Finite Group Databasebeta

← #10592869⋊C3 #13143373

Subgroups — #10592869⋊C3 #13143373

Order 1728.

621 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#64147⋊C3C211yesmaximal, characteristic
4C2×#3754⋊C3C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#63643⋊C3C211yesmaximal, characteristic, complemented
8#10592869C311yesmaximal, characteristic, complemented
9C3×#5170—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13Q8×(C3⋊D3)⋊C3—22no—
14Q8×(C3⋊D3)⋊C3—22no—
15C2×C4×(C3⋊D3)⋊C3—21no—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3—21no—
18C2×#634⋊C3C2211yescharacteristic
19SD16×H(3,𝔽3)—22no—
20C2×C8×H(3,𝔽3)C2211yescharacteristic
21#3757⋊C3—21no—
22#3754⋊C3—22no—
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
25C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
26C2×C3×D3×Q8—31no—
27C2×C32×SD16—21no—
28C2×C32×SD16D311yescharacteristic, complemented
29C2×C32×SD16C2×C311yescharacteristic, complemented
30#64147C2×C311yescharacteristic
31C2×#3754C2×C311yescharacteristic, complemented
32C3×#938—31no—
33C2×C3×#214—31no—
34#63662C2×C311yescharacteristic, complemented
35#63647C2×C311yescharacteristic, complemented
36#63643C2×C311yescharacteristic, complemented
37C3×#835—31no—
38C3×(C2×D3)⋊C8—31no—
39C3×#816—31no—
40C23×H(3,𝔽3)—21no—
41C22×(C3⋊D3)⋊C3D411yescharacteristic
42Q8×H(3,𝔽3)—21no—
43Q8×H(3,𝔽3)D412yescomplemented
44D4×H(3,𝔽3)—21no—
45D4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48#638⋊C3—42no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3D411yescharacteristic
52C4×(C3⋊D3)⋊C3—24no—
53C4×(C3⋊D3)⋊C3—21no—
54C4×(C3⋊D3)⋊C3—21no—
55#634⋊C3—24no—
56#634⋊C3—21no—
57#634⋊C3D412yescomplemented
58C8×H(3,𝔽3)—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60C3×#448—91no—
61#5170—31no—
62#5170—91no—
63C2×C32×Q8—21no—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×Q8C22×C311yescharacteristic
66C2×C32×D4—21no—
67C2×C32×D4C2×D311yescharacteristic
68C2×C32×D4C22×C311yescharacteristic
69C2×#638—21no—
70Q8×C3⋊D3—22no—
71Q8×C3⋊D3—22no—
72C2×C4×C3⋊D3—21no—
73C2×C4×C3⋊D3C22×C311yescharacteristic
74C2×#634—21no—
75C2×#634C22×C311yescharacteristic
76C2×C3×#67—61no—
77C3×D3×Q8—62no—
78C3×D3×Q8—62no—
79C2×C3×C4×D3—31no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—61no—
83C32×SD16—24no—
84C32×SD16—22no—
85C32×SD16—22no—
86C2×C32×C8—21no—
87C2×C32×C8C2×D311yescharacteristic
88C2×C32×C8C22×C311yescharacteristic
89#3757—21no—
90#3754—22no—
91C4⋊(C3⋊Dic3)C22×C311yescharacteristic
92C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
93C3×#217—61no—
94C3×#214—62no—
95C3×C4⋊Dic3—31no—
96C2×C3×C3⋊C8—31no—
97C22×H(3,𝔽3)—41no—
98C22×H(3,𝔽3)—41no—
99C22×H(3,𝔽3)—41no—
100C22×H(3,𝔽3)C2×D411yescharacteristic
