Finite Group Databasebeta

← #13159900

Subgroups — #13159900

Order 1728.

728 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 148 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
2#11723410C211yesmaximal, characteristic
3#11722184C211yesmaximal, characteristic
4C3×#63643C211yesmaximal, characteristic
5C32×#818C211yesmaximal, characteristic
6#11722056C211yesmaximal, characteristic
7#11722039C211yesmaximal, characteristic
8#10592574—31nomaximal
9#10590817—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×#6293—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—
19#88806—21no—
20Dic3×C3⋊Dic3—21no—
21#64097—31no—
22#64096—31no—
23#63919—34no—
24#63843—34no—
25C32×#107—31no—
26#63643D311yescharacteristic
27#63641—31no—
28#63641—34no—
29#63641—64no—
30#63637—31no—
31#63634—31no—
32C3×#818D314yes—
33C3×#816—34no—
34#63576—34no—
35#63565—34no—
36C2×C33×C4C2311yescharacteristic
37C2×C3⋊(C3⋊Dic3)D411yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C3×#634—21no—
42C3×#634Q812yescomplemented
43C2×C32×Dic3—21no—
44C2×C32×Dic3—21no—
45#6293—42no—
46C33×C8—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48#4875—94no—
49#4793—91no—
50C2×#634—61no—
51C2×#634C2×D311yescharacteristic
52C2×C3×Dic6—34no—
53C2×#627—64no—
54C2×C32×C8—24no—
55C2×C32×C8C2×D314yes—
56C2×C32×C8C2×D311yescharacteristic
57C32×C4⋊C4—31no—
58C32×C4⋊C4—31no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60#3750—61no—
61#3750—61no—
62C4×C3⋊Dic3—31no—
63C4×C3⋊Dic3—34no—
64C4×C3⋊Dic3—61no—
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—
72#3724—64no—
73Dic3⋊Dic3—64no—
74Dic32—64no—
75C22×C33C2×D411yescharacteristic
76C33×C4C2×Q811yescharacteristic
77C33×C4#4111yescharacteristic
78C3⋊(C3⋊Dic3)—21no—
79C3⋊(C3⋊Dic3)Q1612yes—
80C3×C3⋊Dic3—22no—
81C3×C3⋊Dic3—41no—
82C32×Dic3—41no—
83C32×Dic3—41no—
84Q8×Dic3—91no—
85#888—94no—
86#888—91no—
87#887—94no—
88C3×#107—34no—
89C3×#103—91no—
90#835—91no—
91#818C3⋊D311yescharacteristic
92#816—34no—
93Dic3⋊C8—94no—
94Dic3⋊C8—91no—
95Dic3⋊C8—184no—
96#810—94no—
97#807—94no—
98#807—91no—
99C2×C32×C4—24no—
100C2×C32×C4—61no—
101C2×C32×C4—61no—
102C2×C32×C4C22×D314yes—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—34no—
109C2×C3⋊Dic3—34no—
110C2×C3⋊Dic3—64no—
111C2×C3⋊Dic3—64no—
112#634—21no—
113#634—122no—
114#634Dic612yescomplemented
115C2×C3×Dic3—24no—
116C2×C3×Dic3—24no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3—64no—
119C3×Dic6—38no—
120C3×Dic6—64no—
121#627—128no—
122C32×C8—21no—
123C32×C8—24no—
124C32×C8—44no—
125C3⋊(C3⋊C8)—61no—
126C3⋊(C3⋊C8)—64no—
127C3⋊(C3⋊C8)—124no—
128#477—271noSylow
129C2×C33C22⋊Q811yescharacteristic
130C2×C33C2×Q1611yescharacteristic
131C2×C33AGL(1,ℤ/8)11yescharacteristic
132C2×C3×Q8—91no—
133C2×Dic6—34no—
134C2×Dic6—181no—
135C2×Dic6—184no—
136C2×C3×C8—24no—
137C2×C3×C8D3214yes—
138C2×C3×C8C2×C3⋊D311yescharacteristic
139C3×C4⋊C4—34no—
140C3×C4⋊C4—34no—
141C3×C4⋊C4—91no—
142C4⋊Dic3—34no—
143C4⋊Dic3—181no—
144C4⋊Dic3C2×C3⋊D311yescharacteristic
145C4⋊Dic3C2×C3⋊D311yescharacteristic
146Dic3⋊C4—181no—
147Dic3⋊C4—184no—
148Dic3⋊C4—184no—
149C4×Dic3—94no—
150C4×Dic3—91no—
151C4×Dic3—184no—
152C4×Dic3—181no—
153C4×Dic3—184no—
154C2×C3⋊C8—94no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—184no—
