Finite Group Databasebeta

← C3×#68130 #11778058

Subgroups — C3×#68130 #11778058

Order 1152.

583 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5515C211yesmaximal, characteristic, complemented
2C3×#4991C211yesmaximal, characteristic, complemented
3C3×#4732C211yesmaximal, characteristic
4C3×#4731C211yesmaximal, characteristic, complemented
5C32×#324C211yesmaximal, characteristic, complemented
6C3×#4470C211yesmaximal, characteristic, complemented
7C3×#4467C211yesmaximal, characteristic, complemented
8C3×#1597—31nomaximal
9#68130C311yesmaximal, characteristic, direct product factor
10C22×C3×D12C2211yescharacteristic
11C2×C32×C4⋊C4C2211yescharacteristic
12C3×#930C2211yescharacteristic
13C3×#929—21no—
14C3×C4×#67—21no—
15C2×C3×(C2×D3)⋊C4—21no—
16C2×C3×(C2×D3)⋊C4—21no—
17C3×#924C2211yescharacteristic
18C2×C3×D24—21no—
19C2×C3×#204—21no—
20C3×#898—21no—
21C3×#896—21no—
22C3×#894—21no—
23C3×#889—21no—
24C32×C22⋊C8C2211yescharacteristic
25C32×C22⋊C8C2211yescharacteristic
26C3×C3⋊#95C2211yescharacteristic
27C3×#821—21no—
28C3×#821—21no—
29C3×#539—31no—
30#5515C2×C311yescharacteristic, complemented
31C3×#450—31no—
32C3×#447—31no—
33C3×#394—31no—
34#4991C2×C311yescharacteristic, complemented
35#4732C2×C311yescharacteristic
36#4731C2×C311yescharacteristic, complemented
37C3×#328—31no—
38C3×#327—31no—
39C3×#324—21no—
40C3×#324D311yescharacteristic, complemented
41C3×#324C2×C311yescharacteristic, complemented
42#4470C2×C311yescharacteristic, complemented
43#4467C2×C311yescharacteristic, complemented
44C23×C3×D3—21no—
45C22×C32×C4D411yescharacteristic
46C22×C32×C4D411yescharacteristic
47C22×C32×C4C2311yescharacteristic
48C2×C3×#67—21no—
49C22×C3×Dic3—21no—
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12—42no—
53C2×C3×D12D411yescharacteristic
54C2×C3×C4×D3—21no—
55C2×C3×Dic6D411yescharacteristic
56C2×C32×C8—21no—
57C2×C32×C8—21no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4—21no—
60C3×#217—21no—
61C3×(C2×D3)⋊C4—21no—
62C3×(C2×D3)⋊C4—21no—
63C3×(C2×D3)⋊C4—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—42no—
66C3×(C2×D3)⋊C4—42no—
67C3×C4⋊Dic3D411yescharacteristic
68C3×C4⋊Dic3D411yescharacteristic
69C3×Dic3⋊C4—21no—
70C3×Dic3⋊C4—21no—
71C3×Dic3⋊C4—21no—
72C3×Dic3⋊C4—21no—
73C3×C4×Dic3—21no—
74C3×D24—42no—
75C3×#204—42no—
76#1597—31noSylow
77C22×C3×D4—31no—
78C22×D12C22×C311yescharacteristic
79C2×C3×SD16—61no—
80C2×C3×D8—61no—
81C3×#126—61no—
82C3×#125—61no—
83C3×#123—61no—
84C3×#123—61no—
85C3×C22⋊Q8—31no—
86C3×C22⋊Q8—61no—
87C3×#121—31no—
88C3×C4×D4—61no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2×D311yescharacteristic
91C2×C3×C4⋊C4C22×C311yescharacteristic
92C2×C3×C22⋊C4—61no—
93C2×C3×C22⋊C4—61no—
94#930C22×C311yescharacteristic
95#929—21no—
96C4×#67—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99#924C22×C311yescharacteristic
100C2×D24—21no—
101C2×#204—21no—
102#898—21no—
103#896—21no—
104#894—21no—
105#889—21no—
106C3×#102—61no—
107C3×#102—61no—
108C3×C22⋊C8—21no—
109C3×C22⋊C8—21no—
110C3×C22⋊C8C2×D311yescharacteristic
111C3×C22⋊C8C2×D311yescharacteristic
112C3×C22⋊C8C22×C311yescharacteristic
113C3×C22⋊C8C22×C311yescharacteristic
114C3×#95—31no—
115C3⋊#95C22×C311yescharacteristic
116#821—21no—
117#821—21no—
118C23×C32C2×D411yescharacteristic
119C22×C3×D3—22no—
120C22×C3×D3—21no—
121C22×C3×D3—41no—
122C22×C3×D3—41no—
123C22×C3×D3—41no—
124C22×C3×D3—41no—
125C22×C3×D3—41no—
126C22×C3×D3—41no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C3×#67—41no—
136C3×#67—41no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—41no—
141C2×C3×Dic3—41no—
142C2×C3×Dic3—41no—
143C3×D12—22no—
144C3×D12—22no—
145C3×D12—41no—
146C3×D12—41no—
