Finite Group Databasebeta

← C22×D3×#208 #11918445

Subgroups — C22×D3×#208 #11918445

Order 1152.

3589 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×D3C211yesmaximal, characteristic
2C22×#3748C211yesmaximal, characteristic, complemented
3C22×C3×#208C211yesmaximal, characteristic, complemented
4C22×#3711C212yesmaximal, complemented
5C2×D3×#208C2124yesmaximal, direct product factor
6C22×D3×C3⋊C8C212yesmaximal, complemented
7C23×#208—31nomaximal
8C22×D3×M16—31nomaximal
9C24×C3×D3C411yescharacteristic
10C23×C32×C4C2211yescharacteristic
11C23×C3×Dic3C411yescharacteristic
12C22×C3×C4×D3C411yescharacteristic
13C22×C3×C4×D3C4112yes—
14C22×C3×C4×D3C411yescharacteristic
15C22×C3×C4×D3C2211yescharacteristic
16C22×C3×C4×D3C22112yes—
17C22×C3×C4×D3C2211yescharacteristic
18C2×#3748C22112yescomplemented
19C22×C3⋊(C3⋊C8)C2212yescomplemented
20C2×C3×#208C22112yescomplemented
21C22×C3×C3⋊C8C2212yescomplemented
22C2×#3711C22124yescomplemented
23D3×#208C22164yesdirect product factor
24C2×D3×C3⋊C8C22124yescomplemented
25C24×C3×C4—31no—
26C23×C4×D3D311yescharacteristic
27C22×C3×M16—31no—
28C22×#208—31no—
29C22×#208—31no—
30C22×#208—324no—
31C22×#208—32no—
32C22×#208—61no—
33C22×#208D311yescharacteristic, direct product factor
34C23×C3⋊C8—32no—
35C2×D3×M16—324no—
36C22×ΓL(1,𝔽25)—32no—
37C22×D3×C8—32no—
38C24×C32C2×C411yescharacteristic
39C23×C3×D3—28no—
40C23×C3×D3C2×C411yescharacteristic
41C23×C3×D3C2×C4112yes—
42C23×C3×D3C2×C411yescharacteristic
43C22×C32×C4C2×C416yes—
44C22×C32×C4C2×C411yescharacteristic
45C22×C32×C4C2311yescharacteristic
46C22×C32×C4C2316yes—
47C22×C3×Dic3C2×C4112yes—
48C22×C3×Dic3C2×C411yescharacteristic
49C22×C3×Dic3C2×C411yescharacteristic
50C2×C3×C4×D3C2×C4112yes—
51C2×C3×C4×D3C2×C4148yes—
52C2×C3×C4×D3C2×C416yes—
53C2×C3×C4×D3C2×C416yes—
54C2×C3×C4×D3C2×C4112yes—
55C2×C3×C4×D3C2316yes—
56C2×C3×C4×D3C23116yes—
57C2×C3×C4×D3C2316yes—
58#3748C23116yescomplemented
59C2×C3⋊(C3⋊C8)C23112yescomplemented
60C3×#208C23116yescomplemented
61C2×C3×C3⋊C8C23112yescomplemented
62#3711C23132yescomplemented
63D3×C3⋊C8C23132yescomplemented
64C23×M16—91noSylow
65C25×C3—31no—
66C24×D3Dic311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4—31no—
69C23×C3×C4—31no—
70C23×C3×C4—312no—
71C23×C3×C4—31no—
72C23×C3×C4—31no—
73C23×C3×C4—312no—
74C23×C3×C4—31no—
75C23×C3×C4C2×D311yescharacteristic
76C23×C3×C4C2×D311yescharacteristic
77C23×Dic3Dic311yescharacteristic
78C22×C4×D3Dic3112yes—
79C22×C4×D3Dic311yescharacteristic
80C22×C4×D3Dic311yescharacteristic
81C22×C4×D3C2×D311yescharacteristic
82C22×C4×D3C2×D311yescharacteristic
83C22×C4×D3C2×D3112yes—
84C2×C3×M16—312no—
85C22×C3×C8—32no—
86C2×#208—312no—
87C2×#208—312no—
88C2×#208—364no—
89C2×#208—324no—
90C2×#208—612no—
91C2×#208C2×D3112yesdirect product factor
92C22×C3⋊C8—32no—
93C22×C3⋊C8—32no—
