Finite Group Databasebeta

← D3×#63780 #13166500

Subgroups — D3×#63780 #13166500

Order 1728.

680 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×D3×#634C211yesmaximal, characteristic
2D3×#3792C212yesmaximal
3#11723105C211yesmaximal, characteristic, complemented
4C3×#63780C211yesmaximal, characteristic, complemented
5#11722730C212yesmaximal, complemented
6C2×#63780—31nomaximal
7C2×#6308C411yescharacteristic
8C2×C3×#634C2211yescharacteristic
9C2×C32×C4×D3C411yescharacteristic
10C2×#6293C411yescharacteristic
11C2×#6293C2211yescharacteristic
12C2×D3×C3⋊Dic3C411yescharacteristic
13C2×D3×C3⋊Dic3C2211yescharacteristic
14D3×#634—22no—
15D3×#634—22no—
16#88993C2212yes—
17C3×#3792C2212yes—
18#88925—22no—
19D3×#622—22no—
20C22×#634—31no—
21C2×D3×Dic6—62no—
22C2×#3792—32no—
23#63780—31no—
24#63780—32no—
25#63780D311yescharacteristic, direct product factor
26C22×C32×D3C2×C411yescharacteristic
27C2×C33×C4C2×C411yescharacteristic
28C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
29C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
30#6308—22no—
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C3×#634—22no—
34C2×C32×Dic3C2×C411yescharacteristic
35C32×C4×D3—21no—
36C32×C4×D3—21no—
37#6293—22no—
38#6293—22no—
39D3×C3⋊Dic3—21no—
40D3×C3⋊Dic3—21no—
41D3×C3⋊Dic3D412yes—
42D3×C3⋊Dic3D412yes—
43C3⋊#622—22no—
44C3×#622—22no—
45D3×#101—91nomaximal
46C22×C32×C4—31no—
47C22×C3⋊Dic3—31no—
48C22×C3⋊Dic3—31no—
49C2×#634—31no—
50C2×#634—31no—
51C2×#634—31no—
52C2×#634—62no—
53C2×#634—62no—
54C2×#634—62no—
55C2×#634—122no—
56C2×#634C2×D311yescharacteristic
57C2×C3×C4×D3—22no—
58C2×C3×Dic6—62no—
59C2×#627—62no—
60C2×#627—62no—
61C2×D3×Dic3—62no—
62C2×D3×Dic3—62no—
63D3×Dic6—68no—
64D3×Dic6—68no—
65#3792—32no—
66#3792—62no—
67#3792C2×D312yes—
68C2×#622—62no—
69C22×C33C22×C411yescharacteristic
70C2×C32×D3—22no—
71C2×C32×D3C22⋊C411yescharacteristic
72C2×C32×D3C22⋊C411yescharacteristic
73C33×C4—21no—
74C3⋊(C3⋊Dic3)—21no—
75C3⋊(C3⋊Dic3)C22⋊C412yes—
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3C2×D412yes—
78C32×Dic3—21no—
79C2×D3×Q8—91no—
80C22×Dic6—182no—
81D3×M16—92no—
82C3×#101—91no—
83#836—91no—
84#824—92no—
85C23×C32—31no—
86C22×C3×D3—22no—
87C2×C32×C4—22no—
88C2×C32×C4—31no—
89C2×C32×C4—42no—
90C2×C32×C4—61no—
91C2×C32×C4—61no—
92C2×C32×C4C4×D311yescharacteristic
93C2×C3⋊Dic3—31no—
94C2×C3⋊Dic3—32no—
95C2×C3⋊Dic3—32no—
96C2×C3⋊Dic3—31no—
97C2×C3⋊Dic3—62no—
98C2×C3⋊Dic3—62no—
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—61no—
101C2×C3⋊Dic3—122no—
102C2×C3⋊Dic3—122no—
103C2×C3⋊Dic3C4×D311yescharacteristic
104C2×C3⋊Dic3C22×D311yescharacteristic
105#634—22no—
106#634—62no—
107#634—62no—
108#634—62no—
109#634—68no—
110#634—128no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C3×C4×D3—24no—
115C3×C4×D3—24no—
116C3×Dic6—68no—
117#627—68no—
118#627—68no—
119D3×Dic3—64no—
120D3×Dic3—64no—
121D3×Dic3—64no—
122D3×Dic3—64no—
123#622—22no—
124#622—62no—
125C2×#101—271noSylow
126C2×C33—21no—
127C2×C33C2×C22⋊C411yescharacteristic
128C32×D3—21no—
129C32×D3#10112yescomplemented
130C2×C3×Q8—91no—
131C22×C3×C4—62no—
132C22×Dic3—182no—
133C22×Dic3—182no—
134D3×Q8—182no—
135D3×Q8—182no—
136C2×C4×D3—91no—
137C2×C4×D3—91no—
138C2×C4×D3#15711yescharacteristic
139C2×Dic6—62no—
140C2×Dic6—91no—
141C2×Dic6—91no—
142C2×Dic6—91no—
143C2×Dic6—182no—
144C2×Dic6—182no—
145C2×Dic6—182no—
146C2×Dic6—188no—
147C2×Dic6—188no—
148C2×Dic6—362no—
149C3×M16—92no—
150#208—92no—
151ΓL(1,𝔽25)—182no—
152D3×C8—182no—
153C22×C32—22no—