147C3×D12—41no—
148C3×C4×D3—81no—
149C3×Dic6—41no—
150C32×C8—41no—
151C32×C8—41no—
152#539—31no—
153#450—31no—
154#447—31no—
155#394—31no—
156#328—31no—
157#327—31no—
158#324C3×D311yescharacteristic, complemented
159C24×C3—61no—
160C23×D3—21no—
161C2×C3×Q8—31no—
162C2×C3×D4—31no—
163C2×C3×D4—61no—
164C2×C3×D4—61no—
165C2×C3×D4—61no—
166C2×C3×D4—122no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—61no—
171C22×C3×C4—61no—
172C22×C3×C4D1211yescharacteristic
173C22×C3×C4D1211yescharacteristic
174C22×C3×C4C3×D411yescharacteristic
175C22×C3×C4C3×D411yescharacteristic
176C22×C3×C4C22×D311yescharacteristic
177C22×C3×C4C23×C311yescharacteristic
178C2×#67—21no—
179C22×Dic3—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—42no—
183C2×D12C3×D411yescharacteristic
184C2×C4×D3—21no—
185C2×Dic6C3×D411yescharacteristic
186C3×SD16—122no—
187C3×D8—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—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—31no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—41no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C22⋊C4—61no—
207C3×C22⋊C4—61no—
208C3×C22⋊C4—61no—
209C3×C22⋊C4—61no—
210C3×C22⋊C4—61no—
211C3×C22⋊C4—122no—
212C3×C22⋊C4—122no—
213C3×C42—61no—
214#217—21no—
215(C2×D3)⋊C4—21no—
216(C2×D3)⋊C4—21no—
217(C2×D3)⋊C4—21no—
218(C2×D3)⋊C4—21no—
219(C2×D3)⋊C4—42no—
220(C2×D3)⋊C4—42no—
221C4⋊Dic3C3×D411yescharacteristic
222C4⋊Dic3C3×D411yescharacteristic
223Dic3⋊C4—21no—
224Dic3⋊C4—21no—
225Dic3⋊C4—21no—
226Dic3⋊C4—21no—
227C4×Dic3—21no—
228D24—42no—
229#204—42no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32#12011yescharacteristic
233C22×C32#13511yescharacteristic
234C22×C32AGL(1,ℤ/8)11yescharacteristic
235C2×C3×D3—42no—
236C2×C3×D3—42no—
237C2×C3×D3—41no—
238C2×C3×D3—41no—
239C2×C3×D3—41no—
240C2×C3×D3—42no—
241C2×C3×D3—81no—
242C2×C3×D3—81no—
243C2×C3×D3—81no—
244C2×C3×D3—81no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—41no—
248C32×C4—41no—
249C3×Dic3—41no—
250C3×Dic3—41no—
251C3×Dic3—41no—
252C3×Dic3—81no—
253C22×D4—31no—
254C2×SD16—61no—
255C2×D8—61no—
256#126—61no—
257#125—61no—
258#123—61no—
259#123—61no—
260C22⋊Q8—31no—
261C22⋊Q8—61no—
262#121—31no—
263C4×D4—61no—
264C2×C4⋊C4C2×C3×D311yescharacteristic
265C2×C22⋊C4—61no—
266C2×C22⋊C4—61no—
267#102—61no—
268#102—61no—
269C22⋊C8C2×C3×D311yescharacteristic
270C22⋊C8C2×C3×D311yescharacteristic
271#95—31no—
272C23×C3—21no—
273C23×C3—62no—
274C23×C3—61no—
275C23×C3—121no—
276C23×C3—121no—
277C23×C3—121no—
278C23×C3—121no—
279C23×C3—121no—
280C23×C3—121no—
281C23×C3C2×D1211yescharacteristic
282C23×C3C2×C3×D411yescharacteristic
283C22×D3—22no—
284C22×D3—21no—
285C22×D3—41no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C3×Q8—121no—
292C3×D4—62no—
293C3×D4—62no—
294C3×D4—121no—
295C3×D4—121no—
296C3×D4—121no—
297C3×D4—121no—
298C3×D4—121no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—81no—
321C2×C3×C4—81no—
322C2×C3×C4—121no—
323C2×C3×C4—121no—
324C2×C3×C4—121no—
325C2×C3×C4—241no—
326C2×C3×C4D3×D411yescharacteristic
327C2×C3×C4D3×D411yescharacteristic
328C2×C3×C4C2×C3×D411yescharacteristic
329C2×C3×C4C2×C3×D411yescharacteristic
330#67—41no—
331#67—41no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338D12—22no—
339D12—22no—
340D12—41no—
341D12—41no—
342D12—41no—
343C4×D3—81no—
344Dic6—41no—
345C3×C8—41no—
346C3×C8—41no—
347C3×C8—41no—
348C3×C8—41no—
349C3×C8—81no—
350C3×C8—81no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32#44911yescharacteristic
354C2×C32#45411yescharacteristic
355C2×C32B(4,𝔽2)11yescharacteristic
356C3×D3—81no—
357C3×D3—82no—
358C24—61no—
359C2×Q8—31no—