94C22×C3⋊C8—324no—
95C22×C3⋊C8—62no—
96C22×C3⋊C8C2×D312yescomplemented
97D3×M16—364no—
98C2×ΓL(1,𝔽25)—324no—
99C2×D3×C8—324no—
100C23×C32—24no—
101C23×C32C22×C416yes—
102C23×C32C22×C411yescharacteristic
103C22×C3×D3—24no—
104C22×C3×D3—248no—
105C22×C3×D3M1618yescomplemented
106C22×C3×D3C22×C416yes—
107C22×C3×D3C22×C4116yes—
108C22×C3×D3C22×C416yes—
109C2×C32×C4C22×C4112yes—
110C2×C32×C4C22×C413yes—
111C2×C32×C4C22×C416yes—
112C2×C32×C4C2413yes—
113C2×C32×C4C2414yes—
114C2×C3×Dic3C22×C4116yes—
115C2×C3×Dic3C22×C416yes—
116C2×C3×Dic3C22×C416yes—
117C3×C4×D3C22×C4116yes—
118C3×C4×D3C22×C4112yes—
119C3×C4×D3C22×C4112yes—
120C3×C4×D3C22×C4116yes—
121C3×C4×D3C2414yes—
122C3×C4×D3C2414yes—
123C3⋊(C3⋊C8)C2418yescomplemented
124C3×C3⋊C8C2418yescomplemented
125C24×C4—31no—
126C22×M16—31no—
127C22×M16—92no—
128C22×M16—924no—
129C22×M16—91no—
130C23×C8—92no—
131C24×C3—21no—
132C24×C3—31no—
133C24×C3—312no—
134C24×C3—31no—
135C24×C3—68no—
136C24×C3C4×D311yescharacteristic
137C24×C3C2×Dic311yescharacteristic
138C23×D3—28no—
139C23×D3C2×Dic311yescharacteristic
140C23×D3C2×Dic3112yes—
141C23×D3C2×Dic311yescharacteristic
142C22×C3×C4—21no—
143C22×C3×C4—26no—
144C22×C3×C4—26no—
145C22×C3×C4—21no—
146C22×C3×C4—312no—
147C22×C3×C4—31no—
148C22×C3×C4—31no—
149C22×C3×C4—36no—
150C22×C3×C4—312no—
151C22×C3×C4—348no—
152C22×C3×C4—312no—
153C22×C3×C4—36no—
154C22×C3×C4—36no—
155C22×C3×C4—316no—
156C22×C3×C4—36no—
157C22×C3×C4C4×D316yes—
158C22×C3×C4C4×D311yescharacteristic
159C22×C3×C4C2×Dic316yes—
160C22×C3×C4C2×Dic311yescharacteristic
161C22×C3×C4C22×D316yes—
162C22×C3×C4C22×D316yes—
163C22×C3×C4C22×D311yescharacteristic
164C22×C3×C4C22×D311yescharacteristic
165C22×Dic3C2×Dic3112yes—
166C22×Dic3C2×Dic311yescharacteristic
167C22×Dic3C2×Dic311yescharacteristic
168C2×C4×D3C2×Dic316yes—
169C2×C4×D3C2×Dic3148yes—
170C2×C4×D3C2×Dic3112yes—
171C2×C4×D3C2×Dic316yes—
172C2×C4×D3C2×Dic3112yes—
173C2×C4×D3C22×D316yes—
174C2×C4×D3C22×D316yes—
175C2×C4×D3C22×D3116yes—
176C3×M16—316no—
177C2×C3×C8—312no—
178#208—316no—
179#208—316no—
180#208—332no—
181#208—616no—
182#208C22×D3116yesdirect product factor
183C2×C3⋊C8—312no—
184C2×C3⋊C8—312no—
185C2×C3⋊C8—332no—
186C2×C3⋊C8—612no—
187C2×C3⋊C8C22×D3112yescomplemented
188ΓL(1,𝔽25)—332no—
189D3×C8—332no—
190C22×C32—212no—
191C22×C32C2×M1614yescomplemented
192C22×C32C23×C414yes—
193C22×C32C23×C413yes—
194C2×C3×D3—212no—
195C2×C3×D3—232no—
196C2×C3×D3C2×M16124yescomplemented
197C2×C3×D3C23×C414yes—
198C2×C3×D3C23×C414yes—
199C32×C4C23×C413yes—
200C32×C4C23×C414yes—
201C32×C4C2511yescharacteristic
202C3×Dic3C23×C414yes—
203C3×Dic3C23×C414yes—
204C25—31no—
205C23×C4—31no—
206C23×C4—31no—
207C23×C4—312no—
208C23×C4—31no—
209C23×C4—312no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4D3211yescharacteristic
213C2×M16—312no—
214C2×M16—924no—
