Finite Group Databasebeta

← #13159292

Subgroups — #13159292

Order 1728.

732 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 182 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
1C32×#888C211yesmaximal, characteristic
2#11723534C211yesmaximal, characteristic
3#11722177C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic
5#11722057C211yesmaximal, characteristic
6#11722056C211yesmaximal, characteristic
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
8#10592777—31nomaximal
9#10590411—34nomaximal
10C2×C32×Dic6—21no—
11C2×#6293—21no—
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21C32×C4⋊Q8—31no—
22#64078—31no—
23C3×#888D314yes—
24#63918—34no—
25#63634—31no—
26#63634—31no—
27#63634—31no—
28#63634—34no—
29#63634—64no—
30#63634D311yescharacteristic
31C4×C3⋊(C3⋊C8)—31no—
32C3×#807—34no—
33#63576—34no—
34#63575—34no—
35Dic3×C3⋊C8—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3×C3⋊Dic3—21no—
40C2×C32×Dic3—21no—
41C2×C32×Dic3D411yescharacteristic
42C2×C32×Dic3D411yescharacteristic
43C32×Dic6—42no—
44#6293—42no—
45C3⋊(C3⋊(C3⋊C8))Q812yescomplemented
46C3×C3⋊(C3⋊C8)Q812yescomplemented
47#5078—94no—
48#4874—91no—
49C2×C32×Q8—61no—
50C2×#634—61no—
51C2×C3×Dic6—24no—
52C2×#627—64no—
53C32×C4⋊C4—24no—
54C32×C4⋊C4—31no—
55C32×C4⋊C4—61no—
56C32×C4⋊C4C2×D314yes—
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C42—31no—
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—34no—
61C4⋊(C3⋊Dic3)—61no—
62C4⋊(C3⋊Dic3)—64no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)—31no—
65C2×C3⋊(C3⋊C8)—34no—
66C2×C3⋊(C3⋊C8)—64no—
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×C4⋊Dic3—34no—
69C3×C4⋊Dic3C2×D314yes—
70C3×Dic3⋊C4—24no—
71C3×C4×Dic3C2×D314yes—
72C2×C3×C3⋊C8—34no—
73Dic3⋊Dic3—64no—
74C22×C33C2×D411yescharacteristic
75C33×C4—41no—
76C33×C4C2×Q811yescharacteristic
77C33×C4C2×Q811yescharacteristic
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—41no—
80C32×Dic3—41no—
81C32×Dic3D812yes—
82C32×Dic3Q1612yes—
83C3×C4⋊Q8—34no—
84#888—91no—
85#888C3⋊D311yescharacteristic
86#869—94no—
87C3×#107—91no—
88#818—91no—
89C8×Dic3—91no—
90#807—34no—
91#807—94no—
92#807—94no—
93#807—94no—
94#807—91no—
95#807—91no—
96#807—184no—
97C4×C3⋊C8—94no—
98C32×Q8—122no—
99C2×C32×C4—21no—
100C2×C32×C4—24no—
101C2×C32×C4—24no—
102C2×C32×C4—31no—
103C2×C32×C4—31no—
104C2×C32×C4—44no—
105C2×C32×C4—61no—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—64no—
111C2×C3⋊Dic3—124no—
112#634—122no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3#6714yes—
116C2×C3×Dic3#6714yes—
117C3×Dic6—48no—
118#627—128no—
119C3⋊(C3⋊C8)—32no—
120C3⋊(C3⋊C8)—38no—
121C3⋊(C3⋊C8)—68no—
122C3⋊(C3⋊C8)Dic612yescomplemented
123C3×C3⋊C8—38no—
124#500—271noSylow
125C2×C33C4⋊Q811yescharacteristic
126C2×C33C2×D811yescharacteristic
127C2×C33C2×Q1611yescharacteristic
128C2×C3×Q8—64no—
129C2×Dic6—21no—
130C2×Dic6—184no—
131C2×Dic6—181no—
132C2×C3×C8—91no—
133C3×C4⋊C4—24no—
134C3×C4⋊C4—34no—
135C3×C4⋊C4—64no—
136C3×C4⋊C4—91no—
137C3×C4⋊C4D3214yes—
138C3×C4⋊C4C2×C3⋊D311yescharacteristic
139C3×C42—34no—
140C4⋊Dic3—34no—
141C4⋊Dic3—94no—
142C4⋊Dic3—91no—
143C4⋊Dic3—184no—
144C4⋊Dic3—184no—
145C4⋊Dic3C2×C3⋊D311yescharacteristic
146Dic3⋊C4—21no—
147Dic3⋊C4—181no—
148C4×Dic3C2×C3⋊D311yescharacteristic
149C2×C3⋊C8—34no—
150C2×C3⋊C8—94no—
151C2×C3⋊C8—91no—
152C2×C3⋊C8—184no—
153C22×C32—24no—
