Finite Group Databasebeta

← C3×#68176 #11774728

Subgroups — C3×#68176 #11774728

Order 1152.

593 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5548C211yesmaximal, characteristic
2C3×#4837C211yesmaximal, characteristic, complemented
3C3×C4×D3×C8C211yesmaximal, characteristic, complemented
4C32×D4⋊C8C211yesmaximal, characteristic, complemented
5C3×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×#4483C211yesmaximal, characteristic, complemented
7C3×#4459C211yesmaximal, characteristic, complemented
8C3×#1492—31nomaximal
9#68176C311yesmaximal, characteristic, direct product factor
10C2×C3×#235—21no—
11C32×C4×D4C2211yescharacteristic
12C3×#940C411yescharacteristic
13C3×#938C411yescharacteristic
14C3×C4×#67—21no—
15C2×C3×ΓL(1,𝔽25)—21no—
16C2×C3×D3×C8—21no—
17C3×#895C411yescharacteristic
18C3×#888C411yescharacteristic
19C3×#885—21no—
20C3×#884—21no—
21C3×C4×D12C2211yescharacteristic
22C3×C42×D3C2211yescharacteristic
23C32×C4⋊C8C2211yescharacteristic
24C32×C4×C8C2211yescharacteristic
25C3×Dic3⋊C8—21no—
26C3×C8×Dic3—21no—
27C3×C4⋊(C3⋊C8)C2211yescharacteristic
28C3×C4×C3⋊C8C2211yescharacteristic
29C3×#532—31no—
30#5548C2×C311yescharacteristic
31C3×#423—31no—
32C2×C3×C4×C8—31no—
33#4837C2×C311yescharacteristic, complemented
34C4×D3×C8C2×C311yescharacteristic, complemented
35C3×D4⋊C8—21no—
36C3×D4⋊C8—31no—
37C3×D4⋊C8—31no—
38C3×D4⋊C8—31no—
39C3×D4⋊C8D311yescharacteristic, complemented
40C3×D4⋊C8C2×C311yescharacteristic, complemented
41D4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
42#4483C2×C311yescharacteristic, complemented
43#4459C2×C311yescharacteristic, complemented
44C2×C32×D4C2×C411yescharacteristic
45C22×C32×C4—21no—
46C2×C3×#67—21no—
47C2×C3×#67—21no—
48C2×C3×#67—21no—
49C3×#235—42no—
50C3×#235—42no—
51C2×C3×D12—21no—
52C2×C3×D12C2×C411yescharacteristic
53C2×C3×C4×D3—21no—
54C2×C3×C4×D3C2×C411yescharacteristic
55C2×C3×C4×D3D411yescharacteristic
56C2×C3×C4×D3D411yescharacteristic
57C2×C3×Dic6—21no—
58C2×C32×C8—21no—
59C2×C32×C8—21no—
60C32×C4⋊C4C2×C411yescharacteristic
61C32×C22⋊C4—21no—
62C32×C42C2311yescharacteristic
63C3×#217—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—21no—
66C3×C4⋊Dic3C2×C411yescharacteristic
67C3×Dic3⋊C4—21no—
68C3×C4×Dic3C2×C411yescharacteristic
69C3×C4×Dic3D411yescharacteristic
70C3×C4×Dic3D411yescharacteristic
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8—21no—
73C3×ΓL(1,𝔽25)—42no—
74C3×D3×C8—24no—
75#1492—31noSylow
76C2×C3×#41—61no—
77C2×#235—21no—
78C2×C3×M16—61no—
79C22×C3×C8—61no—
80C3×C4⋊Q8—31no—
81C3×#127—31no—
82C3×#124—61no—
83C3×#121—31no—
84C3×#121—31no—
85C3×#121—61no—
86C3×C4×D4—21no—
87C3×C4×D4—31no—
88C3×C4×D4—61no—
89C3×C4×D4C2×D311yescharacteristic
90C3×C4×D4C22×C311yescharacteristic
91C2×C3×C42—31no—
92#940C3×C411yescharacteristic
93#938C3×C411yescharacteristic
94C4×#67—21no—
95C2×ΓL(1,𝔽25)—21no—
96C2×D3×C8—21no—
97#895C3×C411yescharacteristic
98#888C3×C411yescharacteristic
99#885—21no—
100#884—21no—
101C4×D12C22×C311yescharacteristic
