Finite Group Databasebeta

← C3×#10591514 #13158541

Subgroups — C3×#10591514 #13158541

Order 1728.

950 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C3×#64361C211yesmaximal, characteristic
2C32×#936C211yesmaximal, characteristic, complemented
3C3×#63833C212yesmaximal, complemented
4C32×#833C211yesmaximal, characteristic, complemented
5C3×#63570C212yesmaximal, complemented
6C3×#5061—31nomaximal
7C3×#4746—31nomaximal
8#10591514C311yesmaximal, characteristic, direct product factor
9C2×C33×D4C2211yescharacteristic
10C2×C3×#638—22no—
11C3×#3830—22no—
12C3×D4×C3⋊D3—22no—
13C3×#3828C2212yes—
14C2×C32×Dic6C2211yescharacteristic, complemented
15C32×#217—22no—
16C32×C4×Dic3C2211yescharacteristic, complemented
17C32×#208C2212yescomplemented
18C3×#3719—22no—
19C3×#3717—22no—
20#64361C2×C311yescharacteristic
21C3×#1004—31no—
22C3×#1004—31no—
23C3×#1004—61no—
24C32×#124—31no—
25C3×#942—32no—
26C3×#936—21no—
27C3×#936D311yescharacteristic, complemented
28C3×#936C2×C311yescharacteristic, complemented
29C3×#909—32no—
30#63833C2×C312yescomplemented
31C32×#100—31no—
32C3×#833—21no—
33C3×#833D311yescharacteristic, complemented
34C3×#833C2×C311yescharacteristic, complemented
35C3×#825—32no—
36C3×#805—32no—
37#63570C2×C312yescomplemented
38C23×C33D412yes—
39C22×C3×C3⋊D3—22no—
40C33×D4—22no—
41C2×C33×C4C2311yescharacteristic
42C3×#638—24no—
43C3×#638—22no—
44C2×C3×C3⋊Dic3—22no—
45C3×C4⋊(C3⋊D3)D412yes—
46C3×C4×C3⋊D3—22no—
47C3×#634D412yes—
48C2×C32×Dic3—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—22no—
51C32×C3⋊C8—22no—
52C3×#455—91no—
53#5061—31no—
54#4746—31no—
55C2×C32×Q8—31no—
56C2×C32×D4—21no—
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4—41no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C2×D311yescharacteristic
62C2×C32×D4C22×C311yescharacteristic
63C2×#638—22no—
64#3830—22no—
65D4×C3⋊D3—22no—
66#3828C22×C312yes—
67C2×C3×#67—64no—
68C2×C3×#67—62no—
69C2×C3×#67—62no—
70C3×#237—62no—
71C3×#237—62no—
72C3×#237—64no—
73C3×D3×D4—62no—
74C3×D3×D4—62no—
75C3×D3×D4—64no—
76C3×#235—32no—
77C3×#235—32no—
78C3×#235—62no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6C2×D311yescharacteristic, complemented
81C2×C3×Dic6C22×C311yescharacteristic, complemented
82C32×M16—32no—
83C32×C22⋊C4—62no—
84C32×C42—31no—
85C3×#217—24no—
86C3×#217—22no—
87C3×#217—22no—
88C3×#216—62no—
89C3×#215—62no—
90C3×C4×Dic3—21no—
91C3×C4×Dic3C2×D311yescharacteristic, complemented
92C3×C4×Dic3C22×C311yescharacteristic, complemented
93C3×#208—22no—
94C3×#208C2×D312yescomplemented
95C3×#208C22×C312yescomplemented
96C3×Dic12—62no—
97C3×#204—62no—
98#3719—22no—
99#3717—22no—
100C22×C33—22no—
101C22×C33—42no—
102C22×C33C2×D411yescharacteristic
103C2×C3×C3⋊D3—22no—
104C2×C3×C3⋊D3—82no—
105C33×C4C2×D412yes—
106C3×C3⋊Dic3—22no—
107C32×Dic3—41no—
108C32×Dic3—41no—
109C3×H(5,𝔽2)—91no—
110#1004—31no—
111#1004—31no—
112#1004—61no—
113C3×#137—92no—
114C3×#124—31no—
115C3×#124—31no—
116C3×#124—61no—
117#942—32no—
118#936C3×D311yescharacteristic, complemented
119#909—32no—
120C3×#104—92no—
121C3×#100—31no—
122C3×#100—31no—
123C3×#100—61no—
124#833C3×D311yescharacteristic, complemented
125#825—32no—
126#805—32no—
127C23×C32—22no—
128C23×C32—22no—
129C23×C32—22no—
130C23×C32—42no—
131C23×C32D1212yes—
132C23×C32#6712yes—
133C23×C32C3×D412yes—
134C22×C3⋊D3—22no—
135C22×C3×D3—64no—
136C22×C3×D3—62no—
137C22×C3×D3—62no—
138C32×Q8—62no—
139C32×D4—22no—
140C32×D4—24no—
141C32×D4—22no—
142C32×D4—22no—
143C32×D4—42no—
144C32×D4—42no—
145C32×D4—44no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—41no—
150C2×C32×C4—61no—
151C2×C32×C4—61no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C23×C311yescharacteristic
155#638—24no—
