Finite Group Databasebeta

← #10592427⋊C3 #13141613

Subgroups — #10592427⋊C3 #13141613

Order 1728.

484 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#638⋊C3C211yesmaximal, characteristic
2C2×#3748⋊C3C211yesmaximal, characteristic, complemented
3C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic, complemented
6#63642⋊C3C211yesmaximal, characteristic, complemented
7#63641⋊C3C211yesmaximal, characteristic, complemented
8#10592427C311yesmaximal, characteristic, complemented
9C3×#4728—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3C411yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#3757⋊C3C411yescharacteristic
16#3752⋊C3C411yescharacteristic
17#3750⋊C3C411yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19#3748⋊C3—22no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C8×(C3⋊D3)⋊C3—22no—
23C4×#638C2×C311yescharacteristic
24C2×#3748C2×C311yescharacteristic, complemented
25C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
26C3×C4×#67—31no—
27C2×C3×#208—31no—
28C2×C3×D3×C8—31no—
29C32×C22⋊C8—21no—
30C32×C22⋊C8D311yescharacteristic, complemented
31C32×C22⋊C8C2×C311yescharacteristic, complemented
32#63647C2×C311yescharacteristic, complemented
33#63642C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35C3×(C2×D3)⋊C8—31no—
36C3×#815—31no—
37C3×Dic3⋊C8—31no—
38C23×H(3,𝔽3)C2×C411yescharacteristic
39C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#638⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
46C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
47C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
48C4×(C3⋊D3)⋊C3—21no—
49C4×(C3⋊D3)⋊C3—21no—
50C8×H(3,𝔽3)—21no—
51C8×H(3,𝔽3)—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54C3×#435—91no—
55#4728—31no—
56#4728—91no—
57C22×C32×C4—21no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C22×C311yescharacteristic
60C2×#638C3×C411yescharacteristic
61C2×C4×C3⋊D3C22×C311yescharacteristic
62C2×C3×#67—31no—
63C2×C3×C4×D3—31no—
64C2×C32×C8—21no—
65C2×C32×C8—21no—
66C2×C32×C8C2×D311yescharacteristic
67C2×C32×C8C2×D311yescharacteristic
68C2×C32×C8C22×C311yescharacteristic
69C2×C32×C8C22×C311yescharacteristic
70#3757C3×C411yescharacteristic
71#3752C3×C411yescharacteristic
72#3750C3×C411yescharacteristic
73C4×C3⋊Dic3C22×C311yescharacteristic
74#3748—22no—
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
77C8×C3⋊D3—22no—
78C3×#217—31no—
79C3×(C2×D3)⋊C4—31no—
80C3×Dic3⋊C4—31no—
81C3×C4×Dic3—31no—
82C3×#208—62no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—31no—
85C3×D3×C8—62no—
86C22×H(3,𝔽3)—21no—
87C22×H(3,𝔽3)—21no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)C22×C411yescharacteristic
90C2×(C3⋊D3)⋊C3—21no—
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3M1612yescomplemented
93C4×H(3,𝔽3)—21no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96(C3⋊Dic3)⋊C3—21no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—21no—
99C2×C3×M16—91no—
100C22×C3×C8—91no—
101C3×C4×D4—91no—
102C4×#67—31no—
103C4×#67—91no—
104C2×#208—31no—
105C2×#208—91no—
106C2×D3×C8—31no—
107C2×D3×C8—91no—
108C3×C4⋊C8—91no—
109C3×C22⋊C8—31no—
110C3×C22⋊C8—31no—
111C3×C22⋊C8—61no—
112C3×C22⋊C8—91no—
113C3×C22⋊C8C3×D311yescharacteristic
114C3×#97—91no—
115(C2×D3)⋊C8—31no—
116(C2×D3)⋊C8—91no—
117#815—31no—
118#815—91no—
119Dic3⋊C8—31no—
120Dic3⋊C8—91no—
121C23×C32—21no—
122C23×C32C4×D311yescharacteristic
123C23×C32C2×C3×C411yescharacteristic
124C22×C3⋊D3C2×C3×C411yescharacteristic
