Finite Group Databasebeta

← #10592426⋊C3 #13141612

Subgroups — #10592426⋊C3 #13141612

Order 1728.

603 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2C2×#3743⋊C3C211yesmaximal, characteristic, complemented
3C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
5#63657⋊C3C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#63647⋊C3C211yesmaximal, characteristic, complemented
8#10592426C311yesmaximal, characteristic, complemented
9C3×#4727—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3C411yescharacteristic
12C2×#638⋊C3C411yescharacteristic
13#3828⋊C3—22no—
14#3828⋊C3—22no—
15C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C411yescharacteristic
19C2×C8×H(3,𝔽3)C2211yescharacteristic
20C2×C8×H(3,𝔽3)C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#3743⋊C3—22no—
24C8×(C3⋊D3)⋊C3—22no—
25C2×#3828C2×C311yescharacteristic
26C2×C3×#235—31no—
27C2×#3743C2×C311yescharacteristic, complemented
28C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
29C2×C3×ΓL(1,𝔽25)—31no—
30C2×C3×D3×C8—31no—
31C32×C22⋊C8—21no—
32C32×C22⋊C8D311yescharacteristic, complemented
33C32×C22⋊C8C2×C311yescharacteristic, complemented
34#63657C2×C311yescharacteristic, complemented
35#63647C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37C3×#830—31no—
38C3×(C2×D3)⋊C8—31no—
39C3×(C2×D3)⋊C8—31no—
40C23×H(3,𝔽3)C2×C411yescharacteristic
41C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
42C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2×C411yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47#638⋊C3—22no—
48#638⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
50C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
51C4⋊((C3⋊D3)⋊C3)—22no—
52C4×(C3⋊D3)⋊C3—21no—
53C4×(C3⋊D3)⋊C3—21no—
54C4×(C3⋊D3)⋊C3D412yes—
55C4×(C3⋊D3)⋊C3D412yes—
56#634⋊C3—22no—
57C8×H(3,𝔽3)—21no—
58C8×H(3,𝔽3)—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60(C3⋊(C3⋊C8))⋊C3—21no—
61C3×#408—91no—
62#4727—31no—
63#4727—91no—
64C22×C32×C4—21no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C22×C311yescharacteristic
67C2×#638C3×C411yescharacteristic
68C2×#638C3×C411yescharacteristic
69#3828—22no—
70#3828—22no—
71C2×C4⋊(C3⋊D3)C3×C411yescharacteristic
72C2×C4×C3⋊D3C22×C311yescharacteristic
73C2×C4×C3⋊D3C22×C311yescharacteristic
74C2×#634C3×C411yescharacteristic
75C2×C3×#67—31no—
76C2×C3×#67—31no—
77C3×#235—62no—
78C3×#235—62no—
79C2×C3×D12—31no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—31no—
82C2×C3×Dic6—31no—
83C2×C32×C8—21no—
84C2×C32×C8—21no—
85C2×C32×C8C2×D311yescharacteristic
86C2×C32×C8C2×D311yescharacteristic
87C2×C32×C8C22×C311yescharacteristic
88C2×C32×C8C22×C311yescharacteristic
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
91#3743—22no—
92C8×C3⋊D3—22no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—31no—
95C3×ΓL(1,𝔽25)—62no—
96C3×D3×C8—62no—
97C22×H(3,𝔽3)—21no—
