Finite Group Databasebeta

← #10592872⋊C3 #13143375

Subgroups — #10592872⋊C3 #13143375

Order 1728.

600 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 100 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×#3830⋊C3C211yesmaximal, characteristic
2C2×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#64155⋊C3C211yesmaximal, characteristic, complemented
4C2×#3756⋊C3C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#63643⋊C3C211yesmaximal, characteristic, complemented
8#10592872C311yesmaximal, characteristic, complemented
9C3×#5173—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13C22×(C3⋊Dic3)⋊C3—21no—
14#3830⋊C3—22no—
15#3830⋊C3—22no—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18SD16×H(3,𝔽3)—22no—
19C2×C8×H(3,𝔽3)C2211yescharacteristic
20#3756⋊C3—22no—
21#3752⋊C3—21no—
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23#3750⋊C3—21no—
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
25C2×#3830C2×C311yescharacteristic
26C2×C3×#237—31no—
27C2×C32×SD16—21no—
28C2×C32×SD16D311yescharacteristic, complemented
29C2×C32×SD16C2×C311yescharacteristic, complemented
30#64155C2×C311yescharacteristic, complemented
31C2×#3756C2×C311yescharacteristic, complemented
32C3×#946—31no—
33C2×C3×#216—31no—
34#63659C2×C311yescharacteristic, complemented
35#63647C2×C311yescharacteristic, complemented
36#63643C2×C311yescharacteristic, complemented
37C3×#832—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)—21no—
44D4×H(3,𝔽3)—21no—
45D4×H(3,𝔽3)D412yes—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48#638⋊C3—24no—
49C2×(C3⋊Dic3)⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C2×(C3⋊Dic3)⋊C3D411yescharacteristic
53C4×(C3⋊D3)⋊C3—21no—
54C4×(C3⋊D3)⋊C3—21no—
55#634⋊C3—21no—
56#634⋊C3D412yes—
57C8×H(3,𝔽3)—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59C3×#452—91no—
60#5173—31no—
61#5173—91no—
62C2×C32×Q8—21no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C22×C311yescharacteristic
65C2×C32×D4—21no—
66C2×C32×D4C2×D311yescharacteristic
67C2×C32×D4C22×C311yescharacteristic
68C2×#638—21no—
69C22×C3⋊Dic3—21no—
70#3830—22no—
71#3830—22no—
72C2×C4×C3⋊D3C22×C311yescharacteristic
73C2×#634C22×C311yescharacteristic
74C2×C3×#67—61no—
75C22×C3×Dic3—61no—
76C3×#237—62no—
77C3×#237—62no—
78C2×C3×C4×D3—31no—
79C2×C3×Dic6—31no—
80C32×SD16—24no—
81C32×SD16—22no—
82C32×SD16—22no—
83C2×C32×C8—21no—
84C2×C32×C8C2×D311yescharacteristic
85C2×C32×C8C22×C311yescharacteristic
86#3756—22no—
87#3752—21no—
88C4⋊(C3⋊Dic3)C22×C311yescharacteristic
89#3750—21no—
90C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
91C3×#216—62no—
92C3×(C2×D3)⋊C4—61no—
93C3×C4⋊Dic3—31no—
94C3×Dic3⋊C4—61no—
95C2×C3×C3⋊C8—31no—
96C22×H(3,𝔽3)—22no—
97C22×H(3,𝔽3)—41no—
98C22×H(3,𝔽3)—41no—
99C22×H(3,𝔽3)C2×D411yescharacteristic
100C2×(C3⋊D3)⋊C3—21no—
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—21no—
103C4×H(3,𝔽3)—41no—
104C4×H(3,𝔽3)C2×D411yescharacteristic
105C4×H(3,𝔽3)C2×D411yescharacteristic
106(C3⋊Dic3)⋊C3—22no—
107(C3⋊Dic3)⋊C3—21no—
108(C3⋊Dic3)⋊C3—41no—
109C2×C3×#41—91no—
110C2×#237—31no—
111C2×#237—91no—
112C2×C3×Q16—91no—
113C2×C3×SD16—31no—
114C2×C3×SD16—31no—
115C2×C3×SD16—61no—
116C2×C3×SD16C3×D311yescharacteristic
117C3×C22⋊Q8—91no—
118#946—31no—
119#946—91no—
120C2×#216—31no—
121C2×#216—91no—
122C3×#103—91no—
123C3×#102—91no—
124C3×C22⋊C8—91no—
125#832—31no—
126#832—91no—
127(C2×D3)⋊C8—31no—
128(C2×D3)⋊C8—91no—
129#816—31no—
130#816—91no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—41no—
134C22×C3⋊D3C3×D411yescharacteristic
135C22×C3×D3—31no—
136C32×Q8—22no—
137C32×Q8—21no—
138C32×Q8—21no—
139C32×Q8—21no—
