Finite Group Databasebeta

← C3×#68240 #11774757

Subgroups — C3×#68240 #11774757

Order 1152.

499 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5578C211yesmaximal, characteristic
2C3×#4837C211yesmaximal, characteristic
3C3×C4×D3×C8C211yesmaximal, characteristic
4C32×Q8⋊C8C211yesmaximal, characteristic, complemented
5C3×#4535C211yesmaximal, characteristic, complemented
6C3×#4484C211yesmaximal, characteristic, complemented
7C3×#4456C211yesmaximal, characteristic, complemented
8C3×#1493—31nomaximal
9#68240C311yesmaximal, characteristic, direct product factor
10C32×C4×Q8C2211yescharacteristic
11C3×#946C411yescharacteristic
12C3×#944C411yescharacteristic
13C2×C3×ΓL(1,𝔽25)—21no—
14C2×C3×D3×C8—21no—
15C3×#897C411yescharacteristic
16C3×#896—21no—
17C3×#889—21no—
18C3×#888C411yescharacteristic
19C3×#872—21no—
20C3×C42×D3C2211yescharacteristic
21C3×C4×Dic6—21no—
22C3×C4×Dic6C2211yescharacteristic
23C32×C4⋊C8C2211yescharacteristic
24C32×C4×C8C2211yescharacteristic
25C3×Dic3⋊C8—21no—
26C3×C8×Dic3—21no—
27C3×C4⋊(C3⋊C8)C2211yescharacteristic
28C3×C4×C3⋊C8C2211yescharacteristic
29C3×#533—31no—
30#5578C2×C311yescharacteristic
31C3×#423—31no—
32C2×C3×C4×C8—31no—
33#4837C2×C311yescharacteristic
34C4×D3×C8C2×C311yescharacteristic
35C3×Q8⋊C8—21no—
36C3×Q8⋊C8—31no—
37C3×Q8⋊C8—31no—
38C3×Q8⋊C8—31no—
39C3×Q8⋊C8D311yescharacteristic, complemented
40C3×Q8⋊C8C2×C311yescharacteristic, complemented
41#4535C2×C311yescharacteristic, complemented
42#4484C2×C311yescharacteristic, complemented
43#4456C2×C311yescharacteristic, complemented
44C2×C32×Q8C2×C411yescharacteristic
45C2×C3×C4×D3C2×C411yescharacteristic
46C2×C3×C4×D3D411yescharacteristic
47C2×C3×C4×D3D411yescharacteristic
48C2×C3×Dic6—21no—
49C2×C3×Dic6C2×C411yescharacteristic
50C2×C32×C8—21no—
51C2×C32×C8—21no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4C2×C411yescharacteristic
54C32×C42—21no—
55C32×C42C2311yescharacteristic
56C3×(C2×D3)⋊C4—21no—
57C3×(C2×D3)⋊C4—21no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3C2×C411yescharacteristic
60C3×Dic3⋊C4—21no—
61C3×Dic3⋊C4—21no—
62C3×Dic3⋊C4—21no—
63C3×Dic3⋊C4—21no—
64C3×Dic3⋊C4—21no—
65C3×C4×Dic3—21no—
66C3×C4×Dic3C2×C411yescharacteristic
67C3×C4×Dic3D411yescharacteristic
68C3×C4×Dic3D411yescharacteristic
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—21no—
71C3×ΓL(1,𝔽25)—42no—
72C3×D3×C8—24no—
73#1493—31noSylow
74C2×C3×M16—61no—
75C22×C3×C8—61no—
76C3×C4⋊Q8—31no—
77C3×C4⋊Q8—31no—
78C3×#125—61no—
79C3×C22⋊Q8—31no—
80C3×C22⋊Q8—31no—
81C3×C22⋊Q8—61no—
82C3×C4×Q8—21no—
83C3×C4×Q8—31no—
84C3×C4×Q8—61no—
85C3×C4×Q8C2×D311yescharacteristic
86C3×C4×Q8C22×C311yescharacteristic
87C3×#117—61no—
88C2×C3×C42—31no—
89#946C3×C411yescharacteristic
90#944C3×C411yescharacteristic
91C2×ΓL(1,𝔽25)—21no—
92C2×D3×C8—21no—
93#897C3×C411yescharacteristic
94#896—21no—
95#889—21no—
96#888C3×C411yescharacteristic
97#872—21no—
98C42×D3C22×C311yescharacteristic
99C4×Dic6—21no—
100C4×Dic6C22×C311yescharacteristic
101C3×C4⋊C8—21no—
102C3×C4⋊C8—31no—
103C3×C4⋊C8—61no—
104C3×C4⋊C8C2×D311yescharacteristic
105C3×C4⋊C8C22×C311yescharacteristic
106C3×C4×C8—21no—
107C3×C4×C8—31no—
108C3×C4×C8—61no—
109C3×C4×C8C2×D311yescharacteristic
110C3×C4×C8C22×C311yescharacteristic
111Dic3⋊C8—21no—
112C8×Dic3—21no—
113C4⋊(C3⋊C8)C22×C311yescharacteristic
114C4×C3⋊C8C22×C311yescharacteristic
115C22×C3×D3C22⋊C411yescharacteristic
116C32×Q8—21no—
117C32×Q8—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4C22×C411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3C22⋊C411yescharacteristic
126C2×C3×Dic3C22⋊C411yescharacteristic
