Finite Group Databasebeta

← #10595995⋊C3 #13152518

Subgroups — #10595995⋊C3 #13152518

Order 1728.

1231 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic, complemented
2C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
4#64106⋊C3C212yesmaximal, complemented
5#64104⋊C3C212yesmaximal, complemented
6#64095⋊C3C212yesmaximal, complemented
7#64091⋊C3C212yesmaximal, complemented
8C4×C4⋊((C3⋊D3)⋊C3)C214yesmaximal, complemented
9#10595995C311yesmaximal, characteristic, complemented
10C3×#5524—31nomaximal
11C23×(C3⋊D3)⋊C3C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#638⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15#3828⋊C3—24no—
16C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
18C2×C4×(C3⋊D3)⋊C3—24no—
19C2×C4×(C3⋊D3)⋊C3C2212yes—
20C2×C4×(C3⋊D3)⋊C3C2212yes—
21C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
22C2×#634⋊C3C2211yescharacteristic, complemented
23C4⋊C4×H(3,𝔽3)C2212yescomplemented
24C22⋊C4×H(3,𝔽3)C2212yes—
25C42×H(3,𝔽3)C2212yescomplemented
26#3752⋊C3C2214yes—
27#3752⋊C3C2214yescomplemented
28C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
29C2×#3828C2×C311yescharacteristic, complemented
30C22×C4×C3⋊D3C2×C311yescharacteristic, complemented
31C2×C3×#235—31no—
32C22×C3×C4×D3—31no—
33C32×#117—21no—
34C32×#117D311yescharacteristic, complemented
35C32×#117C2×C311yescharacteristic, complemented
36#64106C2×C312yescomplemented
37#64104C2×C312yescomplemented
38#64095C2×C312yescomplemented
39#64091C2×C312yescomplemented
40C4×C4⋊(C3⋊D3)C2×C314yescomplemented
41C3×#897—32no—
42C3×#895—32no—
43C3×#886—32no—
44C3×#882—32no—
45C3×C4×D12—34no—
46C23×H(3,𝔽3)C2311yescharacteristic
47C22×(C3⋊D3)⋊C3—22no—
48C22×(C3⋊D3)⋊C3—24no—
49C22×(C3⋊D3)⋊C3C2312yes—
50C22×(C3⋊D3)⋊C3C2312yes—
51C2×C4×H(3,𝔽3)D414yescomplemented
52C2×C4×H(3,𝔽3)C2314yes—
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55#638⋊C3—24no—
56C2×(C3⋊Dic3)⋊C3—22no—
57C2×(C3⋊Dic3)⋊C3C2312yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C4⋊((C3⋊D3)⋊C3)—28no—
60C4⋊((C3⋊D3)⋊C3)—22no—
61C4×(C3⋊D3)⋊C3—24no—
62C4×(C3⋊D3)⋊C3—24no—
63C4×(C3⋊D3)⋊C3—24no—
64#634⋊C3—22no—
65C3×#525—91no—
66#5524—31no—
67#5524—91no—
68C23×C3⋊D3C22×C311yescharacteristic
69C23×C3×D3—31no—
70C22×C32×C4—21no—
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C22×C311yescharacteristic
73C2×#638C22×C312yes—
74C22×C3⋊Dic3C22×C311yescharacteristic
75#3828—24no—
76C2×C4⋊(C3⋊D3)C22×C314yes—
77C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
78C2×C4×C3⋊D3—24no—
79C2×C4×C3⋊D3C22×C312yes—
80C2×C4×C3⋊D3C22×C312yes—
