Finite Group Databasebeta

← D3×#5184 #11860113

Subgroups — D3×#5184 #11860113

Order 1152.

1014 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Q16C211yesmaximal, characteristic
2D3×Q8×Dic3C212yesmaximal
3#10592883C211yesmaximal, characteristic, complemented
4C3×#5184C211yesmaximal, characteristic, complemented
5#10591582C212yesmaximal, complemented
6#10591576C212yesmaximal, complemented
7D3×#835C212yesmaximal
8#10590819C211yesmaximal, characteristic, complemented
9#10590812C211yesmaximal, characteristic, complemented
10D3×#817C211yesmaximal, characteristic
11D3×#815C211yesmaximal, characteristic
12C2×#5184—31nomaximal
13D3×#441—31nomaximal
14C2×C3×D3×Q8C2212yes—
15C2×C32×Q16C2211yescharacteristic
16Q8×C3⋊Dic3C2212yes—
17C3×Q8×Dic3C2212yes—
18C2×C3×#216C2212yes—
19C3×D3×Q16C418yescomplemented
20C2×C3×Dic12C2211yescharacteristic
21C2×C3×D3×C8C2211yescharacteristic
22D3×C4⋊Dic3—22no—
23D3×C4⋊Dic3C2212yes—
24C4×D3×Dic3—22no—
25C4×D3×Dic3C2211yescharacteristic
26#63843—22no—
27#63843C2212yes—
28Dic3×Dic6—22no—
29Dic3×Dic6C2212yes—
30C2×D3×C3⋊C8C2211yescharacteristic
31#63662C2212yes—
32#63644C2211yescharacteristic
33#63642C2211yescharacteristic
34C3×#835C2212yes—
35C3×#817C2211yescharacteristic
36C3×#815C2211yescharacteristic
37#63573C2212yes—
38#63566C2212yes—
39#63565C2212yes—
40#63555C2211yescharacteristic
41#63554C2211yescharacteristic
42C22×C3×Q16—31no—
43C2×Q8×Dic3—32no—
44C2×D3×Q16D311yescharacteristic
45C4×D3×Q8—32no—
46C3×#441—31no—
47C2×#835—32no—
48#5184—31no—
49#5184—31no—
50#5184—32no—
51#5184—32no—
52#5184—31no—
53#5184—31no—
54#5184—61no—
55#5184D311yescharacteristic, direct product factor
56C2×#817—31no—
57C2×#815—31no—
58#5030—32no—
59D3×#106—31no—
60#4853—31no—
61D3×#103—32no—
62#4789—32no—
63#4716—31no—
64D3×#97—31no—
65C2×C32×Q8C2312yes—
66C3×D3×Q8—24no—
67C3×D3×Q8C2×C418yes—
68C2×C3×C4×D3—22no—
69C2×C3×C4×D3C2311yescharacteristic
70C2×C3×Dic6—22no—
71C2×C3×Dic6C2312yes—
72C2×D3×Dic3—22no—
73C2×D3×Dic3D412yes—
74C32×Q16C2×C414yescomplemented
75C2×C32×C8C2311yescharacteristic
76C4⋊(C3⋊Dic3)—22no—
77C4⋊(C3⋊Dic3)C2312yes—
78C4×C3⋊Dic3—22no—
79C4×C3⋊Dic3C2311yescharacteristic
80C2×C3⋊(C3⋊C8)C2311yescharacteristic
81C3×#216C2×C418yescomplemented
82C3×C4⋊Dic3—22no—
83C3×C4⋊Dic3C2312yes—
84C3×C4×Dic3—22no—
85C3×C4×Dic3C2311yescharacteristic
86C2×C3×C3⋊C8C2311yescharacteristic
87C2×C3×C3⋊C8C2311yescharacteristic
88C3×Dic12C2×C414yescomplemented
89C3×D3×C8C2×C414yes—
90Dic3⋊Dic3—22no—
91Dic3⋊Dic3—22no—
92Dic3⋊Dic3—22no—
93Dic32—22no—
94Dic32—22no—
95Dic32D412yes—
96D3×C3⋊C8—22no—
97C2×#441—91noSylow
98C22×C3×Q8—32no—
99C2×D3×Q8C2×D312yes—
100C2×C3×Q16—21no—
101C2×C3×Q16—32no—
102C2×C3×Q16—38no—
103C2×C3×Q16—31no—
104C2×C3×Q16C2×D311yescharacteristic
