Finite Group Databasebeta

← C3×#81056 #11897633

Subgroups — C3×#81056 #11897633

Order 1152.

734 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5722C211yesmaximal, characteristic, complemented
2C3×#5543C211yesmaximal, characteristic
3C32×#476C211yesmaximal, characteristic
4C3×#5078C211yesmaximal, characteristic, complemented
5C3×C4×#214C211yesmaximal, characteristic
6C3×#4857C212yesmaximal, complemented
7C3×#4856C212yesmaximal, complemented
8C3×#4850C211yesmaximal, characteristic, complemented
9C3×#4831C211yesmaximal, characteristic, complemented
10C3×#4759C212yesmaximal
11C3×#4757C212yesmaximal
12C3×#3385—31nomaximal
13#81056C311yesmaximal, characteristic, direct product factor
14C32×C4⋊Q8C2211yescharacteristic
15C32×C4×D4C2211yescharacteristic
16C3×#944—22no—
17C3×D4×Dic3C2212yes—
18C2×C3×#214C2211yescharacteristic
19C2×C3×C4⋊Dic3—21no—
20C3×#924—21no—
21C3×#890C2212yes—
22C3×#888—21no—
23C3×#888C2212yes—
24C3×#887—21no—
25C3×#887C2212yes—
26C3×#878—22no—
27C3×#869C2211yescharacteristic
28C3×C4×Dic6C2211yescharacteristic
29C3×C4×Dic6C2211yescharacteristic
30C32×#106C2212yes—
31C32×C4⋊C8C2211yescharacteristic
32C32×#102C2212yes—
33C3×#832C2211yescharacteristic
34C3×#818C2212yes—
35C3×#816C2212yes—
36C3×#815C2212yes—
37C3×Dic3⋊C8C2212yes—
38C3×#810C2211yescharacteristic
39C3×#808C2211yescharacteristic
40C3×#807C2212yes—
41C3×#807C2212yes—
42C3×C4×C3⋊C8C2211yescharacteristic
43C3×Q82—31no—
44C3×#554—31no—
45#5722C2×C311yescharacteristic, complemented
46#5543C2×C311yescharacteristic
47C3×#501—32no—
48C3×#500—31no—
49C3×#479—32no—
50C3×#477—31no—
51C3×#476—21no—
52C3×#476D311yescharacteristic
53C3×#476C2×C311yescharacteristic
54C3×#474—32no—
55C3×#446—31no—
56C3×#440—32no—
57C3×C4×SD16—31no—
58#5078C2×C311yescharacteristic, complemented
59C4×#214C2×C311yescharacteristic
60#4857C2×C312yescomplemented
61#4856C2×C312yescomplemented
62#4850C2×C311yescharacteristic, complemented
63#4831C2×C311yescharacteristic, complemented
64#4759C2×C312yes—
65#4757C2×C312yes—
66C2×C32×Q8—21no—
67C2×C32×D4C2311yescharacteristic
68C22×C32×C4—21no—
69C22×C3×Dic3—22no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6—22no—
72C2×C3×Dic6D411yescharacteristic
73C2×C3×Dic6C2311yescharacteristic
74C2×C32×C8C2312yes—
75C32×C4⋊C4—21no—
76C32×C4⋊C4C2312yes—
77C32×C4⋊C4C2311yescharacteristic
78C32×C22⋊C4—21no—
79C32×C42C2311yescharacteristic
80C3×#217—22no—
81C3×#214Q814yescomplemented
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—22no—
84C3×C4⋊Dic3D411yescharacteristic
85C3×C4⋊Dic3C2311yescharacteristic
86C3×C4⋊Dic3C2312yes—
87C3×Dic3⋊C4—21no—
88C3×Dic3⋊C4—22no—
89C3×Dic3⋊C4—22no—
90C3×Dic3⋊C4—22no—
91C3×Dic3⋊C4D412yes—
92C3×C4×Dic3—21no—
93C3×C4×Dic3—22no—
94C3×C4×Dic3C2312yes—
95C2×C3×C3⋊C8C2311yescharacteristic
96C2×C3×C3⋊C8C2311yescharacteristic
97#3385—31noSylow
98C2×C3×SD16—31no—
99C3×C4⋊Q8—21no—
100C3×C4⋊Q8—32no—
101C3×C4⋊Q8—31no—
102C3×C4⋊Q8—61no—
103C3×C4⋊Q8—62no—