154C22×C32—31no—
155C22×C32—31no—
156C22×C32—42no—
157C22×C32—62no—
158C22×C32C2×C4×D311yescharacteristic
159C2×C3×D3—28no—
160C2×C3×D3—22no—
161C2×C3×D3—22no—
162C32×C4—21no—
163C32×C4—24no—
164C32×C4—44no—
165C32×C4—61no—
166C3⋊Dic3—21no—
167C3⋊Dic3—32no—
168C3⋊Dic3—64no—
169C3⋊Dic3—61no—
170C3⋊Dic3—64no—
171C3⋊Dic3—124no—
172C3⋊Dic3—124no—
173C3⋊Dic3D3×D412yes—
174C3×Dic3—24no—
175C3×Dic3—64no—
176C3×Dic3—64no—
177C22×Q8—271no—
178C2×M16—272no—
179#101—91no—
180#101—272no—
181#101—271no—
182C33C2×#10111yescharacteristic, complemented, Sylow
183C23×C3—62no—
184C22×D3C2×#15711yescharacteristic
185C3×Q8—182no—
186C2×C3×C4—22no—
187C2×C3×C4—42no—
188C2×C3×C4—62no—
189C2×C3×C4—64no—
190C2×C3×C4—64no—
191C2×C3×C4—91no—
192C2×C3×C4—91no—
193C2×C3×C4C2×#15711yescharacteristic
194C2×Dic3—62no—
195C2×Dic3—62no—
196C2×Dic3—91no—
197C2×Dic3—91no—
198C2×Dic3—184no—
199C2×Dic3—184no—
200C2×Dic3—182no—
201C2×Dic3—182no—
202C2×Dic3—184no—
203C2×Dic3—184no—
204C2×Dic3—362no—
205C2×Dic3—362no—
206C2×Dic3C2×#15711yescharacteristic
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—92no—
210C4×D3—92no—
211C4×D3—181no—
212C4×D3—181no—
213Dic6—68no—
214Dic6—182no—
215Dic6—182no—
216Dic6—188no—
217Dic6—188no—
218Dic6—182no—
219Dic6—188no—
220Dic6—368no—
221C3×C8—182no—
222C3⋊C8—182no—
223C2×C32—21no—
224C2×C32—24no—
225C2×C32—22no—
226C2×C32—32no—
227C2×C32—44no—
228C2×C32—42no—
229C2×C32—61no—
230C2×C32D3×C22⋊C411yescharacteristic
231C3×D3—24no—
232C3×D3—24no—
233C2×Q8—91no—
234C2×Q8—271no—
235C2×Q8—271no—
236C2×Q8—271no—
237C2×Q8—542no—
238C2×Q8—542no—
239C22×C4—31no—
240C22×C4—271no—
241C22×C4—271no—
242M16—92no—
243M16—272no—
244M16—542no—
245C2×C8—542no—
246C22×C3—22no—
247C22×C3—42no—
248C22×C3—62no—
249C22×C3—62no—
250C22×C3—68no—
251C22×C3C22×#15711yescharacteristic
252C2×D3—22no—
253C2×D3C22⋊#15711yescharacteristic
254C2×D3C22⋊#15711yescharacteristic
255C3×C4—24no—
256C3×C4—21no—
257C3×C4—44no—
258C3×C4—64no—
259C3×C4—92no—
260C3×C4—181no—
261Dic3—21no—
262Dic3—64no—
263Dic3—64no—
264Dic3—92no—
265Dic3—184no—
266Dic3—184no—
267Dic3—181no—
268Dic3—364no—
269Dic3—364no—
270C32—22no—
271C32—42no—
272C32D3×#10111yescharacteristic, complemented
273C23—31no—
274Q8—182no—
275Q8—542no—
276Q8—542no—
277Q8—542no—
278C2×C4—31no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—91no—
282C2×C4—91no—
283C2×C4—272no—
284C2×C4—272no—
285C2×C4—271no—
286C2×C4—271no—
287C2×C4—541no—
288C2×C4—541no—
289C2×C4D3×#15711yescharacteristic
290C8—182no—
291C8—542no—
292C2×C3—24no—
293C2×C3—21no—
294C2×C3—22no—
295C2×C3—44no—
296C2×C3—42no—
297C2×C3—64no—
298C2×C3—64no—
299C2×C3C2×C22⋊#15711yescharacteristic
300D3—21no—
301D3#6378012yesdirect product factor
302C22—31no—
303C22—31no—
304C22—62no—
305C22C2×D3×#15711yescharacteristic
306C4—21no—
307C4—61no—
308C4—92no—
309C4—181no—
310C4—272no—
311C4—541no—
312C3—22no—
313C3—42no—
314C3C2×#6378011yescharacteristic, complemented
315C2—21no—
316C2—32no—
317C2—61no—
318C2D3×C22⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C22⋊#157
commutatorC22×C33C22×C4
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#101
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723105C2semidirect
4. C3×#63780C2semidirect
5. #11722730C2semidirect
25. #63780D3direct (N × H)
129. C32×D3#101semidirect
182. C33C2×#101semidirect
272. C32D3×#101semidirect
301. D3#63780direct (N × H)
314. C3C2×#63780semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6342
D3×#37922
#117231052
C3×#637802
#117227302
C2×#637803
D3×#1019