81C2×C4×C3⋊D3C22×C312yescomplemented
82C2×#634C22×C311yescharacteristic, complemented
83C2×C3×#67—32no—
84C22×C3×Dic3—31no—
85C3×#235—64no—
86C2×C3×D12—34no—
87C2×C3×D12—31no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—64no—
92C2×C3×Dic6—31no—
93C32×C4⋊C4—22no—
94C32×C4⋊C4C2×D312yescomplemented
95C32×C4⋊C4C22×C312yescomplemented
96C32×C22⋊C4—22no—
97C32×C22⋊C4C2×D312yes—
98C32×C22⋊C4C22×C312yes—
99C32×C42—22no—
100C32×C42C2×D312yescomplemented
101C32×C42C22×C312yescomplemented
102#3752C22×C314yes—
103#3752C22×C314yescomplemented
104C4⋊(C3⋊Dic3)C22×C314yescomplemented
105C3×(C2×D3)⋊C4—34no—
106C3×(C2×D3)⋊C4—34no—
107C3×C4⋊Dic3—34no—
108C22×H(3,𝔽3)—22no—
109C22×H(3,𝔽3)C2×D412yes—
110C22×H(3,𝔽3)C2411yescharacteristic
111C2×(C3⋊D3)⋊C3—28no—
112C2×(C3⋊D3)⋊C3—22no—
113C2×(C3⋊D3)⋊C3—22no—
114C2×(C3⋊D3)⋊C3—22no—
115C2×(C3⋊D3)⋊C3—24no—
116C2×(C3⋊D3)⋊C3#4114yescomplemented
117C4×H(3,𝔽3)—24no—
118C4×H(3,𝔽3)C2×D412yes—
119C4×H(3,𝔽3)C2×D412yes—
120(C3⋊Dic3)⋊C3—22no—
121(C3⋊Dic3)⋊C3—22no—
122C2×C3×#41—91no—
123C23×C3×C4—91no—
124C2×#235—31no—
125C2×#235—91no—
126C22×C4×D3—31no—
127C22×C4×D3—91no—
128C3×#123—92no—
129C3×C22⋊Q8—92no—
130C3×#121—92no—
131C3×#120—92no—
132C3×C4×D4—94no—
133C3×#117—31no—
134C3×#117—31no—
135C3×#117—61no—
136C3×#117C3×D311yescharacteristic
137#897—32no—
138#897—92no—
139#895—32no—
140#895—92no—
141#886—32no—
142#886—92no—
143#882—32no—
144#882—92no—
145C4×D12—34no—
146C4×D12—94no—
147C23×C32—21no—
148C23×C32C22×D311yescharacteristic
149C23×C32C23×C311yescharacteristic
150C22×C3⋊D3—22no—
151C22×C3⋊D3—24no—
152C22×C3⋊D3C23×C312yes—
153C22×C3⋊D3C23×C312yes—
154C22×C3×D3—32no—
155C22×C3×D3—32no—
156C22×C3×D3—64no—
157C22×C3×D3—62no—
158C2×C32×C4—24no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—24no—
162C2×C32×C4D1214yescomplemented
163C2×C32×C4C3×D414yescomplemented
164C2×C32×C4C22×D314yes—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C23×C314yes—
168C2×C32×C4C23×C311yescharacteristic
169C2×C32×C4C23×C311yescharacteristic
170#638—24no—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3C23×C312yes—
173C2×C3⋊Dic3C23×C312yes—
174C4⋊(C3⋊D3)—28no—
175C4⋊(C3⋊D3)—22no—
176C4×C3⋊D3—24no—
177C4×C3⋊D3—24no—
178C4×C3⋊D3—24no—
179#634—22no—
180C3×#67—64no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—62no—
184C3×D12—68no—
185C3×D12—62no—
186C3×C4×D3—64no—
187C3×C4×D3—64no—
188C3×C4×D3—64no—
189C3×Dic6—62no—
190#525—91noSylow
191C2×H(3,𝔽3)—22no—
192C2×H(3,𝔽3)C22×D411yescharacteristic
193C2×H(3,𝔽3)C2×#4112yes—
194(C3⋊D3)⋊C3—24no—
195(C3⋊D3)⋊C3—42no—
196C24×C3—91no—