104C3×C4⋊Q8C2×D311yescharacteristic
105C3×C4⋊Q8C22×C311yescharacteristic
106C3×#125—32no—
107C3×C22⋊Q8—62no—
108C3×C22⋊Q8—61no—
109C3×C4×Q8—31no—
110C3×C4×Q8—32no—
111C3×C4×Q8—31no—
112C3×C4×Q8—61no—
113C3×C4×D4—21no—
114C3×C4×D4—32no—
115C3×C4×D4C2×D311yescharacteristic
116C3×C4×D4C22×C311yescharacteristic
117C2×C3×C4⋊C4—61no—
118#944—22no—
119D4×Dic3C22×C312yes—
120C2×#214C22×C311yescharacteristic
121C2×C4⋊Dic3—21no—
122#924—21no—
123#890C22×C312yes—
124#888—21no—
125#888C22×C312yes—
126#887—21no—
127#887C22×C312yes—
128#878—22no—
129#869C22×C311yescharacteristic
130C4×Dic6C22×C311yescharacteristic
131C4×Dic6C22×C311yescharacteristic
132C3×#107—32no—
133C3×#107—32no—
134C3×#107—32no—
135C3×#106—22no—
136C3×#106—31no—
137C3×#106C2×D312yes—
138C3×#106C22×C312yes—
139C3×C4⋊C8—21no—
140C3×C4⋊C8—32no—
141C3×C4⋊C8C2×D311yescharacteristic
142C3×C4⋊C8C22×C311yescharacteristic
143C3×#103—32no—
144C3×#103—31no—
145C3×#102—22no—
146C3×#102—31no—
147C3×#102C2×D312yes—
148C3×#102C22×C312yes—
149C3×#97—32no—
150C3×C4×C8—31no—
151#832C22×C311yescharacteristic
152#818C22×C312yes—
153#816C22×C312yes—
154#815C22×C312yes—
155Dic3⋊C8C22×C312yes—
156#810C22×C311yescharacteristic
157#808C22×C311yescharacteristic
158#807C22×C312yes—
159#807C22×C312yes—
160C4×C3⋊C8C22×C311yescharacteristic
161C23×C32—21no—
162C32×Q8—81no—
163C32×D4—21no—
164C32×D4C2×Q812yes—
165C2×C32×C4—22no—
166C2×C32×C4—21no—
167C2×C32×C4—42no—
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2411yescharacteristic
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—82no—
175C2×C3×Dic3C2×D412yes—
176C2×C3×Dic3C2×D412yes—
177C3×Dic6—22no—
178C3×Dic6—21no—
179C3×Dic6—82no—
180C3×Dic6—81no—
181C3×Dic6C2×Q812yes—
182C32×C8—22no—
183C3×C3⋊C8—21no—
184C3×C3⋊C8C2×Q812yes—
185Q82—31no—
186#554—31no—
187#501—32no—
188#500—31no—
189#479—32no—
190#477—31no—
191#476C3×D311yescharacteristic
192#474—32no—
193#446—31no—
194#440—32no—
195C4×SD16—31no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—31no—
199C2×C3×Q8—31no—
200C2×C3×Q8—41no—
201C2×C3×Q8—62no—
202C2×C3×Q8—61no—
203C2×C3×D4—21no—
204C2×C3×D4C22×D311yescharacteristic
205C2×C3×D4C23×C311yescharacteristic
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—41no—
209C22×C3×C4—62no—
210C22×Dic3—22no—
211C2×Dic6—21no—
212C2×Dic6—22no—
213C2×Dic6C3×D411yescharacteristic
214C2×Dic6C23×C311yescharacteristic
215C3×SD16—34no—
216C2×C3×C8—22no—
217C2×C3×C8—31no—
218C2×C3×C8—31no—
219C2×C3×C8C22×D312yes—
220C2×C3×C8C23×C312yes—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—32no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—41no—
230C3×C4⋊C4—62no—
231C3×C4⋊C4—62no—
232C3×C4⋊C4—62no—
233C3×C4⋊C4—62no—
234C3×C4⋊C4—61no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4C22×D312yes—
237C3×C4⋊C4C22×D311yescharacteristic
238C3×C4⋊C4C23×C312yes—
239C3×C4⋊C4C23×C311yescharacteristic
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—41no—
243C3×C22⋊C4—62no—
244C3×C42—21no—