215C2×M16—964no—
216C2×M16—912no—
217C22×C8—32no—
218C22×C8—924no—
219C22×C8—92no—
220C23×C3—24no—
221C23×C3—24no—
222C23×C3—21no—
223C23×C3—26no—
224C23×C3—38no—
225C23×C3—36no—
226C23×C3—316no—
227C23×C3—36no—
228C23×C3—44no—
229C23×C3—648no—
230C23×C3—64no—
231C23×C3C2×C4×D316yes—
232C23×C3C2×C4×D311yescharacteristic
233C23×C3C22×Dic316yes—
234C23×C3C22×Dic311yescharacteristic
235C22×D3—24no—
236C22×D3—248no—
237C22×D3#20818yesdirect product factor
238C22×D3C22×Dic316yes—
239C22×D3C22×Dic3116yes—
240C22×D3C22×Dic316yes—
241C2×C3×C4—23no—
242C2×C3×C4—24no—
243C2×C3×C4—26no—
244C2×C3×C4—212no—
245C2×C3×C4—23no—
246C2×C3×C4—316no—
247C2×C3×C4—36no—
248C2×C3×C4—36no—
249C2×C3×C4—312no—
250C2×C3×C4—316no—
251C2×C3×C4—316no—
252C2×C3×C4—312no—
253C2×C3×C4—34no—
254C2×C3×C4—34no—
255C2×C3×C4C2×C4×D3112yes—
256C2×C3×C4C2×C4×D316yes—
257C2×C3×C4C2×C4×D313yes—
258C2×C3×C4C22×Dic3112yes—
259C2×C3×C4C22×Dic316yes—
260C2×C3×C4C22×Dic313yes—
261C2×C3×C4C23×D313yes—
262C2×C3×C4C23×D314yes—
263C2×C3×C4C23×D313yes—
264C2×C3×C4C23×D314yes—
265C2×Dic3C22×Dic3116yes—
266C2×Dic3C22×Dic316yes—
267C2×Dic3C22×Dic316yes—
268C4×D3C22×Dic3112yes—
269C4×D3C22×Dic3116yes—
270C4×D3C22×Dic3112yes—
271C4×D3C22×Dic3116yes—
272C4×D3C23×D314yes—
273C4×D3C23×D314yes—
274C3×C8—38no—
275C3⋊C8—38no—
276C3⋊C8—38no—
277C3⋊C8—68no—
278C3⋊C8C23×D318yescomplemented
279C2×C32—24no—
280C2×C32C22×M1616yescomplemented
281C2×C32C24×C411yescharacteristic
282C3×D3—24no—
283C3×D3C22×M1618yescomplemented
284C24—31no—
285C24—312no—
286C24—31no—
287C24—68no—
288C24D3×Dic311yescharacteristic
289C22×C4—312no—
290C22×C4—31no—
291C22×C4—31no—
292C22×C4—36no—
293C22×C4—316no—
294C22×C4—36no—
295C22×C4—312no—
296C22×C4—348no—
297C22×C4—312no—
298C22×C4—36no—
299C22×C4—36no—
300C22×C4D3×Dic311yescharacteristic
301C22×C4D3×Dic316yes—
302C22×C4C2×D3216yes—
303C22×C4C2×D3211yescharacteristic
304M16—316no—
305M16—932no—
306M16—916no—
307C2×C8—312no—
308C2×C8—932no—
309C2×C8—912no—
310C22×C3—212no—
311C22×C3—212no—
312C22×C3—23no—
313C22×C3—24no—
314C22×C3—24no—
315C22×C3—34no—
316C22×C3—324no—
317C22×C3—34no—
318C22×C3—412no—
319C22×C3—632no—
320C22×C3—612no—
321C22×C3D3×M1614yescomplemented
322C22×C3C2×#20814yescomplemented
323C22×C3C22×C4×D314yes—
324C22×C3C22×C4×D313yes—
325C22×C3C23×Dic314yes—
326C22×C3C23×Dic313yes—
327C2×D3—212no—
328C2×D3—232no—
329C2×D3C2×#208124yesdirect product factor
330C2×D3C23×Dic314yes—
331C2×D3C23×Dic314yes—
332C3×C4—23no—
333C3×C4—24no—
334C3×C4—21no—
335C3×C4—34no—
336C3×C4—34no—
337C3×C4C22×C4×D313yes—
338C3×C4C22×C4×D314yes—
339C3×C4C23×Dic313yes—
340C3×C4C23×Dic314yes—
341C3×C4C24×D311yescharacteristic
342C3×C4C24×D311yescharacteristic
343Dic3C23×Dic314yes—
344Dic3C23×Dic314yes—
345C32C23×M1611yescharacteristic, complemented, Sylow
346C23—24no—
347C23—36no—
