3⋊Dic3—61no—
53C2×#634—61no—
54C2×C3×D12C2×D314yes—
55C2×#627—64no—
56C2×D3×Dic3—64no—
57C2×C32×C8—24no—
58C2×C32×C8C2×D314yes—
59C2×C32×C8C2×D311yescharacteristic
60C32×C4⋊C4—31no—
61#3757—61no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64#3750—61no—
65C4×C3⋊Dic3—31no—
66C4×C3⋊Dic3—34no—
67C4×C3⋊Dic3—64no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×C4⋊Dic3—34no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74#3724—64no—
75(C2×D3)⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C2×C32×D3—22no—
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C33×C4C2×Q811yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—21no—
83C3⋊(C3⋊Dic3)SD1612yes—
84C3×C3⋊Dic3—41no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—41no—
87D4×Dic3—94no—
88#893—91no—
89#888—94no—
90#888—91no—
91C3×#106—91no—
92C3×#102—34no—
93#832—94no—
94#821C3⋊D311yescharacteristic
95#817—34no—
96Dic3⋊C8—94no—
97Dic3⋊C8—91no—
98Dic3⋊C8—184no—
99#809—91no—
100#808—94no—
101#808—91no—
102C23×C32—61no—
103C22×C3×D3—24no—
104C32×D4—32no—
105C32×D4—61no—
106C2×C32×C4—24no—
107C2×C32×C4—61no—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—34no—
115C2×C3⋊Dic3—34no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—122no—
119#634—122no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—64no—
123C3×D12—24no—
124C3×D12Dic618yescomplemented
125#627—128no—
126D3×Dic3—128no—
127C32×C8—21no—
128C32×C8—24no—
129C32×C8—44no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—124no—
133#476—271noSylow
134C2×C33C22⋊Q811yescharacteristic
135C2×C33C2×SD1611yescharacteristic
136C2×C33AGL(1,ℤ/8)11yescharacteristic
137C32×D3—42no—
138C2×C3×D4—34no—
139C22×Dic3—184no—
140C2×D12C2×C3⋊D311yescharacteristic
141C2×C4×D3—181no—
142C2×Dic6—181no—
143C2×Dic6—184no—
144C2×C3×C8—24no—
145C2×C3×C8D3214yes—
146C2×C3×C8C2×C3⋊D311yescharacteristic
147C3×C4⋊C4—34no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—91no—
150#217—184no—
151(C2×D3)⋊C4—181no—
152C4⋊Dic3—34no—
153C4⋊Dic3—34no—
154C4⋊Dic3C2×C3⋊D311yescharacteristic
155Dic3⋊C4—181no—
156Dic3⋊C4—184no—
157C4×Dic3—94no—
158C4×Dic3—91no—
159C4×Dic3—184no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—184no—
163C22×C32—24no—
164C22×C32—62no—
165C22×C32—121no—
166C22×C32—121no—
167C22×C32D3×D414yes—
168C22×C32D3×D411yescharacteristic
169C2×C3×D3—28no—
170C2×C3×D3—44no—
171C2×C3×D3—44no—
172C32×C4—24no—
173C32×C4—24no—
174C32×C4—121no—
175C32×C4C2×Dic614yes—
176C32×C4#23511yescharacteristic
177C32×C4#23714yes—
178C32×C4D3×Q811yescharacteristic
179C3⋊Dic3—32no—
180C3⋊Dic3—38no—
181C3⋊Dic3—41no—
182C3⋊Dic3—41no—
183C3⋊Dic3—68no—
184C3⋊Dic3—61no—
185C3⋊Dic3—64no—
186C3⋊Dic3—124no—
187C3×Dic3—44no—
188C3×Dic3—124no—
189C3×Dic3—124no—
190C4⋊Q8—271no—
191C4×D4—271no—
192#106—91no—
193#106—271no—
194C4⋊C8—271no—
195#102—31no—
196#102—271no—
197C33#47611yescharacteristic, complemented, Sylow
198C23×C3—64no—
199C22×D3—21no—
200C3×D4—38no—
201C3×D4—64no—
