Finite Group Databasebeta

← C3×#69002 #11786684

Subgroups — C3×#69002 #11786684

Order 1152.

653 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5504C211yesmaximal, characteristic
2C2×C3×#825C211yesmaximal, characteristic
3C3×#5133C211yesmaximal, characteristic
4C3×#5121C211yesmaximal, characteristic, complemented
5C3×#4975C211yesmaximal, characteristic, complemented
6C32×#343C211yesmaximal, characteristic, complemented
7C3×#4555C211yesmaximal, characteristic, complemented
8C3×#1859—31nomaximal
9#69002C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C32×C4⋊C4C2211yescharacteristic
14C3×C4×#67—22no—
15C3×#926C2211yescharacteristic
16C2×C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×Dic3⋊C4—22no—
18C2×C3×C4×Dic3C2211yescharacteristic
19C3×#893—21no—
20C3×#892—21no—
21C3×D3×C4⋊C4—21no—
22C3×#887—21no—
23C3×#825C414yescomplemented
24C3×#821—21no—
25C3×#818—21no—
26C3×#817—21no—
27C3×#816—21no—
28C3×#520—31no—
29#5504C2×C311yescharacteristic
30C3×#428—31no—
31C2×C3×#104—31no—
32C3×#417—31no—
33C3×#384—31no—
34C2×#825C2×C311yescharacteristic
35#5133C2×C311yescharacteristic
36#5121C2×C311yescharacteristic, complemented
37#4975C2×C311yescharacteristic, complemented
38C3×#343—21no—
39C3×#343D311yescharacteristic, complemented
40C3×#343C2×C311yescharacteristic, complemented
41C3×#339—31no—
42#4555C2×C311yescharacteristic, complemented
43C22×C32×C4D412yes—
44C22×C32×C4C2311yescharacteristic
45C2×C3×#67—21no—
46C22×C3×Dic3—21no—
47C22×C3×Dic3—21no—
48C3×#235—21no—
49C3×#235—21no—
50C3×#235C2×C412yes—
51C3×#235Q812yes—
52C2×C3×D12D411yescharacteristic
53C2×C3×C4×D3—21no—
54C2×C3×C4×D3—22no—
55C2×C3×Dic6D411yescharacteristic
56C32×M16—21no—
57C32×M16—21no—
58C32×M16—21no—
59C32×M16C2×C412yescomplemented
60C2×C32×C8—21no—
61C2×C32×C8—21no—
62C32×C4⋊C4—21no—
63C32×C4⋊C4—21no—
64C3×#217—21no—
65C3×(C2×D3)⋊C4—22no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3D411yescharacteristic
69C3×C4⋊Dic3D411yescharacteristic
70C3×Dic3⋊C4—22no—
71C3×Dic3⋊C4—44no—
72C3×C4×Dic3—22no—
73C3×C4×Dic3—21no—
74C3×C4×Dic3—21no—
75C3×C4×Dic3C2×C412yescomplemented
76#1859—31noSylow
77C2×C3×#41—31no—
78C2×#235C22×C311yescharacteristic
79C2×C3×M16—21no—
80C2×C3×M16—21no—
81C2×C3×M16C2×D311yescharacteristic
82C2×C3×M16C2×D311yescharacteristic
83C2×C3×M16C22×C311yescharacteristic
84C2×C3×M16C22×C311yescharacteristic
85C3×C4×Q8—61no—
86C3×C4×D4—61no—
87C3×C4×D4—62no—
88C3×#117—31no—
89C3×#117—61no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—31no—
92C2×C3×C4⋊C4—61no—
93C2×C3×C4⋊C4—62no—
94C2×C3×C4⋊C4C2×D311yescharacteristic
95C2×C3×C4⋊C4C22×C311yescharacteristic
96C2×C3×C42—31no—
97C4×#67—22no—
98#926C22×C311yescharacteristic
99C2×C4⋊Dic3C22×C311yescharacteristic
100C2×Dic3⋊C4—22no—
101C2×C4×Dic3C22×C311yescharacteristic
102#893—21no—
103#892—21no—
104D3×C4⋊C4—21no—
105#887—21no—
