Finite Group Databasebeta

← C3⋊D3×#888 #13175997

Subgroups — C3⋊D3×#888 #13175997

Order 1728.

2012 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×Dic6C212yesmaximal
2#11723929C211yesmaximal, characteristic, complemented
3C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
4C32×#888C211yesmaximal, characteristic, complemented
5#11723536C211yesmaximal, characteristic, complemented
6#11723534C211yesmaximal, characteristic, complemented
7C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic
8C3⋊D3×Dic3⋊C4C212yesmaximal
9C4×Dic3×C3⋊D3C211yesmaximal, characteristic
10#11723395C212yesmaximal, complemented
11#11723393C212yesmaximal, complemented
12C3⋊D3×C4⋊Q8—31nomaximal
13D3×#888—34nomaximal
14C2×#6308C2212yes—
15C2×C3×C4×C3⋊D3C2212yes—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×Dic6C2212yes—
18C2×#6293C2212yes—
19C2×#6293C2212yes—
20C2×Dic3×C3⋊D3C2212yes—
21C2×Dic3×C3⋊D3C2211yescharacteristic
22C2×Dic3×C3⋊D3C2211yescharacteristic
23C3⋊D3×Dic6—28no—
24C33×C4⋊C4C2211yescharacteristic
25C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
26#88857C2212yes—
27C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C3×#3750C2212yes—
30C32×C4⋊Dic3C2211yescharacteristic
31C32×Dic3⋊C4C2212yes—
32C32×C4×Dic3C2211yescharacteristic
33#88807C2212yes—
34#88806C2212yes—
35Dic3⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2211yescharacteristic
37Dic3×C3⋊Dic3C2211yescharacteristic
38C2×Q8×C3⋊D3—32no—
39C2×D3×Dic6—38no—
40C32×C4⋊Q8—31no—
41#64153—32no—
42C4⋊C4×C3⋊D3—32no—
43C4⋊C4×C3⋊D3—31no—
44C4⋊C4×C3⋊D3D311yescharacteristic
45#64097—31no—
46#64097—31no—
47#64097—32no—
48#64097—34no—
49#64097—64no—
50C42×C3⋊D3—31no—
51#64078—31no—
52C3×D3×C4⋊C4—34no—
53C3×#888D314yescomplemented
54#63919—34no—
55#63918—34no—
56D3×C4⋊Dic3—34no—
57D3×Dic3⋊C4—38no—
58C4×D3×Dic3—34no—
59#63840—38no—
60#63839—38no—
61C22×C3×C3⋊D3C2311yescharacteristic
62C2×C33×C4C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2312yes—
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67#6308—24no—
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C3×C4×C3⋊D3—24no—
71C3×C4×C3⋊D3Q814yescomplemented
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2311yescharacteristic
75C32×Dic6—24no—
76#6293—24no—
77#6293—24no—
78Dic3×C3⋊D3—24no—
79Dic3×C3⋊D3D414yes—
80Dic3×C3⋊D3Q814yescomplemented
81D3×C4⋊Q8—94no—
82C2×#888—91no—
83C2×C32×Q8—32no—
84Q8×C3⋊D3—68no—
85C2×C4×C3⋊D3—31no—
86C2×C4×C3⋊D3—31no—
87C2×C4×C3⋊D3—32no—
88C2×C4×C3⋊D3C2×D312yes—
89C2×C4×C3⋊D3C2×D311yescharacteristic
90C2×#634—32no—
91C2×#634—38no—
92C2×#634—32no—
93C2×#634—32no—
94C2×#634—68no—
95C2×C3×C4×D3—34no—
96C2×C3×C4×D3—38no—
97C2×C3×Dic6C2×D318yes—
98C2×#627—38no—
99C2×#627—38no—
100C2×D3×Dic3—34no—
101C2×D3×Dic3—34no—
102C2×D3×Dic3—38no—
103D3×Dic6—632no—
104C32×C4⋊C4—24no—
105C32×C4⋊C4—32no—
106C32×C4⋊C4—31no—
107C32×C4⋊C4C2×D314yes—
108C32×C4⋊C4C2×D311yescharacteristic
109C32×C42—31no—
110C4⋊(C3⋊Dic3)—31no—