125C22×C3×D3—31no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
134C2×C32×C4C4×D311yescharacteristic
135C2×C32×C4C2×C3×C411yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C23×C311yescharacteristic
138#638—22no—
139C2×C3⋊Dic3C2×C3×C411yescharacteristic
140C2×C3⋊Dic3C2×C3×C411yescharacteristic
141C2×C3⋊Dic3C2×C3×C411yescharacteristic
142C4×C3⋊D3—21no—
143C4×C3⋊D3—21no—
144C3×#67—62no—
145C2×C3×Dic3—31no—
146C2×C3×Dic3—31no—
147C2×C3×Dic3—31no—
148C3×C4×D3—61no—
149C3×C4×D3—61no—
150C32×C8—21no—
151C32×C8—21no—
152C32×C8—21no—
153C32×C8—21no—
154C32×C8—41no—
155C32×C8—41no—
156C3⋊(C3⋊C8)—21no—
157C3⋊(C3⋊C8)C3×D412yescomplemented
158C3×C3⋊C8—32no—
159C3×C3⋊C8—61no—
160#435—91noSylow
161C2×H(3,𝔽3)—41no—
162C2×H(3,𝔽3)C4×D411yescharacteristic
163C2×H(3,𝔽3)C2×M1611yescharacteristic
164C2×H(3,𝔽3)#13111yescharacteristic
165(C3⋊D3)⋊C3—22no—
166C2×C3×D4—91no—
167C22×C3×C4—31no—
168C22×C3×C4—31no—
169C22×C3×C4—61no—
170C22×C3×C4—91no—
171C22×C3×C4C2×C3×D311yescharacteristic
172C2×#67—31no—
173C2×#67—91no—
174C2×C4×D3—31no—
175C2×C4×D3—91no—
176C3×M16—182no—
177C2×C3×C8—31no—
178C2×C3×C8—31no—
179C2×C3×C8—31no—
180C2×C3×C8—31no—
181C2×C3×C8—61no—
182C2×C3×C8—61no—
183C2×C3×C8—91no—
184C2×C3×C8—91no—
185C2×C3×C8—182no—
186C2×C3×C8C2×C3×D311yescharacteristic
187C2×C3×C8C2×C3×D311yescharacteristic
188C3×C4⋊C4—91no—
189C3×C22⋊C4—91no—
190C3×C22⋊C4—91no—
191C3×C42—91no—
192#217—31no—
193#217—91no—
194(C2×D3)⋊C4—31no—
195(C2×D3)⋊C4—91no—
196Dic3⋊C4—31no—
197Dic3⋊C4—91no—
198C4×Dic3—31no—
199C4×Dic3—91no—
200#208—62no—
201#208—182no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—91no—
206D3×C8—62no—
207D3×C8—182no—
208C22×C32—22no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32C2×C4×D311yescharacteristic
219C22×C32C22×C3×C411yescharacteristic
220C2×C3⋊D3—21no—
221C2×C3⋊D3—21no—
222C2×C3⋊D3C3×M1612yescomplemented
223C2×C3×D3—32no—
224C2×C3×D3—61no—
225C2×C3×D3—61no—
226C32×C4—21no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—21no—
230C32×C4—41no—
231C32×C4D3×D411yescharacteristic
232C32×C4#23711yescharacteristic
233C32×C4C2×C3×D411yescharacteristic
234C32×C4C3×#4111yescharacteristic
235C3⋊Dic3—21no—
236C3⋊Dic3—21no—
237C3⋊Dic3—21no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240C3×Dic3—61no—
241C2×M16—91no—
242C22×C8—91no—
243C4×D4—91no—
244C4⋊C8—91no—
245C22⋊C8—91no—
246C22⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
247#97—91no—
248H(3,𝔽3)#43511yescharacteristic, complemented, Sylow
249C23×C3—31no—
250C23×C3—31no—
251C23×C3—61no—
252C23×C3—91no—
253C23×C3C3×C4×D311yescharacteristic
254C22×D3—31no—
255C22×D3—91no—
256C3×D4—182no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—121no—
273C2×C3×C4—121no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4C3×C4×D311yescharacteristic
277C2×C3×C4C22×C3×D311yescharacteristic
278#67—62no—
279#67—182no—
280C2×Dic3—31no—
281C2×Dic3—31no—
282C2×Dic3—31no—
283C2×Dic3—91no—
284C2×Dic3—91no—
285C2×Dic3—91no—
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—181no—
289C4×D3—181no—
290C3×C8—21no—
291C3×C8—21no—
292C3×C8—61no—
293C3×C8—61no—
294C3×C8—61no—
295C3×C8—61no—
296C3×C8—92no—
297C3×C8—121no—
298C3×C8—121no—
299C3×C8—181no—
300C3⋊C8—32no—
301C3⋊C8—61no—
302C3⋊C8—92no—
303C3⋊C8—181no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—42no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32#67⋊C411yescharacteristic
311C2×C32#90111yescharacteristic
312C2×C32D3×M1611yescharacteristic
313C2×C32C3×C4×D411yescharacteristic