127C2×C3×Dic3C22⋊C411yescharacteristic
128C3×C4×D3—21no—
129C3×C4×D3—21no—
130C3×C4×D3—22no—
131C3×C4×D3SD1612yes—
132C3×C4×D3Q1612yes—
133C3×Dic6—21no—
134C3×Dic6—21no—
135C3×Dic6—42no—
136C32×C8—22no—
137C32×C8—41no—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—41no—
140#533—31no—
141#423—31no—
142C2×C4×C8—31no—
143Q8⋊C8—31no—
144Q8⋊C8—31no—
145Q8⋊C8—31no—
146Q8⋊C8C3×D311yescharacteristic, complemented
147C2×C3×Q8—21no—
148C2×C3×Q8—31no—
149C2×C3×Q8—61no—
150C2×C3×Q8C4×D311yescharacteristic
151C2×C3×Q8C2×C3×C411yescharacteristic
152C22×C3×C4—31no—
153C22×C3×C4—31no—
154C22×C3×C4—31no—
155C2×C4×D3C2×C3×C411yescharacteristic
156C2×C4×D3C3×D411yescharacteristic
157C2×C4×D3C3×D411yescharacteristic
158C2×Dic6—21no—
159C2×Dic6C2×C3×C411yescharacteristic
160C3×M16—122no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—41no—
166C2×C3×C8—41no—
167C2×C3×C8—64no—
168C2×C3×C8—61no—
169C2×C3×C8—61no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—41no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4—61no—
177C3×C4⋊C4—61no—
178C3×C4⋊C4—61no—
179C3×C4⋊C4—61no—
180C3×C4⋊C4—61no—
181C3×C4⋊C4C4×D311yescharacteristic
182C3×C4⋊C4C2×C3×C411yescharacteristic
183C3×C22⋊C4—61no—
184C3×C22⋊C4—61no—
185C3×C42—21no—
186C3×C42—21no—
187C3×C42—21no—
188C3×C42—31no—
189C3×C42—31no—
190C3×C42—31no—
191C3×C42—41no—
192C3×C42—61no—
193C3×C42C22×D311yescharacteristic
194C3×C42C23×C311yescharacteristic
195(C2×D3)⋊C4—21no—
196(C2×D3)⋊C4—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3C2×C3×C411yescharacteristic
199Dic3⋊C4—21no—
200Dic3⋊C4—21no—
201Dic3⋊C4—21no—
202Dic3⋊C4—21no—
203Dic3⋊C4—21no—
204C4×Dic3—21no—
205C4×Dic3C2×C3×C411yescharacteristic
206C4×Dic3C3×D411yescharacteristic
207C4×Dic3C3×D411yescharacteristic
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210ΓL(1,𝔽25)—42no—
211D3×C8—24no—
212C22×C32C2×C22⋊C411yescharacteristic
213C2×C3×D3—21no—
214C2×C3×D3—21no—
215C2×C3×D3#10312yes—
216C32×C4—21no—
217C32×C4—41no—
218C32×C4—41no—
219C32×C4#13111yescharacteristic
220C32×C4#13111yescharacteristic
221C32×C4C2×SD1611yescharacteristic
222C32×C4C2×Q1611yescharacteristic
223C3×Dic3—21no—
224C3×Dic3—41no—
225C3×Dic3—41no—
226C3×Dic3#10312yes—
227C3×Dic3#10412yes—
228C2×M16—61no—
229C22×C8—61no—
230C4⋊Q8—31no—
231C4⋊Q8—31no—
232#125—61no—
233C22⋊Q8—31no—
234C22⋊Q8—31no—
235C22⋊Q8—61no—
236C4×Q8—31no—
237C4×Q8—61no—
238C4×Q8C2×C3×D311yescharacteristic
239#117—61no—
240C2×C42—31no—
241C4⋊C8—31no—
242C4⋊C8—61no—
243C4⋊C8C2×C3×D311yescharacteristic
244C4×C8—31no—
245C4×C8—61no—
246C4×C8C2×C3×D311yescharacteristic
247C23×C3—31no—
248C22×D3C3×C22⋊C411yescharacteristic
249C3×Q8—22no—
250C3×Q8—21no—
251C3×Q8—21no—
252C3×Q8—21no—
253C3×Q8—21no—
254C3×Q8—41no—
255C3×Q8—61no—
256C3×Q8—61no—
257C3×Q8—122no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—32no—
268C2×C3×C4—32no—
269C2×C3×C4—31no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—62no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4C2×C4×D311yescharacteristic
278C2×C3×C4D3×D411yescharacteristic
279C2×C3×C4D3×D411yescharacteristic
280C2×C3×C4C22×C3×C411yescharacteristic
281C2×C3×C4C2×C3×D411yescharacteristic
282C2×C3×C4C2×C3×D411yescharacteristic
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3C3×C22⋊C411yescharacteristic
286C2×Dic3C3×C22⋊C411yescharacteristic
287C2×Dic3C3×C22⋊C411yescharacteristic
288C4×D3—21no—
289C4×D3—21no—
290C4×D3—22no—
291C4×D3C3×SD1612yes—
292C4×D3C3×Q1612yes—
293Dic6—21no—
294Dic6—21no—
295Dic6—42no—
296C3×C8—22no—
297C3×C8—22no—
298C3×C8—42no—
299C3×C8—41no—
300C3×C8—41no—
301C3×C8—62no—
