Finite Group Databasebeta

← #102×(C3⋊D3)⋊C3 #13141786

Subgroups — #102×(C3⋊D3)⋊C3 #13141786

Order 1728.

757 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic, complemented
8C3⋊D3×#102C311yesmaximal, characteristic, complemented
9C3×D3×#102—31nomaximal
10C23×(C3⋊D3)⋊C3—21no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13D4×(C3⋊D3)⋊C3—22no—
14D4×(C3⋊D3)⋊C3C414yescomplemented
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3—21no—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
21#3750⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
23C8×(C3⋊D3)⋊C3—22no—
24C2×D4×C3⋊D3C2×C311yescharacteristic
25C2×C3×D3×D4—31no—
26C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
27C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
28C2×C3×D3×C8—31no—
29C3×D3×C4⋊C4—31no—
30C32×#102—21no—
31C32×#102D311yescharacteristic, complemented
32C32×#102C2×C311yescharacteristic, complemented
33#63659C2×C311yescharacteristic, complemented
34#63648C2×C311yescharacteristic, complemented
35#63636C2×C311yescharacteristic, complemented
36C3×#832—31no—
37C3×#821—31no—
38C3×#809—31no—
39C23×H(3,𝔽3)—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—42no—
43C22×(C3⋊D3)⋊C3—42no—
44C22×(C3⋊D3)⋊C3D411yescharacteristic
45D4×H(3,𝔽3)—21no—
46D4×H(3,𝔽3)C2×C412yescomplemented
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49#638⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3D411yescharacteristic
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
54C4×(C3⋊D3)⋊C3—42no—
55C4×(C3⋊D3)⋊C3C2×C412yes—
56C4×(C3⋊D3)⋊C3D412yes—
57C8×H(3,𝔽3)—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59C2×C3×#102—91no—
60D3×#102—31no—
61D3×#102—91no—
62C23×C3⋊D3—21no—
63C23×C3×D3—61no—
64C2×C32×D4—21no—
65C2×C32×D4C2×D311yescharacteristic
66C2×C32×D4C22×C311yescharacteristic
67C2×#638—21no—
68D4×C3⋊D3—22no—
69D4×C3⋊D3C3×C414yescomplemented
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×C4×C3⋊D3C22×C311yescharacteristic
73C2×C3×#67—61no—
74C3×D3×D4—34no—
75C3×D3×D4—62no—
76C2×C3×D12—31no—
77C2×C3×C4×D3—31no—
78C2×C3×C4×D3—61no—
79C2×C32×C8—21no—
80C2×C32×C8C2×D311yescharacteristic, complemented
81C2×C32×C8C22×C311yescharacteristic, complemented
82C32×C4⋊C4—21no—
83C32×C4⋊C4C2×D311yescharacteristic, complemented
84C32×C4⋊C4C22×C311yescharacteristic, complemented
85C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
86#3750—21no—
87C2×C3⋊(C3⋊C8)C22×C311yescharacteristic, complemented
88C8×C3⋊D3—22no—
89C3×C4⋊Dic3—31no—
90C3×Dic3⋊C4—61no—
91C2×C3×C3⋊C8—31no—
92C3×D3×C8—62no—
93C22×H(3,𝔽3)—22no—
94C22×H(3,𝔽3)—41no—
95C22×H(3,𝔽3)—41no—
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—41no—
99C2×(C3⋊D3)⋊C3—41no—
100C2×(C3⋊D3)⋊C3—48no—
101C2×(C3⋊D3)⋊C3C22⋊C412yes—
102C2×(C3⋊D3)⋊C3D812yes—
103C2×(C3⋊D3)⋊C3SD1612yes—
104C4×H(3,𝔽3)—41no—
105C4×H(3,𝔽3)C22×C411yescharacteristic
106C4×H(3,𝔽3)C2×D411yescharacteristic
107(C3⋊Dic3)⋊C3—41no—
108(C3⋊Dic3)⋊C3C22⋊C412yes—
109C22×C3×D4—91no—
110C2×D3×D4—31no—
111C2×D3×D4—91no—
112C22×C3×C8—91no—
113C2×C3×C4⋊C4—91no—
114C2×D3×C8—31no—
