Finite Group Databasebeta

← C3×#68725 #11788374

Subgroups — C3×#68725 #11788374

Order 1152.

651 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic
2C3×#5745C211yesmaximal, characteristic
3C3×#5139C211yesmaximal, characteristic
4C2×C3×#824C211yesmaximal, characteristic, complemented
5C3×#5120C211yesmaximal, characteristic, complemented
6C32×#338C211yesmaximal, characteristic, complemented
7C3×#4556C211yesmaximal, characteristic, complemented
8C3×#1902—31nomaximal
9#68725C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C3×#926C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#909—22no—
18C3×#909—22no—
19C3×#907—22no—
20C3×D3×M16—22no—
21C2×C3×Dic12C412yescomplemented
22C2×C3×#204C412yescomplemented
23C3×#901—22no—
24C3×#825—22no—
25C3×#824—22no—
26C3×#816—22no—
27C2×C3×#137—31no—
28C3×#568—31no—
29C2×#909C2×C311yescharacteristic
30#5745C2×C311yescharacteristic
31C3×#421—31no—
32C3×#419—31no—
33C2×C3×#101—31no—
34#5139C2×C311yescharacteristic
35C2×#824C2×C311yescharacteristic, complemented
36#5120C2×C311yescharacteristic, complemented
37C3×#344—31no—
38C3×#338—21no—
39C3×#338D311yescharacteristic, complemented
40C3×#338C2×C311yescharacteristic, complemented
41#4556C2×C311yescharacteristic, complemented
42C22×C32×C4C2311yescharacteristic
43C2×C3×#67—21no—
44C22×C3×Dic3—21no—
45C3×#235—21no—
46C3×#235—21no—
47C3×#235D412yes—
48C3×#235D412yes—
49C2×C3×D12C2×C411yescharacteristic
50C2×C3×C4×D3—21no—
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—
60C32×M16—21no—
61C32×M16—21no—
62C32×M16—21no—
63C32×M16—21no—
64C2×C32×C8—21no—
65C2×C32×C8C2×C412yes—
66C2×C32×C8D412yes—
67C3×#217—21no—
68C3×C4⋊Dic3—21no—
69C3×C4×Dic3—21no—
70C3×#208—21no—
71C3×#208D412yes—
72C2×C3×C3⋊C8—21no—
73C3×Dic12—24no—
74C3×#204—24no—
75C3×ΓL(1,𝔽25)—24no—
76C3×D3×C8—24no—
77#1902—31noSylow
78C2×C3×#41—31no—
79C22×C3×Q8—31no—
80C2×#235C22×C311yescharacteristic
81C22×Dic6C22×C311yescharacteristic
82C3×#137—62no—
83C3×#137—62no—
84C2×C3×Q16—32no—
85C2×C3×SD16—32no—
86C3×#131—62no—
87C3×#131—62no—
88C2×C3×M16—21no—
89C2×C3×M16—21no—
90C2×C3×M16—21no—
91C2×C3×M16—31no—
92C2×C3×M16—62no—
93C2×C3×M16C2×D311yescharacteristic
94C2×C3×M16C2×D311yescharacteristic
95C2×C3×M16C2×D311yescharacteristic
96C2×C3×M16C22×C311yescharacteristic
97C2×C3×M16C22×C311yescharacteristic
98C2×C3×M16C22×C311yescharacteristic
99C3×#117—31no—
100#926C22×C311yescharacteristic
101C2×#208C22×C311yescharacteristic
102#909—22no—
103#909—22no—
104#907—22no—
105D3×M16—22no—
106C2×Dic12C3×C412yescomplemented
107C2×#204C3×C412yescomplemented
108#901—22no—
109C3×#104—62no—
110C3×#103—62no—
111C3×#101—62no—
112#825—22no—
113#824—22no—
114#816—22no—
115C23×C32#4111yescharacteristic
116C22×C3×D3—21no—
117C2×C32×C4C22×C411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4#4111yescharacteristic
123C3×#67—22no—
124C3×#67—22no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—41no—
128C2×C3×Dic3—41no—
129C2×C3×Dic3—41no—
130C3×D12—21no—
131C3×D12C22⋊C412yes—
132C3×C4×D3—22no—
133C3×C4×D3—22no—
134C3×Dic6—21no—
135C3×Dic6—22no—
136C3×Dic6—22no—
137C3×Dic6—41no—
138C3×Dic6—41no—
139C3×Dic6C22⋊C412yes—
140C32×C8—22no—
141C32×C8—22no—
142C32×C8—41no—
143C3×C3⋊C8—22no—
144C2×#137—31no—
145#568—31no—
146#421—31no—
147#419—31no—
148C2×#101—31no—
149#344—31no—
150#338C3×D311yescharacteristic, complemented
151C3×#41—32no—
152C3×#41—32no—
