Finite Group Databasebeta

← #103×(C3⋊D3)⋊C3 #13141914

Subgroups — #103×(C3⋊D3)⋊C3 #13141914

Order 1728.

609 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
4H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#63643⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8C3⋊D3×#103C311yesmaximal, characteristic, complemented
9C3×D3×#103—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11Q8×(C3⋊D3)⋊C3—22no—
12Q8×(C3⋊D3)⋊C3C414yescomplemented
13C2×C4×(C3⋊D3)⋊C3—21no—
14C2×C4×(C3⋊D3)⋊C3—21no—
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3—21no—
17C2×#634⋊C3C2211yescharacteristic
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21#3750⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C8×(C3⋊D3)⋊C3—22no—
24C2×Q8×C3⋊D3C2×C311yescharacteristic
25C2×C3×D3×Q8—31no—
26C2×C8×C3⋊D3C2×C311yescharacteristic
27C4⋊C4×C3⋊D3C2×C311yescharacteristic
28C2×C3×D3×C8—31no—
29C3×D3×C4⋊C4—31no—
30C32×#103—21no—
31C32×#103D311yescharacteristic, complemented
32C32×#103C2×C311yescharacteristic, complemented
33#63662C2×C311yescharacteristic, complemented
34#63643C2×C311yescharacteristic, complemented
35#63637C2×C311yescharacteristic, complemented
36C3×#835—31no—
37C3×#816—31no—
38C3×#810—31no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)C2×C412yescomplemented
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4×(C3⋊D3)⋊C3—24no—
49C4×(C3⋊D3)⋊C3—42no—
50C4×(C3⋊D3)⋊C3C2×C412yes—
51C4×(C3⋊D3)⋊C3D412yes—
52#634⋊C3—21no—
53#634⋊C3—24no—
54#634⋊C3C2×C412yescomplemented
55C8×H(3,𝔽3)—21no—
56(C3⋊(C3⋊C8))⋊C3—21no—
57C2×C3×#103—91no—
58D3×#103—31no—
59D3×#103—91no—
60C2×C32×Q8—21no—
61C2×C32×Q8C2×D311yescharacteristic
62C2×C32×Q8C22×C311yescharacteristic
63Q8×C3⋊D3—22no—
64Q8×C3⋊D3C3×C414yescomplemented
65C2×C4×C3⋊D3—21no—
66C2×C4×C3⋊D3—21no—
67C2×C4×C3⋊D3C22×C311yescharacteristic
68C2×#634—21no—
69C2×#634C22×C311yescharacteristic
70C3×D3×Q8—34no—
71C3×D3×Q8—62no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3—61no—
75C2×C3×Dic6—31no—
76C2×C3×Dic6—61no—
77C2×C32×C8—21no—
78C2×C32×C8C2×D311yescharacteristic
79C2×C32×C8C22×C311yescharacteristic
80C32×C4⋊C4—21no—
81C32×C4⋊C4C2×D311yescharacteristic
82C32×C4⋊C4C22×C311yescharacteristic
83C4⋊(C3⋊Dic3)C22×C311yescharacteristic
84#3750—21no—
85C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
86C8×C3⋊D3—22no—
87C3×C4⋊Dic3—31no—
88C3×Dic3⋊C4—61no—
89C2×C3×C3⋊C8—31no—
90C3×D3×C8—62no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3C22⋊C412yes—
93C2×(C3⋊D3)⋊C3SD1612yes—
94C2×(C3⋊D3)⋊C3Q1612yes—
95C4×H(3,𝔽3)—22no—
96C4×H(3,𝔽3)—41no—
97C4×H(3,𝔽3)C22×C411yescharacteristic
98C4×H(3,𝔽3)C2×D411yescharacteristic
99(C3⋊Dic3)⋊C3—22no—
100(C3⋊Dic3)⋊C3—41no—
101(C3⋊Dic3)⋊C3C22⋊C412yes—
102C22×C3×Q8—91no—
103C2×D3×Q8—31no—
104C2×D3×Q8—91no—
105C22×C3×C8—91no—
106C2×C3×C4⋊C4—91no—
107C2×D3×C8—31no—
108C2×D3×C8—91no—
109D3×C4⋊C4—31no—
110D3×C4⋊C4—91no—
111C3×#103—31no—
112C3×#103—31no—
113C3×#103—61no—
114C3×#103—91no—
115C3×#103—91no—
116C3×#103—91no—
117C3×#103C3×D311yescharacteristic
118#835—31no—
119#835—91no—
120#816—31no—
121#816—91no—
122#810—31no—
123#810—91no—
124C22×C3⋊D3C3×D411yescharacteristic
125C22×C3×D3—31no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—22no—
129C32×Q8—41no—
130C32×Q8C4×D312yescomplemented
131C32×Q8C2×C3×C412yescomplemented
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—41no—
138C2×C32×C4—41no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C23×C311yescharacteristic
