Finite Group Databasebeta

← #10592469⋊C3 #13141787

Subgroups — #10592469⋊C3 #13141787

Order 1728.

715 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#3743⋊C3C211yesmaximal, characteristic, complemented
3#64101⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic, complemented
8#10592469C311yesmaximal, characteristic, complemented
9C3×#4770—31nomaximal
10C23×(C3⋊D3)⋊C3—21no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13D4×(C3⋊D3)⋊C3—22no—
14D4×(C3⋊D3)⋊C3C414yescomplemented
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
19#3752⋊C3—21no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
21C4×(C3⋊Dic3)⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
23#3743⋊C3—22no—
24C2×D4×C3⋊D3C2×C311yescharacteristic
25C2×C3×D3×D4—31no—
26C2×#3743C2×C311yescharacteristic, complemented
27#64101C2×C311yescharacteristic, complemented
28C2×C3×ΓL(1,𝔽25)—31no—
29C3×#892—31no—
30C32×#102—21no—
31C32×#102D311yescharacteristic, complemented
32C32×#102C2×C311yescharacteristic, complemented
33#63659C2×C311yescharacteristic, complemented
34#63648C2×C311yescharacteristic, complemented
35#63636C2×C311yescharacteristic, complemented
36C3×#832—31no—
37C3×#821—31no—
38C3×#809—31no—
39C23×H(3,𝔽3)—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—42no—
43C22×(C3⋊D3)⋊C3—42no—
44C22×(C3⋊D3)⋊C3D411yescharacteristic
45D4×H(3,𝔽3)—21no—
46D4×H(3,𝔽3)C2×C412yes—
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49#638⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3D411yescharacteristic
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)C2×C412yes—
54C4×(C3⋊D3)⋊C3C2×C412yes—
55C4×(C3⋊D3)⋊C3D412yes—
56C8×H(3,𝔽3)—21no—
57(C3⋊(C3⋊C8))⋊C3—21no—
58C3×#418—91no—
59#4770—31no—
60#4770—91no—
61C23×C3⋊D3—21no—
62C23×C3×D3—61no—
63C2×C32×D4—21no—
64C2×C32×D4C2×D311yescharacteristic
65C2×C32×D4C22×C311yescharacteristic
66C2×#638—21no—
67D4×C3⋊D3—22no—
68D4×C3⋊D3C3×C414yescomplemented
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×C4×C3⋊D3C22×C311yescharacteristic
71C2×C3×#67—61no—
72C3×D3×D4—34no—
73C3×D3×D4—62no—
74C2×C3×D12—31no—
75C2×C3×C4×D3—31no—
76C2×C32×C8—21no—
77C2×C32×C8C2×D311yescharacteristic, complemented
78C2×C32×C8C22×C311yescharacteristic, complemented
79C32×C4⋊C4—21no—
80C32×C4⋊C4C2×D311yescharacteristic, complemented
81C32×C4⋊C4C22×C311yescharacteristic, complemented
82#3752—21no—
83C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
84C4×C3⋊Dic3—21no—
85C2×C3⋊(C3⋊C8)C22×C311yescharacteristic, complemented
86#3743—22no—
87C3×(C2×D3)⋊C4—61no—
88C3×C4⋊Dic3—31no—
89C3×C4×Dic3—61no—
90C2×C3×C3⋊C8—31no—
91C3×ΓL(1,𝔽25)—62no—
92C22×H(3,𝔽3)—22no—
93C22×H(3,𝔽3)—41no—
94C22×H(3,𝔽3)—41no—
95C22×H(3,𝔽3)C2×D411yescharacteristic
96C2×(C3⋊D3)⋊C3—21no—
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—48no—
100C2×(C3⋊D3)⋊C3—41no—
101C2×(C3⋊D3)⋊C3—41no—
102C2×(C3⋊D3)⋊C3C22⋊C412yes—
103C4×H(3,𝔽3)—41no—
104C4×H(3,𝔽3)C22×C411yescharacteristic