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3SD1612yescomplemented
104C4×H(3,𝔽3)—22no—
105C4×H(3,𝔽3)C2×D411yescharacteristic
106C4×H(3,𝔽3)C2×D411yescharacteristic
107(C3⋊Dic3)⋊C3—21no—
108(C3⋊Dic3)⋊C3—22no—
109(C3⋊Dic3)⋊C3—41no—
110C22×C3×Q8—91no—
111C2×D3×Q8—31no—
112C2×D3×Q8—91no—
113C2×C3×SD16—31no—
114C2×C3×SD16—31no—
115C2×C3×SD16—61no—
116C2×C3×SD16—91no—
117C2×C3×SD16C3×D311yescharacteristic
118C3×#121—91no—
119#938—31no—
120#938—91no—
121C2×#214—31no—
122C2×#214—91no—
123C3×#103—91no—
124C3×#103—91no—
125C3×C22⋊C8—91no—
126#835—31no—
127#835—91no—
128(C2×D3)⋊C8—31no—
129(C2×D3)⋊C8—91no—
130#816—31no—
131#816—91no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—41no—
135C22×C3⋊D3C3×D411yescharacteristic
136C22×C3×D3—31no—
137C32×Q8—21no—
138C32×Q8—21no—
139C32×Q8—22no—
140C32×Q8—41no—
141C32×Q8#6712yescomplemented
142C32×Q8C3×D412yescomplemented
143C32×D4—22no—
144C32×D4—21no—
145C32×D4—21no—
146C32×D4—21no—
147C32×D4—21no—
148C32×D4—41no—
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—42no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3C3×D411yescharacteristic
159C4×C3⋊D3—24no—
160C4×C3⋊D3—21no—
161C4×C3⋊D3—21no—
162#634—24no—
163#634—21no—
164#634C3×D412yescomplemented
165C3×#67—122no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C3×C4×D3—64no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172C3×Dic6—32no—
173C3×Dic6—64no—
174C3×Dic6—61no—
175C32×C8—22no—
176C32×C8—21no—
177C32×C8—21no—
178C3⋊(C3⋊C8)—21no—
179C3×C3⋊C8—61no—
180#448—91noSylow
181C2×H(3,𝔽3)—81no—
182C2×H(3,𝔽3)#12011yescharacteristic
183C2×H(3,𝔽3)C2×SD1611yescharacteristic
184C2×H(3,𝔽3)#13711yescharacteristic
185(C3⋊D3)⋊C3—22no—
186C2×C3×Q8—31no—
187C2×C3×Q8—31no—
188C2×C3×Q8—61no—
189C2×C3×Q8—91no—
190C2×C3×Q8—181no—
191C2×C3×Q8—182no—
192C2×C3×Q8—182no—
193C2×C3×Q8C2×C3×D311yescharacteristic
194C2×C3×D4—31no—
195C2×C3×D4—31no—
196C2×C3×D4—61no—
197C2×C3×D4—181no—
198C2×C3×D4C2×C3×D311yescharacteristic
199C22×C3×C4—91no—
200C22×C3×C4—181no—
201C2×#67—61no—
202C2×#67—181no—
203D3×Q8—62no—
204D3×Q8—62no—
205D3×Q8—182no—
206D3×Q8—182no—
207C2×C4×D3—31no—
208C2×C4×D3—61no—
209C2×C4×D3—91no—
210C2×C4×D3—181no—
211C2×Dic6—31no—
212C2×Dic6—61no—
213C2×Dic6—91no—
214C2×Dic6—181no—
215C3×SD16—22no—
216C3×SD16—62no—
217C3×SD16—62no—
218C3×SD16—64no—
219C3×SD16—182no—
220C2×C3×C8—31no—
221C2×C3×C8—31no—
222C2×C3×C8—61no—
223C2×C3×C8—91no—
224C2×C3×C8C2×C3×D311yescharacteristic
225C3×C4⋊C4—91no—
226C3×C22⋊C4—181no—
227#217—61no—
228#217—181no—
229#214—62no—
230#214—182no—
231C4⋊Dic3—31no—
232C4⋊Dic3—91no—
233C2×C3⋊C8—31no—
234C2×C3⋊C8—91no—
235C22×C32—21no—
236C22×C32—42no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32—41no—
240C22×C32—41no—
241C22×C32—41no—
242C22×C32—41no—
243C22×C32—81no—
