Finite Group Databasebeta

← C3×#68109 #11778397

Subgroups — C3×#68109 #11778397

Order 1152.

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

normal subgroups only

This group has 569 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×#5494C211yesmaximal, characteristic
2C3×#4947C211yesmaximal, characteristic, complemented
3C3×#4739C211yesmaximal, characteristic, complemented
4C3×#4738C211yesmaximal, characteristic, complemented
5C32×#323C211yesmaximal, characteristic, complemented
6C3×#4470C211yesmaximal, characteristic, complemented
7C3×#4467C211yesmaximal, characteristic, complemented
8C3×#1606—31nomaximal
9#68109C311yesmaximal, characteristic, direct product factor
10C22×C3×Dic6C2211yescharacteristic
11C2×C32×C22⋊C4C2211yescharacteristic
12C3×#953—21no—
13C2×C3×#217—21no—
14C3×#930C2211yescharacteristic
15C3×C4×#67—21no—
16C3×#924C2211yescharacteristic
17C2×C3×Dic12—21no—
18C2×C3×#204—21no—
19C3×#885—21no—
20C3×#884—21no—
21C3×#883—21no—
22C3×#879—21no—
23C32×C22⋊C8C2211yescharacteristic
24C32×C22⋊C8C2211yescharacteristic
25C3×C3⋊#95—21no—
26C3×C3⋊#95C2211yescharacteristic
27C3×#816—21no—
28C3×#816—21no—
29C3×#538—31no—
30#5494C2×C311yescharacteristic
31C3×#451—31no—
32C3×#448—31no—
33C3×#393—31no—
34#4947C2×C311yescharacteristic, complemented
35#4739C2×C311yescharacteristic, complemented
36#4738C2×C311yescharacteristic, complemented
37C3×#328—31no—
38C3×#327—31no—
39C3×#323—21no—
40C3×#323D311yescharacteristic, complemented
41C3×#323C2×C311yescharacteristic, complemented
42#4470C2×C311yescharacteristic, complemented
43#4467C2×C311yescharacteristic, complemented
44C24×C32D411yescharacteristic
45C22×C32×C4D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C2×C3×#67—21no—
48C2×C3×#67—21no—
49C2×C3×#67—21no—
50C22×C3×Dic3—21no—
51C22×C3×Dic3—21no—
52C2×C3×D12D411yescharacteristic
53C2×C3×C4×D3—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—42no—
57C2×C3×Dic6D411yescharacteristic
58C2×C32×C8—21no—
59C2×C32×C8—21no—
60C32×C22⋊C4—21no—
61C32×C22⋊C4—21no—
62C3×#217—21no—
63C3×#217—21no—
64C3×#217—21no—
65C3×#217—42no—
66C3×(C2×D3)⋊C4—21no—
67C3×(C2×D3)⋊C4—21no—
68C3×C4⋊Dic3D411yescharacteristic
69C3×C4⋊Dic3D411yescharacteristic
70C3×Dic3⋊C4—21no—
71C3×Dic3⋊C4—21no—
72C3×C4×Dic3—21no—
73C3×Dic12—42no—
74C3×#204—42no—
75#1606—31noSylow
76C22×C3×Q8—31no—
77C22×Dic6C22×C311yescharacteristic
78C2×C3×Q16—61no—
79C2×C3×SD16—61no—
80C3×#126—61no—
81C3×#124—61no—
82C3×#123—61no—
83C3×C22⋊Q8—31no—
84C3×#121—31no—
85C3×#121—61no—
86C3×#120—61no—
87C3×C4×D4—61no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—61no—
90C2×C3×C22⋊C4C2×D311yescharacteristic
91C2×C3×C22⋊C4C22×C311yescharacteristic
92#953—21no—
93C2×#217—21no—
94#930C22×C311yescharacteristic
95C4×#67—21no—
96#924C22×C311yescharacteristic
97C2×Dic12—21no—
98C2×#204—21no—
99#885—21no—
100#884—21no—
101#883—21no—
102#879—21no—
103C3×#103—61no—
104C3×#103—61no—
105C3×C22⋊C8—21no—
106C3×C22⋊C8—21no—
107C3×C22⋊C8C2×D311yescharacteristic
108C3×C22⋊C8C2×D311yescharacteristic
109C3×C22⋊C8C22×C311yescharacteristic
110C3×C22⋊C8C22×C311yescharacteristic
111C3×#95—31no—
112C3×#95—61no—
113C3⋊#95—21no—
114C3⋊#95C22×C311yescharacteristic
115#816—21no—
116#816—21no—
117C23×C32—21no—
118C23×C32—21no—
119C23×C32—21no—
120C23×C32—41no—
121C23×C32—41no—
122C23×C32C2×D411yescharacteristic
123C22×C3×D3—21no—
124C2×C32×C4—21no—
125C2×C32×C4—41no—
126C2×C32×C4—41no—
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C3×#67—41no—
130C3×#67—41no—
131C3×#67—81no—
132C3×#67—81no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—21no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—41no—
