Finite Group Databasebeta

← #10596242⋊C3 #13153485

Subgroups — #10596242⋊C3 #13153485

Order 1728.

930 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722639C212yesmaximal, complemented
2H(3,𝔽3)×#135C211yesmaximal, characteristic, complemented
3#64160⋊C3C212yesmaximal, complemented
4Q16×(C3⋊D3)⋊C3C212yesmaximal, complemented
5#64124⋊C3C214yesmaximal, complemented
6#64121⋊C3C212yesmaximal, complemented
7#64110⋊C3C211yesmaximal, characteristic, complemented
8C2×#3746⋊C3C211yesmaximal, characteristic, complemented
9#10596242C311yesmaximal, characteristic, complemented
10C3×#5771—31nomaximal
11#41×H(3,𝔽3)C2212yes—
12Q8×(C3⋊D3)⋊C3C2214yes—
13#3830⋊C3—24no—
14#3830⋊C3C2214yes—
15#3828⋊C3—22no—
16#3828⋊C3C2211yescharacteristic
17C2×#634⋊C3—22no—
18C2×#634⋊C3C2212yes—
19Q16×H(3,𝔽3)C2211yescharacteristic
20SD16×H(3,𝔽3)C2212yes—
21D8×H(3,𝔽3)C2211yescharacteristic
22C2×C8×H(3,𝔽3)C2211yescharacteristic
23#3756⋊C3C2214yes—
24#3754⋊C3C2214yes—
25#3748⋊C3C2211yescharacteristic
26#3743⋊C3C2212yes—
27C8×(C3⋊D3)⋊C3C2212yes—
28#3746⋊C3C2212yes—
29#3746⋊C3C2212yes—
30#64367C2×C312yescomplemented
31C3×#1010—32no—
32C32×#135—21no—
33C32×#135D311yescharacteristic, complemented
34C32×#135C2×C311yescharacteristic, complemented
35#64160C2×C312yescomplemented
36Q16×C3⋊D3C2×C312yescomplemented
37#64124C2×C314yescomplemented
38#64121C2×C312yescomplemented
39C2×#3746C2×C311yescharacteristic, complemented
40#64110C2×C311yescharacteristic, complemented
41C3×#951—32no—
42C3×D3×Q16—32no—
43C3×#915—34no—
44C3×#912—32no—
45C2×C3×Dic12—31no—
46C3×#901—31no—
47Q8×H(3,𝔽3)C2312yes—
48D4×H(3,𝔽3)—22no—
49D4×H(3,𝔽3)C2312yes—
50C2×C4×H(3,𝔽3)—22no—
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52#638⋊C3—24no—
53#638⋊C3D412yes—
54C2×(C3⋊Dic3)⋊C3—22no—
55C2×(C3⋊Dic3)⋊C3—24no—
56C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
57C4×(C3⋊D3)⋊C3—24no—
58C4×(C3⋊D3)⋊C3C2312yes—
59#634⋊C3—22no—
60#634⋊C3—24no—
61#634⋊C3D411yescharacteristic
62#634⋊C3C2314yes—
63C8×H(3,𝔽3)C2311yescharacteristic
64C8×H(3,𝔽3)C2311yescharacteristic
65(C3⋊(C3⋊C8))⋊C3C2312yes—
66C3×#578—91no—
67#5771—31no—
68#5771—91no—
69C32×#41—22no—
70C32×#41C2×D312yes—
71C32×#41C22×C312yes—
72Q8×C3⋊D3C22×C314yes—
73#3830—24no—
74#3830C22×C314yes—
75#3828—22no—
76#3828C22×C311yescharacteristic
77C2×#634—22no—
78C2×#634C22×C312yes—
79C3×D3×Q8—34no—
80C3×#237—34no—
81C3×#237—64no—
82C3×#235—31no—
83C3×#235—62no—
84C2×C3×Dic6—32no—
85C2×C3×Dic6—62no—
86C32×Q16—21no—
87C32×Q16C2×D311yescharacteristic
88C32×Q16C22×C311yescharacteristic
89C32×SD16—22no—
90C32×SD16C2×D312yes—
91C32×SD16C22×C312yes—
92C32×D8—21no—
93C32×D8C2×D311yescharacteristic
94C32×D8C22×C311yescharacteristic
95C2×C32×C8—21no—
96C2×C32×C8C2×D311yescharacteristic
