Finite Group Databasebeta

← D3×#63578 #13155501

Subgroups — D3×#63578 #13155501

Order 1728.

582 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 136 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×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C3×D3×#208C212yesmaximal
3#11722081C211yesmaximal, characteristic, complemented
4#11722059C212yesmaximal, complemented
5C3×#63578C211yesmaximal, characteristic, complemented
6C2×#63578—31nomaximal
7D3×#822—32nomaximal
8C2×C32×C4×D3C2211yescharacteristic
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C3×#3748C2212yes—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×#208C2212yes—
13D3×C3⋊(C3⋊C8)C412yes—
14D3×C3⋊(C3⋊C8)C412yes—
15C3×#3711—22no—
16C3×D3×C3⋊C8—22no—
17C22×C3⋊(C3⋊C8)—31no—
18C2×C3×#208—32no—
19C3×D3×M16—32no—
20D3×#208D312yes—
21C2×D3×C3⋊C8—32no—
22C2×D3×C3⋊C8—61no—
23#63649—32no—
24C3×#822—32no—
25#63578—31no—
26#63578—32no—
27#63578—32no—
28#63578—62no—
29#63578D311yescharacteristic, direct product factor
30#63577—32no—
31C22×C32×D3Q811yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C32×Dic3Q811yescharacteristic
34C32×C4×D3C2×C411yescharacteristic
35C32×C4×D3C2×C411yescharacteristic
36C32×C4×D3D411yescharacteristic
37C32×C4×D3D411yescharacteristic
38C3⋊(C3⋊(C3⋊C8))C2×C412yes—
39C3×C3⋊(C3⋊C8)—22no—
40C3×C3⋊(C3⋊C8)C2×C412yes—
41C32×C3⋊C8—22no—
42C2×#822—92no—
43D3×#108—91no—
44C22×C32×C4—31no—
45C2×C3×C4×D3—21no—
46C2×C3×C4×D3C2×D312yes—
47C32×M16—32no—
48#3748C2×D312yes—
49C2×C3⋊(C3⋊C8)—32no—
50C2×C3⋊(C3⋊C8)—31no—
51C2×C3⋊(C3⋊C8)—32no—
52C2×C3⋊(C3⋊C8)—32no—
53C2×C3⋊(C3⋊C8)—61no—
54C2×C3⋊(C3⋊C8)—62no—
55C2×C3⋊(C3⋊C8)—121no—
56C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
57C3×#208—22no—
58C3×#208—32no—
59C3×#208—32no—
60C3×#208—62no—
61C3×#208—62no—
62C3×#208C2×D312yes—
63C3×#208C2×D312yes—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—62no—
66C2×C3×C3⋊C8—61no—
67C3×ΓL(1,𝔽25)—62no—
68C3×D3×C8—62no—
69#3711—22no—
70D3×C3⋊C8—22no—
71D3×C3⋊C8—34no—
72D3×C3⋊C8—34no—
73D3×C3⋊C8—62no—
74D3×C3⋊C8—62no—
75C22×C33C2×Q811yescharacteristic
76C2×C32×D3—22no—
77C2×C32×D3C4⋊C411yescharacteristic
78C2×C32×D3C4⋊C411yescharacteristic
79C33×C4C22×C411yescharacteristic
80C33×C4C2×D411yescharacteristic
81C32×Dic3C4⋊C411yescharacteristic
82C32×Dic3C4⋊C411yescharacteristic
83C2×C3×M16—92no—
84C2×#208—32no—
85C22×C3⋊C8—92no—
86C22×C3⋊C8—181no—
87D3×M16—32no—
88C2×D3×C8—91no—
89C3×#108—91no—
90#822—32no—
91#822—92no—
92#822—92no—
93#822—92no—
94#822—92no—
95#822—182no—
96#819—91no—
97C23×C32—31no—
98C22×C3×D3—21no—
99C22×C3×D3Dic612yes—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—31no—
103C2×C32×C4—31no—
104C2×C32×C4—31no—