302C3×C8—81no—
303C3×C8—121no—
304C3⋊C8—22no—
305C3⋊C8—41no—
306C2×C32#40811yescharacteristic
307C2×C32C2×#10311yescharacteristic
308C2×C32C2×#10411yescharacteristic
309C3×D3—22no—
310C2×Q8—31no—
311C2×Q8—61no—
312C2×Q8C3×C4×D311yescharacteristic
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—31no—
316M16—122no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—64no—
320C2×C8—61no—
321C2×C8—61no—
322C4⋊C4—21no—
323C4⋊C4—31no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4C3×C4×D311yescharacteristic
331C22⋊C4—61no—
332C22⋊C4—61no—
333C42—21no—
334C42—31no—
335C42—31no—
336C42—31no—
337C42—61no—
338C42C22×C3×D311yescharacteristic
339C22×C3—21no—
340C22×C3—32no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3D3×C22⋊C411yescharacteristic
344C22×C3C2×C3×C22⋊C411yescharacteristic
345C2×D3—21no—
346C2×D3—21no—
347C2×D3C3×#10312yes—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—32no—
355C3×C4—32no—
356C3×C4—41no—
357C3×C4—42no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—61no—
363C3×C4—81no—
364C3×C4—121no—
365C3×C4—121no—
366C3×C4#90111yescharacteristic
367C3×C4#90711yescharacteristic
368C3×C4D3×SD1611yescharacteristic
369C3×C4D3×Q1611yescharacteristic
370C3×C4C3×#13111yescharacteristic
371C3×C4C3×#13111yescharacteristic
372C3×C4C2×C3×SD1611yescharacteristic
373C3×C4C2×C3×Q1611yescharacteristic
374Dic3—21no—
375Dic3—41no—
376Dic3—41no—
377Dic3C3×#10312yes—
378Dic3C3×#10412yes—
379C32#149311yescharacteristic, complemented, Sylow
380C23—31no—
381Q8—21no—
382Q8—21no—
383Q8—61no—
384Q8—61no—
385Q8—122no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—32no—
391C2×C4—32no—
392C2×C4—31no—
393C2×C4—62no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4C2×C3×C4×D311yescharacteristic
399C2×C4C3×D3×D411yescharacteristic
400C2×C4C3×D3×D411yescharacteristic
401C8—22no—
402C8—41no—
403C8—62no—
404C8—121no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—62no—
409C2×C3#472711yescharacteristic
410C2×C3D3×#10311yescharacteristic
411C2×C3D3×#10411yescharacteristic
412C2×C3C3×#40811yescharacteristic
413C2×C3C2×C3×#10311yescharacteristic
414C2×C3C2×C3×#10411yescharacteristic
415D3—22no—
416C22—32no—
417C22—61no—
418C22—61no—
419C22C3×D3×C22⋊C411yescharacteristic
420C4—21no—
421C4—32no—
422C4—32no—
423C4—41no—
424C4—41no—
425C4—61no—
426C4—121no—
427C4—121no—
428C4C3×#90111yescharacteristic
429C4C3×#90711yescharacteristic
430C4C3×D3×SD1611yescharacteristic
431C4C3×D3×Q1611yescharacteristic
432C3—21no—
433C3#6824011yescharacteristic, direct product factor
434C3C3×#149311yescharacteristic, complemented
435C2—62no—
436C2C3×#472711yescharacteristic
437C2C3×D3×#10311yescharacteristic
438C2C3×D3×#10411yescharacteristic
subgroupG/H
centerC2×C3×C4D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC42C22×C3×D3
FittingC32×Q8⋊C8C2
socleC22×C32C2×C22⋊C4
prime radicalC22×C3×D3C22⋊C4
cosocleC3×C42C23×C3
hypercenterC3×Q8⋊C8D3
hypocenterC3C3×#1493
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C3×C42 ◁ C3×Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. C32×Q8⋊C8C2semidirect
5. C3×#4535C2semidirect
6. C3×#4484C2semidirect
7. C3×#4456C2semidirect
9. #68240C3direct (N × H)
39. C3×Q8⋊C8D3semidirect
40. C3×Q8⋊C8C2×C3semidirect
41. #4535C2×C3semidirect
42. #4484C2×C3semidirect
43. #4456C2×C3semidirect
146. Q8⋊C8C3×D3semidirect
379. C32#1493semidirect
433. C3#68240direct (N × H)
434. C3C3×#1493semidirect

Maximal subgroups

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

subgroupindex
C3×#55782
C3×#48372
C3×C4×D3×C82
C32×Q8⋊C82
C3×#45352
C3×#44842
C3×#44562
C3×#14933
#682403