Finite Group Databasebeta

← #10592710⋊C3 #13142739

Subgroups — #10592710⋊C3 #13142739

Order 1728.

528 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#3748⋊C3C211yesmaximal, characteristic, complemented
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#63657⋊C3C211yesmaximal, characteristic, complemented
5#63657⋊C3C211yesmaximal, characteristic, complemented
6C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
7#63630⋊C3C211yesmaximal, characteristic, complemented
8#10592710C311yesmaximal, characteristic, complemented
9C3×#5011—31nomaximal
10C2×D4×H(3,𝔽3)C411yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C411yescharacteristic
14C22⋊C4×H(3,𝔽3)C411yescharacteristic
15C22⋊C4×H(3,𝔽3)C411yescharacteristic
16C42×H(3,𝔽3)C2211yescharacteristic
17#3748⋊C3—22no—
18C2×(C3⋊(C3⋊C8))⋊C3—22no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C32×C4×D4—21no—
24C32×C4×D4D311yescharacteristic
25C32×C4×D4C2×C311yescharacteristic
26C2×#3748C2×C311yescharacteristic, complemented
27C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
28C2×C3×#208—31no—
29C22×C3×C3⋊C8—31no—
30#63657C2×C311yescharacteristic, complemented
31#63657C2×C311yescharacteristic, complemented
32C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
33#63630C2×C311yescharacteristic, complemented
34C3×#830—31no—
35C3×#830—31no—
36C3×C4⋊(C3⋊C8)—31no—
37C3×#803—31no—
38C23×H(3,𝔽3)C2×C411yescharacteristic
39C23×H(3,𝔽3)C2×C411yescharacteristic
40D4×H(3,𝔽3)—22no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2×C411yescharacteristic
46C2×C4×H(3,𝔽3)C2×C411yescharacteristic
47C2×C4×H(3,𝔽3)C2×C411yescharacteristic
48C2×C4×H(3,𝔽3)C2×C411yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50(C3⋊(C3⋊C8))⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54C3×#435—91no—
55#5011—31no—
56#5011—91no—
57C2×C32×D4—21no—
58C2×C32×D4Dic311yescharacteristic
59C2×C32×D4C3×C411yescharacteristic
60C22×C32×C4—21no—
61C22×C32×C4—21no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C22×C311yescharacteristic
65C22×C32×C4C22×C311yescharacteristic
66C32×C4⋊C4—21no—
67C32×C4⋊C4Dic311yescharacteristic
68C32×C4⋊C4C3×C411yescharacteristic
69C32×C22⋊C4—21no—
70C32×C22⋊C4—21no—
71C32×C22⋊C4Dic311yescharacteristic
72C32×C22⋊C4Dic311yescharacteristic
73C32×C22⋊C4C3×C411yescharacteristic
74C32×C22⋊C4C3×C411yescharacteristic
75C32×C42—21no—
76C32×C42C2×D311yescharacteristic
77C32×C42C22×C311yescharacteristic
78#3748—22no—
79C2×C3⋊(C3⋊C8)—22no—
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
84C3×#208—62no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—62no—
90C22×H(3,𝔽3)—21no—
91C22×H(3,𝔽3)—21no—
92C22×H(3,𝔽3)—21no—
93C22×H(3,𝔽3)—21no—
94C22×H(3,𝔽3)—21no—
95C22×H(3,𝔽3)M1612yescomplemented
96C22×H(3,𝔽3)C22×C411yescharacteristic
97C4×H(3,𝔽3)—21no—
98C4×H(3,𝔽3)—21no—
99C4×H(3,𝔽3)—21no—
100C4×H(3,𝔽3)—21no—
101C4×H(3,𝔽3)C2×D411yescharacteristic
102C4×H(3,𝔽3)#4111yescharacteristic
103C2×C3×M16—91no—
104C22×C3×C8—91no—
105C3×C4×D4—31no—
106C3×C4×D4—31no—
107C3×C4×D4—61no—
108C3×C4×D4C3×D311yescharacteristic
109C2×#208—31no—
110C2×#208—91no—
111C22×C3⋊C8—31no—
112C22×C3⋊C8—91no—
113C3×C4⋊C8—91no—
114C3×C22⋊C8—91no—
115C3×C22⋊C8—91no—
116C3×#97—91no—
117#830—31no—
118#830—31no—
119#830—91no—
120#830—91no—
121C4⋊(C3⋊C8)—31no—
122C4⋊(C3⋊C8)—91no—
123#803—31no—
124#803—91no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32C2×Dic311yescharacteristic
128C23×C32C2×Dic311yescharacteristic
129C23×C32C2×C3×C411yescharacteristic
130C23×C32C2×C3×C411yescharacteristic
131C32×D4—24no—
132C32×D4—22no—
133C32×D4—22no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
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×C4—41no—