105C2×C32×C4—31no—
106C2×C32×C4—31no—
107C2×C32×C4—41no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D312yes—
110C2×C3×Dic3—21no—
111C2×C3×Dic3Dic612yes—
112C3×C4×D3—21no—
113C3×C4×D3—21no—
114C3×C4×D3—21no—
115C3×C4×D3—21no—
116C3×C4×D3C4×D312yes—
117C3×C4×D3C4×D312yes—
118C3×C4×D3#6712yes—
119C3×C4×D3#6712yes—
120C32×C8—62no—
121C3⋊(C3⋊C8)—22no—
122C3⋊(C3⋊C8)—34no—
123C3⋊(C3⋊C8)—32no—
124C3⋊(C3⋊C8)—62no—
125C3⋊(C3⋊C8)—64no—
126C3⋊(C3⋊C8)—122no—
127C3⋊(C3⋊C8)C4×D312yes—
128C3×C3⋊C8—22no—
129C3×C3⋊C8—22no—
130C3×C3⋊C8—34no—
131C3×C3⋊C8—42no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—62no—
135C3×C3⋊C8—122no—
136C2×#108—271noSylow
137C2×C33—21no—
138C2×C33C2×C4⋊C411yescharacteristic
139C32×D3—21no—
140C32×D3#10812yescomplemented
141C22×C3×C4—32no—
142C22×C3×C4—61no—
143C2×C4×D3D3211yescharacteristic
144C3×M16—32no—
145C3×M16—32no—
146C3×M16—62no—
147C3×M16—92no—
148C3×M16—182no—
149C2×C3×C8—91no—
150C2×C3×C8—182no—
151#208—32no—
152#208—32no—
153#208—62no—
154#208—62no—
155#208D3212yes—
156C2×C3⋊C8—32no—
157C2×C3⋊C8—61no—
158C2×C3⋊C8—62no—
159C2×C3⋊C8—94no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—92no—
163C2×C3⋊C8—182no—
164C2×C3⋊C8—182no—
165C2×C3⋊C8—182no—
166C2×C3⋊C8—181no—
167C2×C3⋊C8—361no—
168ΓL(1,𝔽25)—62no—
169D3×C8—62no—
170D3×C8—92no—
171D3×C8—92no—
172C22×C32—22no—
173C22×C32—21no—
174C22×C32—31no—
175C22×C32—31no—
176C22×C32—41no—
177C22×C32—62no—
178C22×C32C2×Dic612yes—
179C22×C32D3×Q811yescharacteristic
180C2×C3×D3—24no—
181C2×C3×D3—24no—
182C2×C3×D3—21no—
183C2×C3×D3—21no—
184C2×C3×D3Dic3⋊C412yes—
185C2×C3×D3Dic3⋊C412yes—
186C32×C4—22no—
187C32×C4—21no—
188C32×C4—21no—
189C32×C4—22no—
190C32×C4—31no—
191C32×C4—31no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4C2×C4×D311yescharacteristic
195C32×C4C2×C4×D312yes—
196C32×C4D3×D411yescharacteristic
197C32×C4C2×#6712yes—
198C3×Dic3—21no—
199C3×Dic3—21no—
200C3×Dic3Dic3⋊C412yes—
201C3×Dic3Dic3⋊C412yes—
202C2×M16—92no—
203C22×C8—271no—
204#108—91no—
205#108—271no—
206#108—272no—
207C33C2×#10811yescharacteristic, complemented, Sylow
208C23×C3—32no—
209C23×C3—61no—
210C22×D3#62711yescharacteristic
211C2×C3×C4—21no—
212C2×C3×C4—22no—
213C2×C3×C4—32no—
214C2×C3×C4—32no—
215C2×C3×C4—32no—
216C2×C3×C4—32no—
217C2×C3×C4—32no—
218C2×C3×C4—41no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4C2×D3212yes—
225C2×C3×C4C2×D3211yescharacteristic
226C2×Dic3#62711yescharacteristic
227C4×D3#62411yescharacteristic
228C4×D3#62411yescharacteristic
229C4×D3#62511yescharacteristic
230C4×D3#62511yescharacteristic
231C3×C8—62no—
232C3×C8—62no—
233C3×C8—92no—
234C3×C8—122no—
235C3×C8—182no—
236C3⋊C8—22no—
237C3⋊C8—34no—
238C3⋊C8—62no—