140C32×Q8—21no—
141C32×Q8—41no—
142C32×D4—21no—
143C32×D4—21no—
144C32×D4—22no—
145C32×D4—41no—
146C32×D4#6712yes—
147C32×D4C3×D412yes—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—41no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C23×C311yescharacteristic
154#638—24no—
155C2×C3⋊Dic3—24no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3C3×D411yescharacteristic
159C4×C3⋊D3—21no—
160C4×C3⋊D3—21no—
161#634—21no—
162#634C3×D412yes—
163C3×#67—64no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—64no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C3×C4×D3—61no—
169C3×C4×D3—61no—
170C3×Dic6—32no—
171C3×Dic6—61no—
172C32×C8—22no—
173C32×C8—21no—
174C32×C8—21no—
175C3⋊(C3⋊C8)—21no—
176C3×C3⋊C8—61no—
177#452—91noSylow
178C2×H(3,𝔽3)—42no—
179C2×H(3,𝔽3)#12011yescharacteristic
180C2×H(3,𝔽3)#13511yescharacteristic
181C2×H(3,𝔽3)#13711yescharacteristic
182(C3⋊D3)⋊C3—41no—
183C3×#41—182no—
184C3×#41—182no—
185C2×C3×Q8—31no—
186C2×C3×Q8—31no—
187C2×C3×Q8—61no—
188C2×C3×Q8—91no—
189C2×C3×Q8C2×C3×D311yescharacteristic
190C2×C3×D4—31no—
191C2×C3×D4—31no—
192C2×C3×D4—61no—
193C2×C3×D4—181no—
194C2×C3×D4C2×C3×D311yescharacteristic
195C22×C3×C4—91no—
196C22×C3×C4—181no—
197C2×#67—61no—
198C2×#67—181no—
199C22×Dic3—61no—
200C22×Dic3—181no—
201#237—62no—
202#237—62no—
203#237—182no—
204#237—182no—
205C2×C4×D3—31no—
206C2×C4×D3—91no—
207C2×Dic6—31no—
208C2×Dic6—91no—
209C3×Q16—182no—
210C3×SD16—22no—
211C3×SD16—62no—
212C3×SD16—62no—
213C3×SD16—64no—
214C2×C3×C8—31no—
215C2×C3×C8—31no—
216C2×C3×C8—61no—
217C2×C3×C8—91no—
218C2×C3×C8C2×C3×D311yescharacteristic
219C3×C4⋊C4—91no—
220C3×C4⋊C4—181no—
221C3×C22⋊C4—181no—
222#216—62no—
223#216—182no—
224(C2×D3)⋊C4—61no—
225(C2×D3)⋊C4—181no—
226C4⋊Dic3—31no—
227C4⋊Dic3—91no—
228Dic3⋊C4—61no—
229Dic3⋊C4—181no—
230C2×C3⋊C8—31no—
231C2×C3⋊C8—91no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—21no—
235C22×C32—42no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32—41no—
240C22×C32—81no—
241C22×C32—81no—
242C22×C32D3×D411yescharacteristic
243C22×C32C2×C3×D411yescharacteristic
244C2×C3⋊D3—21no—
245C2×C3⋊D3—21no—
246C2×C3⋊D3—21no—
247C2×C3×D3—61no—
248C2×C3×D3—61no—
249C2×C3×D3—61no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—42no—
253C32×C4—41no—
254C32×C4—41no—
255C32×C4D3×D411yescharacteristic
256C32×C4C2×#6711yescharacteristic
257C32×C4C2×C3×D411yescharacteristic
258C32×C4C2×C3×D411yescharacteristic
259C3⋊Dic3—22no—
260C3⋊Dic3—21no—
261C3⋊Dic3—41no—
262C3×Dic3—62no—
263C3×Dic3—61no—
264C3×Dic3—121no—
265C2×#41—91no—
266C2×Q16—91no—
267C2×SD16(C3⋊D3)⋊C311yescharacteristic, complemented
268C22⋊Q8—91no—
269#103—91no—
270#102—91no—
271C22⋊C8—91no—
272H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
273C23×C3—21no—
274C23×C3—61no—
275C23×C3—61no—
276C23×C3—91no—
277C23×C3—121no—
278C22×D3—31no—
279C22×D3—91no—
280C3×Q8—21no—
281C3×Q8—21no—
282C3×Q8—61no—
283C3×Q8—61no—
284C3×Q8—61no—
285C3×Q8—61no—
286C3×Q8—62no—
287C3×Q8—92no—
288C3×Q8—121no—
289C3×Q8—181no—
290C3×D4—21no—
291C3×D4—32no—
292C3×D4—32no—
293C3×D4—61no—
294C3×D4—61no—
295C3×D4—62no—
296C3×D4—121no—
297C3×D4—184no—
298C3×D4C3×#6712yes—
299C2×C3×C4—21no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—91no—
306C2×C3×C4—121no—
307C2×C3×C4—181no—
308C2×C3×C4—181no—
309C2×C3×C4—181no—
310C2×C3×C4—184no—
311C2×C3×C4—181no—
312C2×C3×C4C22×C3×D311yescharacteristic
313#67—64no—
314#67—184no—
315C2×Dic3—31no—
316C2×Dic3—64no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—91no—
320C2×Dic3—181no—
321C2×Dic3—184no—
322C2×Dic3—181no—
323C4×D3—61no—
324C4×D3—61no—
325C4×D3—181no—
326C4×D3—181no—