197C23×D3—31no—
198C23×D3—91no—
199C3×#41—184no—
200C2×C3×Q8—91no—
201C2×C3×D4—94no—
202C2×C3×D4—91no—
203C2×C3×D4—92no—
204C22×C3×C4—31no—
205C22×C3×C4—31no—
206C22×C3×C4—61no—
207C22×C3×C4—92no—
208C22×C3×C4—92no—
209C22×C3×C4—92no—
210C22×C3×C4—91no—
211C22×C3×C4—184no—
212C22×C3×C4C2×C3×D311yescharacteristic
213C2×#67—32no—
214C2×#67—92no—
215C22×Dic3—31no—
216C22×Dic3—91no—
217#235—64no—
218#235—184no—
219C2×D12—34no—
220C2×D12—31no—
221C2×D12—94no—
222C2×D12—91no—
223C2×C4×D3—32no—
224C2×C4×D3—32no—
225C2×C4×D3—32no—
226C2×C4×D3—64no—
227C2×C4×D3—92no—
228C2×C4×D3—92no—
229C2×C4×D3—92no—
230C2×C4×D3—184no—
231C2×Dic6—31no—
232C2×Dic6—91no—
233C3×C4⋊C4—32no—
234C3×C4⋊C4—32no—
235C3×C4⋊C4—62no—
236C3×C4⋊C4—94no—
237C3×C4⋊C4C2×C3×D312yes—
238C3×C22⋊C4—32no—
239C3×C22⋊C4—32no—
240C3×C22⋊C4—62no—
241C3×C22⋊C4—94no—
242C3×C22⋊C4—94no—
243C3×C22⋊C4C2×C3×D312yes—
244C3×C42—32no—
245C3×C42—32no—
246C3×C42—62no—
247C3×C42C2×C3×D312yes—
248(C2×D3)⋊C4—34no—
249(C2×D3)⋊C4—34no—
250(C2×D3)⋊C4—94no—
251(C2×D3)⋊C4—94no—
252C4⋊Dic3—34no—
253C4⋊Dic3—94no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—21no—
257C22×C32—22no—
258C22×C32—42no—
259C22×C32C2×D1212yes—
260C22×C32C2×C3×D412yes—
261C22×C32C23×D311yescharacteristic
262C22×C32C24×C311yescharacteristic
263C2×C3⋊D3—28no—
264C2×C3⋊D3—22no—
265C2×C3⋊D3—22no—
266C2×C3⋊D3—22no—
267C2×C3⋊D3—24no—
268C2×C3⋊D3C3×#4114yescomplemented
269C2×C3×D3—34no—
270C2×C3×D3—68no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—64no—
275C32×C4—24no—
276C32×C4—24no—
277C32×C4—22no—
278C32×C4—22no—
279C32×C4—44no—
280C32×C4C2×D1212yes—
281C32×C4C2×D1212yes—
282C32×C4C2×C3×D412yes—
283C32×C4C2×C3×D412yes—
284C3⋊Dic3—22no—
285C3⋊Dic3—22no—
286C3×Dic3—62no—
287C3×Dic3—62no—
288C2×#41—91no—
289C23×C4—91no—
290#123—92no—
291C22⋊Q8—92no—
292#121—92no—
293#120—92no—
294C4×D4—94no—
295#117(C3⋊D3)⋊C311yescharacteristic, complemented
296H(3,𝔽3)#52511yescharacteristic, complemented, Sylow
297C23×C3—31no—
298C23×C3—31no—
299C23×C3—61no—
300C23×C3—92no—
301C23×C3—92no—
302C23×C3—182no—
303C23×C3—184no—
304C23×C3C22×C3×D311yescharacteristic
305C22×D3—32no—
306C22×D3—32no—
307C22×D3—64no—
308C22×D3—62no—
309C22×D3—92no—
310C22×D3—92no—
311C22×D3—184no—
312C22×D3—182no—
313C3×Q8—182no—
314C3×D4—182no—
315C3×D4—188no—
316C3×D4—184no—
317C2×C3×C4—34no—
318C2×C3×C4—34no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—34no—
322C2×C3×C4—34no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—64no—
326C2×C3×C4—61no—
327C2×C3×C4—64no—
328C2×C3×C4—61no—
329C2×C3×C4—92no—
330C2×C3×C4—92no—
