Finite Group Databasebeta

← #13161233

Subgroups — #13161233

Order 1728.

708 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 152 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×#944C211yesmaximal, characteristic
2#11723454C211yesmaximal, characteristic
3#11722205C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic
5#11722038C211yesmaximal, characteristic
6#11722037C211yesmaximal, characteristic
7#11722019C211yesmaximal, characteristic
8#10592781—31nomaximal
9#10591613—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×Dic3⋊C4—21no—
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88807—21no—
20#88806—21no—
21C32×C4⋊Q8—31no—
22#64079—31no—
23C3×#944D314yes—
24#63869—34no—
25#63662—31no—
26#63662—31no—
27#63662—31no—
28#63662—34no—
29#63662—64no—
30#63662D311yescharacteristic
31#63630—31no—
32C3×#835—34no—
33#63566—34no—
34#63565—34no—
35#63554—34no—
36C33×Q8—21no—
37C33×Q8—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C32×Dic3—21no—
43C2×C32×Dic3D411yescharacteristic
44C2×C32×Dic3D411yescharacteristic
45C32×Dic6—21no—
46C32×Dic6—21no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C3×C3⋊(C3⋊C8)—21no—
49#5082—94no—
50#4797—91no—
51C2×C32×Q8—24no—
52C2×C32×Q8—31no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C3×Dic6C2×D314yes—
56C32×C4⋊C4—61no—
57C32×C4⋊C4—61no—
58C32×C42—31no—
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—34no—
61C4⋊(C3⋊Dic3)—64no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63#3750—61no—
64#3750—61no—
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C4⋊Dic3—34no—
70C3×Dic3⋊C4—24no—
71C3×Dic3⋊C4—24no—
72C3×C4×Dic3C2×D314yes—
73C2×C3×C3⋊C8—34no—
74#3724—64no—
75Dic3⋊Dic3—64no—
76C22×C33C2×D411yescharacteristic
77C33×C4—41no—
78C33×C4#4111yescharacteristic
79C33×C4#4111yescharacteristic
80C3⋊(C3⋊Dic3)—41no—
81C3×C3⋊Dic3—41no—
82C32×Dic3—21no—
83C32×Dic3—21no—
84C32×Dic3—41no—
85C3×C4⋊Q8—34no—
86#944C3⋊D311yescharacteristic
87#890—91no—
88#870—94no—
89C3×#103—91no—
90#835—34no—
91#835—94no—
92#835—94no—
93#835—94no—
94#835—91no—
95#835—184no—
96#816—91no—
97#815—91no—
98#810—91no—
99#803—94no—
100C32×Q8—28no—
101C32×Q8—21no—
102C32×Q8—24no—
103C32×Q8—21no—
104C32×Q8—24no—
105C32×Q8—44no—
106C32×Q8—61no—
107C32×Q8—61no—
108C2×C32×C4—21no—
109C2×C32×C4—24no—
110C2×C32×C4—24no—
111C2×C32×C4—31no—
112C2×C32×C4—31no—
113C2×C32×C4—44no—
114C2×C32×C4—61no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—124no—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3#6714yes—
124C2×C3×Dic3#6714yes—
125C3×Dic6—24no—
126C3×Dic6—24no—
127C3⋊(C3⋊C8)—21no—
128C3⋊(C3⋊C8)—61no—
129C3⋊(C3⋊C8)—64no—
130C3⋊(C3⋊C8)—124no—
131C3×C3⋊C8—64no—
132#491—271noSylow
133C2×C33#12411yescharacteristic
134C2×C33#13711yescharacteristic
135C2×C33#13711yescharacteristic
136C2×C3×Q8—24no—
137C2×C3×Q8—34no—
138C2×C3×Q8D3214yes—
139C2×C3×Q8C2×C3⋊D311yescharacteristic
140C2×Dic6C2×C3⋊D311yescharacteristic
141C2×C3×C8—91no—
142C3×C4⋊C4—64no—
143C3×C4⋊C4—64no—
144C3×C4⋊C4—91no—
145C3×C42—34no—
146C4⋊Dic3—34no—
147C4⋊Dic3—94no—
148C4⋊Dic3—91no—
149C4⋊Dic3—184no—
150C4⋊Dic3—181no—
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153Dic3⋊C4—184no—
154Dic3⋊C4—184no—
155Dic3⋊C4—181no—
156C4×Dic3C2×C3⋊D311yescharacteristic
157C2×C3⋊C8—34no—
158C2×C3⋊C8—94no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—184no—