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—31no—
163C2×C3×D4—61no—
164C22×C3×C4—21no—
165C22×C3×C4—61no—
166C22×C3×C4—61no—
167C22×C3×C4C22×D311yescharacteristic
168C22×C3×C4C23×C311yescharacteristic
169C2×#67—21no—
170C22×Dic3—21no—
171#235—21no—
172#235—21no—
173#235C3×D412yes—
174#235C3×D412yes—
175C2×D12C2×C3×C411yescharacteristic
176C2×C4×D3—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6C2×C3×C412yes—
183C2×Dic6C2×C3×C411yescharacteristic
184C3×Q16—64no—
185C3×SD16—64no—
186C3×M16—21no—
187C3×M16—21no—
188C3×M16—21no—
189C3×M16—21no—
190C3×M16—21no—
191C3×M16—21no—
192C3×M16—21no—
193C3×M16—21no—
194C3×M16—21no—
195C3×M16—21no—
196C3×M16—21no—
197C3×M16—21no—
198C3×M16—32no—
199C3×M16—41no—
200C3×M16—41no—
201C3×M16—41no—
202C3×M16—41no—
203C3×M16—41no—
204C3×M16—41no—
205C3×M16—64no—
206C3×M16—61no—
207C2×C3×C8—22no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—64no—
213C2×C3×C8—61no—
214C2×C3×C8C4×D312yes—
215C2×C3×C8D1212yes—
216C2×C3×C8C2×C3×C412yes—
217C2×C3×C8C3×D412yes—
218C3×C4⋊C4—61no—
219C3×C22⋊C4—61no—
220C3×C42—61no—
221#217—21no—
222C4⋊Dic3—21no—
223C4×Dic3—21no—
224#208—21no—
225#208C3×D412yes—
226C2×C3⋊C8—21no—
227Dic12—24no—
228#204—24no—
229ΓL(1,𝔽25)—24no—
230D3×C8—24no—
231C22×C32—41no—
232C22×C32C4×D411yescharacteristic
233C22×C32#12111yescharacteristic
234C22×C32#12311yescharacteristic
235C2×C3×D3—22no—
236C2×C3×D3—81no—
237C32×C4C2×C22⋊C411yescharacteristic
238C32×C4C4×D411yescharacteristic
239C32×C4#12011yescharacteristic
240C32×C4#12111yescharacteristic
241C3×Dic3—22no—
242C3×Dic3—42no—
243C3×Dic3—81no—
244C2×#41—31no—
245C22×Q8—31no—
246#137—62no—
247#137—62no—
248C2×Q16—32no—
249C2×SD16—32no—
250#131—62no—
251#131—62no—
252C2×M16—31no—
253C2×M16—62no—
254C2×M16C2×C3×D311yescharacteristic
255C2×M16C2×C3×D311yescharacteristic
256C2×M16C2×C3×D311yescharacteristic
257#117—31no—
258#104—62no—
259#103—62no—
260#101—62no—
261C23×C3—21no—
262C23×C3—61no—
263C23×C3#23911yescharacteristic
264C23×C3C3×#4111yescharacteristic
265C22×D3—21no—
266C3×Q8—32no—
267C3×Q8—61no—
268C3×Q8—62no—
269C3×Q8—62no—
270C3×Q8—121no—
271C3×Q8—121no—
272C3×D4—32no—
273C3×D4—62no—
274C3×D4—62no—
275C3×D4—61no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—61no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—121no—
287C2×C3×C4—121no—
288C2×C3×C4—121no—
289C2×C3×C4C2×C4×D311yescharacteristic
290C2×C3×C4C2×D1211yescharacteristic
291C2×C3×C4#23511yescharacteristic
292C2×C3×C4D3×D411yescharacteristic
293C2×C3×C4D3×D411yescharacteristic
294C2×C3×C4D3×D411yescharacteristic
295C2×C3×C4C22×C3×C411yescharacteristic
296C2×C3×C4C2×C3×D411yescharacteristic
297C2×C3×C4C2×C3×D411yescharacteristic
298C2×C3×C4C2×C3×D411yescharacteristic
299C2×C3×C4C2×C3×D411yescharacteristic
300C2×C3×C4C3×#4111yescharacteristic
301#67—22no—
302#67—22no—
303C2×Dic3—21no—
304C2×Dic3—22no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308D12—21no—
309D12C3×C22⋊C412yes—
310C4×D3—22no—
311C4×D3—22no—
312Dic6—21no—
313Dic6—22no—
314Dic6—22no—
315Dic6—41no—
316Dic6—41no—
317Dic6C3×C22⋊C412yes—
318C3×C8—22no—
319C3×C8—22no—
320C3×C8—22no—
321C3×C8—22no—
322C3×C8—42no—
323C3×C8—42no—
324C3×C8—42no—
325C3×C8—41no—
326C3×C8—41no—
327C3×C8—62no—
328C3⋊C8—22no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32#38611yescharacteristic
333C3×D3—42no—
334#41—32no—
335#41—32no—
336#41—61no—
