Finite Group Databasebeta

← #101×(C3⋊D3)⋊C3 #13141696

Subgroups — #101×(C3⋊D3)⋊C3 #13141696

Order 1728.

549 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 118 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×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2M16×(C3⋊D3)⋊C3C212yesmaximal
3H(3,𝔽3)×#101C211yesmaximal, characteristic, complemented
4#63663⋊C3C211yesmaximal, characteristic, complemented
5#63651⋊C3C212yesmaximal, complemented
6C3⋊D3×#101C311yesmaximal, characteristic, complemented
7C3×D3×#101—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9Q8×(C3⋊D3)⋊C3—22no—
10Q8×(C3⋊D3)⋊C3—22no—
11C2×C4×(C3⋊D3)⋊C3C412yes—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#634⋊C3C412yes—
14C2×#634⋊C3C2211yescharacteristic
15M16×H(3,𝔽3)C2212yes—
16#3748⋊C3C2212yes—
17#3743⋊C3—22no—
18C8×(C3⋊D3)⋊C3—22no—
19C2×Q8×C3⋊D3C2×C311yescharacteristic
20C2×C3×D3×Q8—31no—
21M16×C3⋊D3C2×C312yes—
22C3×D3×M16—32no—
23C32×#101—21no—
24C32×#101D311yescharacteristic, complemented
25C32×#101C2×C311yescharacteristic, complemented
26#63663C2×C311yescharacteristic, complemented
27#63651C2×C312yescomplemented
28C3×#836—31no—
29C3×#824—32no—
30C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
31Q8×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)C2×C412yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×(C3⋊Dic3)⋊C3C2×C412yes—
35C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
36C4×(C3⋊D3)⋊C3—22no—
37C4×(C3⋊D3)⋊C3—22no—
38C4×(C3⋊D3)⋊C3D412yes—
39C4×(C3⋊D3)⋊C3D412yes—
40#634⋊C3—22no—
41#634⋊C3—24no—
42C8×H(3,𝔽3)—22no—
43(C3⋊(C3⋊C8))⋊C3—22no—
44C2×C3×#101—91no—
45D3×#101—31no—
46D3×#101—91no—
47C2×C32×Q8—21no—
48C2×C32×Q8C2×D311yescharacteristic
49C2×C32×Q8C22×C311yescharacteristic
50Q8×C3⋊D3—22no—
51Q8×C3⋊D3—22no—
52C2×C4×C3⋊D3C3×C412yes—
53C2×C4×C3⋊D3C22×C311yescharacteristic
54C2×#634C3×C412yes—
55C2×#634C22×C311yescharacteristic
56C3×D3×Q8—62no—
57C3×D3×Q8—62no—
58C2×C3×C4×D3—31no—
59C2×C3×C4×D3—32no—
60C2×C3×Dic6—32no—
61C2×C3×Dic6—31no—
62C32×M16—22no—
63C32×M16C2×D312yes—
64C32×M16C22×C312yes—
65#3748C22×C312yes—
66#3743—22no—
67C8×C3⋊D3—22no—
68C3×#208—32no—
69C3×ΓL(1,𝔽25)—62no—
70C3×D3×C8—62no—
71C22×H(3,𝔽3)C22×C411yescharacteristic
72C2×(C3⋊D3)⋊C3—22no—
73C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
74C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
75C4×H(3,𝔽3)—22no—
76C4×H(3,𝔽3)C2×D412yes—
77(C3⋊Dic3)⋊C3—22no—
78(C3⋊Dic3)⋊C3C22⋊C412yes—
79C22×C3×Q8—91no—
80C2×D3×Q8—31no—
81C2×D3×Q8—91no—
82C2×C3×M16—92no—
83D3×M16—32no—
84D3×M16—92no—
85C3×#101—31no—
86C3×#101—31no—
87C3×#101—61no—
88C3×#101—92no—
89C3×#101—91no—
90C3×#101C3×D311yescharacteristic
91#836—31no—
92#836—91no—
93#824—32no—
94#824—92no—
95C22×C3⋊D3C2×C3×C411yescharacteristic