161C22×C32—24no—
162C22×C32D3×D411yescharacteristic
163C22×C32C2×#6714yes—
164C32×C4—24no—
165C32×C4—24no—
166C32×C4—48no—
167C32×C4—41no—
168C32×C4—44no—
169C32×C4—61no—
170C32×C4—61no—
171C32×C4—121no—
172C32×C4#23511yescharacteristic
173C32×C4#23714yes—
174C32×C4#23714yes—
175C32×C4#23711yescharacteristic
176C3⋊Dic3—41no—
177C3⋊Dic3—121no—
178C3⋊Dic3—124no—
179C3⋊Dic3—244no—
180C3×Dic3—24no—
181C3×Dic3—24no—
182C3×Dic3—44no—
183C3×Dic3—124no—
184C4⋊Q8—31no—
185#125—271no—
186#103—91no—
187#103—271no—
188#103—271no—
189#103—271no—
190#97—271no—
191C33#49111yescharacteristic, complemented, Sylow
192C3×Q8—28no—
193C3×Q8—24no—
194C3×Q8—24no—
195C3×Q8—21no—
196C3×Q8—21no—
197C3×Q8—44no—
198C3×Q8—64no—
199C3×Q8—64no—
200C2×C3×C4—24no—
201C2×C3×C4—21no—
202C2×C3×C4—24no—
203C2×C3×C4—34no—
204C2×C3×C4—34no—
205C2×C3×C4—44no—
206C2×C3×C4—64no—
207C2×C3×C4—181no—
208C2×C3×C4C2×D3214yes—
209C2×C3×C4C22×C3⋊D311yescharacteristic
210C2×Dic3—21no—
211C2×Dic3—64no—
212C2×Dic3—184no—
213C2×Dic3—181no—
214C2×Dic3—364no—
215C2×Dic3#63811yescharacteristic
216C2×Dic3#63811yescharacteristic
217Dic6—21no—
218Dic6—21no—
219C3×C8—181no—
220C3⋊C8—64no—
221C3⋊C8—184no—
222C3⋊C8—181no—
223C3⋊C8—364no—
224C2×C32—24no—
225C2×C32—24no—
226C2×C32—24no—
227C2×C32#88511yescharacteristic
228C2×C32#91511yescharacteristic
229C2×C32#91811yescharacteristic
230C2×C32#93614yes—
231C2×C32#94214yes—
232C2×C32#94214yes—
233C2×Q8—31no—
234C2×Q8D3×C3⋊D311yescharacteristic
235C2×C8—91no—
236C2×C8—271no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—91no—
240C4⋊C4—271no—
241C4⋊C4—541no—
242C4⋊C4—541no—
243C42—31no—
244C22×C3—24no—
245C22×C3D3×#6714yes—
246C22×C3C2×#63811yescharacteristic
247C3×C4—24no—
248C3×C4—24no—
249C3×C4—48no—
250C3×C4—41no—
251C3×C4—44no—
252C3×C4—64no—
253C3×C4—64no—
254C3×C4—124no—
255C3×C4—361no—
256C3×C4#380414yes—
257C3×C4#380514yes—
258C3×C4#383011yescharacteristic
259C3×C4#383011yescharacteristic
260Dic3—21no—
261Dic3—21no—
262Dic3—41no—
263Dic3—124no—
264Dic3—361no—
265Dic3—364no—
266Dic3—724no—
267C32—24no—
268C32#479711yescharacteristic, complemented
269C32#508214yescomplemented
270Q8—21no—
271Q8—21no—
272Q8—61no—
273Q8—61no—
274C2×C4—21no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—61no—
278C2×C4—181no—
279C2×C4—541no—
280C2×C4C2×D3×C3⋊D311yescharacteristic
281C8—181no—
282C8—541no—
283C2×C3—24no—
284C2×C3—24no—
285C2×C3—24no—
286C2×C3#6394614yes—
287C2×C3#6394814yes—
288C2×C3#6395914yes—
289C2×C3#6414511yescharacteristic
290C2×C3#6415111yescharacteristic
291C2×C3#6415111yescharacteristic
292C22D3×#63811yescharacteristic
293C4—41no—
294C4—61no—
295C4—61no—
296C4—121no—
297C4—361no—
298C4—1081no—
299C4#8902811yescharacteristic
300C4#8902911yescharacteristic
301C3—24no—
302C3#1059161314yescomplemented
303C3#1059278111yescharacteristic, complemented
304C2#1172357911yescharacteristic
305C2#1172358111yescharacteristic
306C2#1172360011yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#491
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
191. C33#491semidirect
268. C32#4797semidirect
269. C32#5082semidirect
302. C3#10591613semidirect
303. C3#10592781semidirect

Maximal subgroups

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

subgroupindex
C32×#9442
#117234542
#117222052
C3×#636622
#117220382
#117220372
#117220192
#105927813
#105916133