360C2×D4—31no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—122no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4C3×D1211yescharacteristic
368C22×C4C3×D1211yescharacteristic
369C22×C4C22×C3×D311yescharacteristic
370SD16—122no—
371D8—122no—
372C2×C8—21no—
373C2×C8—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—31no—
377C4⋊C4—31no—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C22⋊C4—61no—
383C22⋊C4—61no—
384C22⋊C4—61no—
385C22⋊C4—61no—
386C22⋊C4—61no—
387C22⋊C4—122no—
388C22⋊C4—122no—
389C42—61no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—122no—
400C22×C3—122no—
401C22×C3—121no—
402C22×C3—121no—
403C22×C3—122no—
404C22×C3—121no—
405C22×C3—241no—
406C22×C3—241no—
407C22×C3—241no—
408C22×C3—241no—
409C22×C3#88211yescharacteristic
410C22×C3#90411yescharacteristic
411C22×C3#90811yescharacteristic
412C22×C3C3×#12011yescharacteristic
413C22×C3C3×#13511yescharacteristic
414C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
415C2×D3—42no—
416C2×D3—42no—
417C2×D3—41no—
418C2×D3—41no—
419C2×D3—41no—
420C2×D3—42no—
421C2×D3—81no—
422C2×D3—81no—
423C2×D3—81no—
424C2×D3—81no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—41no—
430C3×C4—41no—
431C3×C4—42no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—81no—
437C3×C4—121no—
438C3×C4—121no—
439C3×C4—121no—
440C3×C4—241no—
441Dic3—41no—
442Dic3—41no—
443Dic3—41no—
444Dic3—81no—
445C32#159711yescharacteristic, complemented, Sylow
446C23—62no—
447C23—61no—
448C23—121no—
449C23—121no—
450C23—121no—
451C23—121no—
452C23—121no—
453C23—121no—
454C23C2×C3×D1211yescharacteristic
455Q8—121no—
456D4—62no—
457D4—62no—
458D4—121no—
459D4—121no—
460D4—121no—
461D4—121no—
462D4—121no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—241no—
476C2×C4C3×D3×D411yescharacteristic
477C2×C4C3×D3×D411yescharacteristic
478C8—41no—
479C8—41no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3—242no—
490C2×C3—241no—
491C2×C3#473411yescharacteristic
492C2×C3#474111yescharacteristic
493C2×C3#475311yescharacteristic
494C2×C3C3×#44911yescharacteristic
495C2×C3C3×#45411yescharacteristic
496C2×C3C3×B(4,𝔽2)11yescharacteristic
497D3—82no—
498D3—81no—
499C22—21no—
500C22—21no—
501C22—122no—
502C22—122no—
503C22—121no—
504C22—121no—
505C22—122no—
506C22—121no—
507C22—241no—
508C22—241no—
509C22—241no—
510C22—241no—
511C22C3×#88211yescharacteristic
512C22C3×#90411yescharacteristic
513C22C3×#90811yescharacteristic
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—121no—
519C4—121no—
520C4—121no—
521C4—241no—
522C3—21no—
523C3#6813011yescharacteristic, direct product factor
524C3C3×#159711yescharacteristic, complemented
525C2—21no—
526C2—21no—
527C2—242no—
528C2—241no—
529C2C3×#473411yescharacteristic
530C2C3×#474111yescharacteristic
531C2C3×#475311yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#324C2
socleC22×C32#120
prime radicalC3×#4732C2
hypercenterC3×#324D3
hypocenterC3C3×#1597
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#324

Complements

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

NH ≅ G/Ntype
1. C3×#5515C2semidirect
2. C3×#4991C2semidirect
4. C3×#4731C2semidirect
5. C32×#324C2semidirect
6. C3×#4470C2semidirect
7. C3×#4467C2semidirect
9. #68130C3direct (N × H)
30. #5515C2×C3semidirect
34. #4991C2×C3semidirect
36. #4731C2×C3semidirect
40. C3×#324D3semidirect
41. C3×#324C2×C3semidirect
42. #4470C2×C3semidirect
43. #4467C2×C3semidirect
158. #324C3×D3semidirect
445. C32#1597semidirect
523. C3#68130direct (N × H)
524. C3C3×#1597semidirect

Maximal subgroups

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

subgroupindex
C3×#55152
C3×#49912
C3×#47322
C3×#47312
C32×#3242
C3×#44702
C3×#44672
C3×#15973
#681303