102C42×D3C22×C311yescharacteristic
103C3×C4⋊C8—21no—
104C3×C4⋊C8—31no—
105C3×C4⋊C8—61no—
106C3×C4⋊C8C2×D311yescharacteristic
107C3×C4⋊C8C22×C311yescharacteristic
108C3×C4×C8—21no—
109C3×C4×C8—31no—
110C3×C4×C8—61no—
111C3×C4×C8C2×D311yescharacteristic
112C3×C4×C8C22×C311yescharacteristic
113Dic3⋊C8—21no—
114C8×Dic3—21no—
115C4⋊(C3⋊C8)C22×C311yescharacteristic
116C4×C3⋊C8C22×C311yescharacteristic
117C23×C32—21no—
118C22×C3×D3—21no—
119C22×C3×D3C22⋊C411yescharacteristic
120C32×D4—21no—
121C32×D4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—41no—
124C2×C32×C4—41no—
125C2×C32×C4C22×C411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C3×#67—42no—
129C3×#67—42no—
130C3×#67—42no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3C22⋊C411yescharacteristic
133C2×C3×Dic3C22⋊C411yescharacteristic
134C2×C3×Dic3C22⋊C411yescharacteristic
135C3×D12—21no—
136C3×D12—21no—
137C3×D12—42no—
138C3×C4×D3—21no—
139C3×C4×D3—21no—
140C3×C4×D3—22no—
141C3×C4×D3—41no—
142C3×C4×D3—41no—
143C3×C4×D3D812yes—
144C3×C4×D3SD1612yes—
145C3×Dic6—42no—
146C32×C8—22no—
147C32×C8—41no—
148C3×C3⋊C8—22no—
149C3×C3⋊C8—41no—
150#532—31no—
151#423—31no—
152C2×C4×C8—31no—
153D4⋊C8—31no—
154D4⋊C8—31no—
155D4⋊C8—31no—
156D4⋊C8C3×D311yescharacteristic, complemented
157C3×#41—122no—
158C3×#41—122no—
159C2×C3×Q8—61no—
160C2×C3×D4—21no—
161C2×C3×D4—31no—
162C2×C3×D4—61no—
163C2×C3×D4—61no—
164C2×C3×D4—61no—
165C2×C3×D4—61no—
166C2×C3×D4C4×D311yescharacteristic
167C2×C3×D4C2×C3×C411yescharacteristic
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—31no—
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—41no—
174C22×C3×C4—61no—
175C2×#67—21no—
176C2×#67—21no—
177C2×#67—21no—
178#235—42no—
179#235—42no—
180C2×D12—21no—
181C2×D12C2×C3×C411yescharacteristic
182C2×C4×D3—21no—
183C2×C4×D3C2×C3×C411yescharacteristic
184C2×C4×D3C3×D411yescharacteristic
185C2×C4×D3C3×D411yescharacteristic
186C2×Dic6—21no—
187C3×M16—122no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—41no—
193C2×C3×C8—41no—
194C2×C3×C8—64no—
195C2×C3×C8—61no—
196C2×C3×C8—61no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4C4×D311yescharacteristic
201C3×C4⋊C4C2×C3×C411yescharacteristic
202C3×C22⋊C4—21no—
203C3×C22⋊C4—21no—
204C3×C22⋊C4—41no—
205C3×C22⋊C4—61no—
206C3×C22⋊C4—61no—
207C3×C22⋊C4—61no—
208C3×C42—21no—
209C3×C42—31no—
210C3×C42—31no—
211C3×C42—31no—
212C3×C42C22×D311yescharacteristic
213C3×C42C23×C311yescharacteristic
214#217—21no—
215(C2×D3)⋊C4—21no—
216(C2×D3)⋊C4—21no—
217C4⋊Dic3C2×C3×C411yescharacteristic
218Dic3⋊C4—21no—
219C4×Dic3C2×C3×C411yescharacteristic
220C4×Dic3C3×D411yescharacteristic
221C4×Dic3C3×D411yescharacteristic
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224ΓL(1,𝔽25)—42no—
225D3×C8—24no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32C2×C22⋊C411yescharacteristic
230C2×C3×D3—21no—
231C2×C3×D3—21no—
232C2×C3×D3—41no—
233C2×C3×D3—41no—
234C2×C3×D3—41no—
235C2×C3×D3#10212yes—
236C32×C4—21no—
237C32×C4—41no—
238C32×C4#13111yescharacteristic