97C2×C32×C8C22×C311yescharacteristic
98#3756C22×C314yes—
99#3754C22×C314yes—
100#3748C22×C311yescharacteristic
101#3746C22×C312yes—
102#3746C22×C312yes—
103#3743C22×C312yes—
104C8×C3⋊D3C22×C312yes—
105C3×#216—34no—
106C3×#214—34no—
107C3×#208—31no—
108C3×Dic12—32no—
109C3×Dic12—32no—
110C3×ΓL(1,𝔽25)—32no—
111C3×D3×C8—32no—
112C22×H(3,𝔽3)—22no—
113C22×H(3,𝔽3)C2×D411yescharacteristic
114C2×(C3⋊D3)⋊C3C2×D412yes—
115C4×H(3,𝔽3)—22no—
116C4×H(3,𝔽3)C2×D411yescharacteristic
117C4×H(3,𝔽3)C2411yescharacteristic
118(C3⋊Dic3)⋊C3—24no—
119(C3⋊Dic3)⋊C3C2×D412yes—
120C3×2−1+4—92no—
121#1010—32no—
122#1010—92no—
123C3×#137—94no—
124C3×#137—92no—
125C3×#135—31no—
126C3×#135—31no—
127C3×#135—61no—
128C3×#135—92no—
129C3×#135C3×D311yescharacteristic
130C2×C3×Q16—91no—
131C2×C3×Q16—92no—
132C3×#131—91no—
133#951—32no—
134#951—92no—
135D3×Q16—32no—
136D3×Q16—92no—
137#915—34no—
138#915—94no—
139#912—32no—
140#912—92no—
141C2×Dic12—31no—
142C2×Dic12—91no—
143#901—31no—
144#901—91no—
145C32×Q8—22no—
146C32×Q8C22×D312yes—
147C32×Q8C23×C312yes—
148C32×D4—22no—
149C32×D4—22no—
150C32×D4—22no—
151C32×D4—42no—
152C32×D4C22×D312yes—
153C32×D4C23×C312yes—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—42no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C23×C311yescharacteristic
160#638—24no—
161#638C3×D412yes—
162C2×C3⋊Dic3—22no—
163C2×C3⋊Dic3—24no—
164C4⋊(C3⋊D3)C3×D411yescharacteristic
165C4×C3⋊D3—24no—
166C4×C3⋊D3C23×C312yes—
167#634—22no—
168#634—24no—
169#634C3×D411yescharacteristic
170#634C23×C314yes—
171C3×#67—32no—
172C3×#67—64no—
173C2×C3×Dic3—64no—
174C2×C3×Dic3—62no—
175C3×D12—31no—
176C3×C4×D3—32no—
177C3×C4×D3—64no—
178C3×Dic6—31no—
179C3×Dic6—34no—
180C3×Dic6—64no—
181C3×Dic6—62no—
182C32×C8—21no—
183C32×C8—21no—
184C32×C8C22×D311yescharacteristic
185C32×C8C22×D311yescharacteristic
186C32×C8C23×C311yescharacteristic
187C32×C8C23×C311yescharacteristic
188C3⋊(C3⋊C8)C23×C312yes—
189C3×C3⋊C8—32no—
190#578—91noSylow
191C2×H(3,𝔽3)—21no—
192C2×H(3,𝔽3)—42no—
193C2×H(3,𝔽3)C22×D411yescharacteristic
194(C3⋊D3)⋊C3—22no—
195C3×#41—32no—
196C3×#41—32no—
197C3×#41—62no—
198C3×#41—91no—
199C3×#41—94no—
200C3×#41—182no—
201C3×#41—184no—
202C3×#41C2×C3×D312yes—
203C2×C3×Q8—92no—
204C2×C3×Q8—94no—
205C2×C3×Q8—182no—
206D3×Q8—34no—
207D3×Q8—94no—
208#237—34no—
209#237—64no—
210#237—94no—
211#237—184no—
212#235—31no—
213#235—62no—
214#235—91no—
215#235—182no—
216C2×Dic6—32no—
217C2×Dic6—62no—
218C2×Dic6—92no—
219C2×Dic6—182no—
220C3×Q16—31no—
221C3×Q16—31no—
222C3×Q16—61no—
223C3×Q16—92no—
224C3×Q16—92no—
225C3×Q16—94no—
226C3×Q16C2×C3×D311yescharacteristic
227C3×SD16—32no—
228C3×SD16—32no—
229C3×SD16—62no—
230C3×SD16—94no—
231C3×SD16C2×C3×D312yes—
232C3×D8—31no—