314C2×C32C2×C3×M1611yescharacteristic
315C2×C32C3×#13111yescharacteristic
316C3⋊D3—22no—
317C3×D3—62no—
318C2×D4—91no—
319C22×C4—91no—
320C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
321M16—182no—
322C2×C8—91no—
323C2×C8—91no—
324C2×C8—182no—
325C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
326C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
327C4⋊C4—91no—
328C22⋊C4—91no—
329C22⋊C4—91no—
330C42—91no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—31no—
335C22×C3—31no—
336C22×C3—61no—
337C22×C3—61no—
338C22×C3—61no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3—61no—
342C22×C3—62no—
343C22×C3—61no—
344C22×C3—92no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—181no—
348C22×C3—181no—
349C22×C3C2×C3×C4×D311yescharacteristic
350C2×D3—32no—
351C2×D3—61no—
352C2×D3—61no—
353C2×D3—92no—
354C2×D3—181no—
355C2×D3—181no—
356C3×C4—21no—
357C3×C4—31no—
358C3×C4—31no—
359C3×C4—31no—
360C3×C4—31no—
361C3×C4—61no—
362C3×C4—61no—
363C3×C4—61no—
364C3×C4—61no—
365C3×C4—121no—
366C3×C4—181no—
367C3×C4—181no—
368C3×C4—181no—
369C3×C4C3×D3×D411yescharacteristic
370C3×C4C3×#23711yescharacteristic
371Dic3—61no—
372Dic3—61no—
373Dic3—61no—
374Dic3—181no—
375Dic3—181no—
376Dic3—181no—
377C32—21no—
378C32#472811yescharacteristic, complemented
379C32C3×#43511yescharacteristic, complemented
380C23—91no—
381C23C4×(C3⋊D3)⋊C311yescharacteristic
382D4—182no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—91no—
386C2×C4—91no—
387C2×C4—91no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
391C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
392C8—21no—
393C8—21no—
394C8—92no—
395C8—181no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—41no—
403C2×C3—61no—
404C2×C3—61no—
405C2×C3—61no—
406C2×C3—121no—
407C2×C3—121no—
408C2×C3—122no—
409C2×C3—182no—
410C2×C3C3×#67⋊C411yescharacteristic
411C2×C3C3×#90111yescharacteristic
412C2×C3C3×D3×M1611yescharacteristic
413D3—62no—
414D3—182no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—92no—
419C22—181no—
420C22—181no—
421C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
422C4—21no—
423C4—181no—
424C4—181no—
425C4—181no—
426C4D4×(C3⋊D3)⋊C311yescharacteristic
427C4#3830⋊C311yescharacteristic
428C3—31no—
429C3—31no—
430C3—61no—
431C3C3×#472811yescharacteristic
432C2—41no—
433C2—182no—
434C2#64090⋊C311yescharacteristic
435C2#64110⋊C311yescharacteristic
436C2M16×(C3⋊D3)⋊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×#638⋊C3C4
cosocleC2×C32×C4C23×C3
hypercenterC22⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#435
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×#3748⋊C3C2semidirect
3. C2×C8×(C3⋊D3)⋊C3C2semidirect
4. H(3,𝔽3)×C22⋊C8C2semidirect
5. #63647⋊C3C2semidirect
6. #63642⋊C3C2semidirect
7. #63641⋊C3C2semidirect
8. #10592427C3semidirect
24. C2×#3748C2×C3semidirect
25. C2×C8×C3⋊D3C2×C3semidirect
30. C32×C22⋊C8D3semidirect
31. C32×C22⋊C8C2×C3semidirect
32. #63647C2×C3semidirect
33. #63642C2×C3semidirect
34. #63641C2×C3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
92. C2×(C3⋊D3)⋊C3M16semidirect
157. C3⋊(C3⋊C8)C3×D4semidirect
222. C2×C3⋊D3C3×M16semidirect
246. C22⋊C8(C3⋊D3)⋊C3semidirect
248. H(3,𝔽3)#435semidirect
378. C32#4728semidirect
379. C32C3×#435semidirect

Maximal subgroups

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

subgroupindex
C4×#638⋊C32
C2×#3748⋊C32
C2×C8×(C3⋊D3)⋊C32
H(3,𝔽3)×C22⋊C82
#63647⋊C32
#63642⋊C32
#63641⋊C32
#105924273
C3×#47283