337#41—61no—
338C2×Q8—31no—
339C2×Q8—32no—
340C2×Q8—61no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×D4—31no—
346C2×D4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4C22×C3×D311yescharacteristic
350Q16—64no—
351SD16—64no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—21no—
356M16—21no—
357M16—21no—
358M16—32no—
359M16—64no—
360M16—61no—
361C2×C8—21no—
362C2×C8—64no—
363C2×C8—61no—
364C2×C8C3×C4×D312yes—
365C2×C8C3×D1212yes—
366C4⋊C4—61no—
367C22⋊C4—61no—
368C42—61no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—62no—
375C22×C3—81no—
376C22×C3—241no—
377C22×C3#89311yescharacteristic
378C22×C3#89411yescharacteristic
379C22×C3#89511yescharacteristic
380C22×C3C3×C4×D411yescharacteristic
381C22×C3C3×#12111yescharacteristic
382C22×C3C3×#12311yescharacteristic
383C2×D3—22no—
384C2×D3—81no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—62no—
390C3×C4—122no—
391C3×C4—241no—
392C3×C4C4×D1211yescharacteristic
393C3×C4D3×C22⋊C411yescharacteristic
394C3×C4#88211yescharacteristic
395C3×C4#88411yescharacteristic
396C3×C4C2×C3×C22⋊C411yescharacteristic
397C3×C4C3×C4×D411yescharacteristic
398C3×C4C3×#12011yescharacteristic
399C3×C4C3×#12111yescharacteristic
400Dic3—22no—
401Dic3—42no—
402Dic3—81no—
403C32#190211yescharacteristic, complemented, Sylow
404C23—61no—
405C23C3×#23911yescharacteristic
406Q8—32no—
407Q8—61no—
408Q8—62no—
409Q8—62no—
410Q8—121no—
411Q8—121no—
412D4—32no—
413D4—62no—
414D4—62no—
415D4—61no—
416C2×C4—61no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4C2×C3×C4×D311yescharacteristic
424C2×C4C2×C3×D1211yescharacteristic
425C2×C4C3×#23511yescharacteristic
426C2×C4C3×D3×D411yescharacteristic
427C2×C4C3×D3×D411yescharacteristic
428C2×C4C3×D3×D411yescharacteristic
429C8—22no—
430C8—22no—
431C8—41no—
432C8—62no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3—122no—
444C2×C3#466511yescharacteristic
445C2×C3C3×#38611yescharacteristic
446D3—42no—
447C22—41no—
448C22—62no—
449C22—241no—
450C22C3×#89311yescharacteristic
451C22C3×#89411yescharacteristic
452C22C3×#89511yescharacteristic
453C4—62no—
454C4—122no—
455C4—241no—
456C4C3×C4×D1211yescharacteristic
457C4C3×D3×C22⋊C411yescharacteristic
458C4C3×#88211yescharacteristic
459C4C3×#88411yescharacteristic
460C3—21no—
461C3#6872511yescharacteristic, direct product factor
462C3C3×#190211yescharacteristic, complemented
463C2—21no—
464C2—21no—
465C2—21no—
466C2—122no—
467C2C3×#466511yescharacteristic
subgroupG/H
centerC2×C3#4665
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#338C2
socleC2×C32#386
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#338D3
hypocenterC3C3×#1902
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#338

Complements

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

NH ≅ G/Ntype
4. C2×C3×#824C2semidirect
5. C3×#5120C2semidirect
6. C32×#338C2semidirect
7. C3×#4556C2semidirect
9. #68725C3direct (N × H)
21. C2×C3×Dic12C4semidirect
22. C2×C3×#204C4semidirect
35. C2×#824C2×C3semidirect
36. #5120C2×C3semidirect
39. C3×#338D3semidirect
40. C3×#338C2×C3semidirect
41. #4556C2×C3semidirect
106. C2×Dic12C3×C4semidirect
107. C2×#204C3×C4semidirect
150. #338C3×D3semidirect
403. C32#1902semidirect
461. C3#68725direct (N × H)
462. C3C3×#1902semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C3×#57452
C3×#51392
C2×C3×#8242
C3×#51202
C32×#3382
C3×#45562
C3×#19023
#687253