Finite Group Databasebeta

← C3×#68712 #11788708

Subgroups — C3×#68712 #11788708

Order 1152.

671 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5745C211yesmaximal, characteristic
2C3×#5526C211yesmaximal, characteristic
3C3×#5120C211yesmaximal, characteristic, complemented
4C3×#5006C211yesmaximal, characteristic
5C3×#5004C211yesmaximal, characteristic, complemented
6C32×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
7C3×#4531C211yesmaximal, characteristic, complemented
8C3×#1909—31nomaximal
9#68712C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C32×#117C2211yescharacteristic
15C3×#926C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#907—22no—
18C3×D3×M16—22no—
19C3×#901—22no—
20C3×#897—22no—
21C3×#886—22no—
22C3×#875C412yescomplemented
23C3×#869C412yescomplemented
24C3×#816—22no—
25C3×#810—22no—
26C3×#806—22no—
27C3×#568—31no—
28C3×#536—31no—
29#5745C2×C311yescharacteristic
30#5526C2×C311yescharacteristic
31C3×#419—31no—
32C3×#419—31no—
33C3×#410—31no—
34#5120C2×C311yescharacteristic, complemented
35#5006C2×C311yescharacteristic
36#5004C2×C311yescharacteristic, complemented
37C3×H(3,ℤ/4)—21no—
38C3×H(3,ℤ/4)—31no—
39C3×H(3,ℤ/4)D311yescharacteristic, complemented
40C3×H(3,ℤ/4)C2×C311yescharacteristic, complemented
41#4531C2×C311yescharacteristic, complemented
42C22×C32×C4C2311yescharacteristic
43C2×C3×#67D412yes—
44C22×C3×Dic3—21no—
45C3×#235—21no—
46C3×#235—21no—
47C3×#235—21no—
48C3×#235—21no—
49C2×C3×D12C2×C411yescharacteristic
50C2×C3×C4×D3D412yes—
51C2×C3×Dic6—21no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6C2×C412yes—
57C2×C3×Dic6C2×C411yescharacteristic
58C32×M16—21no—
59C32×M16—21no—
60C2×C32×C8D412yes—
61C32×C4⋊C4—21no—
62C32×C4⋊C4—21no—
63C32×C22⋊C4—21no—
64C32×C22⋊C4—21no—
65C32×C42—21no—
66C32×C42C2×C412yes—
67C3×#217—21no—
68C3×(C2×D3)⋊C4—24no—
69C3×C4⋊Dic3—24no—
70C3×C4⋊Dic3—21no—
71C3×C4×Dic3—21no—
72C3×#208—21no—
73C3×#208—21no—
74C2×C3×C3⋊C8D412yes—
75C3×ΓL(1,𝔽25)—24no—
76C3×D3×C8—24no—
77#1909—31noSylow
78C2×C3×#41—31no—
79C22×C3×Q8—31no—
80C2×#235C22×C311yescharacteristic
81C22×Dic6C22×C311yescharacteristic
82C3×#131—62no—
83C3×#131—62no—
84C2×C3×M16—21no—
85C2×C3×M16—31no—
86C2×C3×M16—62no—
87C2×C3×M16C2×D311yescharacteristic
88C2×C3×M16C22×C311yescharacteristic
89C3×C4⋊Q8—32no—
90C3×#124—32no—
91C3×#123—62no—
92C3×C22⋊Q8—62no—
93C3×#117—21no—
94C3×#117—21no—
95C3×#117—31no—
96C3×#117C2×D311yescharacteristic
97C3×#117C2×D311yescharacteristic
98C3×#117C22×C311yescharacteristic
99C3×#117C22×C311yescharacteristic
100#926C22×C311yescharacteristic
101C2×#208C22×C311yescharacteristic
102#907—22no—
103D3×M16—22no—
104#901—22no—
105#897—22no—
106#886—22no—
107#875C3×C412yescomplemented
108#869C3×C412yescomplemented
109C3×#103—62no—
110C3×#103—62no—
111C3×#99—62no—
112#816—22no—
113#810—22no—
114#806—22no—
115C23×C32C2×D411yescharacteristic
116C22×C3×D3C22⋊C412yes—
117C2×C32×C4—22no—
118C2×C32×C4—41no—
119C2×C32×C4C22×C411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4#4111yescharacteristic
124C2×C32×C4#4111yescharacteristic
125C3×#67—22no—
126C3×#67—22no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—41no—
129C2×C3×Dic3—41no—
130C2×C3×Dic3—41no—
131C2×C3×Dic3C22⋊C412yes—
132C3×D12—21no—
133C3×D12—21no—
134C3×C4×D3—22no—
135C3×C4×D3—22no—
136C3×Dic6—21no—
137C3×Dic6—21no—
138C3×Dic6—22no—
139C3×Dic6—22no—
140C3×Dic6—41no—
141C3×Dic6—41no—
142C32×C8—22no—
143C3×C3⋊C8—22no—
144#568—31no—
145#536—31no—
146#419—31no—
147#419—31no—
148#410—31no—