150C2×C32×C4—41no—
151C2×C32×C4C2×Dic311yescharacteristic
152C2×C32×C4C2×Dic311yescharacteristic
153C2×C32×C4C2×Dic311yescharacteristic
154C2×C32×C4C2×Dic311yescharacteristic
155C2×C32×C4C2×C3×C411yescharacteristic
156C2×C32×C4C2×C3×C411yescharacteristic
157C2×C32×C4C2×C3×C411yescharacteristic
158C2×C32×C4C2×C3×C411yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C23×C311yescharacteristic
161C3⋊(C3⋊C8)—21no—
162C3⋊(C3⋊C8)—21no—
163C3⋊(C3⋊C8)—21no—
164C3⋊(C3⋊C8)C3×D412yescomplemented
165C3×C3⋊C8—32no—
166C3×C3⋊C8—61no—
167C3×C3⋊C8—61no—
168C3×C3⋊C8—61no—
169#435—91noSylow
170C2×H(3,𝔽3)—22no—
171C2×H(3,𝔽3)—41no—
172C2×H(3,𝔽3)C4×D411yescharacteristic
173C2×H(3,𝔽3)C2×M1611yescharacteristic
174C2×H(3,𝔽3)#13111yescharacteristic
175C2×C3×D4—31no—
176C2×C3×D4—31no—
177C2×C3×D4—61no—
178C2×C3×D4C3×Dic311yescharacteristic
179C22×C3×C4—31no—
180C22×C3×C4—31no—
181C22×C3×C4—31no—
182C22×C3×C4—31no—
183C22×C3×C4—61no—
184C22×C3×C4—61no—
185C22×C3×C4C2×C3×D311yescharacteristic
186C22×C3×C4C2×C3×D311yescharacteristic
187C3×M16—182no—
188C2×C3×C8—91no—
189C2×C3×C8—91no—
190C2×C3×C8—91no—
191C2×C3×C8—91no—
192C2×C3×C8—182no—
193C3×C4⋊C4—31no—
194C3×C4⋊C4—31no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4C3×Dic311yescharacteristic
197C3×C22⋊C4—31no—
198C3×C22⋊C4—31no—
199C3×C22⋊C4—31no—
200C3×C22⋊C4—31no—
201C3×C22⋊C4—61no—
202C3×C22⋊C4—61no—
203C3×C22⋊C4C3×Dic311yescharacteristic
204C3×C22⋊C4C3×Dic311yescharacteristic
205C3×C42—31no—
206C3×C42—31no—
207C3×C42—61no—
208C3×C42C2×C3×D311yescharacteristic
209#208—62no—
210#208—182no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—31no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—62no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—91no—
218C2×C3⋊C8—91no—
219C2×C3⋊C8—91no—
220C2×C3⋊C8—182no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—22no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—22no—
233C22×C32—21no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32#20812yescomplemented
239C22×C32C3×M1612yescomplemented
240C22×C32C22×Dic311yescharacteristic
241C22×C32C22×C3×C411yescharacteristic
242C32×C4—22no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—41no—
254C32×C4—41no—
255C32×C4—41no—
256C32×C4D3×D411yescharacteristic
257C32×C4#23711yescharacteristic
258C32×C4C2×C3×D411yescharacteristic
259C32×C4C3×#4111yescharacteristic
260C2×M16—91no—
261C22×C8—91no—
262C4×D4(C3⋊D3)⋊C311yescharacteristic
263C4⋊C8—91no—
264C22⋊C8—91no—
265C22⋊C8—91no—
266#97—91no—
267H(3,𝔽3)#43511yescharacteristic, complemented, Sylow
268C23×C3—31no—
269C23×C3—31no—
270C23×C3—31no—
271C23×C3—31no—
272C23×C3—61no—
273C23×C3—61no—
274C23×C3C2×C3×Dic311yescharacteristic
275C23×C3C2×C3×Dic311yescharacteristic
276C3×D4—22no—
277C3×D4—62no—
278C3×D4—62no—
279C3×D4—64no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—121no—
308C2×C3×C4—121no—
309C2×C3×C4—121no—
310C2×C3×C4—121no—
311C2×C3×C4C2×C3×Dic311yescharacteristic
312C2×C3×C4C2×C3×Dic311yescharacteristic
313C2×C3×C4C2×C3×Dic311yescharacteristic
314C2×C3×C4C2×C3×Dic311yescharacteristic
315C2×C3×C4C22×C3×D311yescharacteristic
316C3×C8—92no—
317C3×C8—181no—
318C3×C8—181no—
319C3×C8—181no—
320C3⋊C8—32no—
321C3⋊C8—61no—
322C3⋊C8—61no—
323C3⋊C8—61no—
324C3⋊C8—92no—
325C3⋊C8—181no—
326C3⋊C8—181no—
327C3⋊C8—181no—
328C2×C32—22no—
329C2×C32—22no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—42no—
334C2×C32—42no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32C2×#20811yescharacteristic
338C2×C32D4×Dic311yescharacteristic
339C2×C32#94811yescharacteristic
340C2×C32C3×C4×D411yescharacteristic
341C2×C32C2×C3×M1611yescharacteristic
342C2×C32C3×#13111yescharacteristic