154C22×C32D3×D411yescharacteristic
155C22×C32C2×#6714yes—
156C32×C4—24no—
157C32×C4—24no—
158C32×C4—32no—
159C32×C4—32no—
160C32×C4—41no—
161C32×C4—44no—
162C32×C4—84no—
163C32×C4—121no—
164C32×C4C2×Dic611yescharacteristic
165C32×C4D3×Q814yes—
166C32×C4D3×Q814yes—
167C32×C4D3×Q811yescharacteristic
168C3⋊Dic3—41no—
169C3⋊Dic3—121no—
170C3⋊Dic3—124no—
171C3⋊Dic3—244no—
172C3×Dic3—44no—
173C3×Dic3—124no—
174C3×Dic3#21318yes—
175C3×Dic3#21618yes—
176C4⋊Q8—31no—
177C4⋊Q8—271no—
178#107—91no—
179#107—271no—
180#107—271no—
181#107—271no—
182C4×C8—271no—
183C33#50011yescharacteristic, complemented, Sylow
184C3×Q8—128no—
185C2×C3×C4—24no—
186C2×C3×C4—21no—
187C2×C3×C4—24no—
188C2×C3×C4—34no—
189C2×C3×C4—34no—
190C2×C3×C4—44no—
191C2×C3×C4—64no—
192C2×C3×C4—181no—
193C2×C3×C4C2×D3214yes—
194C2×C3×C4C22×C3⋊D311yescharacteristic
195C2×Dic3—21no—
196C2×Dic3—64no—
197C2×Dic3—184no—
198C2×Dic3—181no—
199C2×Dic3—364no—
200C2×Dic3#63811yescharacteristic
201C2×Dic3#63811yescharacteristic
202Dic6—42no—
203Dic6—368no—
204Dic6—362no—
205C3×C8—92no—
206C3⋊C8—38no—
207C3⋊C8—98no—
208C3⋊C8—92no—
209C3⋊C8—188no—
210C2×C32—24no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32#88811yescharacteristic
214C2×C32D3×D811yescharacteristic
215C2×C32D3×Q1611yescharacteristic
216C2×C32C2×#21314yes—
217C2×C32C2×#21614yes—
218C2×C32#94414yes—
219C2×Q8—61no—
220C2×Q8—541no—
221C2×C8—91no—
222C2×C8—271no—
223C4⋊C4—31no—
224C4⋊C4—61no—
225C4⋊C4—91no—
226C4⋊C4—271no—
227C4⋊C4—541no—
228C4⋊C4D3×C3⋊D311yescharacteristic
229C42—31no—
230C22×C3—24no—
231C22×C3D3×#6714yes—
232C22×C3C2×#63811yescharacteristic
233C3×C4—24no—
234C3×C4—24no—
235C3×C4—38no—
236C3×C4—38no—
237C3×C4—41no—
238C3×C4—44no—
239C3×C4—84no—
240C3×C4—124no—
241C3×C4—361no—
242C3×C4D3×Dic614yes—
243C3×C4#379714yes—
244C3×C4Q8×C3⋊D311yescharacteristic
245C3×C4Q8×C3⋊D311yescharacteristic
246Dic3—41no—
247Dic3—124no—
248Dic3—361no—
249Dic3—364no—
250Dic3—724no—
251Dic3#375312yes—
252Dic3#375612yes—
253C32—24no—
254C32#487411yescharacteristic, complemented
255C32#507814yescomplemented
256Q8—122no—
257Q8—1082no—
258C2×C4—21no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—61no—
262C2×C4—181no—
263C2×C4—541no—
264C2×C4C2×D3×C3⋊D311yescharacteristic
265C8—92no—
266C8—272no—
267C2×C3—24no—
268C2×C3—24no—
269C2×C3—24no—
270C2×C3#6383914yes—
271C2×C3D3×#21314yes—
272C2×C3D3×#21614yes—
273C2×C3C2×#375311yescharacteristic
274C2×C3C2×#375611yescharacteristic
275C2×C3#6415311yescharacteristic
276C22D3×#63811yescharacteristic
277C4—32no—
278C4—32no—
279C4—41no—
280C4—121no—
281C4—361no—
282C4—1081no—
283C4C3⋊D3×Dic611yescharacteristic
284C4#8901911yescharacteristic
285C3—24no—
286C3#1059041114yescomplemented
287C3#1059277711yescharacteristic, complemented
288C2#1172339311yescharacteristic
289C2D3×#375311yescharacteristic
290C2D3×#375611yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#500
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
45. C3⋊(C3⋊(C3⋊C8))Q8semidirect
46. C3×C3⋊(C3⋊C8)Q8semidirect
122. C3⋊(C3⋊C8)Dic6semidirect
183. C33#500semidirect
254. C32#4874semidirect
255. C32#5078semidirect
286. C3#10590411semidirect
287. C3#10592777semidirect

Maximal subgroups

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

subgroupindex
C32×#8882
#117235342
#117221772
C3×#636342
#117220572
#117220562
Dic3×C3⋊(C3⋊C8)2
#105927773
#105904113