149H(3,ℤ/4)—31no—
150H(3,ℤ/4)C3×D311yescharacteristic, complemented
151C3×#41—61no—
152C3×#41—61no—
153C3×#41—61no—
154C3×#41—61no—
155C2×C3×Q8—31no—
156C2×C3×Q8—32no—
157C2×C3×Q8—61no—
158C2×C3×Q8—61no—
159C2×C3×Q8—61no—
160C2×C3×Q8—61no—
161C2×C3×Q8—61no—
162C2×C3×D4—32no—
163C2×C3×D4—31no—
164C22×C3×C4—21no—
165C22×C3×C4—32no—
166C22×C3×C4—61no—
167C22×C3×C4C22×D311yescharacteristic
168C22×C3×C4C23×C311yescharacteristic
169C2×#67C3×D412yes—
170C22×Dic3—21no—
171#235—21no—
172#235—21no—
173#235—21no—
174#235—21no—
175C2×D12C2×C3×C411yescharacteristic
176C2×C4×D3C3×D412yes—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6C2×C3×C412yes—
183C2×Dic6C2×C3×C411yescharacteristic
184C3×M16—21no—
185C3×M16—21no—
186C3×M16—21no—
187C3×M16—21no—
188C3×M16—41no—
189C3×M16—41no—
190C3×M16—64no—
191C3×M16—61no—
192C3×M16—61no—
193C2×C3×C8—22no—
194C2×C3×C8—32no—
195C2×C3×C8—64no—
196C2×C3×C8D1212yes—
197C2×C3×C8C3×D412yes—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4—41no—
204C3×C4⋊C4—64no—
205C3×C4⋊C4—61no—
206C3×C22⋊C4—21no—
207C3×C22⋊C4—21no—
208C3×C22⋊C4—21no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—41no—
211C3×C22⋊C4—41no—
212C3×C22⋊C4—64no—
213C3×C22⋊C4—61no—
214C3×C42—22no—
215C3×C42—21no—
216C3×C42—21no—
217C3×C42—22no—
218C3×C42—61no—
219C3×C42C4×D312yes—
220C3×C42C2×C3×C412yes—
221#217—21no—
222(C2×D3)⋊C4—24no—
223C4⋊Dic3—24no—
224C4⋊Dic3—21no—
225C4×Dic3—21no—
226#208—21no—
227#208—21no—
228C2×C3⋊C8C3×D412yes—
229ΓL(1,𝔽25)—24no—
230D3×C8—24no—
231C22×C32—41no—
232C22×C32C2×C22⋊C411yescharacteristic
233C22×C32#12111yescharacteristic
234C22×C32#12111yescharacteristic
235C2×C3×D3—22no—
236C2×C3×D3—42no—
237C32×C4—42no—
238C32×C4—81no—
239C32×C4C4×D411yescharacteristic
240C32×C4C4×D411yescharacteristic
241C32×C4#12011yescharacteristic
242C32×C4#12311yescharacteristic
243C3×Dic3—22no—
244C3×Dic3—42no—
245C3×Dic3—81no—
246C2×#41—31no—
247C22×Q8—31no—
248#131—62no—
249#131—62no—
250C2×M16—31no—
251C2×M16—62no—
252C2×M16C2×C3×D311yescharacteristic
253C4⋊Q8—32no—
254#124—32no—
255#123—62no—
256C22⋊Q8—62no—
257#117—31no—
258#117C2×C3×D311yescharacteristic
259#117C2×C3×D311yescharacteristic
260#103—62no—
261#103—62no—
262#99—62no—
263C23×C3—21no—
264C23×C3—32no—
265C23×C3D3×D411yescharacteristic
266C23×C3C2×C3×D411yescharacteristic
267C22×D3C3×C22⋊C412yes—
268C3×Q8—61no—
269C3×Q8—61no—
270C3×Q8—62no—
271C3×Q8—62no—
272C3×Q8—121no—
273C3×Q8—121no—
274C3×D4—62no—
275C3×D4—62no—
276C3×D4—61no—
277C3×D4—61no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—32no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—121no—
295C2×C3×C4—121no—
296C2×C3×C4—121no—
297C2×C3×C4C2×C4×D311yescharacteristic
298C2×C3×C4C2×D1211yescharacteristic
299C2×C3×C4#23511yescharacteristic
300C2×C3×C4D3×D411yescharacteristic
301C2×C3×C4D3×D411yescharacteristic
302C2×C3×C4#23911yescharacteristic
303C2×C3×C4C22×C3×C411yescharacteristic
304C2×C3×C4C2×C3×D411yescharacteristic
305C2×C3×C4C2×C3×D411yescharacteristic
306C2×C3×C4C2×C3×D411yescharacteristic
307C2×C3×C4C3×#4111yescharacteristic
308C2×C3×C4C3×#4111yescharacteristic
309#67—22no—
310#67—22no—
311C2×Dic3—22no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3C3×C22⋊C412yes—
316D12—21no—
317D12—21no—
318C4×D3—22no—
319C4×D3—22no—
320Dic6—21no—
321Dic6—21no—
322Dic6—22no—
323Dic6—22no—
324Dic6—41no—
325Dic6—41no—
326C3×C8—22no—
327C3×C8—22no—
328C3×C8—42no—
329C3×C8—62no—
330C3⋊C8—22no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32#38611yescharacteristic
