Finite Group Databasebeta

← C3×#68987 #11793943

Subgroups — C3×#68987 #11793943

Order 1152.

573 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic
2C2×C3×#824C211yesmaximal, characteristic, complemented
3C3×#5129C211yesmaximal, characteristic
4C2×C3×#822C211yesmaximal, characteristic, complemented
5C2×C3×#810C211yesmaximal, characteristic, complemented
6C3×#4966C211yesmaximal, characteristic
7C32×#343C211yesmaximal, characteristic, complemented
8C3×#2052—31nomaximal
9#68987C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C3×#909—22no—
18C3×#909—22no—
19C2×C3×Dic12—21no—
20C2×C3×#204—21no—
21C3×#824—22no—
22C3×#822—22no—
23C3×(C2×D3)⋊C8—22no—
24C3×#810—21no—
25C3×#810—22no—
26C3×#809—21no—
27C2×C3×#137—31no—
28C2×#909C2×C311yescharacteristic
29C2×C3×#108—31no—
30C3×#417—31no—
31C2×C3×#103—31no—
32C2×C3×#101—31no—
33C3×#408—31no—
34C2×#824C2×C311yescharacteristic, complemented
35#5129C2×C311yescharacteristic
36C2×#822C2×C311yescharacteristic, complemented
37C2×#810C2×C311yescharacteristic, complemented
38#4966C2×C311yescharacteristic
39C3×#343—21no—
40C3×#343D311yescharacteristic, complemented
41C3×#343C2×C311yescharacteristic, complemented
42C22×C32×C4—21no—
43C22×C32×C4C2311yescharacteristic
44C2×C3×#67—21no—
45C22×C3×Dic3—21no—
46C3×#235—21no—
47C3×#235—21no—
48C3×#235—21no—
49C3×#235—21no—
50C2×C3×D12D411yescharacteristic
51C2×C3×C4×D3—21no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6D411yescharacteristic
58C2×C3×Dic6D412yes—
59C32×M16—21no—
60C32×M16—21no—
61C32×M16—21no—
62C32×M16D412yes—
63C2×C32×C8—21no—
64C2×C32×C8—21no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C3×#208—21no—
68C3×#208—21no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—21no—
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8D412yes—
73C3×Dic12—24no—
74C3×#204—24no—
75#2052—31noSylow
76C2×C3×#41—31no—
77C22×C3×Q8—31no—
78C2×#235C22×C311yescharacteristic
79C22×Dic6C22×C311yescharacteristic
80C3×#137—62no—
81C3×#137—62no—
82C2×C3×Q16—61no—
83C2×C3×SD16—61no—
84C2×C3×M16—21no—
85C2×C3×M16—21no—
86C2×C3×M16—31no—
87C2×C3×M16C2×D311yescharacteristic
88C2×C3×M16C2×D311yescharacteristic
89C2×C3×M16C22×C311yescharacteristic
90C2×C3×M16C22×C311yescharacteristic
91C22×C3×C8—31no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2×D311yescharacteristic
94C2×C3×C4⋊C4C22×C311yescharacteristic
95C2×#208C22×C311yescharacteristic
96C22×C3⋊C8C22×C311yescharacteristic
97#909—22no—
98#909—22no—
99C2×Dic12—21no—
100C2×#204—21no—
101C3×#108—62no—
102C3×#103—62no—
103C3×#103—61no—
104C3×#102—61no—
105C3×#101—62no—
106C3×C22⋊C8—62no—
107#824—22no—
108#822—22no—
109(C2×D3)⋊C8—22no—
110#810—21no—
111#810—22no—
112#809—21no—
113C23×C32#4111yescharacteristic
114C22×C3×D3—21no—
115C2×C32×C4—41no—
116C2×C32×C4—81no—
117C2×C32×C4C2×D411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C32×C4#4111yescharacteristic
123C3×#67—41no—
124C3×#67—41no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—41no—
128C2×C3×Dic3—41no—
129C3×D12—21no—
130C3×D12—21no—
131C3×C4×D3—41no—
132C3×C4×D3—41no—
133C3×Dic6—22no—
134C3×Dic6—21no—
135C3×Dic6—22no—
136C3×Dic6—21no—
137C3×Dic6—41no—
138C3×Dic6—41no—
139C32×C8—22no—
140C32×C8—41no—
141C3×C3⋊C8—22no—
142C3×C3⋊C8—41no—
143C2×#137—31no—
144C2×#108—31no—
145#417—31no—
146C2×#103—31no—
147C2×#101—31no—
148#408—31no—
149#343C3×D311yescharacteristic, complemented
150C3×#41—61no—
151C3×#41—61no—
152C3×#41—61no—
153C3×#41—61no—
154C2×C3×Q8—32no—