115C2×D3×C8—91no—
116D3×C4⋊C4—31no—
117D3×C4⋊C4—91no—
118C3×#102—31no—
119C3×#102—31no—
120C3×#102—61no—
121C3×#102—91no—
122C3×#102—91no—
123C3×#102—91no—
124C3×#102C3×D311yescharacteristic
125#832—31no—
126#832—91no—
127#821—31no—
128#821—91no—
129#809—31no—
130#809—91no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—41no—
134C22×C3⋊D3—21no—
135C22×C3⋊D3—24no—
136C22×C3⋊D3—42no—
137C22×C3⋊D3—42no—
138C22×C3⋊D3C3×D411yescharacteristic
139C22×C3×D3—31no—
140C22×C3×D3—64no—
141C22×C3×D3—61no—
142C22×C3×D3—122no—
143C22×C3×D3—122no—
144C32×D4—21no—
145C32×D4—21no—
146C32×D4—22no—
147C32×D4—41no—
148C32×D4C4×D312yescomplemented
149C32×D4C2×C3×C412yescomplemented
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—41no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C23×C311yescharacteristic
156#638—24no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3C3×D411yescharacteristic
159C4⋊(C3⋊D3)—21no—
160C4⋊(C3⋊D3)C2×C3×C412yescomplemented
161C4×C3⋊D3—42no—
162C4×C3⋊D3C2×C3×C412yes—
163C4×C3⋊D3C3×D412yes—
164C3×#67—64no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—61no—
167C3×D12—32no—
168C3×D12—61no—
169C3×C4×D3—32no—
170C3×C4×D3—32no—
171C3×C4×D3—122no—
172C32×C8—21no—
173C32×C8—21no—
174C32×C8—41no—
175C3⋊(C3⋊C8)—21no—
176C3×C3⋊C8—61no—
177C2×#102—91noSylow
178C2×H(3,𝔽3)—42no—
179C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
180C2×H(3,𝔽3)C2×D811yescharacteristic
181C2×H(3,𝔽3)C2×SD1611yescharacteristic
182(C3⋊D3)⋊C3—42no—
183(C3⋊D3)⋊C3#10214yesdirect product factor
184C24×C3—181no—
185C23×D3—61no—
186C23×D3—181no—
187C2×C3×D4—31no—
188C2×C3×D4—31no—
189C2×C3×D4—61no—
190C2×C3×D4—91no—
191C2×C3×D4—94no—
192C2×C3×D4—182no—
193C2×C3×D4—181no—
194C2×C3×D4C2×C3×D311yescharacteristic
195C22×C3×C4—91no—
196C22×C3×C4—181no—
197C2×#67—61no—
198C2×#67—181no—
199D3×D4—34no—
200D3×D4—62no—
201D3×D4—94no—
202D3×D4—182no—
203C2×D12—31no—
204C2×D12—91no—
205C2×C4×D3—31no—
206C2×C4×D3—61no—
207C2×C4×D3—91no—
208C2×C4×D3—181no—
209C2×C3×C8—31no—
210C2×C3×C8—31no—
211C2×C3×C8—61no—
212C2×C3×C8—91no—
213C2×C3×C8—182no—
214C2×C3×C8C2×C3×D311yescharacteristic
215C3×C4⋊C4—31no—
216C3×C4⋊C4—31no—
217C3×C4⋊C4—61no—
218C3×C4⋊C4—91no—
219C3×C4⋊C4—181no—
220C3×C4⋊C4C2×C3×D311yescharacteristic
221C4⋊Dic3—31no—
222C4⋊Dic3—91no—
223Dic3⋊C4—61no—
224Dic3⋊C4—181no—
225C2×C3⋊C8—31no—
226C2×C3⋊C8—91no—
227D3×C8—62no—
228D3×C8—182no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—21no—
232C22×C32—42no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—81no—
238C22×C32—81no—
239C22×C32D3×D411yescharacteristic
240C22×C32C2×C3×D411yescharacteristic
241C2×C3⋊D3—22no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3—41no—
244C2×C3⋊D3—48no—
245C2×C3⋊D3C3×C22⋊C412yes—
246C2×C3⋊D3C3×D812yes—
247C2×C3⋊D3C3×SD1612yes—
248C2×C3×D3—32no—
249C2×C3×D3—32no—
250C2×C3×D3—32no—
251C2×C3×D3—62no—
252C2×C3×D3—128no—
253C2×C3×D3—121no—
254C2×C3×D3—121no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—41no—
258C32×C4—41no—
259C32×C4—81no—
260C32×C4C2×C4×D311yescharacteristic