327Dic6—32no—
328Dic6—61no—
329Dic6—92no—
330Dic6—181no—
331C3×C8—21no—
332C3×C8—61no—
333C3×C8—61no—
334C3×C8—62no—
335C3×C8—181no—
336C3⋊C8—61no—
337C3⋊C8—181no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—42no—
342C2×C32—42no—
343C2×C32—82no—
344C2×C32#91511yescharacteristic
345C2×C32#91611yescharacteristic
346C2×C32#93711yescharacteristic
347C2×C32C3×#12011yescharacteristic
348C2×C32C3×#13511yescharacteristic
349C2×C32C3×#13711yescharacteristic
350C3⋊D3—41no—
351C3×D3—121no—
352#41—182no—
353#41—182no—
354C2×Q8—91no—
355C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
356C2×D4—181no—
357C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
358C22×C4—91no—
359C22×C4—181no—
360Q16—182no—
361SD16—22no—
362C2×C8—91no—
363C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
364C4⋊C4—91no—
365C4⋊C4—181no—
366C22⋊C4—181no—
367C22×C3—22no—
368C22×C3—31no—
369C22×C3—31no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—62no—
373C22×C3—62no—
374C22×C3—61no—
375C22×C3—122no—
376C22×C3—121no—
377C22×C3—121no—
378C22×C3—121no—
379C22×C3—121no—
380C22×C3—181no—
381C22×C3—181no—
382C22×C3—181no—
383C22×C3—241no—
384C22×C3—241no—
385C22×C3C3×D3×D411yescharacteristic
386C2×D3—61no—
387C2×D3—61no—
388C2×D3—61no—
389C2×D3—181no—
390C2×D3—181no—
391C2×D3—181no—
392C3×C4—31no—
393C3×C4—31no—
394C3×C4—31no—
395C3×C4—31no—
396C3×C4—41no—
397C3×C4—61no—
398C3×C4—61no—
399C3×C4—121no—
400C3×C4—121no—
401C3×C4—122no—
402C3×C4—181no—
403C3×C4—182no—
404C3×C4—361no—
405C3×C4C3×D3×D411yescharacteristic
406C3×C4C2×C3×#6711yescharacteristic
407Dic3—62no—
408Dic3—61no—
409Dic3—121no—
410Dic3—182no—
411Dic3—181no—
412Dic3—361no—
413C32—21no—
414C32#517311yescharacteristic, complemented
415C32C3×#45211yescharacteristic, complemented
416C23—21no—
417C23—91no—
418Q8—21no—
419Q8—21no—
420Q8—92no—
421Q8—181no—
422D4—21no—
423D4—184no—
424D4#638⋊C312yes—
425C2×C4—21no—
426C2×C4—91no—
427C2×C4—184no—
428C2×C4—181no—
429C2×C4—181no—
430C2×C4—181no—
431C2×C4—181no—
432C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
433C8—21no—
434C8—181no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—42no—
442C2×C3—61no—
443C2×C3—61no—
444C2×C3—61no—
445C2×C3—122no—
446C2×C3—122no—
447C2×C3—242no—
448C2×C3—361no—
449C2×C3C3×#91511yescharacteristic
450C2×C3C3×#91611yescharacteristic
451C2×C3C3×#93711yescharacteristic
452D3—121no—
453D3—361no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—181no—
458C22—181no—
459C22—181no—
460C22D4×(C3⋊D3)⋊C311yescharacteristic
461C4—41no—
462C4—181no—
463C4—182no—
464C4—361no—
465C4D4×(C3⋊D3)⋊C311yescharacteristic
466C4C2×#638⋊C311yescharacteristic
467C3—31no—
468C3—31no—
469C3—61no—
470C3C3×#517311yescharacteristic
471C2—42no—
472C2—361no—
473C2#64124⋊C311yescharacteristic
474C2#64125⋊C311yescharacteristic
475C2#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 radicalC2×#3830⋊C3C2
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#452
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
2. C2×SD16×H(3,𝔽3)C2semidirect
3. #64155⋊C3C2semidirect
4. C2×#3756⋊C3C2semidirect
5. #63659⋊C3C2semidirect
6. #63647⋊C3C2semidirect
7. #63643⋊C3C2semidirect
8. #10592872C3semidirect
28. C2×C32×SD16D3semidirect
29. C2×C32×SD16C2×C3semidirect
30. #64155C2×C3semidirect
31. C2×#3756C2×C3semidirect
34. #63659C2×C3semidirect
35. #63647C2×C3semidirect
36. #63643C2×C3semidirect
267. C2×SD16(C3⋊D3)⋊C3semidirect
272. H(3,𝔽3)#452semidirect
414. C32#5173semidirect
415. C32C3×#452semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
C2×SD16×H(3,𝔽3)2
#64155⋊C32
C2×#3756⋊C32
#63659⋊C32
#63647⋊C32
#63643⋊C32
#105928723
C3×#51733