98C22×H(3,𝔽3)—21no—
99C22×H(3,𝔽3)—21no—
100C22×H(3,𝔽3)C22×C411yescharacteristic
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3—21no—
104C2×(C3⋊D3)⋊C3—21no—
105C2×(C3⋊D3)⋊C3—21no—
106C2×(C3⋊D3)⋊C3C22⋊C412yes—
107C4×H(3,𝔽3)—21no—
108C4×H(3,𝔽3)C2×D411yescharacteristic
109C4×H(3,𝔽3)C2×D411yescharacteristic
110(C3⋊Dic3)⋊C3—21no—
111(C3⋊Dic3)⋊C3C22⋊C412yes—
112C2×C3×#41—91no—
113C2×#235—31no—
114C2×#235—91no—
115C2×C3×M16—91no—
116C22×C3×C8—91no—
117C2×ΓL(1,𝔽25)—31no—
118C2×ΓL(1,𝔽25)—91no—
119C2×D3×C8—31no—
120C2×D3×C8—91no—
121C3×C22⋊C8—31no—
122C3×C22⋊C8—31no—
123C3×C22⋊C8—61no—
124C3×C22⋊C8—91no—
125C3×C22⋊C8—91no—
126C3×C22⋊C8—91no—
127C3×C22⋊C8C3×D311yescharacteristic
128#830—31no—
129#830—91no—
130(C2×D3)⋊C8—31no—
131(C2×D3)⋊C8—31no—
132(C2×D3)⋊C8—91no—
133(C2×D3)⋊C8—91no—
134C23×C32—21no—
135C23×C32C4×D311yescharacteristic
136C23×C32C2×C3×C411yescharacteristic
137C22×C3⋊D3C2×C3×C411yescharacteristic
138C22×C3⋊D3C2×C3×C411yescharacteristic
139C22×C3×D3—31no—
140C22×C3×D3—31no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—41no—
148C2×C32×C4—41no—
149C2×C32×C4C4×D311yescharacteristic
150C2×C32×C4C2×C3×C411yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C23×C311yescharacteristic
153#638—22no—
154#638—22no—
155C2×C3⋊Dic3C2×C3×C411yescharacteristic
156C2×C3⋊Dic3C2×C3×C411yescharacteristic
157C4⋊(C3⋊D3)—22no—
158C4×C3⋊D3—21no—
159C4×C3⋊D3—21no—
160C4×C3⋊D3C3×D412yes—
161C4×C3⋊D3C3×D412yes—
162#634—22no—
163C3×#67—62no—
164C3×#67—62no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—31no—
167C3×D12—62no—
168C3×C4×D3—32no—
169C3×C4×D3—32no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172C3×Dic6—62no—
173C32×C8—21no—
174C32×C8—21no—
175C32×C8—21no—
176C32×C8—21no—
177C32×C8—41no—
178C32×C8—41no—
179C3⋊(C3⋊C8)—21no—
180C3⋊(C3⋊C8)—21no—
181C3×C3⋊C8—61no—
182C3×C3⋊C8—61no—
183#408—91noSylow
184C2×H(3,𝔽3)—41no—
185C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
186C2×H(3,𝔽3)#13111yescharacteristic
187C2×H(3,𝔽3)#13111yescharacteristic
188(C3⋊D3)⋊C3—22no—
189(C3⋊D3)⋊C3—41no—
190C3×#41—182no—
191C3×#41—182no—
192C2×C3×Q8—91no—
193C2×C3×D4—91no—
194C2×C3×D4—91no—
195C2×C3×D4—91no—
196C22×C3×C4—31no—
197C22×C3×C4—31no—
198C22×C3×C4—61no—
199C22×C3×C4—91no—
200C22×C3×C4—91no—
201C22×C3×C4C2×C3×D311yescharacteristic
202C2×#67—31no—
203C2×#67—31no—
204C2×#67—91no—
205C2×#67—91no—
206#235—62no—
207#235—62no—
208#235—182no—
209#235—182no—
210C2×D12—31no—
211C2×D12—91no—
212C2×C4×D3—31no—
213C2×C4×D3—31no—
214C2×C4×D3—91no—
215C2×C4×D3—91no—
216C2×Dic6—31no—
217C2×Dic6—91no—
218C3×M16—182no—
219C2×C3×C8—31no—
220C2×C3×C8—31no—
221C2×C3×C8—31no—
222C2×C3×C8—31no—
223C2×C3×C8—61no—
224C2×C3×C8—61no—
225C2×C3×C8—91no—
226C2×C3×C8—91no—
227C2×C3×C8—182no—
228C2×C3×C8C2×C3×D311yescharacteristic
229C2×C3×C8C2×C3×D311yescharacteristic
230C2×C3⋊C8—31no—
231C2×C3⋊C8—31no—
232C2×C3⋊C8—91no—