105C2×C3×Q16C2×D311yescharacteristic
106C22×C3×C8—31no—
107C3×C4×Q8—32no—
108Q8×Dic3—32no—
109Q8×Dic3—32no—
110Q8×Dic3—32no—
111Q8×Dic3—32no—
112Q8×Dic3—62no—
113Q8×Dic3—62no—
114Q8×Dic3—62no—
115Q8×Dic3C2×D312yes—
116C2×#216C2×D312yes—
117C2×C4⋊Dic3—32no—
118C2×C4⋊Dic3—62no—
119C2×C4×Dic3—31no—
120C2×C4×Dic3—62no—
121C22×C3⋊C8—31no—
122D3×Q16Dic318yescomplemented
123C2×Dic12C2×D311yescharacteristic
124C2×D3×C8—31no—
125C2×D3×C8C2×D311yescharacteristic
126D3×C4⋊C4—32no—
127D3×C4⋊C4—62no—
128#887—32no—
129#887—62no—
130C42×D3—31no—
131C42×D3—62no—
132C4×Dic6—32no—
133C4×Dic6—62no—
134C3×#106—31no—
135C3×#103—32no—
136C3×#97—31no—
137#835—32no—
138#835—32no—
139#835—32no—
140#835—32no—
141#835—62no—
142#835C2×D312yes—
143#817—31no—
144#817—31no—
145#817—32no—
146#817—61no—
147#817C2×D311yescharacteristic
148#816—32no—
149#815—31no—
150#815—31no—
151#815—31no—
152#815—31no—
153#815—31no—
154#815—61no—
155#815C2×D311yescharacteristic
156#810—32no—
157#808—32no—
158#803—31no—
159C22×C3×D3C2×D411yescharacteristic
160C32×Q8—22no—
161C32×Q8C22×C414yes—
162C2×C32×C4—22no—
163C2×C32×C4C2411yescharacteristic
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3C2×D412yes—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3C2×D412yes—
169C2×C3×Dic3C2×D411yescharacteristic
170C3×C4×D3—28no—
171C3×C4×D3C22×C412yes—
172C3×C4×D3#4112yes—
173C3×Dic6—22no—
174C3×Dic6—28no—
175C3×Dic6C22×C414yes—
176D3×Dic3—24no—
177D3×Dic3—44no—
178C32×C8C22×C412yes—
179C3⋊(C3⋊C8)—21no—
180C3×C3⋊C8—21no—
181C3×C3⋊C8C22×C412yes—
182C22×Q16—31no—
183C2×C4×Q8—92no—
184#441—31no—
185#441—91no—
186#441—92no—
187#441—91no—
188#441—92no—
189#441—91no—
190C2×#106—91no—
191C2×#103—92no—
192C2×#97—91no—
193C2×C3×Q8—22no—
194C2×C3×Q8—38no—
195C2×C3×Q8—32no—
196C2×C3×Q8—62no—
197C2×C3×Q8—64no—
198C2×C3×Q8C22×D312yes—
199C2×C3×Q8C22×D312yes—
200C22×C3×C4—31no—
201C22×C3×C4—62no—
202C22×Dic3—32no—
203C22×Dic3—62no—
204D3×Q8—24no—
205D3×Q8C2×Dic318yes—
206C2×C4×D3—22no—
207C2×C4×D3—32no—
208C2×C4×D3—62no—
209C2×C4×D3C22×D311yescharacteristic
210C2×Dic6—22no—
211C2×Dic6C22×D312yes—
212C3×Q16—24no—
213C3×Q16—38no—
214C3×Q16—34no—
215C3×Q16C4×D314yescomplemented
216C3×Q16C2×Dic314yescomplemented
217C2×C3×C8—21no—
218C2×C3×C8—31no—
219C2×C3×C8—34no—
220C2×C3×C8—31no—
221C2×C3×C8C22×D311yescharacteristic
222C2×C3×C8C22×D311yescharacteristic
223C3×C4⋊C4—32no—
224C3×C4⋊C4—62no—
225C3×C42—31no—
226C3×C42—62no—
227#216C2×Dic318yescomplemented
228C4⋊Dic3—22no—
229C4⋊Dic3—32no—
230C4⋊Dic3—32no—
231C4⋊Dic3—62no—
232C4⋊Dic3—62no—
233C4⋊Dic3—62no—
234C4⋊Dic3—62no—
235C4⋊Dic3—62no—
236C4⋊Dic3—62no—
237C4⋊Dic3—122no—
238C4⋊Dic3C22×D312yes—
239Dic3⋊C4—62no—
240Dic3⋊C4—62no—
241Dic3⋊C4—62no—
242C4×Dic3—22no—
243C4×Dic3—31no—