348C23—38no—
349C23—316no—
350C23—36no—
351C23—648no—
352C23—64no—
353C23C2×D3×Dic316yes—
354C23C2×D3×Dic311yescharacteristic
355C2×C4—316no—
356C2×C4—36no—
357C2×C4—36no—
358C2×C4—312no—
359C2×C4—316no—
360C2×C4—316no—
361C2×C4—312no—
362C2×C4—34no—
363C2×C4—34no—
364C2×C4C2×D3×Dic3112yes—
365C2×C4C2×D3×Dic316yes—
366C2×C4C2×D3×Dic313yes—
367C2×C4C22×D3213yes—
368C2×C4C22×D3214yes—
369C8—38no—
370C8—98no—
371C2×C3—24no—
372C2×C3—24no—
373C2×C3—21no—
374C2×C3—26no—
375C2×C3—38no—
376C2×C3—44no—
377C2×C3—64no—
378C2×C3C2×D3×M1616yescomplemented
379C2×C3C22×#20816yescomplemented
380C2×C3C23×C4×D311yescharacteristic
381C2×C3C24×Dic311yescharacteristic
382D3—24no—
383D3C22×#20818yesdirect product factor
384C22—212no—
385C22—34no—
386C22—324no—
387C22—34no—
388C22—632no—
389C22—612no—
390C22D3×#20814yesdirect product factor
391C22C22×D3×Dic314yes—
392C22C22×D3×Dic313yes—
393C4—34no—
394C4—34no—
395C4C22×D3×Dic313yes—
396C4C22×D3×Dic314yes—
397C4C23×D3211yescharacteristic
398C3—21no—
399C3C22×D3×M1611yescharacteristic, complemented
400C3C23×#20811yescharacteristic, complemented
401C2—24no—
402C2—38no—
403C2—64no—
404C2C2×D3×#20816yesdirect product factor
405C2C23×D3×Dic311yescharacteristic
subgroupG/H
centerC22×C4C2×D32
commutatorC2×C32C24×C4
FrattiniC4C23×D32
FittingC23×C32×C4C22
socleC23×C32C22×C4
prime radicalC24×C3×D3C4
cosocleC32×C4C25
hypercenterC23×C4D32
hypocenterC32C23×M16
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C22×#3748C2semidirect
3. C22×C3×#208C2semidirect
4. C22×#3711C2semidirect
5. C2×D3×#208C2direct (N × H)
6. C22×D3×C3⋊C8C2semidirect
18. C2×#3748C22semidirect
19. C22×C3⋊(C3⋊C8)C22semidirect
20. C2×C3×#208C22semidirect
21. C22×C3×C3⋊C8C22semidirect
22. C2×#3711C22semidirect
23. D3×#208C22direct (N × H)
24. C2×D3×C3⋊C8C22semidirect
33. C22×#208D3direct (N × H)
58. #3748C23semidirect
59. C2×C3⋊(C3⋊C8)C23semidirect
60. C3×#208C23semidirect
61. C2×C3×C3⋊C8C23semidirect
62. #3711C23semidirect
63. D3×C3⋊C8C23semidirect
91. C2×#208C2×D3direct (N × H)
96. C22×C3⋊C8C2×D3semidirect
105. C22×C3×D3M16semidirect
123. C3⋊(C3⋊C8)C24semidirect
124. C3×C3⋊C8C24semidirect
182. #208C22×D3direct (N × H)
187. C2×C3⋊C8C22×D3semidirect
191. C22×C32C2×M16semidirect
196. C2×C3×D3C2×M16semidirect
237. C22×D3#208direct (N × H)
278. C3⋊C8C23×D3semidirect
280. C2×C32C22×M16semidirect
283. C3×D3C22×M16semidirect
321. C22×C3D3×M16semidirect
322. C22×C3C2×#208semidirect
329. C2×D3C2×#208direct (N × H)
345. C32C23×M16semidirect
378. C2×C3C2×D3×M16semidirect
379. C2×C3C22×#208semidirect
383. D3C22×#208direct (N × H)
390. C22D3×#208direct (N × H)
399. C3C22×D3×M16semidirect
400. C3C23×#208semidirect
404. C2C2×D3×#208direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×D32
C22×#37482
C22×C3×#2082
C22×#37112
C2×D3×#2082
C22×D3×C3⋊C82
C23×#2083
C22×D3×M163