245C3×C42—32no—
246C3×C42—62no—
247C3×C42—61no—
248C3×C42C22×D311yescharacteristic
249C3×C42C23×C311yescharacteristic
250#217—22no—
251#214C3×Q814yescomplemented
252C4⋊Dic3—21no—
253C4⋊Dic3—22no—
254C4⋊Dic3C3×D411yescharacteristic
255C4⋊Dic3C23×C312yes—
256C4⋊Dic3C23×C311yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—22no—
259Dic3⋊C4—22no—
260Dic3⋊C4—22no—
261Dic3⋊C4C3×D412yes—
262C4×Dic3—21no—
263C4×Dic3—22no—
264C4×Dic3C23×C312yes—
265C2×C3⋊C8C23×C311yescharacteristic
266C2×C3⋊C8C23×C311yescharacteristic
267C22×C32—22no—
268C22×C32—41no—
269C22×C32—41no—
270C22×C32C22×D411yescharacteristic
271C32×C4—21no—
272C32×C4—41no—
273C32×C4—42no—
274C32×C4#13712yes—
275C32×C4C22×Q811yescharacteristic
276C32×C42−1+411yescharacteristic
277C3×Dic3—22no—
278C3×Dic3—22no—
279C3×Dic3—22no—
280C3×Dic3—41no—
281C3×Dic3—42no—
282C2×SD16—31no—
283C4⋊Q8—32no—
284C4⋊Q8—31no—
285C4⋊Q8—61no—
286C4⋊Q8—62no—
287C4⋊Q8C2×C3×D311yescharacteristic
288#125—32no—
289C22⋊Q8—62no—
290C22⋊Q8—61no—
291C4×Q8—31no—
292C4×Q8—32no—
293C4×Q8—31no—
294C4×Q8—61no—
295C4×D4—32no—
296C4×D4C2×C3×D311yescharacteristic
297C2×C4⋊C4—61no—
298#107—32no—
299#107—32no—
300#107—32no—
301#106—31no—
302#106C2×C3×D312yes—
303C4⋊C8—32no—
304C4⋊C8C2×C3×D311yescharacteristic
305#103—32no—
306#103—31no—
307#102—31no—
308#102C2×C3×D312yes—
309#97—32no—
310C4×C8—31no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—41no—
314C3×Q8—32no—
315C3×Q8—62no—
316C3×Q8—61no—
317C3×Q8—81no—
318C3×Q8—81no—
319C3×Q8—82no—
320C3×Q8—241no—
321C3×Q8—242no—
322C3×D4—21no—
323C3×D4—21no—
324C3×D4—22no—
325C3×D4—41no—
326C3×D4D3×Q812yes—
327C3×D4C2×C3×Q812yes—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—32no—
336C2×C3×C4—32no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—62no—
344C2×C3×C4—82no—
345C2×C3×C4—242no—
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4C2×C3×D411yescharacteristic
349C2×C3×C4C2×C3×D411yescharacteristic
350C2×C3×C4C23×D311yescharacteristic
351C2×C3×C4C24×C311yescharacteristic
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—22no—
355C2×Dic3—82no—
356C2×Dic3C2×C3×D412yes—
357C2×Dic3C2×C3×D412yes—
358Dic6—22no—
359Dic6—21no—
360Dic6—82no—
361Dic6—81no—
362Dic6C2×C3×Q812yes—
363C3×C8—22no—
364C3×C8—22no—
365C3×C8—32no—
366C3×C8—42no—
367C3×C8—61no—
368C3⋊C8—21no—
369C3⋊C8C2×C3×Q812yes—
370C2×C32—42no—
371C2×C32D4×Q811yescharacteristic
372C2×C32C2×#13711yescharacteristic
373C2×C32#57611yescharacteristic
374C2×Q8—21no—
375C2×Q8—31no—
376C2×Q8—31no—
377C2×Q8—62no—
378C2×Q8—61no—
379C2×D4C22×C3×D311yescharacteristic
380C22×C4—21no—
381C22×C4—62no—
382SD16—34no—
383C2×C8—31no—
384C2×C8—31no—
385C2×C8C22×C3×D312yes—
386C4⋊C4—21no—
387C4⋊C4—31no—
388C4⋊C4—32no—
389C4⋊C4—31no—
390C4⋊C4—32no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4—62no—
394C4⋊C4—62no—
395C4⋊C4—61no—
396C4⋊C4—61no—
397C4⋊C4C22×C3×D312yes—
398C4⋊C4C22×C3×D311yescharacteristic
399C22⋊C4—21no—
400C22⋊C4—62no—
401C42—32no—
402C42—62no—