111C4⋊(C3⋊Dic3)—34no—
112C4⋊(C3⋊Dic3)—32no—
113C4⋊(C3⋊Dic3)—64no—
114C4⋊(C3⋊Dic3)C2×D311yescharacteristic
115#3750—32no—
116#3750—38no—
117#3750—32no—
118#3750—32no—
119#3750—68no—
120#3750C2×D312yes—
121C4×C3⋊Dic3—31no—
122C4×C3⋊Dic3—34no—
123C4×C3⋊Dic3—31no—
124C4×C3⋊Dic3—31no—
125C4×C3⋊Dic3—64no—
126C3×C4⋊Dic3—34no—
127C3×C4⋊Dic3C2×D314yes—
128C3×Dic3⋊C4—38no—
129C3×Dic3⋊C4C2×D318yes—
130C3×C4×Dic3C2×D314yes—
131#3724—38no—
132Dic3⋊Dic3—38no—
133Dic3⋊Dic3—38no—
134Dic32—34no—
135Dic32—34no—
136C22×C33C2411yescharacteristic
137C2×C3×C3⋊D3C2×D412yes—
138C2×C3×C3⋊D3C2×Q812yes—
139C2×C3×C3⋊D3C2×Q812yes—
140C33×C4—22no—
141C33×C4C2×Q812yescomplemented
142C3⋊(C3⋊Dic3)—22no—
143C3⋊(C3⋊Dic3)C2×D412yes—
144C3⋊(C3⋊Dic3)C2×Q812yescomplemented
145C3×C3⋊Dic3—22no—
146C3×C3⋊Dic3C2×Q812yescomplemented
147C32×Dic3—22no—
148C32×Dic3C2×D412yes—
149C32×Dic3C2×Q812yescomplemented
150C2×D3×Q8—98no—
151C22×Dic6—92no—
152C3×C4⋊Q8—34no—
153C2×C3×C4⋊C4—91no—
154#944—98no—
155C2×C4⋊Dic3—91no—
156C2×Dic3⋊C4—92no—
157C2×C4×Dic3—91no—
158D3×C4⋊C4—34no—
159D3×C4⋊C4—98no—
160D3×C4⋊C4—94no—
161#888—98no—
162#888—94no—
163#888—92no—
164#888—94no—
165#888—91no—
166#888—91no—
167#888—92no—
168#888—91no—
169#888—184no—
170#888C3⋊D311yescharacteristic, direct product factor
171C42×D3—94no—
172#869—94no—
173C22×C3⋊D3C22×D311yescharacteristic
174C22×C3×D3—34no—
175C32×Q8—64no—
176C2×C32×C4—24no—
177C2×C32×C4—28no—
178C2×C32×C4—31no—
179C2×C32×C4—31no—
180C2×C32×C4—32no—
181C2×C32×C4C22×D318yes—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—34no—
190C2×C3⋊Dic3—38no—
191C2×C3⋊Dic3—64no—
192C2×C3⋊Dic3—64no—
193C2×C3⋊Dic3—68no—
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D311yescharacteristic
196C4×C3⋊D3—24no—
197C4×C3⋊D3—34no—
198C4×C3⋊D3—34no—
199C4×C3⋊D3—64no—
200C4×C3⋊D3Dic614yescomplemented
201#634—64no—
202#634—64no—
203#634—64no—
204#634—616no—
205#634—1216no—
206C2×C3×Dic3—34no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3C22×D318yes—
209C2×C3×Dic3C22×D314yes—
210C2×C3×Dic3C22×D314yes—
211C3×C4×D3—316no—
212C3×C4×D3—616no—
213C3×Dic6—216no—
214#627—616no—
215#627—616no—
216D3×Dic3—316no—
217D3×Dic3—316no—
218D3×Dic3—616no—
219C2×C4⋊Q8—271noSylow
220C2×C33C22×D411yescharacteristic
221C2×C33C22×Q811yescharacteristic
222C2×C33C22×Q811yescharacteristic
223C3×C3⋊D3C4⋊Q814yescomplemented
224C2×C3×Q8—38no—
225C22×C3×C4—91no—
226C22×C3×C4—92no—
227C22×Dic3—91no—
228C22×Dic3—91no—
229C22×Dic3—92no—
230D3×Q8—1832no—
231C2×C4×D3—34no—
232C2×C4×D3—38no—
233C2×C4×D3—94no—
234C2×C4×D3—94no—
235C2×C4×D3—98no—
236C2×Dic6—98no—
237C2×Dic6—98no—
238C2×Dic6—98no—
239C2×Dic6—92no—
240C2×Dic6—92no—
241C2×Dic6—92no—
242C2×Dic6—188no—
243C2×Dic6—188no—
244C2×Dic6C2×C3⋊D312yes—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—38no—
247C3×C4⋊C4—34no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4D3214yes—
251C3×C4⋊C4C2×C3⋊D311yescharacteristic
252C3×C42—34no—
253C4⋊Dic3—34no—