106C3×#107—61no—
107C3×#106—61no—
108C3×#104—34no—
109C3×#103—61no—
110C3×#102—61no—
111#825C3×C414yescomplemented
112#821—21no—
113#818—21no—
114#817—21no—
115#816—21no—
116C23×C32C2×D411yescharacteristic
117C22×C3×D3—21no—
118C2×C32×C4—22no—
119C2×C32×C4—42no—
120C2×C32×C4C22×C411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×Q811yescharacteristic
124C2×C32×C4#4111yescharacteristic
125C2×C32×C4#4111yescharacteristic
126C3×#67—22no—
127C3×#67—22no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—42no—
135C2×C3×Dic3—41no—
136C3×D12—21no—
137C3×D12C4⋊C412yescomplemented
138C3×C4×D3—22no—
139C3×C4×D3—22no—
140C3×C4×D3—82no—
141C3×Dic6—21no—
142C3×Dic6C4⋊C412yescomplemented
143C32×C8—22no—
144C32×C8—41no—
145#520—31no—
146#428—31no—
147C2×#104—31no—
148#417—31no—
149#384—31no—
150#343C3×D311yescharacteristic, complemented
151#339—31no—
152C3×#41—32no—
153C3×#41—32no—
154C3×#41—61no—
155C3×#41—61no—
156C2×C3×Q8—31no—
157C2×C3×D4—31no—
158C2×C3×D4—61no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—62no—
162C22×C3×C4—61no—
163C22×C3×C4—61no—
164C22×C3×C4—61no—
165C22×C3×C4D1212yes—
166C22×C3×C4C3×D412yes—
167C22×C3×C4C22×D311yescharacteristic
168C22×C3×C4C23×C311yescharacteristic
169C2×#67—21no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172#235—21no—
173#235—21no—
174#235C2×C3×C412yes—
175#235C3×Q812yes—
176C2×D12C3×D411yescharacteristic
177C2×C4×D3—21no—
178C2×C4×D3—22no—
179C2×Dic6C3×D411yescharacteristic
180C3×M16—21no—
181C3×M16—21no—
182C3×M16—22no—
183C3×M16—21no—
184C3×M16—21no—
185C3×M16—21no—
186C3×M16—21no—
187C3×M16—22no—
188C3×M16—41no—
189C3×M16—41no—
190C3×M16C4×D312yescomplemented
191C3×M16C2×C3×C412yescomplemented
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—21no—
196C2×C3×C8—41no—
197C2×C3×C8—41no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—31no—
203C3×C4⋊C4—31no—
204C3×C4⋊C4—41no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—62no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—61no—
209C3×C4⋊C4—124no—
210C3×C22⋊C4—62no—
211C3×C22⋊C4—61no—
212C3×C42—32no—
213C3×C42—62no—
214C3×C42—61no—
215C3×C42—61no—
216#217—21no—
217(C2×D3)⋊C4—22no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3C3×D411yescharacteristic
221C4⋊Dic3C3×D411yescharacteristic
222Dic3⋊C4—22no—
223Dic3⋊C4—44no—
224C4×Dic3—22no—
225C4×Dic3—21no—
226C4×Dic3—21no—
227C4×Dic3C2×C3×C412yescomplemented
228C22×C32—21no—
229C22×C32—21no—
230C22×C32C4×D411yescharacteristic
231C22×C32#12011yescharacteristic
232C22×C32C22⋊Q811yescharacteristic
233C2×C3×D3—22no—
234C2×C3×D3—41no—
235C2×C3×D3—41no—
236C32×C4—42no—
237C32×C4C2×C4⋊C411yescharacteristic
238C32×C4C4×D411yescharacteristic
239C32×C4C22⋊Q811yescharacteristic
240C32×C4#12311yescharacteristic
241C3×Dic3—24no—
242C3×Dic3—22no—
243C3×Dic3—41no—
244C3×Dic3—41no—
245C3×Dic3—81no—