233C3×D8—31no—
234C3×D8—61no—
235C3×D8C2×C3×D311yescharacteristic
236C3×M16—92no—
237C3×M16—91no—
238C2×C3×C8—31no—
239C2×C3×C8—31no—
240C2×C3×C8—61no—
241C2×C3×C8—92no—
242C2×C3×C8C2×C3×D311yescharacteristic
243#216—34no—
244#216—94no—
245#214—34no—
246#214—94no—
247#208—31no—
248#208—91no—
249Dic12—32no—
250Dic12—32no—
251Dic12—92no—
252Dic12—92no—
253ΓL(1,𝔽25)—32no—
254ΓL(1,𝔽25)—92no—
255D3×C8—32no—
256D3×C8—92no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—21no—
260C22×C32—42no—
261C22×C32D3×D411yescharacteristic
262C22×C32C2×C3×D411yescharacteristic
263C2×C3⋊D3C2×C3×D412yes—
264C2×C3×D3—32no—
265C32×C4—22no—
266C32×C4—22no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—42no—
270C32×C4D3×D411yescharacteristic
271C32×C4C2×C3×D411yescharacteristic
272C32×C4C23×D311yescharacteristic
273C32×C4C24×C311yescharacteristic
274C3⋊Dic3—24no—
275C3⋊Dic3C2×C3×D412yes—
276C3×Dic3—32no—
277C3×Dic3—64no—
2782−1+4—92no—
279#137—92no—
280#137—94no—
281#135—92no—
282#135(C3⋊D3)⋊C311yescharacteristic, complemented
283C2×Q16—92no—
284C2×Q16—91no—
285#131—91no—
286H(3,𝔽3)#57811yescharacteristic, complemented, Sylow
287C3×Q8—32no—
288C3×Q8—32no—
289C3×Q8—62no—
290C3×Q8—91no—
291C3×Q8—94no—
292C3×Q8—182no—
293C3×Q8—184no—
294C3×Q8C22×C3×D312yes—
295C3×D4—22no—
296C3×D4—32no—
297C3×D4—32no—
298C3×D4—62no—
299C3×D4—62no—
300C3×D4—62no—
301C3×D4—91no—
302C3×D4—92no—
303C3×D4—122no—
304C3×D4—184no—
305C3×D4C22×C3×D312yes—
306C2×C3×C4—22no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—61no—
312C2×C3×C4—92no—
313C2×C3×C4—122no—
314C2×C3×C4—184no—
315C2×C3×C4—184no—
316C2×C3×C4—182no—
317C2×C3×C4C22×C3×D311yescharacteristic
318#67—32no—
319#67—64no—
320#67—92no—
321#67—184no—
322C2×Dic3—64no—
323C2×Dic3—62no—
324C2×Dic3—184no—
325C2×Dic3—182no—
326D12—31no—
327D12—91no—
328C4×D3—32no—
329C4×D3—64no—
330C4×D3—92no—
331C4×D3—184no—
332Dic6—31no—
333Dic6—34no—
334Dic6—64no—
335Dic6—62no—
336Dic6—91no—
337Dic6—94no—
338Dic6—184no—
339Dic6—182no—
340C3×C8—31no—
341C3×C8—31no—
342C3×C8—31no—
343C3×C8—31no—
344C3×C8—61no—
345C3×C8—61no—
346C3×C8—92no—
347C3×C8C22×C3×D311yescharacteristic
348C3×C8C22×C3×D311yescharacteristic
349C3⋊C8—32no—
350C3⋊C8—92no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—41no—
355C2×C32—42no—
356C2×C32—42no—
357C2×C32—82no—
358C2×C32C2×D3×D411yescharacteristic
359C2×C32C22×C3×D411yescharacteristic
360C3⋊D3—22no—
361C3×D3—62no—
362#41—91no—
363#41—94no—
364#41—182no—
365#41—184no—
366#41C2×(C3⋊D3)⋊C312yes—
367C2×Q8—94no—
368C2×Q8—92no—
369C2×Q8—182no—
370Q16—94no—
371Q16—92no—
372Q16—92no—
373Q16C2×(C3⋊D3)⋊C311yescharacteristic
374SD16—94no—
375SD16C2×(C3⋊D3)⋊C312yes—
376D8C2×(C3⋊D3)⋊C311yescharacteristic
377M16—91no—
