Finite Group Databasebeta

← #13159308

Subgroups — #13159308

Order 1728.

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

normal subgroups only

This group has 764 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#11723931C211yesmaximal, characteristic
2#11723536C211yesmaximal, characteristic
3C3×#63634C211yesmaximal, characteristic
4C32×#807C211yesmaximal, characteristic
5#11722054C211yesmaximal, characteristic
6#11722052C211yesmaximal, characteristic
7#11722021C211yesmaximal, characteristic
8#10592575—31nomaximal
9#10590563—34nomaximal
10C2×#6293—21no—
11C33×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14#88857—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#64099—31no—
22#64099—34no—
23#64099—64no—
24#64097—31no—
25#63919—34no—
26C32×#107—31no—
27#63644—31no—
28#63642—31no—
29#63635—31no—
30#63634D311yescharacteristic
31C3×#807—34no—
32C3×#807D314yes—
33#63573—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
41C2×C3×C3⋊Dic3—21no—
42C2×C32×Dic3—21no—
43#6293—42no—
44C3×C3⋊(C3⋊C8)Q812yescomplemented
45C32×C3⋊C8Q812yescomplemented
46#4876—94no—
47#4876—91no—
48C2×#634—61no—
49C2×#627—64no—
50C2×C32×C8—31no—
51C32×C4⋊C4—24no—
52C32×C4⋊C4—31no—
53C32×C4⋊C4C2×D314yes—
54C32×C4⋊C4C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)—31no—
56C4⋊(C3⋊Dic3)—34no—
57C4⋊(C3⋊Dic3)—61no—
58C4⋊(C3⋊Dic3)—64no—
59C4⋊(C3⋊Dic3)—64no—
60C4⋊(C3⋊Dic3)—68no—
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—61no—
63#3750—61no—
64#3750—64no—
65#3750—68no—
66C4×C3⋊Dic3—31no—
67C4×C3⋊Dic3—34no—
68C4×C3⋊Dic3—64no—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×C4⋊Dic3—34no—
71C3×C4⋊Dic3C2×D314yes—
72C2×C3×C3⋊C8—34no—
73C2×C3×C3⋊C8C2×D314yes—
74#3724—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#890—184no—
87#888—94no—
88#888—91no—
89C3×#107—34no—
90C3×#107—91no—
91#817—94no—
92#817—91no—
93#815—94no—
94#815—91no—
95#808—94no—
96#808—91no—
97#807—34no—
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—122no—
117C2×C3×Dic3—24no—
118C2×C3×Dic3—64no—
119#627—128no—
120C32×C8—32no—
121C3⋊(C3⋊C8)Dic612yescomplemented
122C3×C3⋊C8—38no—
123C3×C3⋊C8Dic618yescomplemented
124#501—271noSylow
125C2×C33C4⋊Q811yescharacteristic
126C2×C33AGL(1,ℤ/8)11yescharacteristic
127C2×C33#13711yescharacteristic
128C2×Dic6—181no—
129C2×Dic6—184no—
130C2×C3×C8—34no—
131C2×C3×C8—91no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4—34no—
134C3×C4⋊C4—91no—
135C3×C4⋊C4D3214yes—
136C3×C4⋊C4C2×C3⋊D311yescharacteristic
137C4⋊Dic3—34no—
138C4⋊Dic3—94no—
139C4⋊Dic3—91no—
140C4⋊Dic3—184no—
141C4⋊Dic3—184no—
142C4⋊Dic3—181no—
143C4⋊Dic3—188no—
144C4⋊Dic3C2×C3⋊D311yescharacteristic
145Dic3⋊C4—181no—
146Dic3⋊C4—184no—
147Dic3⋊C4—184no—
148Dic3⋊C4—181no—
149Dic3⋊C4—188no—
150C4×Dic3—94no—
151C4×Dic3—91no—
152C4×Dic3—184no—
153C2×C3⋊C8—34no—
154C2×C3⋊C8C2×C3⋊D311yescharacteristic
155C22×C32—24no—
156C22×C32D3×D414yes—
157C22×C32D3×D411yescharacteristic
158C32×C4—24no—
159C32×C4—24no—
160C32×C4—48no—
161C32×C4—44no—
162C32×C4—41no—
163C32×C4—121no—
164C32×C4C2×Dic614yes—
165C32×C4C2×Dic611yescharacteristic