335C3×D3—42no—
336#41—61no—
337#41—61no—
338#41—61no—
339#41—61no—
340C2×Q8—31no—
341C2×Q8—32no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×D4—32no—
348C2×D4—31no—
349C22×C4—32no—
350C22×C4—61no—
351C22×C4C22×C3×D311yescharacteristic
352M16—21no—
353M16—21no—
354M16—64no—
355M16—61no—
356M16—61no—
357C2×C8—32no—
358C2×C8—64no—
359C2×C8C3×D1212yes—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—64no—
363C4⋊C4—61no—
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—64no—
367C22⋊C4—61no—
368C42—21no—
369C42—61no—
370C42C3×C4×D312yes—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—62no—
377C22×C3—81no—
378C22×C3—122no—
379C22×C3D3×C22⋊C411yescharacteristic
380C22×C3#88411yescharacteristic
381C22×C3#89511yescharacteristic
382C22×C3C2×C3×C22⋊C411yescharacteristic
383C22×C3C3×#12111yescharacteristic
384C22×C3C3×#12111yescharacteristic
385C2×D3—22no—
386C2×D3—42no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—62no—
394C3×C4—82no—
395C3×C4—81no—
396C3×C4—81no—
397C3×C4—82no—
398C3×C4—122no—
399C3×C4—241no—
400C3×C4C4×D1211yescharacteristic
401C3×C4#88211yescharacteristic
402C3×C4#89311yescharacteristic
403C3×C4#89411yescharacteristic
404C3×C4C3×C4×D411yescharacteristic
405C3×C4C3×C4×D411yescharacteristic
406C3×C4C3×#12011yescharacteristic
407C3×C4C3×#12311yescharacteristic
408Dic3—22no—
409Dic3—42no—
410Dic3—81no—
411C32#190911yescharacteristic, complemented, Sylow
412C23—32no—
413C23C3×D3×D411yescharacteristic
414Q8—61no—
415Q8—61no—
416Q8—62no—
417Q8—62no—
418Q8—121no—
419Q8—121no—
420D4—62no—
421D4—62no—
422D4—61no—
423D4—61no—
424C2×C4—22no—
425C2×C4—32no—
426C2×C4—41no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4C2×C3×C4×D311yescharacteristic
434C2×C4C2×C3×D1211yescharacteristic
435C2×C4C3×#23511yescharacteristic
436C2×C4C3×D3×D411yescharacteristic
437C2×C4C3×D3×D411yescharacteristic
438C2×C4C3×#23911yescharacteristic
439C8—22no—
440C8—62no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—122no—
452C2×C3#466511yescharacteristic
453C2×C3C3×#38611yescharacteristic
454D3—42no—
455C22—41no—
456C22—62no—
457C22—122no—
458C22C3×D3×C22⋊C411yescharacteristic
459C22C3×#88411yescharacteristic
460C22C3×#89511yescharacteristic
461C4—42no—
462C4—62no—
463C4—81no—
464C4—122no—
465C4—241no—
466C4C3×C4×D1211yescharacteristic
467C4C3×#88211yescharacteristic
468C4C3×#89311yescharacteristic
469C4C3×#89411yescharacteristic
470C3—21no—
471C3#6871211yescharacteristic, direct product factor
472C3C3×#190911yescharacteristic, complemented
473C2—21no—
474C2—21no—
475C2—21no—
476C2—122no—
477C2C3×#466511yescharacteristic
subgroupG/H
centerC2×C3#4665
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×H(3,ℤ/4)C2
socleC2×C32#386
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×H(3,ℤ/4)D3
hypocenterC3C3×#1909
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
3. C3×#5120C2semidirect
5. C3×#5004C2semidirect
6. C32×H(3,ℤ/4)C2semidirect
7. C3×#4531C2semidirect
9. #68712C3direct (N × H)
22. C3×#875C4semidirect
23. C3×#869C4semidirect
34. #5120C2×C3semidirect
36. #5004C2×C3semidirect
39. C3×H(3,ℤ/4)D3semidirect
40. C3×H(3,ℤ/4)C2×C3semidirect
41. #4531C2×C3semidirect
107. #875C3×C4semidirect
108. #869C3×C4semidirect
150. H(3,ℤ/4)C3×D3semidirect
411. C32#1909semidirect
471. C3#68712direct (N × H)
472. C3C3×#1909semidirect

Maximal subgroups

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

subgroupindex
C3×#57452
C3×#55262
C3×#51202
C3×#50062
C3×#50042
C32×H(3,ℤ/4)2
C3×#45312
C3×#19093
#687123