343C2×D4(C3⋊Dic3)⋊C311yescharacteristic
344C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
345C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
346M16—182no—
347C2×C8—91no—
348C2×C8—91no—
349C2×C8—91no—
350C2×C8—91no—
351C2×C8—182no—
352C4⋊C4(C3⋊Dic3)⋊C311yescharacteristic
353C22⋊C4(C3⋊Dic3)⋊C311yescharacteristic
354C22⋊C4(C3⋊Dic3)⋊C311yescharacteristic
355C42C2×(C3⋊D3)⋊C311yescharacteristic
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—31no—
362C22×C3—31no—
363C22×C3—32no—
364C22×C3—32no—
365C22×C3—61no—
366C22×C3—61no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3—61no—
373C22×C3—61no—
374C22×C3—61no—
375C22×C3—62no—
376C22×C3—61no—
377C22×C3—62no—
378C22×C3—121no—
379C22×C3—121no—
380C22×C3—121no—
381C22×C3—121no—
382C22×C3C3×#20812yes—
383C22×C3C22×C3×Dic311yescharacteristic
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—31no—
389C3×C4—31no—
390C3×C4—31no—
391C3×C4—31no—
392C3×C4—61no—
393C3×C4—61no—
394C3×C4—61no—
395C3×C4—61no—
396C3×C4—61no—
397C3×C4—61no—
398C3×C4—61no—
399C3×C4—61no—
400C3×C4—62no—
401C3×C4—61no—
402C3×C4—61no—
403C3×C4—121no—
404C3×C4—121no—
405C3×C4—121no—
406C3×C4C3×D3×D411yescharacteristic
407C3×C4C3×#23711yescharacteristic
408C32—21no—
409C32#501111yescharacteristic, complemented
410C32C3×#43511yescharacteristic, complemented
411C23C2×(C3⋊Dic3)⋊C311yescharacteristic
412C23C2×(C3⋊Dic3)⋊C311yescharacteristic
413D4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4C2×(C3⋊Dic3)⋊C311yescharacteristic
419C2×C4C2×(C3⋊Dic3)⋊C311yescharacteristic
420C2×C4C2×(C3⋊Dic3)⋊C311yescharacteristic
421C2×C4C2×(C3⋊Dic3)⋊C311yescharacteristic
422C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
423C8—92no—
424C8—181no—
425C8—181no—
426C8—181no—
427C2×C3—22no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—41no—
435C2×C3—62no—
436C2×C3—62no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—61no—
440C2×C3—122no—
441C2×C3—121no—
442C2×C3—121no—
443C2×C3—122no—
444C2×C3C2×C3×#20811yescharacteristic
445C2×C3C3×D4×Dic311yescharacteristic
446C2×C3C3×#94811yescharacteristic
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22#3748⋊C312yescomplemented
453C22C22×(C3⋊Dic3)⋊C311yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4D4×(C3⋊D3)⋊C311yescharacteristic
459C4#3830⋊C311yescharacteristic
460C3—31no—
461C3—31no—
462C3—61no—
463C3C3×#501111yescharacteristic
464C2—22no—
465C2—41no—
466C2C2×#3748⋊C311yescharacteristic
467C2D4×(C3⋊Dic3)⋊C311yescharacteristic
468C2#1172168911yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC4×D4×H(3,𝔽3)C2
socleC22×C3C22×C3×Dic3
prime radicalC2×D4×H(3,𝔽3)C4
cosocleC2×C32×C4C23×C3
hypercenterC4×D4(C3⋊D3)⋊C3
hypocenterC32C3×#435
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#3748⋊C3C2semidirect
3. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. #63657⋊C3C2semidirect
5. #63657⋊C3C2semidirect
6. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
7. #63630⋊C3C2semidirect
8. #10592710C3semidirect
26. C2×#3748C2×C3semidirect
27. C22×C3⋊(C3⋊C8)C2×C3semidirect
30. #63657C2×C3semidirect
31. #63657C2×C3semidirect
32. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
33. #63630C2×C3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
95. C22×H(3,𝔽3)M16semidirect
164. C3⋊(C3⋊C8)C3×D4semidirect
238. C22×C32#208semidirect
239. C22×C32C3×M16semidirect
267. H(3,𝔽3)#435semidirect
409. C32#5011semidirect
410. C32C3×#435semidirect
452. C22#3748⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×D4×H(3,𝔽3)2
C2×#3748⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
#63657⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#63630⋊C32
#105927103
C3×#50113