105C4×H(3,𝔽3)C2×D411yescharacteristic
106(C3⋊Dic3)⋊C3—41no—
107(C3⋊Dic3)⋊C3C22⋊C412yes—
108C22×C3×D4—91no—
109C2×D3×D4—31no—
110C2×D3×D4—91no—
111C2×C3×M16—91no—
112C3×#117—91no—
113C2×ΓL(1,𝔽25)—31no—
114C2×ΓL(1,𝔽25)—91no—
115#892—31no—
116#892—91no—
117C3×#102—31no—
118C3×#102—31no—
119C3×#102—61no—
120C3×#102—91no—
121C3×#102—91no—
122C3×#102—91no—
123C3×#102C3×D311yescharacteristic
124#832—31no—
125#832—91no—
126#821—31no—
127#821—91no—
128#809—31no—
129#809—91no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—41no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3—24no—
135C22×C3⋊D3—42no—
136C22×C3⋊D3—42no—
137C22×C3⋊D3C3×D411yescharacteristic
138C22×C3×D3—31no—
139C22×C3×D3—64no—
140C22×C3×D3—61no—
141C22×C3×D3—122no—
142C22×C3×D3—122no—
143C32×D4—21no—
144C32×D4—21no—
145C32×D4—22no—
146C32×D4—41no—
147C32×D4C4×D312yes—
148C32×D4C2×C3×C412yes—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—41no—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C23×C311yescharacteristic
155#638—24no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3C3×D411yescharacteristic
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)C2×C3×C412yes—
160C4×C3⋊D3C2×C3×C412yes—
161C4×C3⋊D3C3×D412yes—
162C3×#67—64no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—61no—
165C3×D12—32no—
166C3×D12—61no—
167C3×C4×D3—32no—
168C3×C4×D3—32no—
169C32×C8—21no—
170C32×C8—21no—
171C32×C8—41no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—61no—
174#418—91noSylow
175C2×H(3,𝔽3)—42no—
176C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
177C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
178C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
179(C3⋊D3)⋊C3—22no—
180(C3⋊D3)⋊C3—42no—
181C24×C3—181no—
182C23×D3—61no—
183C23×D3—181no—
184C2×C3×D4—31no—
185C2×C3×D4—31no—
186C2×C3×D4—61no—
187C2×C3×D4—91no—
188C2×C3×D4—94no—
189C2×C3×D4—182no—
190C2×C3×D4—181no—
191C2×C3×D4C2×C3×D311yescharacteristic
192C22×C3×C4—91no—
193C2×#67—61no—
194C2×#67—181no—
195D3×D4—34no—
196D3×D4—62no—
197D3×D4—94no—
198D3×D4—182no—
199C2×D12—31no—
200C2×D12—91no—
201C2×C4×D3—31no—
202C2×C4×D3—91no—
203C3×M16—182no—
204C2×C3×C8—31no—
205C2×C3×C8—31no—
206C2×C3×C8—61no—
207C2×C3×C8—91no—
208C2×C3×C8C2×C3×D311yescharacteristic
209C3×C4⋊C4—31no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—91no—
213C3×C4⋊C4C2×C3×D311yescharacteristic
214C3×C22⋊C4—181no—
215C3×C42—181no—
216(C2×D3)⋊C4—61no—
217(C2×D3)⋊C4—181no—
218C4⋊Dic3—31no—
219C4⋊Dic3—91no—
220C4×Dic3—61no—
221C4×Dic3—181no—
222C2×C3⋊C8—31no—
223C2×C3⋊C8—91no—
224ΓL(1,𝔽25)—62no—
225ΓL(1,𝔽25)—182no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—21no—
229C22×C32—42no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—81no—
235C22×C32—81no—
236C22×C32D3×D411yescharacteristic
237C22×C32C2×C3×D411yescharacteristic
238C2×C3⋊D3—21no—
239C2×C3⋊D3—21no—
240C2×C3⋊D3—22no—
241C2×C3⋊D3—48no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3—41no—