331C2×C3×C4—184no—
332C2×C3×C4—184no—
333C2×C3×C4—184no—
334C2×C3×C4—182no—
335C2×C3×C4C3×D1214yes—
336C2×C3×C4C22×C3×D314yes—
337C2×C3×C4C22×C3×D311yescharacteristic
338C2×C3×C4C22×C3×D311yescharacteristic
339#67—64no—
340#67—184no—
341C2×Dic3—32no—
342C2×Dic3—32no—
343C2×Dic3—62no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—182no—
347D12—68no—
348D12—62no—
349D12—182no—
350D12—188no—
351C4×D3—64no—
352C4×D3—64no—
353C4×D3—64no—
354C4×D3—184no—
355C4×D3—184no—
356C4×D3—184no—
357Dic6—62no—
358Dic6—182no—
359C2×C32—22no—
360C2×C32—22no—
361C2×C32—21no—
362C2×C32—22no—
363C2×C32—42no—
364C2×C32C22×D1211yescharacteristic
365C2×C32D3×#4112yes—
366C2×C32C22×C3×D411yescharacteristic
367C2×C32C2×C3×#4112yes—
368C3⋊D3—24no—
369C3⋊D3—42no—
370C3×D3—64no—
371C3×D3—122no—
372C24—91no—
373#41—184no—
374C2×Q8—91no—
375C2×D4—92no—
376C2×D4—94no—
377C2×D4—91no—
378C22×C4—91no—
379C22×C4—92no—
380C22×C4—92no—
381C22×C4—92no—
382C22×C4—184no—
383C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
384C4⋊C4—94no—
385C4⋊C4C2×(C3⋊D3)⋊C312yescomplemented
386C22⋊C4—94no—
387C22⋊C4—94no—
388C22⋊C4C2×(C3⋊D3)⋊C312yes—
389C42C2×(C3⋊D3)⋊C312yescomplemented
390C22×C3—22no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—32no—
394C22×C3—32no—
395C22×C3—62no—
396C22×C3—62no—
397C22×C3—61no—
398C22×C3—62no—
399C22×C3—94no—
400C22×C3—122no—
401C22×C3—182no—
402C22×C3—182no—
403C22×C3—182no—
404C22×C3—184no—
405C22×C3—188no—
406C22×C3C2×C3×D1212yes—
407C22×C3C23×C3×D311yescharacteristic
408C2×D3—34no—
409C2×D3—68no—
410C2×D3—62no—
411C2×D3—62no—
412C2×D3—62no—
413C2×D3—64no—
414C2×D3—94no—
415C2×D3—182no—
416C2×D3—182no—
417C2×D3—188no—
418C2×D3—182no—
419C2×D3—184no—
420C3×C4—24no—
421C3×C4—32no—
422C3×C4—32no—
423C3×C4—32no—
424C3×C4—32no—
425C3×C4—64no—
426C3×C4—64no—
427C3×C4—62no—
428C3×C4—62no—
429C3×C4—124no—
430C3×C4—182no—
431C3×C4—182no—
432C3×C4C2×C3×D1212yes—
433C3×C4C2×C3×D1212yes—
434Dic3—62no—
435Dic3—62no—
436Dic3—182no—
437Dic3—182no—
438C32—21no—
439C32#552411yescharacteristic, complemented
440C32C3×#52511yescharacteristic, complemented
441C23—92no—
442C23—92no—
443C23—184no—
444C23—182no—
445C23C22×(C3⋊D3)⋊C311yescharacteristic
446Q8—182no—
447D4—188no—
448D4—184no—
449D4—182no—
450C2×C4—92no—
451C2×C4—92no—
452C2×C4—182no—
453C2×C4—184no—
454C2×C4—184no—
455C2×C4—184no—
456C2×C4C4⋊((C3⋊D3)⋊C3)14yescomplemented
457C2×C4C22×(C3⋊D3)⋊C314yes—
458C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
459C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