244C22×C32—81no—
245C22×C32D3×D411yescharacteristic
246C22×C32C2×C3×D411yescharacteristic
247C2×C3⋊D3—21no—
248C2×C3⋊D3—21no—
249C2×C3⋊D3C3×SD1612yescomplemented
250C2×C3×D3—32no—
251C2×C3×D3—61no—
252C2×C3×D3—61no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—42no—
258C32×C4D3×D411yescharacteristic
259C32×C4C2×#6711yescharacteristic
260C32×C4C2×C3×D411yescharacteristic
261C32×C4C2×C3×D411yescharacteristic
262C3⋊Dic3—21no—
263C3⋊Dic3—22no—
264C3⋊Dic3—41no—
265C3×Dic3—62no—
266C3×Dic3—61no—
267C3×Dic3—121no—
268C22×Q8—91no—
269C2×SD16—91no—
270C2×SD16(C3⋊D3)⋊C311yescharacteristic, complemented
271#121—91no—
272#103—91no—
273#103—91no—
274C22⋊C8—91no—
275H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
276C23×C3—21no—
277C23×C3—61no—
278C23×C3—61no—
279C23×C3—91no—
280C23×C3—121no—
281C22×D3—31no—
282C22×D3—91no—
283C3×Q8—21no—
284C3×Q8—32no—
285C3×Q8—32no—
286C3×Q8—61no—
287C3×Q8—61no—
288C3×Q8—62no—
289C3×Q8—92no—
290C3×Q8—121no—
291C3×Q8—181no—
292C3×Q8—184no—
293C3×Q8C3×#6712yes—
294C3×D4—21no—
295C3×D4—21no—
296C3×D4—61no—
297C3×D4—61no—
298C3×D4—61no—
299C3×D4—61no—
300C3×D4—62no—
301C3×D4—121no—
302C3×D4—362no—
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—91no—
310C2×C3×C4—121no—
311C2×C3×C4—181no—
312C2×C3×C4—181no—
313C2×C3×C4—184no—
314C2×C3×C4—181no—
315C2×C3×C4—181no—
316C2×C3×C4C22×C3×D311yescharacteristic
317#67—122no—
318#67—362no—
319C2×Dic3—31no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—91no—
323C2×Dic3—181no—
324C2×Dic3—181no—
325C4×D3—64no—
326C4×D3—61no—
327C4×D3—61no—
328C4×D3—181no—
329C4×D3—181no—
330C4×D3—184no—
331Dic6—32no—
332Dic6—64no—
333Dic6—61no—
334Dic6—92no—
335Dic6—181no—
336Dic6—184no—
337C3×C8—21no—
338C3×C8—61no—
339C3×C8—61no—
340C3×C8—62no—
341C3×C8—181no—
342C3⋊C8—61no—
343C3⋊C8—181no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—82no—
348C2×C32—81no—
349C2×C32—81no—
350C2×C32D3×SD1611yescharacteristic
351C2×C32#91511yescharacteristic
352C2×C32#93711yescharacteristic
353C2×C32C3×#12011yescharacteristic
354C2×C32C2×C3×SD1611yescharacteristic
355C2×C32C3×#13711yescharacteristic
356C3⋊D3—22no—
357C3×D3—62no—
358C2×Q8—91no—
359C2×Q8—181no—
360C2×Q8—182no—
361C2×Q8—182no—
362C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
363C2×D4—181no—
364C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
365C22×C4—91no—
366C22×C4—181no—
367SD16—22no—
368SD16—182no—
369C2×C8—91no—
370C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
371C4⋊C4—91no—
372C22⋊C4—181no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—61no—
379C22×C3—92no—
380C22×C3—121no—
381C22×C3—121no—
382C22×C3—121no—
383C22×C3—121no—
384C22×C3—121no—
385C22×C3—121no—
386C22×C3—122no—
387C22×C3—181no—
388C22×C3—181no—
389C22×C3—241no—