96C22×C3×D3—31no—
97C32×Q8—22no—
98C32×Q8—22no—
99C32×Q8—42no—
100C2×C32×C4—22no—
101C2×C32×C4—21no—
102C2×C32×C4C4×D312yes—
103C2×C32×C4C2×C3×C412yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C32×C4C23×C311yescharacteristic
106C2×C3⋊Dic3C2×C3×C412yes—
107C2×C3⋊Dic3C2×C3×C411yescharacteristic
108C4×C3⋊D3—22no—
109C4×C3⋊D3—22no—
110C4×C3⋊D3C3×D412yes—
111C4×C3⋊D3C3×D412yes—
112#634—22no—
113#634—24no—
114C2×C3×Dic3—31no—
115C2×C3×Dic3—32no—
116C3×C4×D3—32no—
117C3×C4×D3—32no—
118C3×C4×D3—62no—
119C3×C4×D3—62no—
120C3×Dic6—64no—
121C3×Dic6—62no—
122C32×C8—22no—
123C32×C8—22no—
124C32×C8—42no—
125C3⋊(C3⋊C8)—22no—
126C3×C3⋊C8—62no—
127C2×#101—91noSylow
128C2×H(3,𝔽3)—21no—
129C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
130(C3⋊D3)⋊C3—21no—
131(C3⋊D3)⋊C3#10112yesdirect product factor
132C2×C3×Q8—31no—
133C2×C3×Q8—31no—
134C2×C3×Q8—61no—
135C2×C3×Q8—91no—
136C2×C3×Q8—92no—
137C2×C3×Q8—182no—
138C2×C3×Q8—182no—
139C2×C3×Q8C2×C3×D311yescharacteristic
140C22×C3×C4—91no—
141C22×C3×C4—92no—
142D3×Q8—62no—
143D3×Q8—62no—
144D3×Q8—182no—
145D3×Q8—182no—
146C2×C4×D3—31no—
147C2×C4×D3—32no—
148C2×C4×D3—91no—
149C2×C4×D3—92no—
150C2×Dic6—32no—
151C2×Dic6—31no—
152C2×Dic6—91no—
153C2×Dic6—92no—
154C3×M16—32no—
155C3×M16—32no—
156C3×M16—62no—
157C3×M16—92no—
158C3×M16—182no—
159C3×M16C2×C3×D312yes—
160C2×C3×C8—182no—
161#208—32no—
162#208—92no—
163ΓL(1,𝔽25)—62no—
164ΓL(1,𝔽25)—182no—
165D3×C8—62no—
166D3×C8—182no—
167C22×C32—21no—
168C22×C32C2×C4×D311yescharacteristic
169C22×C32C22×C3×C411yescharacteristic
170C2×C3⋊D3—22no—
171C2×C3⋊D3C3×C22⋊C411yescharacteristic
172C2×C3⋊D3C3×C22⋊C411yescharacteristic
173C2×C3×D3—31no—
174C2×C3×D3—31no—
175C2×C3×D3—62no—
176C32×C4—22no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—42no—
180C32×C4D3×D412yes—
181C32×C4C2×C3×D412yes—
182C3⋊Dic3—22no—
183C3⋊Dic3C3×C22⋊C412yes—
184C3×Dic3—32no—
185C3×Dic3—62no—
186C22×Q8—91no—
187C2×M16—92no—
188#101—91no—
189#101—92no—
190#101(C3⋊D3)⋊C311yescharacteristic, direct product factor
191H(3,𝔽3)C2×#10111yescharacteristic, complemented, Sylow
192C23×C3—91no—
193C22×D3—31no—
194C22×D3—91no—
195C3×Q8—22no—
196C3×Q8—62no—
197C3×Q8—62no—
198C3×Q8—122no—
199C3×Q8—182no—
200C3×Q8—184no—
201C2×C3×C4—32no—
202C2×C3×C4—32no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—62no—
206C2×C3×C4—61no—
207C2×C3×C4—92no—
208C2×C3×C4—92no—
209C2×C3×C4—91no—
210C2×C3×C4—92no—
211C2×C3×C4—182no—
212C2×C3×C4—182no—
213C2×C3×C4C3×C4×D312yes—
214C2×C3×C4C22×C3×D311yescharacteristic
215C2×Dic3—31no—
216C2×Dic3—32no—
217C2×Dic3—91no—
218C2×Dic3—92no—
219C4×D3—32no—
220C4×D3—32no—
221C4×D3—62no—
222C4×D3—62no—
223C4×D3—92no—
224C4×D3—92no—
225C4×D3—182no—
226C4×D3—182no—
227Dic6—64no—