254C4⋊Dic3—98no—
255C4⋊Dic3—94no—
256C4⋊Dic3—91no—
257C4⋊Dic3—92no—
258C4⋊Dic3—184no—
259C4⋊Dic3C2×C3⋊D311yescharacteristic
260Dic3⋊C4—38no—
261Dic3⋊C4—98no—
262Dic3⋊C4—98no—
263Dic3⋊C4—98no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267Dic3⋊C4—188no—
268Dic3⋊C4C2×C3⋊D312yes—
269C4×Dic3—94no—
270C4×Dic3—94no—
271C4×Dic3—94no—
272C4×Dic3—91no—
273C4×Dic3—91no—
274C4×Dic3—91no—
275C4×Dic3—184no—
276C4×Dic3C2×C3⋊D311yescharacteristic
277C22×C32—24no—
278C22×C32C23×D314yes—
279C22×C32C23×D311yescharacteristic
280C2×C3⋊D3C2×Dic612yes—
281C2×C3⋊D3D3×D412yes—
282C2×C3⋊D3D3×Q812yes—
283C2×C3×D3—38no—
284C2×C3×D3—38no—
285C2×C3×D3—38no—
286C32×C4—22no—
287C32×C4—28no—
288C32×C4—28no—
289C32×C4—32no—
290C32×C4—32no—
291C32×C4—48no—
292C32×C4—62no—
293C32×C4C2×Dic612yescomplemented
294C32×C4D3×Q818yescomplemented
295C3⋊Dic3—22no—
296C3⋊Dic3—32no—
297C3⋊Dic3—32no—
298C3⋊Dic3—38no—
299C3⋊Dic3—38no—
300C3⋊Dic3—68no—
301C3⋊Dic3—68no—
302C3⋊Dic3—62no—
303C3⋊Dic3—68no—
304C3⋊Dic3—128no—
305C3⋊Dic3C2×Dic612yescomplemented
306C3×Dic3—28no—
307C3×Dic3—38no—
308C3×Dic3—68no—
309C3×Dic3D3×D418yes—
310C3×Dic3D3×Q818yescomplemented
311C22×Q8—272no—
312C4⋊Q8—31no—
313C4⋊Q8—271no—
314C4⋊Q8—272no—
315C4⋊Q8—271no—
316C4⋊Q8—272no—
317C4⋊Q8—271no—
318C2×C4⋊C4—91no—
319C2×C4⋊C4—272no—
320C2×C4⋊C4—271no—
321C2×C42—271no—
322C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
323C23×C3—91no—
324C22×D3—34no—
325C3×Q8—616no—
326C2×C3×C4—24no—
327C2×C3×C4—28no—
328C2×C3×C4—34no—
329C2×C3×C4—34no—
330C2×C3×C4—38no—
331C2×C3×C4—91no—
332C2×C3×C4—94no—
333C2×C3×C4—92no—
334C2×C3×C4—184no—
335C2×C3×C4C2×D3218yes—
336C2×C3×C4C2×D3214yes—
337C2×C3×C4C22×C3⋊D312yes—
338C2×C3×C4C22×C3⋊D311yescharacteristic
339C2×Dic3—34no—
340C2×Dic3—38no—
341C2×Dic3—94no—
342C2×Dic3—94no—
343C2×Dic3—91no—
344C2×Dic3—91no—
345C2×Dic3—94no—
346C2×Dic3—94no—
347C2×Dic3—98no—
348C2×Dic3—92no—
349C2×Dic3—184no—
350C2×Dic3—184no—
351C2×Dic3—188no—
352C2×Dic3—184no—
353C2×Dic3C22×C3⋊D312yes—
354C2×Dic3C22×C3⋊D311yescharacteristic
355C2×Dic3C22×C3⋊D311yescharacteristic
356C4×D3—316no—
357C4×D3—616no—
358C4×D3—916no—
359C4×D3—916no—
360C4×D3—1816no—
361Dic6—24no—
362Dic6—184no—
363Dic6—1816no—
364Dic6—1816no—
365Dic6—184no—
366Dic6—1816no—
367Dic6—184no—
368Dic6—3616no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32C22×Dic611yescharacteristic
373C2×C32C2×D3×D414yes—
374C2×C32C2×D3×D411yescharacteristic
375C2×C32C2×D3×Q814yes—
376C2×C32C2×D3×Q814yes—
377C2×C32C2×D3×Q811yescharacteristic
378C3⋊D3#88814yesdirect product factor
379C3×D3—316no—
380C2×Q8—32no—
381C2×Q8—272no—
382C2×Q8—272no—
383C2×Q8—272no—
384C2×Q8—548no—
385C22×C4—91no—
386C22×C4—92no—
387C22×C4—271no—
388C22×C4—271no—
389C22×C4—272no—
390C4⋊C4—32no—
391C4⋊C4—31no—
392C4⋊C4—92no—
393C4⋊C4—91no—
394C4⋊C4—272no—
395C4⋊C4—272no—
396C4⋊C4—272no—
397C4⋊C4—272no—
398C4⋊C4—271no—
399C4⋊C4D3×C3⋊D311yescharacteristic
400C42—31no—
401C42—271no—