233C2×C3⋊C8—91no—
234ΓL(1,𝔽25)—62no—
235ΓL(1,𝔽25)—182no—
236D3×C8—62no—
237D3×C8—182no—
238C22×C32—22no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32C2×C4×D311yescharacteristic
249C22×C32C22×C3×C411yescharacteristic
250C2×C3⋊D3—21no—
251C2×C3⋊D3—21no—
252C2×C3⋊D3—21no—
253C2×C3⋊D3—21no—
254C2×C3⋊D3—21no—
255C2×C3⋊D3C3×C22⋊C412yes—
256C2×C3×D3—32no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—41no—
267C32×C4D3×D411yescharacteristic
268C32×C4D3×D411yescharacteristic
269C32×C4C2×C3×D411yescharacteristic
270C32×C4C2×C3×D411yescharacteristic
271C3⋊Dic3—21no—
272C3⋊Dic3C3×C22⋊C412yes—
273C3×Dic3—32no—
274C3×Dic3—61no—
275C2×#41—91no—
276C2×M16—91no—
277C22×C8—91no—
278C22⋊C8—91no—
279C22⋊C8—91no—
280C22⋊C8—91no—
281C22⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
282H(3,𝔽3)#40811yescharacteristic, complemented, Sylow
283C23×C3—31no—
284C23×C3—31no—
285C23×C3—61no—
286C23×C3—91no—
287C23×C3—91no—
288C23×C3C3×C4×D311yescharacteristic
289C22×D3—31no—
290C22×D3—31no—
291C22×D3—91no—
292C22×D3—91no—
293C3×Q8—182no—
294C3×D4—182no—
295C3×D4—182no—
296C3×D4—182no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—92no—
310C2×C3×C4—92no—
311C2×C3×C4—91no—
312C2×C3×C4—91no—
313C2×C3×C4—121no—
314C2×C3×C4—121no—
315C2×C3×C4—181no—
316C2×C3×C4—181no—
317C2×C3×C4C3×C4×D311yescharacteristic
318C2×C3×C4C22×C3×D311yescharacteristic
319#67—62no—
320#67—62no—
321#67—182no—
322#67—182no—
323C2×Dic3—31no—
324C2×Dic3—31no—
325C2×Dic3—91no—
326C2×Dic3—91no—
327D12—62no—
328D12—182no—
329C4×D3—32no—
330C4×D3—32no—
331C4×D3—61no—
332C4×D3—61no—
333C4×D3—92no—
334C4×D3—92no—
335C4×D3—181no—
336C4×D3—181no—
337Dic6—62no—
338Dic6—182no—
339C3×C8—21no—
340C3×C8—21no—
341C3×C8—61no—
342C3×C8—61no—
343C3×C8—61no—
344C3×C8—61no—
345C3×C8—121no—
346C3×C8—121no—
347C3×C8—181no—
348C3×C8—181no—
349C3⋊C8—61no—
350C3⋊C8—61no—
351C3⋊C8—181no—
352C3⋊C8—181no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—42no—
357C2×C32—41no—
358C2×C32—41no—
359C2×C32D3×C22⋊C411yescharacteristic
360C2×C32#90111yescharacteristic
361C2×C32#90711yescharacteristic
362C2×C32C2×C3×C22⋊C411yescharacteristic
363C2×C32C3×#13111yescharacteristic
364C2×C32C3×#13111yescharacteristic
365C3⋊D3—22no—
366C3⋊D3—41no—
367C3×D3—62no—
368C3×D3—121no—
369#41—182no—
370#41—182no—
371C2×Q8—91no—
372C2×D4—91no—
373C2×D4—91no—
374C2×D4—91no—
375C22×C4—91no—
376C22×C4—91no—
377C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
378M16—182no—
379C2×C8—91no—
380C2×C8—91no—
381C2×C8—182no—
382C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
383C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—61no—
390C22×C3—61no—
391C22×C3—61no—
392C22×C3—61no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—62no—
396C22×C3—61no—
397C22×C3—92no—
398C22×C3—121no—
399C22×C3—121no—
400C22×C3—181no—
401C22×C3—181no—
402C22×C3—181no—