239C32×C4#13111yescharacteristic
240C32×C4C2×D811yescharacteristic
241C32×C4C2×SD1611yescharacteristic
242C3×Dic3—21no—
243C3×Dic3—41no—
244C3×Dic3#10212yes—
245C3×Dic3#10412yes—
246C2×#41—61no—
247C2×M16—61no—
248C22×C8—61no—
249C4⋊Q8—31no—
250#127—31no—
251#124—61no—
252#121—31no—
253#121—31no—
254#121—61no—
255C4×D4—31no—
256C4×D4—61no—
257C4×D4C2×C3×D311yescharacteristic
258C2×C42—31no—
259C4⋊C8—31no—
260C4⋊C8—61no—
261C4⋊C8C2×C3×D311yescharacteristic
262C4×C8—31no—
263C4×C8—61no—
264C4×C8C2×C3×D311yescharacteristic
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—31no—
268C23×C3—41no—
269C23×C3—61no—
270C22×D3—21no—
271C22×D3C3×C22⋊C411yescharacteristic
272C3×Q8—122no—
273C3×D4—22no—
274C3×D4—21no—
275C3×D4—21no—
276C3×D4—21no—
277C3×D4—21no—
278C3×D4—41no—
279C3×D4—61no—
280C3×D4—61no—
281C3×D4—122no—
282C3×D4—122no—
283C3×D4—122no—
284C3×D4—122no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—32no—
293C2×C3×C4—32no—
294C2×C3×C4—31no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—62no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—81no—
305C2×C3×C4—81no—
306C2×C3×C4—121no—
307C2×C3×C4—121no—
308C2×C3×C4C2×C4×D311yescharacteristic
309C2×C3×C4D3×D411yescharacteristic
310C2×C3×C4D3×D411yescharacteristic
311C2×C3×C4C22×C3×C411yescharacteristic
312C2×C3×C4C2×C3×D411yescharacteristic
313C2×C3×C4C2×C3×D411yescharacteristic
314#67—42no—
315#67—42no—
316#67—42no—
317C2×Dic3—21no—
318C2×Dic3C3×C22⋊C411yescharacteristic
319C2×Dic3C3×C22⋊C411yescharacteristic
320C2×Dic3C3×C22⋊C411yescharacteristic
321D12—21no—
322D12—21no—
323D12—42no—
324C4×D3—21no—
325C4×D3—21no—
326C4×D3—22no—
327C4×D3—41no—
328C4×D3—41no—
329C4×D3C3×D812yes—
330C4×D3C3×SD1612yes—
331Dic6—42no—
332C3×C8—22no—
333C3×C8—22no—
334C3×C8—42no—
335C3×C8—41no—
336C3×C8—41no—
337C3×C8—62no—
338C3×C8—81no—
339C3×C8—121no—
340C3⋊C8—22no—
341C3⋊C8—41no—
342C2×C32—81no—
343C2×C32#40811yescharacteristic
344C2×C32C2×#10211yescharacteristic
345C2×C32C2×#10411yescharacteristic
346C3×D3—22no—
347C3×D3—81no—
348#41—122no—
349#41—122no—
350C2×Q8—61no—
351C2×D4—31no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4C3×C4×D311yescharacteristic
357C22×C4—21no—
358C22×C4—31no—
359C22×C4—31no—
360C22×C4—31no—
361C22×C4—61no—
362M16—122no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—64no—
366C2×C8—61no—
367C2×C8—61no—
368C4⋊C4—31no—
369C4⋊C4—61no—
370C4⋊C4C3×C4×D311yescharacteristic
371C22⋊C4—21no—
372C22⋊C4—61no—
373C22⋊C4—61no—
374C22⋊C4—61no—
375C42—31no—
376C42—31no—
377C42—31no—
378C42C22×C3×D311yescharacteristic
379C22×C3—21no—
380C22×C3—32no—
381C22×C3—42no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—61no—
389C22×C3—61no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3—121no—
393C22×C3—121no—
394C22×C3—121no—
395C22×C3D3×C22⋊C411yescharacteristic
396C22×C3C2×C3×C22⋊C411yescharacteristic
397C2×D3—21no—
398C2×D3—21no—
399C2×D3—41no—
400C2×D3—41no—
401C2×D3—41no—
402C2×D3C3×#10212yes—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—32no—
410C3×C4—32no—