140C2×C3×Dic3—41no—
141C2×C3×Dic3—41no—
142C3×D12—41no—
143C3×C4×D3—81no—
144C3×Dic6—22no—
145C3×Dic6—22no—
146C3×Dic6—41no—
147C3×Dic6—41no—
148C3×Dic6—41no—
149C32×C8—41no—
150C32×C8—41no—
151#538—31no—
152#451—31no—
153#448—31no—
154#393—31no—
155#328—31no—
156#327—31no—
157#323C3×D311yescharacteristic, complemented
158C24×C3—21no—
159C24×C3D1211yescharacteristic
160C24×C3C3×D411yescharacteristic
161C2×C3×Q8—31no—
162C2×C3×Q8—61no—
163C2×C3×Q8—61no—
164C2×C3×Q8—122no—
165C2×C3×D4—31no—
166C2×C3×D4—61no—
167C2×C3×D4—61no—
168C2×C3×D4—61no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—61no—
172C22×C3×C4—61no—
173C22×C3×C4—61no—
174C22×C3×C4D1211yescharacteristic
175C22×C3×C4C3×D411yescharacteristic
176C22×C3×C4C22×D311yescharacteristic
177C22×C3×C4C23×C311yescharacteristic
178C2×#67—21no—
179C2×#67—21no—
180C2×#67—21no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C2×D12C3×D411yescharacteristic
184C2×C4×D3—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—42no—
188C2×Dic6C3×D411yescharacteristic
189C3×Q16—122no—
190C3×SD16—122no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—41no—
196C2×C3×C8—41no—
197C3×C4⋊C4—31no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4—61no—
201C3×C22⋊C4—21no—
202C3×C22⋊C4—21no—
203C3×C22⋊C4—21no—
204C3×C22⋊C4—21no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4—41no—
207C3×C22⋊C4—61no—
208C3×C22⋊C4—61no—
209C3×C22⋊C4—61no—
210C3×C22⋊C4—61no—
211C3×C22⋊C4—61no—
212C3×C22⋊C4—122no—
213C3×C42—61no—
214#217—21no—
215#217—21no—
216#217—21no—
217#217—42no—
218(C2×D3)⋊C4—21no—
219(C2×D3)⋊C4—21no—
220C4⋊Dic3C3×D411yescharacteristic
221C4⋊Dic3C3×D411yescharacteristic
222Dic3⋊C4—21no—
223Dic3⋊C4—21no—
224C4×Dic3—21no—
225Dic12—42no—
226#204—42no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—41no—
230C22×C32—42no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32—81no—
234C22×C32#12011yescharacteristic
235C22×C32#13511yescharacteristic
236C22×C32#13711yescharacteristic
237C2×C3×D3—41no—
238C2×C3×D3—41no—
239C2×C3×D3—41no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—41no—
243C3×Dic3—41no—
244C3×Dic3—41no—
245C3×Dic3—42no—
246C3×Dic3—41no—
247C3×Dic3—81no—
248C22×Q8—31no—
249C2×Q16—61no—
250C2×SD16—61no—
251#126—61no—
252#124—61no—
253#123—61no—
254C22⋊Q8—31no—
255#121—31no—
256#121—61no—
257#120—61no—
258C4×D4—61no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4C2×C3×D311yescharacteristic
261#103—61no—
262#103—61no—
263C22⋊C8C2×C3×D311yescharacteristic
264C22⋊C8C2×C3×D311yescharacteristic
265#95—31no—
266#95—61no—
267C23×C3—22no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—61no—
282C23×C3—81no—
283C23×C3—81no—
284C23×C3C2×D1211yescharacteristic
285C23×C3C2×C3×D411yescharacteristic
286C22×D3—21no—
287C3×Q8—62no—
288C3×Q8—62no—
289C3×Q8—121no—
290C3×Q8—121no—
291C3×Q8—121no—
292C3×D4—121no—
293C3×D4—121no—
294C3×D4—121no—
295C3×D4—241no—
296C3×D4—241no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—62no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—81no—
311C2×C3×C4—81no—
312C2×C3×C4—121no—
313C2×C3×C4—121no—
314C2×C3×C4—121no—
315C2×C3×C4—121no—
316C2×C3×C4—121no—
317C2×C3×C4—241no—
318C2×C3×C4D3×D411yescharacteristic
319C2×C3×C4D3×D411yescharacteristic
320C2×C3×C4C2×C3×D411yescharacteristic
321C2×C3×C4C2×C3×D411yescharacteristic
322#67—41no—
323#67—41no—
324#67—81no—
325#67—81no—
326C2×Dic3—22no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335D12—41no—
336C4×D3—81no—
337Dic6—22no—
338Dic6—22no—
339Dic6—41no—
340Dic6—41no—
341Dic6—41no—
342C3×C8—41no—
343C3×C8—41no—
344C3×C8—41no—
345C3×C8—41no—
346C3×C8—81no—
