Finite Group Databasebeta

← #10592501⋊C3 #13141915

Subgroups — #10592501⋊C3 #13141915

Order 1728.

567 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#3743⋊C3C211yesmaximal, characteristic
3#64101⋊C3C211yesmaximal, characteristic
4H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#63643⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592501C311yesmaximal, characteristic, complemented
9C3×#4802—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)⋊C3C2211yescharacteristic
15C2×#634⋊C3—21no—
16C2×#634⋊C3C2211yescharacteristic
17C2×C8×H(3,𝔽3)C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19#3752⋊C3—21no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C4×(C3⋊Dic3)⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#3743⋊C3—22no—
24C2×Q8×C3⋊D3C2×C311yescharacteristic
25C2×C3×D3×Q8—31no—
26C2×#3743C2×C311yescharacteristic
27#64101C2×C311yescharacteristic
28C2×C3×ΓL(1,𝔽25)—31no—
29C3×#892—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×C412yes—
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)⋊C3C2×C412yes—
50C4×(C3⋊D3)⋊C3D412yes—
51#634⋊C3—21no—
52#634⋊C3—24no—
53#634⋊C3C2×C412yes—
54C8×H(3,𝔽3)—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56C3×#419—91no—
57#4802—31no—
58#4802—91no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C22×C311yescharacteristic
62Q8×C3⋊D3—22no—
63Q8×C3⋊D3C3×C414yescomplemented
64C2×C4×C3⋊D3—21no—
65C2×C4×C3⋊D3C22×C311yescharacteristic
66C2×#634—21no—
67C2×#634C22×C311yescharacteristic
68C3×D3×Q8—34no—
69C3×D3×Q8—62no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—61no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×C32×C8—21no—
75C2×C32×C8C2×D311yescharacteristic
76C2×C32×C8C22×C311yescharacteristic
77C32×C4⋊C4—21no—
78C32×C4⋊C4C2×D311yescharacteristic
79C32×C4⋊C4C22×C311yescharacteristic
80#3752—21no—
81C4⋊(C3⋊Dic3)C22×C311yescharacteristic
82C4×C3⋊Dic3—21no—
83C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
84#3743—22no—
85C3×(C2×D3)⋊C4—61no—
86C3×C4⋊Dic3—31no—
87C3×C4×Dic3—61no—
88C2×C3×C3⋊C8—31no—
89C3×ΓL(1,𝔽25)—62no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3C22⋊C412yes—
94C4×H(3,𝔽3)—22no—
95C4×H(3,𝔽3)—41no—
96C4×H(3,𝔽3)C22×C411yescharacteristic
97C4×H(3,𝔽3)C2×D411yescharacteristic
98(C3⋊Dic3)⋊C3—22no—
99(C3⋊Dic3)⋊C3—41no—
100(C3⋊Dic3)⋊C3C22⋊C412yes—
101C22×C3×Q8—91no—
102C2×D3×Q8—31no—
103C2×D3×Q8—91no—
104C2×C3×M16—91no—
105C3×#117—91no—
106C2×ΓL(1,𝔽25)—31no—
107C2×ΓL(1,𝔽25)—91no—
108#892—31no—
109#892—91no—
110C3×#103—31no—
111C3×#103—31no—
112C3×#103—61no—
113C3×#103—91no—
114C3×#103—91no—
115C3×#103—91no—
116C3×#103C3×D311yescharacteristic
117#835—31no—
118#835—91no—
119#816—31no—
120#816—91no—
121#810—31no—
122#810—91no—
123C22×C3⋊D3C3×D411yescharacteristic
124C22×C3×D3—31no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—22no—
128C32×Q8—41no—
129C32×Q8C4×D312yes—
130C32×Q8C2×C3×C412yes—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—41no—
137C2×C32×C4—41no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C23×C311yescharacteristic