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3C3×D411yescharacteristic
144C4×C3⋊D3—24no—
145C4×C3⋊D3—42no—
146C4×C3⋊D3C2×C3×C412yes—
147C4×C3⋊D3C3×D412yes—
148#634—21no—
149#634—24no—
150#634C2×C3×C412yescomplemented
151C2×C3×Dic3—31no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C3×C4×D3—32no—
155C3×C4×D3—32no—
156C3×C4×D3—64no—
157C3×C4×D3—122no—
158C3×Dic6—32no—
159C3×Dic6—64no—
160C3×Dic6—61no—
161C32×C8—21no—
162C32×C8—21no—
163C32×C8—41no—
164C3⋊(C3⋊C8)—21no—
165C3×C3⋊C8—61no—
166C2×#103—91noSylow
167C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
168C2×H(3,𝔽3)C2×SD1611yescharacteristic
169C2×H(3,𝔽3)C2×Q1611yescharacteristic
170(C3⋊D3)⋊C3#10314yesdirect product factor
171C2×C3×Q8—31no—
172C2×C3×Q8—31no—
173C2×C3×Q8—61no—
174C2×C3×Q8—91no—
175C2×C3×Q8—94no—
176C2×C3×Q8—181no—
177C2×C3×Q8—182no—
178C2×C3×Q8C2×C3×D311yescharacteristic
179C22×C3×C4—91no—
180C22×C3×C4—181no—
181C22×C3×C4—181no—
182D3×Q8—34no—
183D3×Q8—62no—
184D3×Q8—94no—
185D3×Q8—182no—
186C2×C4×D3—31no—
187C2×C4×D3—61no—
188C2×C4×D3—61no—
189C2×C4×D3—91no—
190C2×C4×D3—181no—
191C2×C4×D3—181no—
192C2×Dic6—31no—
193C2×Dic6—61no—
194C2×Dic6—91no—
195C2×Dic6—181no—
196C2×C3×C8—31no—
197C2×C3×C8—31no—
198C2×C3×C8—61no—
199C2×C3×C8—91no—
200C2×C3×C8—182no—
201C2×C3×C8C2×C3×D311yescharacteristic
202C3×C4⋊C4—31no—
203C3×C4⋊C4—31no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—91no—
206C3×C4⋊C4—181no—
207C3×C4⋊C4C2×C3×D311yescharacteristic
208C4⋊Dic3—31no—
209C4⋊Dic3—91no—
210Dic3⋊C4—61no—
211Dic3⋊C4—181no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—91no—
214D3×C8—62no—
215D3×C8—182no—
216C22×C32—21no—
217C22×C32D3×D411yescharacteristic
218C22×C32C2×C3×D411yescharacteristic
219C2×C3⋊D3C3×C22⋊C412yes—
220C2×C3⋊D3C3×SD1612yes—
221C2×C3⋊D3C3×Q1612yes—
222C2×C3×D3—32no—
223C2×C3×D3—32no—
224C2×C3×D3—32no—
225C32×C4—22no—
226C32×C4—22no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—42no—
230C32×C4—41no—
231C32×C4—41no—
232C32×C4—81no—
233C32×C4C2×C4×D311yescharacteristic
234C32×C4D3×D411yescharacteristic
235C32×C4C22×C3×C411yescharacteristic
236C32×C4C2×C3×D411yescharacteristic
237C3⋊Dic3—22no—
238C3⋊Dic3—41no—
239C3⋊Dic3C3×C22⋊C412yes—
240C3×Dic3—32no—
241C3×Dic3—62no—
242C3×Dic3—121no—
243C22×Q8—91no—
244C22×C8—91no—
245C2×C4⋊C4—91no—
246#103—91no—
247#103—91no—
248#103—91no—
249#103(C3⋊D3)⋊C311yescharacteristic, direct product factor
250H(3,𝔽3)C2×#10311yescharacteristic, complemented, Sylow
251C23×C3—91no—
252C22×D3—31no—
253C22×D3—91no—
254C3×Q8—21no—
255C3×Q8—32no—
256C3×Q8—32no—
257C3×Q8—61no—
258C3×Q8—61no—
259C3×Q8—62no—
260C3×Q8—92no—
261C3×Q8—121no—
262C3×Q8—181no—
263C3×Q8—184no—
264C3×Q8C3×C4×D312yes—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—92no—
275C2×C3×C4—92no—
276C2×C3×C4—91no—
277C2×C3×C4—121no—
278C2×C3×C4—121no—
279C2×C3×C4—184no—
280C2×C3×C4—181no—
281C2×C3×C4—181no—
282C2×C3×C4—362no—
283C2×C3×C4C22×C3×D311yescharacteristic
284C2×Dic3—31no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—91no—
288C2×Dic3—181no—
289C2×Dic3—181no—
290C4×D3—32no—
291C4×D3—32no—
292C4×D3—64no—
293C4×D3—92no—
294C4×D3—92no—
295C4×D3—122no—
296C4×D3—184no—
297C4×D3—362no—
298Dic6—32no—
299Dic6—64no—
300Dic6—61no—
301Dic6—92no—
302Dic6—184no—
303Dic6—181no—
304C3×C8—21no—
305C3×C8—61no—
306C3×C8—61no—
307C3×C8—121no—
308C3×C8—181no—
309C3⋊C8—61no—
310C3⋊C8—181no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32D3×C22⋊C411yescharacteristic