261C32×C4D3×D411yescharacteristic
262C32×C4C22×C3×C411yescharacteristic
263C32×C4C2×C3×D411yescharacteristic
264C3⋊Dic3—41no—
265C3⋊Dic3C3×C22⋊C412yes—
266C3×Dic3—32no—
267C3×Dic3—121no—
268C22×D4—91no—
269C22×C8—91no—
270C2×C4⋊C4—91no—
271#102—91no—
272#102—91no—
273#102—91no—
274#102(C3⋊D3)⋊C311yescharacteristic, direct product factor
275H(3,𝔽3)C2×#10211yescharacteristic, complemented, Sylow
276C23×C3—21no—
277C23×C3—61no—
278C23×C3—61no—
279C23×C3—91no—
280C23×C3—121no—
281C23×C3—181no—
282C23×C3—184no—
283C23×C3—362no—
284C23×C3—362no—
285C22×D3—31no—
286C22×D3—64no—
287C22×D3—61no—
288C22×D3—91no—
289C22×D3—122no—
290C22×D3—122no—
291C22×D3—184no—
292C22×D3—181no—
293C22×D3—362no—
294C22×D3—362no—
295C3×D4—21no—
296C3×D4—32no—
297C3×D4—32no—
298C3×D4—61no—
299C3×D4—61no—
300C3×D4—62no—
301C3×D4—92no—
302C3×D4—121no—
303C3×D4—181no—
304C3×D4—184no—
305C3×D4C3×C4×D312yes—
306C2×C3×C4—21no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—92no—
313C2×C3×C4—92no—
314C2×C3×C4—91no—
315C2×C3×C4—121no—
316C2×C3×C4—181no—
317C2×C3×C4—362no—
318C2×C3×C4C22×C3×D311yescharacteristic
319#67—64no—
320#67—184no—
321C2×Dic3—31no—
322C2×Dic3—61no—
323C2×Dic3—91no—
324C2×Dic3—181no—
325D12—32no—
326D12—61no—
327D12—92no—
328D12—181no—
329C4×D3—32no—
330C4×D3—32no—
331C4×D3—92no—
332C4×D3—92no—
333C4×D3—122no—
334C4×D3—362no—
335C3×C8—21no—
336C3×C8—61no—
337C3×C8—61no—
338C3×C8—121no—
339C3×C8—181no—
340C3⋊C8—61no—
341C3⋊C8—181no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—42no—
346C2×C32—42no—
347C2×C32—82no—
348C2×C32D3×C22⋊C411yescharacteristic
349C2×C32D3×D811yescharacteristic
350C2×C32D3×SD1611yescharacteristic
351C2×C32C2×C3×C22⋊C411yescharacteristic
352C2×C32C2×C3×D811yescharacteristic
353C2×C32C2×C3×SD1611yescharacteristic
354C3⋊D3—42no—
355C3⋊D3C3×#10214yescomplemented
356C3×D3—34no—
357C3×D3—122no—
358C24—181no—
359C2×D4—94no—
360C2×D4—91no—
361C2×D4—182no—
362C2×D4—181no—
363C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
364C22×C4—91no—
365C22×C4—181no—
366C2×C8—91no—
367C2×C8—182no—
368C2×C8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
369C4⋊C4—91no—
370C4⋊C4—181no—
371C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
372C22×C3—22no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—62no—
378C22×C3—62no—
379C22×C3—61no—
380C22×C3—92no—
381C22×C3—92no—
382C22×C3—92no—
383C22×C3—122no—
384C22×C3—121no—
385C22×C3—121no—
386C22×C3—121no—
387C22×C3—121no—
388C22×C3—182no—
389C22×C3—241no—
390C22×C3—241no—
391C22×C3—361no—
392C22×C3—361no—
393C22×C3—368no—
394C22×C3C3×D3×D411yescharacteristic
395C2×D3—32no—
396C2×D3—32no—
397C2×D3—32no—
398C2×D3—62no—
399C2×D3—92no—
400C2×D3—92no—
401C2×D3—92no—
402C2×D3—128no—
403C2×D3—121no—
404C2×D3—121no—
405C2×D3—182no—
406C2×D3—368no—
407C2×D3—361no—
408C2×D3—361no—
409C3×C4—31no—
410C3×C4—31no—
411C3×C4—31no—
412C3×C4—31no—
413C3×C4—41no—
414C3×C4—61no—
415C3×C4—61no—
416C3×C4—92no—
417C3×C4—121no—
418C3×C4—121no—
419C3×C4—241no—
420C3×C4—361no—
421C3×C4C2×C3×C4×D311yescharacteristic