403C42—61no—
404C42C22×C3×D311yescharacteristic
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—21no—
408C22×C3—42no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3C2×D3×D411yescharacteristic
416C22×C3C22×C3×D411yescharacteristic
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—22no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—42no—
425C3×C4—41no—
426C3×C4—42no—
427C3×C4—62no—
428C3×C4—62no—
429C3×C4—62no—
430C3×C4—81no—
431C3×C4—82no—
432C3×C4—122no—
433C3×C4—121no—
434C3×C4#91512yes—
435C3×C4C3×#13712yes—
436C3×C4C2×D3×Q811yescharacteristic
437C3×C4#101011yescharacteristic
438C3×C4C22×C3×Q811yescharacteristic
439C3×C4C3×2−1+411yescharacteristic
440Dic3—22no—
441Dic3—22no—
442Dic3—22no—
443Dic3—41no—
444Dic3—42no—
445C32#338511yescharacteristic, complemented, Sylow
446C23—21no—
447Q8—32no—
448Q8—62no—
449Q8—61no—
450Q8—81no—
451Q8—241no—
452Q8—242no—
453D4—21no—
454D4C3×D3×Q812yes—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—42no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—62no—
463C2×C4—242no—
464C2×C4C3×D3×D411yescharacteristic
465C2×C4C3×D3×D411yescharacteristic
466C2×C4C23×C3×D311yescharacteristic
467C8—22no—
468C8—32no—
469C8—61no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—42no—
474C2×C3—42no—
475C2×C3—82no—
476C2×C3#563211yescharacteristic
477C2×C3C2×#91511yescharacteristic
478C2×C3#577411yescharacteristic
479C2×C3C3×D4×Q811yescharacteristic
480C2×C3C2×C3×#13711yescharacteristic
481C2×C3C3×#57611yescharacteristic
482C22—22no—
483C22—41no—
484C22—41no—
485C22C2×C3×D3×D411yescharacteristic
486C4—21no—
487C4—41no—
488C4—42no—
489C4—62no—
490C4—62no—
491C4—62no—
492C4—122no—
493C4—121no—
494C4C3×#91512yes—
495C4C2×C3×D3×Q811yescharacteristic
496C4C3×#101011yescharacteristic
497C3—21no—
498C3#8105611yescharacteristic, direct product factor
499C3C3×#338511yescharacteristic, complemented
500C2—42no—
501C2C3×#563211yescharacteristic
502C2C2×C3×#91511yescharacteristic
503C2C3×#577411yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#476C2
socleC22×C32C22×D4
prime radicalC2×C32×D4C23
hypercenterC3×#476D3
hypocenterC3C3×#3385
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#476

Complements

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

NH ≅ G/Ntype
1. C3×#5722C2semidirect
4. C3×#5078C2semidirect
6. C3×#4857C2semidirect
7. C3×#4856C2semidirect
8. C3×#4850C2semidirect
9. C3×#4831C2semidirect
13. #81056C3direct (N × H)
45. #5722C2×C3semidirect
58. #5078C2×C3semidirect
60. #4857C2×C3semidirect
61. #4856C2×C3semidirect
62. #4850C2×C3semidirect
63. #4831C2×C3semidirect
81. C3×#214Q8semidirect
251. #214C3×Q8semidirect
445. C32#3385semidirect
498. C3#81056direct (N × H)
499. C3C3×#3385semidirect

Maximal subgroups

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

subgroupindex
C3×#57222
C3×#55432
C32×#4762
C3×#50782
C3×C4×#2142
C3×#48572
C3×#48562
C3×#48502
C3×#48312
C3×#47592
C3×#47572
C3×#33853
#810563