156#638—22no—
157C2×C3⋊Dic3—22no—
158C4⋊(C3⋊D3)C3×D412yes—
159C4×C3⋊D3—22no—
160#634C3×D412yes—
161C3×#67—64no—
162C3×#67—68no—
163C3×#67—64no—
164C3×#67—64no—
165C3×#67—62no—
166C3×#67—62no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—64no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C3×D12—32no—
177C3×D12—32no—
178C3×D12—62no—
179C3×C4×D3—64no—
180C3×C4×D3—62no—
181C3×C4×D3—62no—
182C3×Dic6—22no—
183C3×Dic6—22no—
184C3×Dic6—32no—
185C3×Dic6—32no—
186C3×Dic6—42no—
187C3×Dic6—62no—
188C32×C8—62no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—22no—
191C3×C3⋊C8—42no—
192#455—91noSylow
193C2×C33—21no—
194C2×C33—42no—
195C2×C33#12011yescharacteristic
196C3×C3⋊D3—42no—
197C3×#41—92no—
198C3×#41—182no—
199C2×C3×Q8—31no—
200C2×C3×Q8—31no—
201C2×C3×Q8—61no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—41no—
206C2×C3×D4—182no—
207C2×C3×D4—182no—
208C2×C3×D4D3211yescharacteristic
209C2×C3×D4C2×C3×D311yescharacteristic
210C2×C3×D4C2×C3×D311yescharacteristic
211C2×#67—64no—
212C2×#67—62no—
213C2×#67—62no—
214#237—62no—
215#237—62no—
216#237—64no—
217D3×D4—62no—
218D3×D4—62no—
219D3×D4—64no—
220#235—32no—
221#235—32no—
222#235—62no—
223C2×Dic6C2×C3×D311yescharacteristic, complemented
224C3×Q16—182no—
225C3×SD16—182no—
226C3×M16—32no—
227C3×M16—32no—
228C3×M16—62no—
229C3×C22⋊C4—64no—
230C3×C22⋊C4—62no—
231C3×C22⋊C4—62no—
232C3×C42—31no—
233C3×C42—31no—
234C3×C42—61no—
235#217—22no—
236#216—62no—
237#215—62no—
238C4×Dic3C2×C3×D311yescharacteristic, complemented
239#208C2×C3×D312yescomplemented
240Dic12—62no—
241#204—62no—
242C22×C32—22no—
243C22×C32—24no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32—42no—
251C22×C32—42no—
252C22×C32—44no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—42no—
256C22×C32—44no—
257C22×C32—82no—
258C22×C32—82no—
259C22×C32—84no—
260C22×C32C2×D1211yescharacteristic
261C22×C32C2×#6711yescharacteristic
262C22×C32C2×C3×D411yescharacteristic
263C2×C3⋊D3—22no—
264C2×C3⋊D3—82no—
265C2×C3×D3—64no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—244no—
269C2×C3×D3—242no—
270C2×C3×D3—242no—
271C32×C4—22no—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4—42no—
275C32×C4—121no—
276C32×C4—121no—
277C32×C4D3×D412yes—
278C32×C4D3×D412yes—
279C32×C4C2×C3×D412yes—
280C3⋊Dic3—22no—
281C3×Dic3—42no—
282C3×Dic3—42no—
283C3×Dic3—41no—
284C3×Dic3—41no—
285C3×Dic3—41no—
286C3×Dic3—41no—
287C3×Dic3—64no—
288C3×Dic3—62no—
289C3×Dic3—62no—
290H(5,𝔽2)—91no—
291#137—92no—
292#124—31no—
293#104—92no—
294#100—31no—
295C33#45511yescharacteristic, complemented, Sylow
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—42no—
300C23×C3—182no—
301C23×C3#62612yes—
302C23×C3C3×D1212yes—
303C23×C3C3×#6712yes—
304C22×D3—64no—
305C22×D3—62no—
306C22×D3—62no—
307C3×Q8—62no—
308C3×Q8—62no—
309C3×Q8—92no—
310C3×Q8—122no—
311C3×D4—24no—
312C3×D4—22no—
313C3×D4—22no—
314C3×D4—22no—
315C3×D4—42no—
316C3×D4—42no—
317C3×D4—44no—
318C3×D4—92no—
319C3×D4—184no—
320C3×D4—182no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—41no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—182no—
332C2×C3×C4—182no—
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C22×C3×D311yescharacteristic
335C2×C3×C4C22×C3×D311yescharacteristic
336#67—64no—
337#67—68no—
338#67—64no—
339#67—64no—
340#67—62no—
341#67—62no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—64no—
345C2×Dic3—62no—
346C2×Dic3—62no—
347D12—32no—
348D12—32no—
349D12—62no—
350C4×D3—64no—
351C4×D3—62no—
352C4×D3—62no—
353Dic6—22no—
354Dic6—32no—
355Dic6—32no—
356Dic6—62no—
357C3×C8—62no—
358C3×C8—62no—
359C3×C8—122no—
360C3⋊C8—22no—
361C2×C32—21no—
362C2×C32—22no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32—41no—