239C3⋊C8—62no—
240C3⋊C8—62no—
241C3⋊C8—94no—
242C3⋊C8—92no—
243C3⋊C8—122no—
244C3⋊C8—184no—
245C3⋊C8—182no—
246C3⋊C8—362no—
247C2×C32—21no—
248C2×C32—22no—
249C2×C32—22no—
250C2×C32—22no—
251C2×C32—21no—
252C2×C32—32no—
253C2×C32—41no—
254C2×C32—42no—
255C2×C32—42no—
256C2×C32—61no—
257C2×C32D3×C4⋊C411yescharacteristic
258C2×C32C2×Dic3⋊C412yes—
259C3×D3—22no—
260C3×D3—22no—
261C3×D3—22no—
262C3×D3#82214yescomplemented
263C22×C4—31no—
264M16—32no—
265M16—92no—
266M16—182no—
267C2×C8—91no—
268C2×C8—182no—
269C2×C8—272no—
270C2×C8—272no—
271C2×C8—271no—
272C22×C3—21no—
273C22×C3—22no—
274C22×C3—32no—
275C22×C3—32no—
276C22×C3—41no—
277C22×C3—61no—
278C22×C3—61no—
279C22×C3—64no—
280C22×C3—64no—
281C22×C3D3×Dic612yes—
282C22×C3C2×#62711yescharacteristic
283C2×D3—22no—
284C2×D3#372411yescharacteristic
285C2×D3#372411yescharacteristic
286C3×C4—21no—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—21no—
290C3×C4—32no—
291C3×C4—32no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4C4×D3212yes—
297C3×C4C2×#62411yescharacteristic
298C3×C4C2×#62511yescharacteristic
299C3×C4D3×#6712yes—
300Dic3#372411yescharacteristic
301Dic3#372411yescharacteristic
302C32—21no—
303C32—22no—
304C32—41no—
305C32D3×#10811yescharacteristic, complemented
306C32C2×#82212yescomplemented
307C23—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4D3311yescharacteristic
314C8—62no—
315C8—92no—
316C8—182no—
317C8—272no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—21no—
322C2×C3—22no—
323C2×C3—34no—
324C2×C3—42no—
325C2×C3—42no—
326C2×C3—41no—
327C2×C3—62no—
328C2×C3—62no—
329C2×C3—62no—
330C2×C3D3×Dic3⋊C412yes—
331C2×C3C2×#372411yescharacteristic
332D3—21no—
333D3#6357812yesdirect product factor
334C22—31no—
335C22—31no—
336C22—62no—
337C22D3×#62711yescharacteristic
338C4—31no—
339C4—31no—
340C4D3×#62411yescharacteristic
341C4D3×#62511yescharacteristic
342C3—21no—
343C3—22no—
344C3—41no—
345C3D3×#82212yescomplemented
346C3C2×#6357811yescharacteristic, complemented
347C2—21no—
348C2—32no—
349C2—61no—
350C2D3×#372411yescharacteristic
subgroupG/H
centerC4D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C2×C4⋊C4
prime radicalC22×C32×D3Q8
hypocenterC33C2×#108
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722081C2semidirect
4. #11722059C2semidirect
5. C3×#63578C2semidirect
29. #63578D3direct (N × H)
140. C32×D3#108semidirect
207. C33C2×#108semidirect
262. C3×D3#822semidirect
305. C32D3×#108semidirect
306. C32C2×#822semidirect
333. D3#63578direct (N × H)
345. C3D3×#822semidirect
346. C3C2×#63578semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C3⋊(C3⋊C8)2
C3×D3×#2082
#117220812
#117220592
C3×#635782
C2×#635783
D3×#8223