422C3×C4C3×D3×D411yescharacteristic
423Dic3—32no—
424Dic3—92no—
425Dic3—121no—
426Dic3—361no—
427C32—21no—
428C32D3×#10211yescharacteristic, complemented
429C32C2×C3×#10211yescharacteristic, complemented
430C23—21no—
431C23—91no—
432C23—184no—
433C23—181no—
434C23—362no—
435C23—362no—
436D4—21no—
437D4—92no—
438D4—184no—
439D4—181no—
440D4C4×(C3⋊D3)⋊C312yescomplemented
441C2×C4—21no—
442C2×C4—91no—
443C2×C4—92no—
444C2×C4—92no—
445C2×C4—181no—
446C2×C4—362no—
447C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
448C8—21no—
449C8—181no—
450C2×C3—31no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—42no—
457C2×C3—61no—
458C2×C3—61no—
459C2×C3—61no—
460C2×C3—94no—
461C2×C3—122no—
462C2×C3—122no—
463C2×C3—242no—
464C2×C3—362no—
465C2×C3C3×D3×C22⋊C411yescharacteristic
466C2×C3C3×D3×D811yescharacteristic
467C2×C3C3×D3×SD1611yescharacteristic
468D3—34no—
469D3—94no—
470D3—122no—
471D3—362no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—92no—
476C22—92no—
477C22—92no—
478C22—182no—
479C22—368no—
480C22—361no—
481C22—361no—
482C22D4×(C3⋊D3)⋊C311yescharacteristic
483C4—41no—
484C4—92no—
485C4—361no—
486C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
487C4D4×(C3⋊D3)⋊C311yescharacteristic
488C3—31no—
489C3—31no—
490C3—61no—
491C3C3×D3×#10211yescharacteristic
492C2—42no—
493C2—94no—
494C2—362no—
495C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
496C2D8×(C3⋊D3)⋊C311yescharacteristic
497C2SD16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radicalC2×D4×(C3⋊D3)⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C2×C3×#102
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×C8×(C3⋊D3)⋊C3C2semidirect
3. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
4. H(3,𝔽3)×#102C2semidirect
5. #63659⋊C3C2semidirect
6. #63648⋊C3C2semidirect
7. #63636⋊C3C2semidirect
8. C3⋊D3×#102C3semidirect
14. D4×(C3⋊D3)⋊C3C4semidirect
18. C2×C8×H(3,𝔽3)C22semidirect
19. C4⋊C4×H(3,𝔽3)C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
26. C2×C8×C3⋊D3C2×C3semidirect
27. C4⋊C4×C3⋊D3C2×C3semidirect
31. C32×#102D3semidirect
32. C32×#102C2×C3semidirect
33. #63659C2×C3semidirect
34. #63648C2×C3semidirect
35. #63636C2×C3semidirect
46. D4×H(3,𝔽3)C2×C4semidirect
53. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
69. D4×C3⋊D3C3×C4semidirect
80. C2×C32×C8C2×D3semidirect
81. C2×C32×C8C22×C3semidirect
83. C32×C4⋊C4C2×D3semidirect
84. C32×C4⋊C4C22×C3semidirect
85. C4⋊(C3⋊Dic3)C22×C3semidirect
87. C2×C3⋊(C3⋊C8)C22×C3semidirect
148. C32×D4C4×D3semidirect
149. C32×D4C2×C3×C4semidirect
160. C4⋊(C3⋊D3)C2×C3×C4semidirect
183. (C3⋊D3)⋊C3#102direct (N × H)
274. #102(C3⋊D3)⋊C3direct (N × H)
275. H(3,𝔽3)C2×#102semidirect
355. C3⋊D3C3×#102semidirect
368. C2×C8C2×(C3⋊D3)⋊C3semidirect
371. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
428. C32D3×#102semidirect
429. C32C2×C3×#102semidirect
440. D4C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
C2×C8×(C3⋊D3)⋊C32
C4⋊C4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1022
#63659⋊C32
#63648⋊C32
#63636⋊C32
C3⋊D3×#1023
C3×D3×#1023