411C3×C4—41no—
412C3×C4—41no—
413C3×C4—41no—
414C3×C4—61no—
415C3×C4—81no—
416C3×C4—121no—
417C3×C4#90111yescharacteristic
418C3×C4#90711yescharacteristic
419C3×C4D3×D811yescharacteristic
420C3×C4D3×SD1611yescharacteristic
421C3×C4C3×#13111yescharacteristic
422C3×C4C3×#13111yescharacteristic
423C3×C4C2×C3×D811yescharacteristic
424C3×C4C2×C3×SD1611yescharacteristic
425Dic3—21no—
426Dic3—41no—
427Dic3C3×#10212yes—
428Dic3C3×#10412yes—
429C32#149211yescharacteristic, complemented, Sylow
430C23—21no—
431C23—31no—
432C23—61no—
433Q8—122no—
434D4—21no—
435D4—21no—
436D4—61no—
437D4—61no—
438D4—122no—
439D4—122no—
440D4—122no—
441D4—122no—
442C2×C4—21no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—32no—
446C2×C4—32no—
447C2×C4—31no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—62no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4C2×C3×C4×D311yescharacteristic
457C2×C4C3×D3×D411yescharacteristic
458C2×C4C3×D3×D411yescharacteristic
459C8—22no—
460C8—41no—
461C8—62no—
462C8—121no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—62no—
467C2×C3—82no—
468C2×C3—81no—
469C2×C3—81no—
470C2×C3—241no—
471C2×C3#472711yescharacteristic
472C2×C3D3×#10211yescharacteristic
473C2×C3D3×#10411yescharacteristic
474C2×C3C3×#40811yescharacteristic
475C2×C3C2×C3×#10211yescharacteristic
476C2×C3C2×C3×#10411yescharacteristic
477D3—22no—
478D3—81no—
479C22—32no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—61no—
484C22—61no—
485C22—121no—
486C22—121no—
487C22—121no—
488C22C3×D3×C22⋊C411yescharacteristic
489C4—21no—
490C4—32no—
491C4—32no—
492C4—41no—
493C4—61no—
494C4—121no—
495C4C3×#90111yescharacteristic
496C4C3×#90711yescharacteristic
497C4C3×D3×D811yescharacteristic
498C4C3×D3×SD1611yescharacteristic
499C3—21no—
500C3#6817611yescharacteristic, direct product factor
501C3C3×#149211yescharacteristic, complemented
502C2—62no—
503C2—81no—
504C2—241no—
505C2C3×#472711yescharacteristic
506C2C3×D3×#10211yescharacteristic
507C2C3×D3×#10411yescharacteristic
subgroupG/H
centerC2×C3×C4D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC42C22×C3×D3
FittingC32×D4⋊C8C2
socleC22×C32C2×C22⋊C4
prime radicalC3×#5548C2
cosocleC3×C42C23×C3
hypercenterC3×D4⋊C8D3
hypocenterC3C3×#1492
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C3×C42 ◁ C3×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C3×#4837C2semidirect
3. C3×C4×D3×C8C2semidirect
4. C32×D4⋊C8C2semidirect
5. C3×D4⋊(C3⋊C8)C2semidirect
6. C3×#4483C2semidirect
7. C3×#4459C2semidirect
9. #68176C3direct (N × H)
33. #4837C2×C3semidirect
34. C4×D3×C8C2×C3semidirect
39. C3×D4⋊C8D3semidirect
40. C3×D4⋊C8C2×C3semidirect
41. D4⋊(C3⋊C8)C2×C3semidirect
42. #4483C2×C3semidirect
43. #4459C2×C3semidirect
156. D4⋊C8C3×D3semidirect
429. C32#1492semidirect
500. C3#68176direct (N × H)
501. C3C3×#1492semidirect

Maximal subgroups

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

subgroupindex
C3×#55482
C3×#48372
C3×C4×D3×C82
C32×D4⋊C82
C3×D4⋊(C3⋊C8)2
C3×#44832
C3×#44592
C3×#14923
#681763