244C2×C3⋊D3C3×C22⋊C412yes—
245C2×C3×D3—32no—
246C2×C3×D3—61no—
247C2×C3×D3—61no—
248C2×C3×D3—62no—
249C2×C3×D3—128no—
250C2×C3×D3—121no—
251C2×C3×D3—121no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—41no—
255C32×C4—41no—
256C32×C4—81no—
257C32×C4C2×C4×D311yescharacteristic
258C32×C4D3×D411yescharacteristic
259C32×C4C22×C3×C411yescharacteristic
260C32×C4C2×C3×D411yescharacteristic
261C3⋊Dic3—41no—
262C3⋊Dic3C3×C22⋊C412yes—
263C3×Dic3—32no—
264C3×Dic3—121no—
265C22×D4—91no—
266C2×M16—91no—
267#117—91no—
268#102—91no—
269#102—91no—
270#102—91no—
271#102(C3⋊D3)⋊C311yescharacteristic, complemented
272H(3,𝔽3)#41811yescharacteristic, complemented, Sylow
273C23×C3—21no—
274C23×C3—61no—
275C23×C3—61no—
276C23×C3—91no—
277C23×C3—121no—
278C23×C3—181no—
279C23×C3—184no—
280C23×C3—362no—
281C23×C3—362no—
282C22×D3—31no—
283C22×D3—64no—
284C22×D3—61no—
285C22×D3—91no—
286C22×D3—122no—
287C22×D3—122no—
288C22×D3—184no—
289C22×D3—181no—
290C22×D3—362no—
291C22×D3—362no—
292C3×D4—21no—
293C3×D4—32no—
294C3×D4—32no—
295C3×D4—61no—
296C3×D4—61no—
297C3×D4—62no—
298C3×D4—92no—
299C3×D4—121no—
300C3×D4—181no—
301C3×D4—184no—
302C3×D4C3×C4×D312yes—
303C2×C3×C4—21no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
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—121no—
313C2×C3×C4—181no—
314C2×C3×C4C22×C3×D311yescharacteristic
315#67—64no—
316#67—184no—
317C2×Dic3—31no—
318C2×Dic3—61no—
319C2×Dic3—91no—
320C2×Dic3—181no—
321D12—32no—
322D12—61no—
323D12—92no—
324D12—181no—
325C4×D3—32no—
326C4×D3—32no—
327C4×D3—92no—
328C4×D3—92no—
329C3×C8—21no—
330C3×C8—61no—
331C3×C8—61no—
332C3×C8—121no—
333C3×C8—181no—
334C3⋊C8—61no—
335C3⋊C8—181no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—42no—
340C2×C32—42no—
341C2×C32—82no—
342C2×C32D3×C22⋊C411yescharacteristic
343C2×C32#91111yescharacteristic
344C2×C32#91411yescharacteristic
345C2×C32C2×C3×C22⋊C411yescharacteristic
346C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
347C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
348C3⋊D3—22no—
349C3⋊D3—42no—
350C3×D3—62no—
351C3×D3—122no—
352C24—181no—
353C2×D4—94no—
354C2×D4—91no—
355C2×D4—182no—
356C2×D4—181no—
357C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
358C22×C4—91no—
359M16—182no—
360C2×C8—91no—
361C2×C8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
362C4⋊C4—91no—
363C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
364C22⋊C4—181no—
365C42—181no—
366C22×C3—22no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—62no—
372C22×C3—62no—
373C22×C3—61no—
374C22×C3—92no—
375C22×C3—122no—
376C22×C3—121no—
377C22×C3—121no—
378C22×C3—121no—
379C22×C3—121no—
380C22×C3—181no—
381C22×C3—181no—
382C22×C3—182no—
383C22×C3—241no—
384C22×C3—241no—
385C22×C3—361no—
386C22×C3—361no—
387C22×C3—368no—
388C22×C3C3×D3×D411yescharacteristic
389C2×D3—32no—
390C2×D3—61no—
391C2×D3—61no—
392C2×D3—62no—
393C2×D3—92no—
394C2×D3—128no—
395C2×D3—121no—
396C2×D3—121no—