246C2×#41—31no—
247C2×M16C2×C3×D311yescharacteristic
248C2×M16C2×C3×D311yescharacteristic
249C4×Q8—61no—
250C4×D4—61no—
251C4×D4—62no—
252#117—31no—
253#117—61no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4C2×C3×D311yescharacteristic
258C2×C42—31no—
259#107—61no—
260#106—61no—
261#104—34no—
262#103—61no—
263#102—61no—
264C23×C3—21no—
265C23×C3—61no—
266C23×C3C2×D1211yescharacteristic
267C23×C3C2×C3×D411yescharacteristic
268C22×D3—21no—
269C3×Q8—32no—
270C3×Q8—61no—
271C3×D4—32no—
272C3×D4—62no—
273C3×D4—62no—
274C3×D4—61no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—62no—
287C2×C3×C4—62no—
288C2×C3×C4—62no—
289C2×C3×C4—61no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—82no—
295C2×C3×C4—122no—
296C2×C3×C4—121no—
297C2×C3×C4—242no—
298C2×C3×C4C2×C4×D311yescharacteristic
299C2×C3×C4#23511yescharacteristic
300C2×C3×C4D3×D411yescharacteristic
301C2×C3×C4D3×D411yescharacteristic
302C2×C3×C4#23711yescharacteristic
303C2×C3×C4D3×Q811yescharacteristic
304C2×C3×C4C22×C3×C411yescharacteristic
305C2×C3×C4C2×C3×D411yescharacteristic
306C2×C3×C4C2×C3×D411yescharacteristic
307C2×C3×C4C2×C3×Q811yescharacteristic
308C2×C3×C4C3×#4111yescharacteristic
309C2×C3×C4C3×#4111yescharacteristic
310#67—22no—
311#67—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—42no—
319C2×Dic3—41no—
320D12—21no—
321D12C3×C4⋊C412yescomplemented
322C4×D3—22no—
323C4×D3—22no—
324C4×D3—82no—
325Dic6—21no—
326Dic6C3×C4⋊C412yescomplemented
327C3×C8—22no—
328C3×C8—22no—
329C3×C8—42no—
330C3×C8—42no—
331C3×C8—41no—
332C3×C8—41no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32#38511yescharacteristic
336C2×C32#45311yescharacteristic
337C2×C32#45611yescharacteristic
338C3×D3—42no—
339#41—32no—
340#41—32no—
341#41—61no—
342#41—61no—
343C2×Q8—31no—
344C2×D4—31no—
345C2×D4—61no—
346C22×C4—62no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4C3×D1212yes—
351C22×C4C22×C3×D311yescharacteristic
352M16—21no—
353M16—21no—
354M16—21no—
355M16C3×C4×D312yescomplemented
356C2×C8—21no—
357C2×C8—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—31no—
361C4⋊C4—31no—
362C4⋊C4—62no—
363C4⋊C4—61no—
364C4⋊C4—61no—
365C4⋊C4—124no—
366C22⋊C4—62no—
367C22⋊C4—61no—
368C42—32no—
369C42—62no—
370C42—61no—
371C42—61no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—62no—
382C22×C3—121no—
383C22×C3—121no—
384C22×C3C4×D1211yescharacteristic
385C22×C3#88211yescharacteristic
386C22×C3#89711yescharacteristic
387C22×C3C3×C4×D411yescharacteristic
388C22×C3C3×#12011yescharacteristic
389C22×C3C3×C22⋊Q811yescharacteristic
390C2×D3—22no—
391C2×D3—41no—
392C2×D3—41no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—64no—
400C3×C4—62no—
401C3×C4—82no—
402C3×C4—121no—
403C3×C4—121no—
404C3×C4—241no—
405C3×C4#67⋊C411yescharacteristic
406C3×C4#88311yescharacteristic
407C3×C4D3×C4⋊C411yescharacteristic