202C2×C3×C4—24no—
203C2×C3×C4—64no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×C3×C4C2×D3214yes—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208C2×Dic3—21no—
209C2×Dic3—64no—
210C2×Dic3—64no—
211C2×Dic3—94no—
212C2×Dic3—91no—
213C2×Dic3—94no—
214C2×Dic3—91no—
215C2×Dic3—184no—
216C2×Dic3—184no—
217C2×Dic3—368no—
218D12—21no—
219D12#63412yescomplemented
220C4×D3—362no—
221Dic6—362no—
222Dic6—368no—
223C3×C8—24no—
224C3×C8—21no—
225C3×C8—44no—
226C3⋊C8—184no—
227C3⋊C8—181no—
228C3⋊C8—364no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32—122no—
233C2×C32#87814yes—
234C2×C32#89611yescharacteristic
235C2×C32#91114yes—
236C2×C32D3×SD1614yes—
237C2×C32D3×SD1611yescharacteristic
238C2×C32#91411yescharacteristic
239C3×D3—48no—
240C2×Q8—541no—
241C2×D4—31no—
242C22×C4—541no—
243C2×C8—271no—
244C2×C8D3×C3⋊D311yescharacteristic
245C4⋊C4—31no—
246C4⋊C4—91no—
247C4⋊C4—91no—
248C4⋊C4—541no—
249C22⋊C4—541no—
250C42—271no—
251C22×C3—24no—
252C22×C3—68no—
253C22×C3—124no—
254C22×C3—124no—
255C22×C3#380214yes—
256C22×C3D4×C3⋊D311yescharacteristic
257C2×D3—22no—
258C2×D3—41no—
259C2×D3—41no—
260C3×C4—24no—
261C3×C4—24no—
262C3×C4—124no—
263C3×C4—361no—
264C3×C4—361no—
265C3×C4D3×Dic614yes—
266C3×C4#380414yes—
267C3×C4C2×#63411yescharacteristic
268C3×C4#383011yescharacteristic
269Dic3—41no—
270Dic3—98no—
271Dic3—92no—
272Dic3—124no—
273Dic3—124no—
274Dic3—188no—
275Dic3—184no—
276Dic3—181no—
277Dic3—364no—
278C32—24no—
279C32#476114yescomplemented
280C32#487011yescharacteristic, complemented
281C23—61no—
282Q8—1082no—
283D4—32no—
284D4—61no—
285C2×C4—61no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—271no—
289C2×C4—271no—
290C2×C4—1082no—
291C2×C4C2×D3×C3⋊D311yescharacteristic
292C8—21no—
293C8—541no—
294C2×C3—24no—
295C2×C3—24no—
296C2×C3—24no—
297C2×C3—128no—
298C2×C3#6379614yes—
299C2×C3#6379814yes—
300C2×C3#6389914yes—
301C2×C3#6408711yescharacteristic
302C2×C3#6412011yescharacteristic
303C2×C3SD16×C3⋊D311yescharacteristic
304D3—42no—
305C22—62no—
306C22—121no—
307C22—121no—
308C22#8902611yescharacteristic
309C4—121no—
310C4—272no—
311C4—361no—
312C4—361no—
313C4—541no—
314C4D3×#63411yescharacteristic
315C4#8902811yescharacteristic
316C3—24no—
317C3#1059077614yescomplemented
318C3#1059246011yescharacteristic, complemented
319C2—122no—
320C2#1172332111yescharacteristic
321C2#1172332411yescharacteristic
322C2#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#476
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
1. #11723536C2semidirect
3. #11722184C2semidirect
4. C3×#63644C2semidirect
6. #11722053C2semidirect
27. #63644D3semidirect
44. C32×D12Q8semidirect
124. C3×D12Dic6semidirect
197. C33#476semidirect
219. D12#634semidirect
279. C32#4761semidirect
280. C32#4870semidirect
317. C3#10590776semidirect
318. C3#10592460semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117234992
#117221842
C3×#636442
C32×#8212
#117220532
#117220312
#105924603
#105907763