228Dic6—62no—
229Dic6—184no—
230Dic6—182no—
231C3×C8—22no—
232C3×C8—62no—
233C3×C8—62no—
234C3×C8—122no—
235C3×C8—182no—
236C3⋊C8—62no—
237C3⋊C8—182no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—41no—
242C2×C32D3×C22⋊C411yescharacteristic
243C2×C32C2×C3×C22⋊C411yescharacteristic
244C3⋊D3—21no—
245C3⋊D3C3×#10112yescomplemented
246C3×D3—32no—
247C3×D3—61no—
248C2×Q8—92no—
249C2×Q8—91no—
250C2×Q8—182no—
251C2×Q8—182no—
252C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
253C22×C4—91no—
254C22×C4—92no—
255M16—92no—
256M16—182no—
257M16C2×(C3⋊D3)⋊C312yes—
258C2×C8—182no—
259C22×C3—31no—
260C22×C3—31no—
261C22×C3—61no—
262C22×C3—91no—
263C22×C3—91no—
264C22×C3—182no—
265C22×C3C2×C3×C4×D311yescharacteristic
266C2×D3—31no—
267C2×D3—31no—
268C2×D3—62no—
269C2×D3—91no—
270C2×D3—91no—
271C2×D3—182no—
272C3×C4—22no—
273C3×C4—32no—
274C3×C4—32no—
275C3×C4—62no—
276C3×C4—62no—
277C3×C4—62no—
278C3×C4—92no—
279C3×C4—122no—
280C3×C4—182no—
281C3×C4C3×D3×D412yes—
282Dic3—32no—
283Dic3—62no—
284Dic3—92no—
285Dic3—182no—
286C32—21no—
287C32D3×#10111yescharacteristic, complemented
288C32C2×C3×#10111yescharacteristic, complemented
289C23—91no—
290Q8—22no—
291Q8—184no—
292Q8—182no—
293C2×C4—91no—
294C2×C4—92no—
295C2×C4—92no—
296C2×C4—92no—
297C2×C4—182no—
298C2×C4—182no—
299C2×C4C4×(C3⋊D3)⋊C312yes—
300C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
301C8—22no—
302C8—182no—
303C2×C3—21no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—61no—
309C2×C3—92no—
310C2×C3—121no—
311C2×C3—181no—
312C2×C3C3×D3×C22⋊C411yescharacteristic
313D3—32no—
314D3—61no—
315D3—92no—
316D3—181no—
317C22—91no—
318C22—91no—
319C22—182no—
320C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
321C4—22no—
322C4—92no—
323C4—182no—
324C4D4×(C3⋊D3)⋊C312yes—
325C3—31no—
326C3—31no—
327C3—61no—
328C3C3×D3×#10111yescharacteristic
329C2—21no—
330C2—92no—
331C2—181no—
332C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2C22⋊C4×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#101C2
socleC2×C3C3×D3×C22⋊C4
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C32×C4C23×C3
hypercenter#101(C3⋊D3)⋊C3
hypocenterC32C2×C3×#101
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
3. H(3,𝔽3)×#101C2semidirect
4. #63663⋊C3C2semidirect
5. #63651⋊C3C2semidirect
6. C3⋊D3×#101C3semidirect
24. C32×#101D3semidirect
25. C32×#101C2×C3semidirect
26. #63663C2×C3semidirect
27. #63651C2×C3semidirect
131. (C3⋊D3)⋊C3#101direct (N × H)
190. #101(C3⋊D3)⋊C3direct (N × H)
191. H(3,𝔽3)C2×#101semidirect
245. C3⋊D3C3×#101semidirect
287. C32D3×#101semidirect
288. C32C2×C3×#101semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
M16×(C3⋊D3)⋊C32
H(3,𝔽3)×#1012
#63663⋊C32
#63651⋊C32
C3⋊D3×#1013
C3×D3×#1013