408C3×C4#89611yescharacteristic
409C3×C4C2×C3×C4⋊C411yescharacteristic
410C3×C4C3×C4×D411yescharacteristic
411C3×C4C3×C22⋊Q811yescharacteristic
412C3×C4C3×#12311yescharacteristic
413Dic3—24no—
414Dic3—22no—
415Dic3—41no—
416Dic3—41no—
417Dic3—81no—
418C32#185911yescharacteristic, complemented, Sylow
419C23—61no—
420C23C2×C3×D1211yescharacteristic
421Q8—32no—
422Q8—61no—
423D4—32no—
424D4—62no—
425D4—62no—
426D4—61no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—61no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—122no—
438C2×C4—121no—
439C2×C4—242no—
440C2×C4C2×C3×C4×D311yescharacteristic
441C2×C4C3×#23511yescharacteristic
442C2×C4C3×D3×D411yescharacteristic
443C2×C4C3×D3×D411yescharacteristic
444C2×C4C3×#23711yescharacteristic
445C2×C4C3×D3×Q811yescharacteristic
446C8—22no—
447C8—41no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—122no—
458C2×C3#466811yescharacteristic
459C2×C3#474711yescharacteristic
460C2×C3#475211yescharacteristic
461C2×C3C3×#38511yescharacteristic
462C2×C3C3×#45311yescharacteristic
463C2×C3C3×#45611yescharacteristic
464D3—42no—
465C22—21no—
466C22—21no—
467C22—62no—
468C22—121no—
469C22—121no—
470C22C3×C4×D1211yescharacteristic
471C22C3×#88211yescharacteristic
472C22C3×#89711yescharacteristic
473C4—42no—
474C4—64no—
475C4—62no—
476C4—121no—
477C4—121no—
478C4—241no—
479C4C3×#67⋊C411yescharacteristic
480C4C3×#88311yescharacteristic
481C4C3×D3×C4⋊C411yescharacteristic
482C4C3×#89611yescharacteristic
483C3—21no—
484C3#6900211yescharacteristic, direct product factor
485C3C3×#185911yescharacteristic, complemented
486C2—21no—
487C2—21no—
488C2—122no—
489C2C3×#466811yescharacteristic
490C2C3×#474711yescharacteristic
491C2C3×#475211yescharacteristic
subgroupG/H
centerC22×C3#882
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#343C2
socleC22×C32#120
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#343D3
hypocenterC3C3×#1859
LCSG ▷ C2×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
4. C3×#5121C2semidirect
5. C3×#4975C2semidirect
6. C32×#343C2semidirect
7. C3×#4555C2semidirect
9. #69002C3direct (N × H)
23. C3×#825C4semidirect
36. #5121C2×C3semidirect
37. #4975C2×C3semidirect
39. C3×#343D3semidirect
40. C3×#343C2×C3semidirect
42. #4555C2×C3semidirect
59. C32×M16C2×C4semidirect
75. C3×C4×Dic3C2×C4semidirect
111. #825C3×C4semidirect
137. C3×D12C4⋊C4semidirect
142. C3×Dic6C4⋊C4semidirect
150. #343C3×D3semidirect
190. C3×M16C4×D3semidirect
191. C3×M16C2×C3×C4semidirect
227. C4×Dic3C2×C3×C4semidirect
321. D12C3×C4⋊C4semidirect
326. Dic6C3×C4⋊C4semidirect
355. M16C3×C4×D3semidirect
418. C32#1859semidirect
484. C3#69002direct (N × H)
485. C3C3×#1859semidirect

Maximal subgroups

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

subgroupindex
C3×#55042
C2×C3×#8252
C3×#51332
C3×#51212
C3×#49752
C32×#3432
C3×#45552
C3×#18593
#690023