157C22×C32—24no—
158C22×C32D3×D414yes—
159C22×C32D3×D411yescharacteristic
160C32×C4—24no—
161C32×C4—24no—
162C32×C4—121no—
163C32×C4—121no—
164C32×C4C2×Dic611yescharacteristic
165C32×C4#23514yes—
166C32×C4#23711yescharacteristic
167C32×C4D3×Q814yes—
168C3⋊Dic3—22no—
169C3⋊Dic3—32no—
170C3⋊Dic3—38no—
171C3⋊Dic3—41no—
172C3⋊Dic3—68no—
173C3⋊Dic3—61no—
174C3⋊Dic3—64no—
175C3⋊Dic3—124no—
176C3×Dic3—44no—
177C3×Dic3—44no—
178C3×Dic3—68no—
179C3×Dic3—124no—
180C4⋊Q8—271no—
181C4×Q8—271no—
182#107—31no—
183#107—271no—
184C4⋊C8—271no—
185#103—91no—
186#103—271no—
187C33#47711yescharacteristic, complemented, Sylow
188C3×Q8—92no—
189C3×Q8—181no—
190C2×C3×C4—24no—
191C2×C3×C4—64no—
192C2×C3×C4—64no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4C2×D3214yes—
196C2×C3×C4C22×C3⋊D311yescharacteristic
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—64no—
200C2×Dic3—64no—
201C2×Dic3—94no—
202C2×Dic3—91no—
203C2×Dic3—94no—
204C2×Dic3—91no—
205C2×Dic3—184no—
206C2×Dic3—184no—
207Dic6—38no—
208Dic6—64no—
209Dic6—362no—
210Dic6—368no—
211C3×C8—24no—
212C3×C8—21no—
213C3×C8—44no—
214C3⋊C8—184no—
215C3⋊C8—181no—
216C3⋊C8—364no—
217C2×C32—24no—
218C2×C32—24no—
219C2×C32—24no—
220C2×C32#87811yescharacteristic
221C2×C32#89614yes—
222C2×C32#91114yes—
223C2×C32#91411yescharacteristic
224C2×C32D3×Q1614yes—
225C2×C32D3×Q1611yescharacteristic
226C2×Q8—91no—
227C2×Q8—541no—
228C2×C8—271no—
229C2×C8D3×C3⋊D311yescharacteristic
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—91no—
233C4⋊C4—541no—
234C4⋊C4—541no—
235C42—271no—
236C42—541no—
237C22×C3—24no—
238C22×C3#380214yes—
239C22×C3D4×C3⋊D311yescharacteristic
240C3×C4—24no—
241C3×C4—24no—
242C3×C4—124no—
243C3×C4—124no—
244C3×C4—182no—
245C3×C4—361no—
246C3×C4D3×Dic614yes—
247C3×C4#380414yes—
248C3×C4#382811yescharacteristic
249C3×C4Q8×C3⋊D311yescharacteristic
250Dic3—41no—
251Dic3—41no—
252Dic3—68no—
253Dic3—98no—
254Dic3—92no—
255Dic3—124no—
256Dic3—188no—
257Dic3—184no—
258Dic3—181no—
259Dic3—364no—
260C32—24no—
261C32#479311yescharacteristic, complemented
262C32#487514yescomplemented
263Q8—92no—
264Q8—181no—
265Q8—1082no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4—271no—
271C2×C4—271no—
272C2×C4C2×D3×C3⋊D311yescharacteristic
273C8—21no—
274C8—541no—
275C2×C3—24no—
276C2×C3—24no—
277C2×C3—24no—
278C2×C3#6379814yes—
279C2×C3#6380214yes—
280C2×C3#6389914yes—
281C2×C3#6410511yescharacteristic
282C2×C3#6412011yescharacteristic
283C2×C3Q16×C3⋊D311yescharacteristic
284C22#8902611yescharacteristic
285C4—121no—
286C4—121no—
287C4—182no—
288C4—272no—
289C4—361no—
290C4—541no—
291C4C3⋊D3×Dic611yescharacteristic
292C4#8903111yescharacteristic
293C3—24no—
294C3#1059081714yescomplemented
295C3#1059257411yescharacteristic, complemented
296C2#1172332411yescharacteristic
297C2#1172333011yescharacteristic
298C2#1172350211yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#477
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
42. C3×#634Q8semidirect
114. #634Dic6semidirect
187. C33#477semidirect
261. C32#4793semidirect
262. C32#4875semidirect
294. C3#10590817semidirect
295. C3#10592574semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117234102
#117221842
C3×#636432
C32×#8182
#117220562
#117220392
#105925743
#105908173