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3C3×D411yescharacteristic
143C4×C3⋊D3—24no—
144C4×C3⋊D3C2×C3×C412yes—
145C4×C3⋊D3C3×D412yes—
146#634—21no—
147#634—24no—
148#634C2×C3×C412yes—
149C2×C3×Dic3—31no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C3×C4×D3—32no—
153C3×C4×D3—32no—
154C3×C4×D3—64no—
155C3×Dic6—32no—
156C3×Dic6—64no—
157C3×Dic6—61no—
158C32×C8—21no—
159C32×C8—21no—
160C32×C8—41no—
161C3⋊(C3⋊C8)—21no—
162C3×C3⋊C8—61no—
163#419—91noSylow
164C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
165C2×H(3,𝔽3)#13711yescharacteristic
166C2×H(3,𝔽3)#13711yescharacteristic
167(C3⋊D3)⋊C3—22no—
168C2×C3×Q8—31no—
169C2×C3×Q8—31no—
170C2×C3×Q8—61no—
171C2×C3×Q8—91no—
172C2×C3×Q8—94no—
173C2×C3×Q8—181no—
174C2×C3×Q8—182no—
175C2×C3×Q8C2×C3×D311yescharacteristic
176C22×C3×C4—91no—
177C22×C3×C4—181no—
178D3×Q8—34no—
179D3×Q8—62no—
180D3×Q8—94no—
181D3×Q8—182no—
182C2×C4×D3—31no—
183C2×C4×D3—61no—
184C2×C4×D3—91no—
185C2×C4×D3—181no—
186C2×Dic6—31no—
187C2×Dic6—61no—
188C2×Dic6—91no—
189C2×Dic6—181no—
190C3×M16—182no—
191C2×C3×C8—31no—
192C2×C3×C8—31no—
193C2×C3×C8—61no—
194C2×C3×C8—91no—
195C2×C3×C8C2×C3×D311yescharacteristic
196C3×C4⋊C4—31no—
197C3×C4⋊C4—31no—
198C3×C4⋊C4—61no—
199C3×C4⋊C4—91no—
200C3×C4⋊C4C2×C3×D311yescharacteristic
201C3×C22⋊C4—181no—
202C3×C42—181no—
203(C2×D3)⋊C4—61no—
204(C2×D3)⋊C4—181no—
205C4⋊Dic3—31no—
206C4⋊Dic3—91no—
207C4×Dic3—61no—
208C4×Dic3—181no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—91no—
211ΓL(1,𝔽25)—62no—
212ΓL(1,𝔽25)—182no—
213C22×C32—21no—
214C22×C32D3×D411yescharacteristic
215C22×C32C2×C3×D411yescharacteristic
216C2×C3⋊D3—21no—
217C2×C3⋊D3—21no—
218C2×C3⋊D3C3×C22⋊C412yes—
219C2×C3×D3—32no—
220C2×C3×D3—61no—
221C2×C3×D3—61no—
222C32×C4—22no—
223C32×C4—22no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—42no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4—81no—
230C32×C4C2×C4×D311yescharacteristic
231C32×C4D3×D411yescharacteristic
232C32×C4C22×C3×C411yescharacteristic
233C32×C4C2×C3×D411yescharacteristic
234C3⋊Dic3—22no—
235C3⋊Dic3—41no—
236C3⋊Dic3C3×C22⋊C412yes—
237C3×Dic3—32no—
238C3×Dic3—62no—
239C3×Dic3—121no—
240C22×Q8—91no—
241C2×M16—91no—
242#117—91no—
243#103—91no—
244#103—91no—
245#103—91no—
246#103(C3⋊D3)⋊C311yescharacteristic, complemented
247H(3,𝔽3)#41911yescharacteristic, complemented, Sylow
248C23×C3—91no—
249C22×D3—31no—
250C22×D3—91no—
251C3×Q8—21no—
252C3×Q8—32no—
253C3×Q8—32no—
254C3×Q8—61no—
255C3×Q8—61no—
256C3×Q8—62no—
257C3×Q8—92no—
258C3×Q8—121no—
259C3×Q8—181no—
260C3×Q8—184no—
261C3×Q8C3×C4×D312yes—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—91no—
274C2×C3×C4—121no—
275C2×C3×C4—121no—
276C2×C3×C4—184no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4C22×C3×D311yescharacteristic
280C2×Dic3—31no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—91no—
284C2×Dic3—181no—
285C2×Dic3—181no—
286C4×D3—32no—
287C4×D3—32no—
288C4×D3—64no—
289C4×D3—92no—
290C4×D3—92no—
291C4×D3—184no—
292Dic6—32no—
293Dic6—64no—
294Dic6—61no—
295Dic6—92no—
296Dic6—184no—