397C2×D3—181no—
398C2×D3—181no—
399C2×D3—182no—
400C2×D3—368no—
401C2×D3—361no—
402C2×D3—361no—
403C3×C4—31no—
404C3×C4—31no—
405C3×C4—31no—
406C3×C4—31no—
407C3×C4—41no—
408C3×C4—61no—
409C3×C4—61no—
410C3×C4—92no—
411C3×C4—121no—
412C3×C4—121no—
413C3×C4—241no—
414C3×C4—361no—
415C3×C4C2×C3×C4×D311yescharacteristic
416C3×C4C3×D3×D411yescharacteristic
417Dic3—32no—
418Dic3—92no—
419Dic3—121no—
420Dic3—361no—
421C32—21no—
422C32#477011yescharacteristic, complemented
423C32C3×#41811yescharacteristic, complemented
424C23—21no—
425C23—91no—
426C23—184no—
427C23—181no—
428C23—362no—
429C23—362no—
430D4—21no—
431D4—92no—
432D4—184no—
433D4—181no—
434D4C4×(C3⋊D3)⋊C312yes—
435C2×C4—21no—
436C2×C4—91no—
437C2×C4—92no—
438C2×C4—92no—
439C2×C4—181no—
440C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
441C8—21no—
442C8—181no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—42no—
450C2×C3—61no—
451C2×C3—61no—
452C2×C3—61no—
453C2×C3—122no—
454C2×C3—122no—
455C2×C3—182no—
456C2×C3—242no—
457C2×C3—362no—
458C2×C3C3×D3×C22⋊C411yescharacteristic
459C2×C3C3×#91111yescharacteristic
460C2×C3C3×#91411yescharacteristic
461D3—62no—
462D3—122no—
463D3—182no—
464D3—362no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—92no—
469C22—181no—
470C22—181no—
471C22—182no—
472C22—368no—
473C22—361no—
474C22—361no—
475C22D4×(C3⋊D3)⋊C311yescharacteristic
476C4—41no—
477C4—92no—
478C4—361no—
479C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
480C4D4×(C3⋊D3)⋊C311yescharacteristic
481C3—31no—
482C3—31no—
483C3—61no—
484C3C3×#477011yescharacteristic
485C2—42no—
486C2—182no—
487C2—362no—
488C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
489C2#64120⋊C311yescharacteristic
490C2#64123⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radicalC2×D4×(C3⋊D3)⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#418
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#3743⋊C3C2semidirect
3. #64101⋊C3C2semidirect
4. H(3,𝔽3)×#102C2semidirect
5. #63659⋊C3C2semidirect
6. #63648⋊C3C2semidirect
7. #63636⋊C3C2semidirect
8. #10592469C3semidirect
14. D4×(C3⋊D3)⋊C3C4semidirect
17. C2×C8×H(3,𝔽3)C22semidirect
18. C4⋊C4×H(3,𝔽3)C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
26. C2×#3743C2×C3semidirect
27. #64101C2×C3semidirect
31. C32×#102D3semidirect
32. C32×#102C2×C3semidirect
33. #63659C2×C3semidirect
34. #63648C2×C3semidirect
35. #63636C2×C3semidirect
68. D4×C3⋊D3C3×C4semidirect
77. C2×C32×C8C2×D3semidirect
78. C2×C32×C8C22×C3semidirect
80. C32×C4⋊C4C2×D3semidirect
81. C32×C4⋊C4C22×C3semidirect
83. C4⋊(C3⋊Dic3)C22×C3semidirect
85. C2×C3⋊(C3⋊C8)C22×C3semidirect
271. #102(C3⋊D3)⋊C3semidirect
272. H(3,𝔽3)#418semidirect
361. C2×C8C2×(C3⋊D3)⋊C3semidirect
363. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
422. C32#4770semidirect
423. C32C3×#418semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
C2×#3743⋊C32
#64101⋊C32
H(3,𝔽3)×#1022
#63659⋊C32
#63648⋊C32
#63636⋊C32
#105924693
C3×#47703