Finite Group Databasebeta

← #13159266

Subgroups — #13159266

Order 1728.

812 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 158 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#11723929C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic
3C3×#63635C211yesmaximal, characteristic
4C32×#807C211yesmaximal, characteristic
5#11722057C211yesmaximal, characteristic
6#11722052C211yesmaximal, characteristic
7#11722021C211yesmaximal, characteristic
8#10592575—31nomaximal
9#10590546—34nomaximal
10C2×#6308—21no—
11C33×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13#88857—21no—
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#64099—31no—
22#64097—31no—
23#64097—34no—
24#64097—64no—
25#63872—34no—
26C32×#107—31no—
27#63644—31no—
28#63642—31no—
29#63635D311yescharacteristic
30#63634—31no—
31C3×#808—34no—
32C3×#807D314yes—
33#63575—34no—
34#63573—34no—
35#63555—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41#6308—42no—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3—21no—
44C3×C3⋊(C3⋊C8)Q812yescomplemented
45C32×C3⋊C8Q812yescomplemented
46#4876—94no—
47#4856—91no—
48C2×#634—61no—
49C2×#634—64no—
50C2×#634—68no—
51C2×C32×C8—31no—
52C32×C4⋊C4—24no—
53C32×C4⋊C4—31no—
54C32×C4⋊C4C2×D314yes—
55C32×C4⋊C4C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)—31no—
57C4⋊(C3⋊Dic3)—34no—
58C4⋊(C3⋊Dic3)—61no—
59C4⋊(C3⋊Dic3)—64no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61#3750—61no—
62#3750—61no—
63#3750—64no—
64#3750—68no—
65C4×C3⋊Dic3—31no—
66C4×C3⋊Dic3—34no—
67C4×C3⋊Dic3—64no—
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C4⋊Dic3—34no—
70C3×C4⋊Dic3C2×D314yes—
71C2×C3×C3⋊C8—34no—
72C2×C3×C3⋊C8C2×D314yes—
73Dic3⋊Dic3—64no—
74Dic3⋊Dic3—64no—
75C22×C33C2×D411yescharacteristic
76C33×C4—41no—
77C33×C4C2×Q811yescharacteristic
78C33×C4C2×Q811yescharacteristic
79C3⋊(C3⋊Dic3)—21no—
80C3⋊(C3⋊Dic3)—21no—
81C3⋊(C3⋊Dic3)—41no—
82C3×C3⋊Dic3—41no—
83C32×Dic3—41no—
84#890—94no—
85#890—91no—
86#888—94no—
87#888—91no—
88#888—184no—
89C3×#107—34no—
90C3×#106—91no—
91#818—91no—
92#817—94no—
93#815—94no—
94#815—91no—
95#808—34no—
96#808—91no—
97#807—94no—
98#807C3⋊D311yescharacteristic
99C2×C32×C4—28no—
100C2×C32×C4—24no—
101C2×C32×C4—21no—
102C2×C32×C4—24no—
103C2×C32×C4—61no—
104C2×C32×C4C22×D314yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—34no—
110C2×C3⋊Dic3—34no—
111C2×C3⋊Dic3—64no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—68no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—61no—
116#634—1216no—
117#634—128no—
118#634—122no—
119C2×C3×Dic3—24no—
120C2×C3×Dic3—64no—
121C32×C8—32no—
122C3⋊(C3⋊C8)Dic612yescomplemented
123C3×C3⋊C8—38no—
124C3×C3⋊C8Dic618yescomplemented
125#501—271noSylow
126C2×C33C4⋊Q811yescharacteristic
127C2×C33AGL(1,ℤ/8)11yescharacteristic
128C2×C33#13711yescharacteristic
129C2×Dic6—184no—
130C2×Dic6—181no—
131C2×Dic6—188no—
132C2×C3×C8—34no—
133C2×C3×C8—91no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—34no—
136C3×C4⋊C4—91no—
137C3×C4⋊C4D3214yes—
138C3×C4⋊C4C2×C3⋊D311yescharacteristic
139C4⋊Dic3—34no—
140C4⋊Dic3—94no—
141C4⋊Dic3—91no—
142C4⋊Dic3—184no—
143C4⋊Dic3—181no—
144C4⋊Dic3—184no—
145C4⋊Dic3C2×C3⋊D311yescharacteristic
146Dic3⋊C4—181no—
147Dic3⋊C4—184no—
148Dic3⋊C4—184no—
149Dic3⋊C4—181no—
150Dic3⋊C4—188no—
151C4×Dic3—94no—
152C4×Dic3—91no—
153C4×Dic3—184no—
154C2×C3⋊C8—34no—
155C2×C3⋊C8C2×C3⋊D311yescharacteristic
156C22×C32—24no—