460C2×C3—22no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—32no—
464C2×C3—32no—
465C2×C3—62no—
466C2×C3—62no—
467C2×C3—61no—
468C2×C3—62no—
469C2×C3—122no—
470C2×C3—184no—
471C2×C3—362no—
472C2×C3C22×C3×D1211yescharacteristic
473C2×C3C3×D3×#4112yes—
474D3—64no—
475D3—122no—
476D3—184no—
477D3—362no—
478C22—22no—
479C22—94no—
480C22—188no—
481C22—182no—
482C22—182no—
483C22—182no—
484C22—184no—
485C22C2×C4⋊((C3⋊D3)⋊C3)12yes—
486C22C23×(C3⋊D3)⋊C311yescharacteristic
487C4—24no—
488C4—182no—
489C4—182no—
490C4C2×C4⋊((C3⋊D3)⋊C3)12yes—
491C4C2×C4⋊((C3⋊D3)⋊C3)12yes—
492C3—31no—
493C3—31no—
494C3—61no—
495C3C3×#552411yescharacteristic
496C2—22no—
497C2—184no—
498C2—362no—
499C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
500C2#41×(C3⋊D3)⋊C312yes—
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#117C2
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#525
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#3828⋊C3C2semidirect
2. C22×C4×(C3⋊D3)⋊C3C2semidirect
3. H(3,𝔽3)×#117C2semidirect
4. #64106⋊C3C2semidirect
5. #64104⋊C3C2semidirect
6. #64095⋊C3C2semidirect
7. #64091⋊C3C2semidirect
8. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
9. #10595995C3semidirect
17. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
21. C2×C4×(C3⋊D3)⋊C3C22semidirect
22. C2×#634⋊C3C22semidirect
23. C4⋊C4×H(3,𝔽3)C22semidirect
25. C42×H(3,𝔽3)C22semidirect
27. #3752⋊C3C22semidirect
28. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
29. C2×#3828C2×C3semidirect
30. C22×C4×C3⋊D3C2×C3semidirect
34. C32×#117D3semidirect
35. C32×#117C2×C3semidirect
36. #64106C2×C3semidirect
37. #64104C2×C3semidirect
38. #64095C2×C3semidirect
39. #64091C2×C3semidirect
40. C4×C4⋊(C3⋊D3)C2×C3semidirect
51. C2×C4×H(3,𝔽3)D4semidirect
77. C2×C4⋊(C3⋊D3)C22×C3semidirect
81. C2×C4×C3⋊D3C22×C3semidirect
82. C2×#634C22×C3semidirect
94. C32×C4⋊C4C2×D3semidirect
95. C32×C4⋊C4C22×C3semidirect
100. C32×C42C2×D3semidirect
101. C32×C42C22×C3semidirect
103. #3752C22×C3semidirect
104. C4⋊(C3⋊Dic3)C22×C3semidirect
116. C2×(C3⋊D3)⋊C3#41semidirect
162. C2×C32×C4D12semidirect
163. C2×C32×C4C3×D4semidirect
268. C2×C3⋊D3C3×#41semidirect
295. #117(C3⋊D3)⋊C3semidirect
296. H(3,𝔽3)#525semidirect
385. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
389. C42C2×(C3⋊D3)⋊C3semidirect
439. C32#5524semidirect
440. C32C3×#525semidirect
456. C2×C4C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C22×C4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1172
#64106⋊C32
#64104⋊C32
#64095⋊C32
#64091⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#105959953
C3×#55243