315C2×C32D3×SD1611yescharacteristic
316C2×C32D3×Q1611yescharacteristic
317C2×C32C2×C3×C22⋊C411yescharacteristic
318C2×C32C2×C3×SD1611yescharacteristic
319C2×C32C2×C3×Q1611yescharacteristic
320C3⋊D3C3×#10314yescomplemented
321C3×D3—34no—
322C2×Q8—94no—
323C2×Q8—91no—
324C2×Q8—181no—
325C2×Q8—182no—
326C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
327C22×C4—91no—
328C22×C4—181no—
329C22×C4—181no—
330C2×C8—91no—
331C2×C8—182no—
332C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
333C4⋊C4—91no—
334C4⋊C4—181no—
335C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—61no—
339C22×C3—92no—
340C22×C3—92no—
341C22×C3—92no—
342C22×C3C3×D3×D411yescharacteristic
343C2×D3—32no—
344C2×D3—32no—
345C2×D3—32no—
346C2×D3—92no—
347C2×D3—92no—
348C2×D3—92no—
349C3×C4—22no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—31no—
353C3×C4—31no—
354C3×C4—41no—
355C3×C4—62no—
356C3×C4—62no—
357C3×C4—61no—
358C3×C4—61no—
359C3×C4—92no—
360C3×C4—122no—
361C3×C4—121no—
362C3×C4—121no—
363C3×C4—182no—
364C3×C4—241no—
365C3×C4—361no—
366C3×C4C2×C3×C4×D311yescharacteristic
367C3×C4C3×D3×D411yescharacteristic
368Dic3—32no—
369Dic3—62no—
370Dic3—92no—
371Dic3—121no—
372Dic3—182no—
373Dic3—361no—
374C32—21no—
375C32D3×#10311yescharacteristic, complemented
376C32C2×C3×#10311yescharacteristic, complemented
377C23—91no—
378Q8—21no—
379Q8—92no—
380Q8—184no—
381Q8—181no—
382Q8C4×(C3⋊D3)⋊C312yescomplemented
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—91no—
386C2×C4—92no—
387C2×C4—92no—
388C2×C4—184no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4—362no—
392C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
393C8—21no—
394C8—181no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—61no—
404C2×C3—94no—
405C2×C3C3×D3×C22⋊C411yescharacteristic
406C2×C3C3×D3×SD1611yescharacteristic
407C2×C3C3×D3×Q1611yescharacteristic
408D3—34no—
409D3—94no—
410C22—92no—
411C22—92no—
412C22—92no—
413C22D4×(C3⋊D3)⋊C311yescharacteristic
414C4—22no—
415C4—41no—
416C4—92no—
417C4—182no—
418C4—361no—
419C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
420C4D4×(C3⋊D3)⋊C311yescharacteristic
421C3—31no—
422C3—31no—
423C3—61no—
424C3C3×D3×#10311yescharacteristic
425C2—94no—
426C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
427C2SD16×(C3⋊D3)⋊C311yescharacteristic
428C2Q16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×(C3⋊D3)⋊C3D4
cosocleC2×C32×C4C23×C3
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C2×C3×#103
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
4. H(3,𝔽3)×#103C2semidirect
5. #63662⋊C3C2semidirect
6. #63643⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. C3⋊D3×#103C3semidirect
12. Q8×(C3⋊D3)⋊C3C4semidirect
31. C32×#103D3semidirect
32. C32×#103C2×C3semidirect
33. #63662C2×C3semidirect
34. #63643C2×C3semidirect
35. #63637C2×C3semidirect
41. Q8×H(3,𝔽3)C2×C4semidirect
54. #634⋊C3C2×C4semidirect
64. Q8×C3⋊D3C3×C4semidirect
130. C32×Q8C4×D3semidirect
131. C32×Q8C2×C3×C4semidirect
150. #634C2×C3×C4semidirect
170. (C3⋊D3)⋊C3#103direct (N × H)
249. #103(C3⋊D3)⋊C3direct (N × H)
250. H(3,𝔽3)C2×#103semidirect
320. C3⋊D3C3×#103semidirect
375. C32D3×#103semidirect
376. C32C2×C3×#103semidirect
382. Q8C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
C2×C8×(C3⋊D3)⋊C32
C4⋊C4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1032
#63662⋊C32
#63643⋊C32
#63637⋊C32
C3⋊D3×#1033
C3×D3×#1033