378M16—92no—
379C2×C8—92no—
380C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
381C22×C3—22no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—62no—
385C22×C3—62no—
386C22×C3—61no—
387C22×C3—92no—
388C22×C3—122no—
389C22×C3C3×D3×D411yescharacteristic
390C2×D3—32no—
391C2×D3—92no—
392C3×C4—22no—
393C3×C4—31no—
394C3×C4—31no—
395C3×C4—31no—
396C3×C4—31no—
397C3×C4—62no—
398C3×C4—62no—
399C3×C4—61no—
400C3×C4—61no—
401C3×C4—92no—
402C3×C4—122no—
403C3×C4—184no—
404C3×C4C3×D3×D411yescharacteristic
405C3×C4C23×C3×D311yescharacteristic
406Dic3—32no—
407Dic3—64no—
408Dic3—92no—
409Dic3—184no—
410C32—21no—
411C32#577111yescharacteristic, complemented
412C32C3×#57811yescharacteristic, complemented
413Q8—91no—
414Q8—94no—
415Q8—184no—
416Q8—182no—
417Q8C22×(C3⋊D3)⋊C312yes—
418D4—22no—
419D4—92no—
420D4—91no—
421D4—184no—
422D4C22×(C3⋊D3)⋊C312yes—
423C2×C4—22no—
424C2×C4—92no—
425C2×C4—184no—
426C2×C4—184no—
427C2×C4—182no—
428C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
429C8—92no—
430C8C22×(C3⋊D3)⋊C311yescharacteristic
431C8C22×(C3⋊D3)⋊C311yescharacteristic
432C2×C3—21no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—42no—
436C2×C3—61no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—121no—
440C2×C3—122no—
441C2×C3—122no—
442C2×C3—182no—
443C2×C3—242no—
444C2×C3C2×C3×D3×D411yescharacteristic
445D3—62no—
446D3—182no—
447C22—22no—
448C22—92no—
449C22D4×(C3⋊D3)⋊C311yescharacteristic
450C4—22no—
451C4—92no—
452C4—184no—
453C4D4×(C3⋊D3)⋊C311yescharacteristic
454C4C23×(C3⋊D3)⋊C311yescharacteristic
455C3—31no—
456C3—31no—
457C3—61no—
458C3C3×#577111yescharacteristic
459C2—21no—
460C2—42no—
461C2—182no—
462C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2C2×D4×(C3⋊D3)⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingH(3,𝔽3)×#135C2
socleC2×C3C2×C3×D3×D4
hypercenter#135(C3⋊D3)⋊C3
hypocenterC32C3×#578
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #11722639C2semidirect
2. H(3,𝔽3)×#135C2semidirect
3. #64160⋊C3C2semidirect
4. Q16×(C3⋊D3)⋊C3C2semidirect
5. #64124⋊C3C2semidirect
6. #64121⋊C3C2semidirect
7. #64110⋊C3C2semidirect
8. C2×#3746⋊C3C2semidirect
9. #10596242C3semidirect
30. #64367C2×C3semidirect
33. C32×#135D3semidirect
34. C32×#135C2×C3semidirect
35. #64160C2×C3semidirect
36. Q16×C3⋊D3C2×C3semidirect
37. #64124C2×C3semidirect
38. #64121C2×C3semidirect
39. C2×#3746C2×C3semidirect
40. #64110C2×C3semidirect
282. #135(C3⋊D3)⋊C3semidirect
286. H(3,𝔽3)#578semidirect
411. C32#5771semidirect
412. C32C3×#578semidirect

Maximal subgroups

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

subgroupindex
#117226392
H(3,𝔽3)×#1352
#64160⋊C32
Q16×(C3⋊D3)⋊C32
#64124⋊C32
#64121⋊C32
#64110⋊C32
C2×#3746⋊C32
#105962423
C3×#57713