297Dic6—181no—
298C3×C8—21no—
299C3×C8—61no—
300C3×C8—61no—
301C3×C8—121no—
302C3×C8—181no—
303C3⋊C8—61no—
304C3⋊C8—181no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32D3×C22⋊C411yescharacteristic
309C2×C32#91511yescharacteristic
310C2×C32#91811yescharacteristic
311C2×C32C2×C3×C22⋊C411yescharacteristic
312C2×C32C3×#13711yescharacteristic
313C2×C32C3×#13711yescharacteristic
314C3⋊D3—22no—
315C3×D3—62no—
316C2×Q8—94no—
317C2×Q8—91no—
318C2×Q8—181no—
319C2×Q8—182no—
320C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
321C22×C4—91no—
322C22×C4—181no—
323M16—182no—
324C2×C8—91no—
325C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
326C4⋊C4—91no—
327C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
328C22⋊C4—181no—
329C42—181no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—61no—
333C22×C3—92no—
334C22×C3—181no—
335C22×C3—181no—
336C22×C3C3×D3×D411yescharacteristic
337C2×D3—32no—
338C2×D3—61no—
339C2×D3—61no—
340C2×D3—92no—
341C2×D3—181no—
342C2×D3—181no—
343C3×C4—22no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—41no—
349C3×C4—62no—
350C3×C4—62no—
351C3×C4—61no—
352C3×C4—61no—
353C3×C4—92no—
354C3×C4—122no—
355C3×C4—121no—
356C3×C4—121no—
357C3×C4—182no—
358C3×C4—241no—
359C3×C4—361no—
360C3×C4C2×C3×C4×D311yescharacteristic
361C3×C4C3×D3×D411yescharacteristic
362Dic3—32no—
363Dic3—62no—
364Dic3—92no—
365Dic3—121no—
366Dic3—182no—
367Dic3—361no—
368C32—21no—
369C32#480211yescharacteristic, complemented
370C32C3×#41911yescharacteristic, complemented
371C23—91no—
372Q8—21no—
373Q8—92no—
374Q8—184no—
375Q8—181no—
376Q8C4×(C3⋊D3)⋊C312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—91no—
380C2×C4—92no—
381C2×C4—92no—
382C2×C4—184no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
386C8—21no—
387C8—181no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—61no—
395C2×C3—61no—
396C2×C3—61no—
397C2×C3—182no—
398C2×C3C3×D3×C22⋊C411yescharacteristic
399C2×C3C3×#91511yescharacteristic
400C2×C3C3×#91811yescharacteristic
401D3—62no—
402D3—182no—
403C22—92no—
404C22—181no—
405C22—181no—
406C22D4×(C3⋊D3)⋊C311yescharacteristic
407C4—22no—
408C4—41no—
409C4—92no—
410C4—182no—
411C4—361no—
412C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
413C4D4×(C3⋊D3)⋊C311yescharacteristic
414C3—31no—
415C3—31no—
416C3—61no—
417C3C3×#480211yescharacteristic
418C2—182no—
419C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
420C2#64124⋊C311yescharacteristic
421C2#64127⋊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
hypocenterC32C3×#419
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. #10592501C3semidirect
12. Q8×(C3⋊D3)⋊C3C4semidirect
31. C32×#103D3semidirect
32. C32×#103C2×C3semidirect
33. #63662C2×C3semidirect
34. #63643C2×C3semidirect
35. #63637C2×C3semidirect
63. Q8×C3⋊D3C3×C4semidirect
246. #103(C3⋊D3)⋊C3semidirect
247. H(3,𝔽3)#419semidirect
369. C32#4802semidirect
370. C32C3×#419semidirect

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×#3743⋊C32
#64101⋊C32
H(3,𝔽3)×#1032
#63662⋊C32
#63643⋊C32
#63637⋊C32
#105925013
C3×#48023