157C22×C32D3×D414yes—
158C22×C32D3×D411yescharacteristic
159C32×C4—24no—
160C32×C4—24no—
161C32×C4—48no—
162C32×C4—44no—
163C32×C4—41no—
164C32×C4—121no—
165C32×C4C2×Dic614yes—
166C32×C4C2×Dic611yescharacteristic
167C32×C4D3×Q814yes—
168C32×C4D3×Q811yescharacteristic
169C3⋊Dic3—41no—
170C3⋊Dic3—68no—
171C3⋊Dic3—68no—
172C3⋊Dic3—64no—
173C3⋊Dic3—61no—
174C3⋊Dic3—61no—
175C3⋊Dic3—64no—
176C3⋊Dic3—128no—
177C3⋊Dic3—124no—
178C3⋊Dic3—121no—
179C3×Dic3—44no—
180C3×Dic3—124no—
181C4⋊Q8—271no—
182#125—271no—
183#107—31no—
184#107—271no—
185#106—91no—
186#106—271no—
187#97—271no—
188C33#50111yescharacteristic, complemented, Sylow
189C2×C3×C4—28no—
190C2×C3×C4—24no—
191C2×C3×C4—21no—
192C2×C3×C4—24no—
193C2×C3×C4—64no—
194C2×C3×C4—181no—
195C2×C3×C4C2×D3214yes—
196C2×C3×C4C22×C3⋊D311yescharacteristic
197C2×Dic3—21no—
198C2×Dic3—64no—
199C2×Dic3—94no—
200C2×Dic3—94no—
201C2×Dic3—91no—
202C2×Dic3—91no—
203C2×Dic3—184no—
204C2×Dic3—184no—
205C2×Dic3—188no—
206C2×Dic3—184no—
207C2×Dic3—181no—
208Dic6—3616no—
209Dic6—362no—
210Dic6—368no—
211C3×C8—38no—
212C3×C8—92no—
213C3⋊C8—38no—
214C3⋊C8#63412yescomplemented
215C2×C32—24no—
216C2×C32—24no—
217C2×C32—24no—
218C2×C32#88814yes—
219C2×C32#88811yescharacteristic
220C2×C32#91114yes—
221C2×C32#91411yescharacteristic
222C2×C32#91511yescharacteristic
223C2×C32#91814yes—
224C2×Q8—541no—
225C2×C8—31no—
226C2×C8—91no—
227C4⋊C4—31no—
228C4⋊C4—91no—
229C4⋊C4—271no—
230C4⋊C4—541no—
231C4⋊C4—541no—
232C4⋊C4—541no—
233C4⋊C4D3×C3⋊D311yescharacteristic
234C42—271no—
235C22×C3—24no—
236C22×C3#381114yes—
237C22×C3D4×C3⋊D311yescharacteristic
238C3×C4—24no—
239C3×C4—24no—
240C3×C4—48no—
241C3×C4—41no—
242C3×C4—44no—
243C3×C4—124no—
244C3×C4—361no—
245C3×C4D3×Dic614yes—
246C3×C4D3×Dic614yes—
247C3×C4C2×#63411yescharacteristic
248C3×C4Q8×C3⋊D311yescharacteristic
249Dic3—41no—
250Dic3—124no—
251Dic3—188no—
252Dic3—188no—
253Dic3—181no—
254Dic3—181no—
255Dic3—184no—
256Dic3—184no—
257Dic3—368no—
258Dic3—361no—
259Dic3—364no—
260C32—24no—
261C32#485611yescharacteristic, complemented
262C32#487614yescomplemented
263Q8—1082no—
264C2×C4—21no—
265C2×C4—61no—
266C2×C4—181no—
267C2×C4—271no—
268C2×C4—271no—
269C2×C4—541no—
270C2×C4C2×D3×C3⋊D311yescharacteristic
271C8—32no—
272C8—92no—
273C2×C3—24no—
274C2×C3—24no—
275C2×C3—24no—
276C2×C3#6384014yes—
277C2×C3#6393714yes—
278C2×C3#6395114yes—
279C2×C3#6409711yescharacteristic
280C2×C3#6412011yescharacteristic
281C2×C3#6412711yescharacteristic
282C22#8903911yescharacteristic
283C4—41no—
284C4—121no—
285C4—361no—
286C4—541no—
287C4—541no—
288C4—1081no—
289C4D3×#63411yescharacteristic
290C4C3⋊D3×Dic611yescharacteristic
291C3—24no—
292C3#1059054614yescomplemented
293C3#1059257511yescharacteristic, complemented
294C2#1172339511yescharacteristic
295C2#1172356411yescharacteristic
296C2#1172359011yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#501
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
44. C3×C3⋊(C3⋊C8)Q8semidirect
45. C32×C3⋊C8Q8semidirect
122. C3⋊(C3⋊C8)Dic6semidirect
124. C3×C3⋊C8Dic6semidirect
188. C33#501semidirect
214. C3⋊C8#634semidirect
261. C32#4856semidirect
262. C32#4876semidirect
292. C3#10590546semidirect
293. C3#10592575semidirect

Maximal subgroups

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

subgroupindex
#117239292
#117234502
C3×#636352
C32×#8072
#117220572
#117220522
#117220212
#105925753
#105905463