347C3×C8—81no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—81no—
351C2×C32#45211yescharacteristic
352C2×C32#45511yescharacteristic
353C2×C32B(4,𝔽2)11yescharacteristic
354C3×D3—81no—
355C24C3×D1211yescharacteristic
356C2×Q8—31no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×Q8—122no—
360C2×D4—31no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4C3×D1211yescharacteristic
368C22×C4C22×C3×D311yescharacteristic
369Q16—122no—
370SD16—122no—
371C2×C8—21no—
372C2×C8—21no—
373C4⋊C4—31no—
374C4⋊C4—31no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—61no—
380C22⋊C4—61no—
381C22⋊C4—61no—
382C22⋊C4—61no—
383C22⋊C4—61no—
384C22⋊C4—122no—
385C42—61no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3—82no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—82no—
410C22×C3—121no—
411C22×C3—121no—
412C22×C3—121no—
413C22×C3#88211yescharacteristic
414C22×C3#90411yescharacteristic
415C22×C3#90911yescharacteristic
416C22×C3C3×#12011yescharacteristic
417C22×C3C3×#13511yescharacteristic
418C22×C3C3×#13711yescharacteristic
419C2×D3—41no—
420C2×D3—41no—
421C2×D3—41no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—81no—
431C3×C4—121no—
432C3×C4—122no—
433C3×C4—121no—
434C3×C4—121no—
435C3×C4—241no—
436Dic3—41no—
437Dic3—41no—
438Dic3—42no—
439Dic3—41no—
440Dic3—81no—
441C32#160611yescharacteristic, complemented, Sylow
442C23—21no—
443C23—21no—
444C23—21no—
445C23—41no—
446C23—41no—
447C23—61no—
448C23C2×C3×D1211yescharacteristic
449Q8—62no—
450Q8—62no—
451Q8—121no—
452Q8—121no—
453Q8—121no—
454D4—121no—
455D4—121no—
456D4—121no—
457D4—241no—
458D4—241no—
459C2×C4—21no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—62no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—241no—
472C2×C4C3×D3×D411yescharacteristic
473C2×C4C3×D3×D411yescharacteristic
474C8—41no—
475C8—41no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—82no—
486C2×C3—81no—
487C2×C3—81no—
488C2×C3—241no—
489C2×C3#473311yescharacteristic
490C2×C3#474111yescharacteristic
491C2×C3#474611yescharacteristic
492C2×C3C3×#45211yescharacteristic
493C2×C3C3×#45511yescharacteristic
494C2×C3C3×B(4,𝔽2)11yescharacteristic
495D3—81no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—81no—
503C22—121no—
504C22—121no—
505C22—121no—
506C22C3×#88211yescharacteristic
507C22C3×#90411yescharacteristic
508C22C3×#90911yescharacteristic
509C4—21no—
510C4—21no—
511C4—41no—
512C4—121no—
513C4—122no—
514C4—121no—
515C4—121no—
516C4—241no—
517C3—21no—
518C3#6810911yescharacteristic, direct product factor
519C3C3×#160611yescharacteristic, complemented
520C2—21no—
521C2—21no—
522C2—81no—
523C2—241no—
524C2C3×#473311yescharacteristic
525C2C3×#474111yescharacteristic
526C2C3×#474611yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#323C2
socleC22×C32#120
prime radicalC3×#5494C2
hypercenterC3×#323D3
hypocenterC3C3×#1606
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#323

Complements

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

NH ≅ G/Ntype
2. C3×#4947C2semidirect
3. C3×#4739C2semidirect
4. C3×#4738C2semidirect
5. C32×#323C2semidirect
6. C3×#4470C2semidirect
7. C3×#4467C2semidirect
9. #68109C3direct (N × H)
34. #4947C2×C3semidirect
35. #4739C2×C3semidirect
36. #4738C2×C3semidirect
40. C3×#323D3semidirect
41. C3×#323C2×C3semidirect
42. #4470C2×C3semidirect
43. #4467C2×C3semidirect
157. #323C3×D3semidirect
441. C32#1606semidirect
518. C3#68109direct (N × H)
519. C3C3×#1606semidirect

Maximal subgroups

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

subgroupindex
C3×#54942
C3×#49472
C3×#47392
C3×#47382
C32×#3232
C3×#44702
C3×#44672
C3×#16063
#681093