155C2×C3×Q8—31no—
156C2×C3×Q8—61no—
157C2×C3×Q8—61no—
158C2×C3×Q8—61no—
159C2×C3×Q8—61no—
160C2×C3×Q8—61no—
161C2×C3×D4—31no—
162C2×C3×D4—61no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—61no—
168C22×C3×C4—61no—
169C22×C3×C4C22×D311yescharacteristic
170C22×C3×C4C23×C311yescharacteristic
171C2×#67—21no—
172C22×Dic3—21no—
173#235—21no—
174#235—21no—
175#235—21no—
176#235—21no—
177C2×D12C3×D411yescharacteristic
178C2×C4×D3—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6C3×D411yescharacteristic
185C2×Dic6C3×D412yes—
186C3×Q16—64no—
187C3×SD16—64no—
188C3×M16—21no—
189C3×M16—21no—
190C3×M16—22no—
191C3×M16—21no—
192C3×M16—21no—
193C3×M16—21no—
194C3×M16—21no—
195C3×M16—22no—
196C3×M16—41no—
197C3×M16—41no—
198C3×M16—61no—
199C3×M16—61no—
200C3×M16D1212yes—
201C3×M16C3×D412yes—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—32no—
207C2×C3×C8—41no—
208C2×C3×C8—41no—
209C2×C3×C8—61no—
210C2×C3×C8—61no—
211C2×C3×C8—61no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—41no—
217C3×C4⋊C4—41no—
218#208—21no—
219#208—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8C3×D412yes—
224Dic12—24no—
225#204—24no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32#12111yescharacteristic
229C22×C32#12311yescharacteristic
230C22×C32#12411yescharacteristic
231C2×C3×D3—41no—
232C2×C3×D3—41no—
233C2×C3×D3—41no—
234C32×C4—81no—
235C32×C4#12011yescharacteristic
236C32×C4#12111yescharacteristic
237C32×C4#12111yescharacteristic
238C32×C4#12311yescharacteristic
239C3×Dic3—42no—
240C3×Dic3—41no—
241C2×#41—31no—
242C22×Q8—31no—
243#137—62no—
244#137—62no—
245C2×Q16—61no—
246C2×SD16—61no—
247C2×M16—31no—
248C2×M16C2×C3×D311yescharacteristic
249C2×M16C2×C3×D311yescharacteristic
250C22×C8—31no—
251C2×C4⋊C4C2×C3×D311yescharacteristic
252#108—62no—
253#103—62no—
254#103—61no—
255#102—61no—
256#101—62no—
257C22⋊C8—62no—
258C23×C3—21no—
259C23×C3—61no—
260C23×C3#23511yescharacteristic
261C23×C3C3×#4111yescharacteristic
262C22×D3—21no—
263C3×Q8—62no—
264C3×Q8—61no—
265C3×Q8—62no—
266C3×Q8—61no—
267C3×Q8—121no—
268C3×Q8—121no—
269C3×D4—61no—
270C3×D4—61no—
271C3×D4—121no—
272C3×D4—121no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—61no—
283C2×C3×C4—62no—
284C2×C3×C4—81no—
285C2×C3×C4—81no—
286C2×C3×C4—82no—
287C2×C3×C4—121no—
288C2×C3×C4—121no—
289C2×C3×C4—121no—
290C2×C3×C4—121no—
291C2×C3×C4C2×D1211yescharacteristic
292C2×C3×C4#23511yescharacteristic
293C2×C3×C4D3×D411yescharacteristic
294C2×C3×C4D3×D411yescharacteristic
295C2×C3×C4D3×D411yescharacteristic
296C2×C3×C4#23911yescharacteristic
297C2×C3×C4C2×C3×D411yescharacteristic
298C2×C3×C4C2×C3×D411yescharacteristic
299C2×C3×C4C2×C3×D411yescharacteristic
300C2×C3×C4C2×C3×D411yescharacteristic
301C2×C3×C4C3×#4111yescharacteristic
302C2×C3×C4C3×#4111yescharacteristic
303#67—41no—
304#67—41no—
305C2×Dic3—21no—
306C2×Dic3—22no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309D12—21no—
310D12—21no—
311C4×D3—41no—
312C4×D3—41no—
313Dic6—22no—
314Dic6—21no—
315Dic6—22no—
316Dic6—21no—
317Dic6—41no—
318Dic6—41no—
319C3×C8—22no—
320C3×C8—22no—
321C3×C8—42no—
322C3×C8—42no—
323C3×C8—41no—
324C3×C8—41no—
325C3×C8—62no—
326C3×C8—121no—
327C3⋊C8—22no—
328C3⋊C8—41no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32#39411yescharacteristic
332C2×C32#47111yescharacteristic
333C2×C32#47311yescharacteristic
334C3×D3—81no—
335#41—61no—
336#41—61no—
337#41—61no—
338#41—61no—
339C2×Q8—32no—