390C22×C3—241no—
391C22×C3C3×D3×D411yescharacteristic
392C2×D3—32no—
393C2×D3—61no—
394C2×D3—61no—
395C2×D3—92no—
396C2×D3—181no—
397C2×D3—181no—
398C3×C4—22no—
399C3×C4—31no—
400C3×C4—31no—
401C3×C4—31no—
402C3×C4—31no—
403C3×C4—62no—
404C3×C4—62no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—122no—
408C3×C4—181no—
409C3×C4—182no—
410C3×C4—361no—
411C3×C4C3×D3×D411yescharacteristic
412C3×C4C2×C3×#6711yescharacteristic
413Dic3—62no—
414Dic3—61no—
415Dic3—121no—
416Dic3—182no—
417Dic3—181no—
418Dic3—361no—
419C32—21no—
420C32#517011yescharacteristic, complemented
421C32C3×#44811yescharacteristic, complemented
422C23—21no—
423C23—91no—
424Q8—21no—
425Q8—92no—
426Q8—184no—
427Q8—181no—
428Q8#638⋊C312yescomplemented
429D4—21no—
430D4—21no—
431D4—362no—
432C2×C4—21no—
433C2×C4—91no—
434C2×C4—184no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
440C8—21no—
441C8—181no—
442C2×C3—31no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—61no—
449C2×C3—61no—
450C2×C3—61no—
451C2×C3—81no—
452C2×C3—182no—
453C2×C3—241no—
454C2×C3—241no—
455C2×C3—242no—
456C2×C3C3×D3×SD1611yescharacteristic
457C2×C3C3×#91511yescharacteristic
458C2×C3C3×#93711yescharacteristic
459D3—62no—
460D3—182no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—92no—
465C22—181no—
466C22—181no—
467C22D4×(C3⋊D3)⋊C311yescharacteristic
468C4—22no—
469C4—181no—
470C4—182no—
471C4—361no—
472C4D4×(C3⋊D3)⋊C311yescharacteristic
473C4C2×#638⋊C311yescharacteristic
474C3—31no—
475C3—31no—
476C3—61no—
477C3C3×#517011yescharacteristic
478C2—81no—
479C2—182no—
480C2SD16×(C3⋊D3)⋊C311yescharacteristic
481C2#64124⋊C311yescharacteristic
482C2#64146⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×SD16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
prime radical#64147⋊C3C2
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#448
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×Q8×(C3⋊D3)⋊C3C2semidirect
2. C2×SD16×H(3,𝔽3)C2semidirect
4. C2×#3754⋊C3C2semidirect
5. #63662⋊C3C2semidirect
6. #63647⋊C3C2semidirect
7. #63643⋊C3C2semidirect
8. #10592869C3semidirect
25. C2×Q8×C3⋊D3C2×C3semidirect
28. C2×C32×SD16D3semidirect
29. C2×C32×SD16C2×C3semidirect
31. C2×#3754C2×C3semidirect
34. #63662C2×C3semidirect
35. #63647C2×C3semidirect
36. #63643C2×C3semidirect
43. Q8×H(3,𝔽3)D4semidirect
57. #634⋊C3D4semidirect
103. C2×(C3⋊D3)⋊C3SD16semidirect
141. C32×Q8#67semidirect
142. C32×Q8C3×D4semidirect
164. #634C3×D4semidirect
249. C2×C3⋊D3C3×SD16semidirect
270. C2×SD16(C3⋊D3)⋊C3semidirect
275. H(3,𝔽3)#448semidirect
420. C32#5170semidirect
421. C32C3×#448semidirect
428. Q8#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
C2×SD16×H(3,𝔽3)2
#64147⋊C32
C2×#3754⋊C32
#63662⋊C32
#63647⋊C32
#63643⋊C32
#105928693
C3×#51703