371C2×C32—42no—
372C2×C32—42no—
373C2×C32—44no—
374C2×C32—42no—
375C2×C32—42no—
376C2×C32—82no—
377C2×C32—82no—
378C2×C32—84no—
379C2×C32#88211yescharacteristic
380C2×C32#93711yescharacteristic
381C2×C32C3×#12011yescharacteristic
382C3⋊D3—42no—
383C3×D3—124no—
384C3×D3—122no—
385C3×D3—122no—
386#41—92no—
387#41—182no—
388C2×Q8—31no—
389C2×D4—182no—
390C2×D4—182no—
391C2×D4C3×D3211yescharacteristic
392Q16—182no—
393SD16—182no—
394M16—32no—
395C22⋊C4—62no—
396C42—31no—
397C22×C3—24no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—44no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—44no—
411C22×C3—41no—
412C22×C3—82no—
413C22×C3—82no—
414C22×C3—84no—
415C22×C3—182no—
416C22×C3—722no—
417C22×C3C2×#62611yescharacteristic
418C22×C3C2×C3×D1211yescharacteristic
419C22×C3C2×C3×#6711yescharacteristic
420C2×D3—64no—
421C2×D3—62no—
422C2×D3—62no—
423C2×D3—244no—
424C2×D3—242no—
425C2×D3—242no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—42no—
430C3×C4—122no—
431C3×C4—122no—
432C3×C4—121no—
433C3×C4—121no—
434C3×C4—121no—
435C3×C4—121no—
436C3×C4—182no—
437C3×C4#381112yes—
438C3×C4C3×D3×D412yes—
439C3×C4C3×D3×D412yes—
440Dic3—41no—
441Dic3—41no—
442Dic3—64no—
443Dic3—62no—
444Dic3—62no—
445C32—21no—
446C32—21no—
447C32—21no—
448C32—41no—
449C32#474611yescharacteristic, complemented
450C32#506111yescharacteristic, complemented
451C32C3×#45511yescharacteristic, complemented
452C23—182no—
453C23C3×#62612yes—
454Q8—62no—
455Q8—92no—
456D4—22no—
457D4—92no—
458D4—184no—
459D4—182no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—182no—
463C2×C4—182no—
464C2×C4C2×C3×D3211yescharacteristic
465C8—62no—
466C2×C3—22no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—42no—
476C2×C3—44no—
477C2×C3—42no—
478C2×C3—42no—
479C2×C3—42no—
480C2×C3—41no—
481C2×C3—82no—
482C2×C3—82no—
483C2×C3—84no—
484C2×C3—362no—
485C2×C3#6398111yescharacteristic
486C2×C3C3×#88211yescharacteristic
487C2×C3C3×#93711yescharacteristic
488D3—124no—
489D3—122no—
490D3—122no—
491C22—22no—
492C22—42no—
493C22—182no—
494C22—722no—
495C22C2×C3×#62611yescharacteristic
496C4—121no—
497C4—121no—
498C4—182no—
499C4C3×#381112yes—
500C3—21no—
501C3—21no—
502C3—21no—
503C3—41no—
504C3#1059151411yescharacteristic, direct product factor
505C3C3×#474611yescharacteristic, complemented
506C3C3×#506111yescharacteristic, complemented
507C2—21no—
508C2—42no—
509C2—362no—
510C2C3×#6398111yescharacteristic
subgroupG/H
centerC2×C3#63981
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC2×C33#120
prime radicalC3×#64361C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#455
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C32×#936C2semidirect
3. C3×#63833C2semidirect
4. C32×#833C2semidirect
5. C3×#63570C2semidirect
8. #10591514C3direct (N × H)
14. C2×C32×Dic6C22semidirect
16. C32×C4×Dic3C22semidirect
17. C32×#208C22semidirect
27. C3×#936D3semidirect
28. C3×#936C2×C3semidirect
30. #63833C2×C3semidirect
33. C3×#833D3semidirect
34. C3×#833C2×C3semidirect
37. #63570C2×C3semidirect
80. C2×C3×Dic6C2×D3semidirect
81. C2×C3×Dic6C22×C3semidirect
91. C3×C4×Dic3C2×D3semidirect
92. C3×C4×Dic3C22×C3semidirect
94. C3×#208C2×D3semidirect
95. C3×#208C22×C3semidirect
118. #936C3×D3semidirect
124. #833C3×D3semidirect
223. C2×Dic6C2×C3×D3semidirect
238. C4×Dic3C2×C3×D3semidirect
239. #208C2×C3×D3semidirect
295. C33#455semidirect
449. C32#4746semidirect
450. C32#5061semidirect
451. C32C3×#455semidirect
504. C3#10591514direct (N × H)
505. C3C3×#4746semidirect
506. C3C3×#5061semidirect

Maximal subgroups

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

subgroupindex
C3×#643612
C32×#9362
C3×#638332
C32×#8332
C3×#635702
C3×#50613
C3×#47463
#105915143