403C22×C3—181no—
404C22×C3—181no—
405C22×C3C2×C3×C4×D311yescharacteristic
406C2×D3—32no—
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C2×D3—61no—
411C2×D3—61no—
412C2×D3—92no—
413C2×D3—181no—
414C2×D3—181no—
415C2×D3—181no—
416C2×D3—181no—
417C2×D3—181no—
418C3×C4—21no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—31no—
422C3×C4—31no—
423C3×C4—61no—
424C3×C4—61no—
425C3×C4—61no—
426C3×C4—61no—
427C3×C4—92no—
428C3×C4—121no—
429C3×C4—181no—
430C3×C4C3×D3×D411yescharacteristic
431C3×C4C3×D3×D411yescharacteristic
432Dic3—32no—
433Dic3—61no—
434Dic3—92no—
435Dic3—181no—
436C32—21no—
437C32#472711yescharacteristic, complemented
438C32C3×#40811yescharacteristic, complemented
439C23—91no—
440C23—91no—
441C23C4×(C3⋊D3)⋊C311yescharacteristic
442Q8—182no—
443D4—182no—
444D4—182no—
445D4—182no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—91no—
449C2×C4—92no—
450C2×C4—92no—
451C2×C4—91no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
455C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
456C8—21no—
457C8—21no—
458C8—181no—
459C8—181no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—31no—
465C2×C3—31no—
466C2×C3—41no—
467C2×C3—61no—
468C2×C3—61no—
469C2×C3—61no—
470C2×C3—121no—
471C2×C3—121no—
472C2×C3—122no—
473C2×C3—182no—
474C2×C3—361no—
475C2×C3C3×D3×C22⋊C411yescharacteristic
476C2×C3C3×#90111yescharacteristic
477C2×C3C3×#90711yescharacteristic
478D3—62no—
479D3—121no—
480D3—182no—
481D3—361no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—92no—
486C22—181no—
487C22—181no—
488C22—181no—
489C22—181no—
490C22—181no—
491C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
492C4—21no—
493C4—92no—
494C4—181no—
495C4D4×(C3⋊D3)⋊C311yescharacteristic
496C4D4×(C3⋊D3)⋊C311yescharacteristic
497C3—31no—
498C3—31no—
499C3—61no—
500C3C3×#472711yescharacteristic
501C2—41no—
502C2—182no—
503C2—361no—
504C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
505C2#64110⋊C311yescharacteristic
506C2#64116⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C3×C4×D3
prime radicalC2×#3828⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenterC22⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#408
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#3743⋊C3C2semidirect
3. C2×C8×(C3⋊D3)⋊C3C2semidirect
4. H(3,𝔽3)×C22⋊C8C2semidirect
5. #63657⋊C3C2semidirect
6. #63647⋊C3C2semidirect
7. #63647⋊C3C2semidirect
8. #10592426C3semidirect
27. C2×#3743C2×C3semidirect
28. C2×C8×C3⋊D3C2×C3semidirect
32. C32×C22⋊C8D3semidirect
33. C32×C22⋊C8C2×C3semidirect
34. #63657C2×C3semidirect
35. #63647C2×C3semidirect
36. #63647C2×C3semidirect
281. C22⋊C8(C3⋊D3)⋊C3semidirect
282. H(3,𝔽3)#408semidirect
437. C32#4727semidirect
438. C32C3×#408semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C2×#3743⋊C32
C2×C8×(C3⋊D3)⋊C32
H(3,𝔽3)×C22⋊C82
#63657⋊C32
#63647⋊C32
#105924263
C3×#47273