244C4×Dic3—31no—
245C4×Dic3—32no—
246C4×Dic3—32no—
247C4×Dic3—61no—
248C4×Dic3—62no—
249C4×Dic3—62no—
250C4×Dic3—62no—
251C4×Dic3—62no—
252C4×Dic3—62no—
253C4×Dic3—62no—
254C4×Dic3—122no—
255C4×Dic3C22×D311yescharacteristic
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—61no—
259C2×C3⋊C8—62no—
260C2×C3⋊C8C22×D311yescharacteristic
261C2×C3⋊C8C22×D311yescharacteristic
262Dic12C2×Dic314yescomplemented
263D3×C8—62no—
264D3×C8C2×Dic314yes—
265C22×C32C22×D411yescharacteristic
266C2×C3×D3C4×D412yes—
267C2×C3×D3#13714yes—
268C32×C4—24no—
269C32×C4C23×C411yescharacteristic
270C32×C4C2×#4111yescharacteristic
271C3⋊Dic3—22no—
272C3⋊Dic3—42no—
273C3×Dic3—24no—
274C3×Dic3—22no—
275C3×Dic3—42no—
276C3×Dic3C4×D412yes—
277C22×Q8—32no—
278C2×Q16—32no—
279C2×Q16—38no—
280C2×Q16—31no—
281C2×Q16D3211yescharacteristic
282C22×C8—31no—
283C22×C8—91no—
284C4×Q8—32no—
285C4×Q8—92no—
286C4×Q8—92no—
287C4×Q8—92no—
288C4×Q8—182no—
289C4×Q8—182no—
290C2×C4⋊C4—92no—
291C2×C4⋊C4—182no—
292C2×C42—91no—
293C2×C42—182no—
294#106—31no—
295#106—92no—
296#106—91no—
297#103—32no—
298#103—92no—
299#103—92no—
300#103—92no—
301#97—31no—
302#97—91no—
303#97—91no—
304#97—91no—
305C23×C3—31no—
306C22×D3D3×D411yescharacteristic
307C3×Q8—22no—
308C3×Q8—22no—
309C3×Q8—24no—
310C3×Q8—34no—
311C3×Q8—42no—
312C3×Q8—68no—
313C3×Q8—62no—
314C3×Q8C2×C4×D314yes—
315C3×Q8C22×Dic314yes—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—31no—
320C2×C3×C4—32no—
321C2×C3×C4—32no—
322C2×C3×C4—32no—
323C2×C3×C4—42no—
324C2×C3×C4—68no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4C23×D311yescharacteristic
328C2×C3×C4C23×D311yescharacteristic
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—32no—
332C2×Dic3—32no—
333C2×Dic3—62no—
334C2×Dic3—64no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—122no—
338C2×Dic3—124no—
339C2×Dic3D3×D412yes—
340C2×Dic3D3×D411yescharacteristic
341C4×D3—28no—
342C4×D3—64no—
343C4×D3—124no—
344C4×D3#23712yes—
345C4×D3C22×Dic312yes—
346Dic6—22no—
347Dic6—28no—
348Dic6C22×Dic314yes—
349C3×C8—22no—
350C3×C8—32no—
351C3×C8—61no—
352C3×C8C2×C4×D312yes—
353C3×C8C22×Dic312yes—
354C3⋊C8—21no—
355C3⋊C8—61no—
356C3⋊C8—61no—
357C3⋊C8—121no—
358C3⋊C8C22×Dic312yes—
359C2×C32C2×C4×D411yescharacteristic
360C2×C32C2×#13712yes—
361C3×D3#44114yescomplemented
362C2×Q8—38no—
363C2×Q8—32no—
364C2×Q8—62no—
365C2×Q8—64no—
366C2×Q8C2×D3212yes—
367C22×C4—31no—
368C22×C4—62no—
369C22×C4—92no—
370C22×C4—182no—
371Q16—38no—
372Q16—34no—
373Q16D3×Dic314yescomplemented
374C2×C8—31no—
375C2×C8—34no—
376C2×C8—31no—
377C2×C8—91no—
378C2×C8—182no—
379C2×C8C2×D3211yescharacteristic
380C4⋊C4—32no—
381C4⋊C4—62no—
382C4⋊C4—92no—
383C4⋊C4—182no—
384C4⋊C4—182no—
385C4⋊C4—182no—
386C4⋊C4—182no—
387C42—31no—