402C42—271no—
403C42—271no—
404C22×C3—24no—
405C22×C3—92no—
406C22×C3—92no—
407C22×C3—92no—
408C22×C3C22×D3214yes—
409C22×C3C23×C3⋊D311yescharacteristic
410C2×D3—38no—
411C2×D3—38no—
412C2×D3—38no—
413C3×C4—28no—
414C3×C4—22no—
415C3×C4—28no—
416C3×C4—38no—
417C3×C4—38no—
418C3×C4—48no—
419C3×C4—68no—
420C3×C4—92no—
421C3×C4—182no—
422C3×C4D3×Dic618yescomplemented
423C3×C4Q8×C3⋊D312yescomplemented
424Dic3—22no—
425Dic3—38no—
426Dic3—68no—
427Dic3—98no—
428Dic3—98no—
429Dic3—92no—
430Dic3—92no—
431Dic3—188no—
432Dic3—188no—
433Dic3—188no—
434Dic3—182no—
435Dic3—368no—
436Dic3D4×C3⋊D312yes—
437Dic3Q8×C3⋊D312yescomplemented
438C32—24no—
439C32C2×#88811yescharacteristic, complemented
440C32D3×C4⋊Q814yescomplemented
441C23—91no—
442Q8—64no—
443Q8—544no—
444Q8—544no—
445Q8—544no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—32no—
449C2×C4—91no—
450C2×C4—94no—
451C2×C4—92no—
452C2×C4—184no—
453C2×C4—274no—
454C2×C4—274no—
455C2×C4—271no—
456C2×C4—271no—
457C2×C4—272no—
458C2×C4—544no—
459C2×C4C2×D3×C3⋊D312yes—
460C2×C4C2×D3×C3⋊D311yescharacteristic
461C2×C3—24no—
462C2×C3—24no—
463C2×C3—24no—
464C2×C3—94no—
465C2×C3C2×D3×Dic614yes—
466C2×C3D32×D414yes—
467C2×C3D32×Q814yes—
468C2×C3C2×D4×C3⋊D311yescharacteristic
469C2×C3C2×Q8×C3⋊D311yescharacteristic
470C2×C3C2×Q8×C3⋊D311yescharacteristic
471D3—316no—
472C22—92no—
473C22—92no—
474C22—92no—
475C22C22×D3×C3⋊D311yescharacteristic
476C4—22no—
477C4—32no—
478C4—32no—
479C4—62no—
480C4—92no—
481C4—182no—
482C4—272no—
483C4—272no—
484C4—542no—
485C4C3⋊D3×Dic612yescomplemented
486C3—24no—
487C3D3×#88814yescomplemented
488C3C3⋊D3×C4⋊Q811yescharacteristic, complemented
489C2—94no—
490C2C2×C3⋊D3×Dic611yescharacteristic
491C2D3×D4×C3⋊D311yescharacteristic
492C2D3×Q8×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×C3×C3⋊D3C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C2×C4⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11723929C2semidirect
4. C32×#888C2semidirect
5. #11723536C2semidirect
6. #11723534C2semidirect
10. #11723395C2semidirect
11. #11723393C2semidirect
53. C3×#888D3semidirect
71. C3×C4×C3⋊D3Q8semidirect
80. Dic3×C3⋊D3Q8semidirect
141. C33×C4C2×Q8semidirect
144. C3⋊(C3⋊Dic3)C2×Q8semidirect
146. C3×C3⋊Dic3C2×Q8semidirect
149. C32×Dic3C2×Q8semidirect
170. #888C3⋊D3direct (N × H)
200. C4×C3⋊D3Dic6semidirect
223. C3×C3⋊D3C4⋊Q8semidirect
293. C32×C4C2×Dic6semidirect
294. C32×C4D3×Q8semidirect
305. C3⋊Dic3C2×Dic6semidirect
310. C3×Dic3D3×Q8semidirect
322. C33C2×C4⋊Q8semidirect
378. C3⋊D3#888direct (N × H)
422. C3×C4D3×Dic6semidirect
423. C3×C4Q8×C3⋊D3semidirect
437. Dic3Q8×C3⋊D3semidirect
439. C32C2×#888semidirect
440. C32D3×C4⋊Q8semidirect
485. C4C3⋊D3×Dic6semidirect
487. C3D3×#888semidirect
488. C3C3⋊D3×C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
#117239292
C3×C4⋊C4×C3⋊D32
C32×#8882
#117235362
#117235342
C3⋊D3×C4⋊Dic32
C3⋊D3×Dic3⋊C42
C4×Dic3×C3⋊D32
#117233952
#117233932
C3⋊D3×C4⋊Q83
D3×#8883