340C2×Q8—31no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×D4—31no—
347C2×D4—61no—
348C22×C4—21no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4C22×C3×D311yescharacteristic
352Q16—64no—
353SD16—64no—
354M16—21no—
355M16—21no—
356M16—21no—
357M16—61no—
358M16—61no—
359M16C3×D1212yes—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—32no—
363C2×C8—61no—
364C2×C8—61no—
365C2×C8—61no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—121no—
378C22×C3—121no—
379C22×C3—121no—
380C22×C3#87511yescharacteristic
381C22×C3#88411yescharacteristic
382C22×C3#89411yescharacteristic
383C22×C3C3×#12111yescharacteristic
384C22×C3C3×#12311yescharacteristic
385C22×C3C3×#12411yescharacteristic
386C2×D3—41no—
387C2×D3—41no—
388C2×D3—41no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—82no—
394C3×C4—81no—
395C3×C4—81no—
396C3×C4—121no—
397C3×C4—122no—
398C3×C4#88211yescharacteristic
399C3×C4#88411yescharacteristic
400C3×C4#89411yescharacteristic
401C3×C4#89511yescharacteristic
402C3×C4C3×#12011yescharacteristic
403C3×C4C3×#12111yescharacteristic
404C3×C4C3×#12111yescharacteristic
405C3×C4C3×#12311yescharacteristic
406Dic3—42no—
407Dic3—41no—
408C32#205211yescharacteristic, complemented, Sylow
409C23—61no—
410C23C3×#23511yescharacteristic
411Q8—62no—
412Q8—61no—
413Q8—62no—
414Q8—61no—
415Q8—121no—
416Q8—121no—
417D4—61no—
418D4—61no—
419D4—121no—
420D4—121no—
421C2×C4—41no—
422C2×C4—61no—
423C2×C4—62no—
424C2×C4—81no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4C2×C3×D1211yescharacteristic
430C2×C4C3×#23511yescharacteristic
431C2×C4C3×D3×D411yescharacteristic
432C2×C4C3×D3×D411yescharacteristic
433C2×C4C3×D3×D411yescharacteristic
434C2×C4C3×#23911yescharacteristic
435C8—22no—
436C8—41no—
437C8—62no—
438C8—121no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—41no—
447C2×C3—41no—
448C2×C3—241no—
449C2×C3#467211yescharacteristic
450C2×C3#474811yescharacteristic
451C2×C3#475511yescharacteristic
452C2×C3C3×#39411yescharacteristic
453C2×C3C3×#47111yescharacteristic
454C2×C3C3×#47311yescharacteristic
455D3—81no—
456C22—21no—
457C22—21no—
458C22—121no—
459C22—121no—
460C22—121no—
461C22C3×#87511yescharacteristic
462C22C3×#88411yescharacteristic
463C22C3×#89411yescharacteristic
464C4—81no—
465C4—121no—
466C4—122no—
467C4C3×#88211yescharacteristic
468C4C3×#88411yescharacteristic
469C4C3×#89411yescharacteristic
470C4C3×#89511yescharacteristic
471C3—21no—
472C3#6898711yescharacteristic, direct product factor
473C3C3×#205211yescharacteristic, complemented
474C2—21no—
475C2—21no—
476C2—241no—
477C2C3×#467211yescharacteristic
478C2C3×#474811yescharacteristic
479C2C3×#475511yescharacteristic
subgroupG/H
centerC22×C3#884
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#343C2
socleC22×C32#121
prime radicalC2×C3×#235C22
hypercenterC3×#343D3
hypocenterC3C3×#2052
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#343

Complements

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

NH ≅ G/Ntype
2. C2×C3×#824C2semidirect
4. C2×C3×#822C2semidirect
5. C2×C3×#810C2semidirect
7. C32×#343C2semidirect
9. #68987C3direct (N × H)
34. C2×#824C2×C3semidirect
36. C2×#822C2×C3semidirect
37. C2×#810C2×C3semidirect
40. C3×#343D3semidirect
41. C3×#343C2×C3semidirect
149. #343C3×D3semidirect
408. C32#2052semidirect
472. C3#68987direct (N × H)
473. C3C3×#2052semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C2×C3×#8242
C3×#51292
C2×C3×#8222
C2×C3×#8102
C3×#49662
C32×#3432
C3×#20523
#689873