388C42—62no—
389C42—92no—
390C42—91no—
391C42—182no—
392C42—182no—
393C42—182no—
394C22×C3—21no—
395C22×C3—34no—
396C22×C3—32no—
397C22×C3C2×D3×D411yescharacteristic
398C22×C3C2×D3×D411yescharacteristic
399C2×D3#91814yes—
400C2×D3D4×Dic312yes—
401C3×C4—24no—
402C3×C4—24no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—32no—
406C3×C4—44no—
407C3×C4—62no—
408C3×C4—64no—
409C3×C4—122no—
410C3×C4C22×C4×D311yescharacteristic
411C3×C4C2×#23711yescharacteristic
412C3×C4D3×#4111yescharacteristic
413C3×C4C23×Dic311yescharacteristic
414Dic3—22no—
415Dic3—24no—
416Dic3—42no—
417Dic3—62no—
418Dic3—62no—
419Dic3—122no—
420Dic3—122no—
421Dic3—122no—
422Dic3—242no—
423Dic3D4×Dic312yes—
424C32C2×#44111yescharacteristic, complemented, Sylow
425C23—31no—
426Q8—22no—
427Q8—34no—
428Q8—68no—
429Q8—62no—
430Q8C2×D3×Dic314yes—
431C2×C4—22no—
432C2×C4—31no—
433C2×C4—32no—
434C2×C4—32no—
435C2×C4—32no—
436C2×C4—68no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—92no—
440C2×C4—184no—
441C2×C4—182no—
442C2×C4—364no—
443C2×C4C22×D3211yescharacteristic
444C8—32no—
445C8—61no—
446C8—181no—
447C8C2×D3×Dic312yes—
448C2×C3—21no—
449C2×C3—22no—
450C2×C3—34no—
451C2×C3C4×D3×D411yescharacteristic
452C2×C3C2×#91812yes—
453C2×C3D3×#13712yes—
454C2×C3C2×D4×Dic311yescharacteristic
455D3#518414yesdirect product factor
456C22—32no—
457C22—34no—
458C22D32×D411yescharacteristic
459C4—24no—
460C4—32no—
461C4—64no—
462C4—62no—
463C4—122no—
464C4—182no—
465C4—362no—
466C4D3×#23711yescharacteristic
467C4C22×D3×Dic311yescharacteristic
468C3—21no—
469C3D3×#44111yescharacteristic, complemented
470C3C2×#518411yescharacteristic, complemented
471C2—34no—
472C2D3×#91812yes—
473C2D3×D4×Dic311yescharacteristic
subgroupG/H
centerC22D32×D4
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC2×C32×Q16C22
socleC22×C32C22×D4
prime radicalC22×C3×D3C2×D4
cosocleC2×C32×C4C24
hypercenterC2×Q16D32
hypocenterC32C2×#441
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
3. #10592883C2semidirect
4. C3×#5184C2semidirect
5. #10591582C2semidirect
6. #10591576C2semidirect
8. #10590819C2semidirect
9. #10590812C2semidirect
19. C3×D3×Q16C4semidirect
55. #5184D3direct (N × H)
74. C32×Q16C2×C4semidirect
81. C3×#216C2×C4semidirect
88. C3×Dic12C2×C4semidirect
122. D3×Q16Dic3semidirect
215. C3×Q16C4×D3semidirect
216. C3×Q16C2×Dic3semidirect
227. #216C2×Dic3semidirect
262. Dic12C2×Dic3semidirect
361. C3×D3#441semidirect
373. Q16D3×Dic3semidirect
424. C32C2×#441semidirect
455. D3#5184direct (N × H)
469. C3D3×#441semidirect
470. C3C2×#5184semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Q162
D3×Q8×Dic32
#105928832
C3×#51842
#105915822
#105915762
D3×#8352
#105908192
#105908122
D3×#8172
D3×#8152
C2×#51843
D3×#4413