166C32×C4D3×Q814yes—
167C32×C4D3×Q811yescharacteristic
168C3⋊Dic3—41no—
169C3⋊Dic3—61no—
170C3⋊Dic3—61no—
171C3⋊Dic3—64no—
172C3⋊Dic3—64no—
173C3⋊Dic3—124no—
174C3⋊Dic3—124no—
175C3⋊Dic3—128no—
176C3⋊Dic3—124no—
177C3⋊Dic3—121no—
178C3×Dic3—44no—
179C3×Dic3—124no—
180C4⋊Q8—271no—
181#125—271no—
182#107—31no—
183#107—91no—
184#106—271no—
185#106—271no—
186#97—271no—
187C33#50111yescharacteristic, complemented, Sylow
188C2×C3×C4—28no—
189C2×C3×C4—24no—
190C2×C3×C4—21no—
191C2×C3×C4—24no—
192C2×C3×C4—64no—
193C2×C3×C4—181no—
194C2×C3×C4C2×D3214yes—
195C2×C3×C4C22×C3⋊D311yescharacteristic
196C2×Dic3—21no—
197C2×Dic3—64no—
198C2×Dic3—94no—
199C2×Dic3—94no—
200C2×Dic3—91no—
201C2×Dic3—91no—
202C2×Dic3—184no—
203C2×Dic3—184no—
204C2×Dic3—188no—
205C2×Dic3—184no—
206C2×Dic3—181no—
207Dic6—362no—
208Dic6—368no—
209C3×C8—38no—
210C3×C8—92no—
211C3⋊C8—38no—
212C3⋊C8#63412yescomplemented
213C2×C32—24no—
214C2×C32—24no—
215C2×C32—24no—
216C2×C32#88814yes—
217C2×C32#88811yescharacteristic
218C2×C32#91114yes—
219C2×C32#91111yescharacteristic
220C2×C32#91814yes—
221C2×C32#91811yescharacteristic
222C2×Q8—541no—
223C2×C8—31no—
224C2×C8—91no—
225C4⋊C4—31no—
226C4⋊C4—91no—
227C4⋊C4—271no—
228C4⋊C4—541no—
229C4⋊C4—541no—
230C4⋊C4—541no—
231C4⋊C4D3×C3⋊D311yescharacteristic
232C42—271no—
233C22×C3—24no—
234C22×C3#381114yes—
235C22×C3D4×C3⋊D311yescharacteristic
236C3×C4—24no—
237C3×C4—24no—
238C3×C4—48no—
239C3×C4—41no—
240C3×C4—44no—
241C3×C4—124no—
242C3×C4—361no—
243C3×C4D3×Dic614yes—
244C3×C4D3×Dic614yes—
245C3×C4C2×#63411yescharacteristic
246C3×C4Q8×C3⋊D311yescharacteristic
247Dic3—41no—
248Dic3—124no—
249Dic3—181no—
250Dic3—181no—
251Dic3—184no—
252Dic3—184no—
253Dic3—364no—
254Dic3—364no—
255Dic3—368no—
256Dic3—361no—
257Dic3—364no—
258C32—24no—
259C32#487614yescomplemented
260C32#487611yescharacteristic, complemented
261Q8—1082no—
262C2×C4—21no—
263C2×C4—61no—
264C2×C4—181no—
265C2×C4—271no—
266C2×C4—271no—
267C2×C4—541no—
268C2×C4C2×D3×C3⋊D311yescharacteristic
269C8—32no—
270C8—92no—
271C2×C3—24no—
272C2×C3—24no—
273C2×C3—24no—
274C2×C3#6384014yes—
275C2×C3#6392714yes—
276C2×C3#6394514yes—
277C2×C3#6409711yescharacteristic
278C2×C3#6412011yescharacteristic
279C2×C3#6412711yescharacteristic
280C22#8903911yescharacteristic
281C4—41no—
282C4—121no—
283C4—361no—
284C4—541no—
285C4—541no—
286C4—1081no—
287C4D3×#63411yescharacteristic
288C4C3⋊D3×Dic611yescharacteristic
289C3—24no—
290C3#1059056314yescomplemented
291C3#1059257511yescharacteristic, complemented
292C2#1172339511yescharacteristic
293C2#1172354811yescharacteristic
294C2#1172357811yescharacteristic
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
121. C3⋊(C3⋊C8)Dic6semidirect
123. C3×C3⋊C8Dic6semidirect
187. C33#501semidirect
212. C3⋊C8#634semidirect
259. C32#4876semidirect
260. C32#4876semidirect
290. C3#10590563semidirect
291. C3#10592575semidirect

Maximal subgroups

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

subgroupindex
#117239312
#117235362
C3×#636342
C32×#8072
#117220542
#117220522
#117220212
#105925753
#105905633