Finite Group Databasebeta

← C3×#70990 #11798552

Subgroups — C3×#70990 #11798552

Order 1152.

592 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 82 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×#5679C211yesmaximal, characteristic, complemented
2C32×#455C211yesmaximal, characteristic, complemented
3C3×#5244C211yesmaximal, characteristic
4C3×#5243C211yesmaximal, characteristic, complemented
5C3×#4541C211yesmaximal, characteristic, complemented
6C3×#4476C211yesmaximal, characteristic, complemented
7C3×#4475C211yesmaximal, characteristic, complemented
8C3×#2197—31nomaximal
9#70990C311yesmaximal, characteristic, direct product factor
10C32×H(5,𝔽2)C2211yescharacteristic
11C32×#137—21no—
12C32×#124C2211yescharacteristic
13C3×#946—21no—
14C3×#937—21no—
15C3×#937C2211yescharacteristic
16C3×#935—21no—
17C3×#935C2211yescharacteristic
18C3×#883—21no—
19C3×#67⋊C4—21no—
20C3×D3×C22⋊C4—21no—
21C3×#879—21no—
22C3×#872—21no—
23C32×#104—21no—
24C32×#100C2211yescharacteristic
25C3×#834—21no—
26C3×#834—21no—
27C3×#834C2211yescharacteristic
28C3×#834C2211yescharacteristic
29C3×#551—31no—
30#5679C2×C311yescharacteristic, complemented
31C3×#458—31no—
32C3×B(4,𝔽2)—31no—
33C3×#455—21no—
34C3×#455D311yescharacteristic, complemented
35C3×#455C2×C311yescharacteristic, complemented
36#5244C2×C311yescharacteristic
37#5243C2×C311yescharacteristic, complemented
38C3×#354—31no—
39C3×#352—31no—
40C3×AΣL(1,𝔽16)—31no—
41#4541C2×C311yescharacteristic, complemented
42#4476C2×C311yescharacteristic, complemented
43#4475C2×C311yescharacteristic, complemented
44C23×C3×D3D411yescharacteristic
45C32×#41—21no—
46C32×#41—41no—
47C2×C32×Q8D411yescharacteristic, complemented
48C2×C32×D4—21no—
49C2×C32×D4—21no—
50C2×C32×D4—41no—
51C2×C32×D4C2311yescharacteristic
52C2×C3×#67—21no—
53C2×C3×#67—41no—
54C22×C3×Dic3D411yescharacteristic, complemented
55C2×C3×C4×D3—21no—
56C2×C3×C4×D3—21no—
57C32×Q16—41no—
58C32×SD16—41no—
59C32×M16—21no—
60C32×C22⋊C4—21no—
61C32×C22⋊C4—21no—
62C32×C42D411yescharacteristic, complemented
63C3×#217—21no—
64C3×#217—21no—
65C3×#217—21no—
66C3×#217—41no—
67C3×#217D411yescharacteristic
68C3×#217D411yescharacteristic, complemented
69C3×(C2×D3)⋊C4—21no—
70C3×(C2×D3)⋊C4—21no—
71C3×(C2×D3)⋊C4—21no—
72C3×(C2×D3)⋊C4—41no—
73C3×C4⋊Dic3—21no—
74C3×Dic3⋊C4—21no—
75C3×Dic3⋊C4—21no—
76C3×Dic3⋊C4—21no—
77C3×Dic3⋊C4—41no—
78C3×C4×Dic3—21no—
79#2197—31noSylow
80C3×H(5,𝔽2)—21no—
81C3×H(5,𝔽2)C2×D311yescharacteristic
82C3×H(5,𝔽2)C22×C311yescharacteristic
83C3×#137—22no—
84C3×#137—21no—
85C3×#137—21no—
86C3×#126—61no—
87C3×#124—21no—
88C3×#124C2×D311yescharacteristic
89C3×#124C22×C311yescharacteristic
90C3×#123—31no—
91C3×#123—61no—
92C3×#123—61no—
93C3×C22⋊Q8—61no—
94C3×#120—31no—
95C3×#120—61no—
96C3×C4×D4—61no—
97C3×#117—61no—
98C2×C3×C22⋊C4—61no—
99#946—21no—
100#937—21no—
101#937C22×C311yescharacteristic
102#935—21no—
103#935C22×C311yescharacteristic
104#883—21no—
105#67⋊C4—21no—
106D3×C22⋊C4—21no—
107#879—21no—
108#872—21no—
109C3×#104—22no—
110C3×#104—21no—
111C3×#104—21no—
112C3×#100—21no—
113C3×#100C2×D311yescharacteristic
114C3×#100C22×C311yescharacteristic
115C3×#99—31no—
116C3×#99—31no—
117C3×#99—61no—
118C3×#99—61no—
119#834—21no—
120#834—21no—
121#834C22×C311yescharacteristic
122#834C22×C311yescharacteristic
123C23×C32—21no—
124C23×C32—21no—
125C23×C32C2×D411yescharacteristic
126C23×C32C2×D411yescharacteristic
127C22×C3×D3—21no—
128C22×C3×D3—42no—
129C22×C3×D3—41no—
130C32×Q8—21no—
131C32×Q8—41no—
132C32×D4—21no—
133C32×D4—41no—
134C32×D4—41no—
135C32×D4—41no—
136C32×D4—41no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—41no—
142C2×C32×C4C2×D411yescharacteristic
143C3×#67—42no—
144C3×#67—82no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—41no—
149C2×C3×Dic3—41no—
150C2×C3×Dic3—41no—
151C3×C4×D3—42no—
152C3×C4×D3—81no—
153C32×C8—41no—
154#551—31no—
155#458—31no—
156B(4,𝔽2)—31no—
157#455C3×D311yescharacteristic, complemented
158#354—31no—
159#352—31no—
160AΣL(1,𝔽16)—31no—
161C24×C3—31no—
162C23×D3C3×D411yescharacteristic
163C3×#41—22no—
164C3×#41—21no—
165C3×#41—21no—
166C3×#41—41no—
167C3×#41—41no—
168C3×#41—42no—
169C2×C3×Q8—21no—
170C2×C3×Q8#6711yescharacteristic, complemented
171C2×C3×Q8C3×D411yescharacteristic, complemented
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—41no—
178C2×C3×D4—41no—
179C2×C3×D4—41no—
180C2×C3×D4—41no—
181C2×C3×D4—42no—
182C2×C3×D4—61no—
183C2×C3×D4—121no—
184C2×C3×D4C22×D311yescharacteristic
185C2×C3×D4C23×C311yescharacteristic
186C22×C3×C4—31no—
187C22×C3×C4—61no—
188C22×C3×C4—61no—
189C2×#67—21no—
190C2×#67—41no—
191C22×Dic3C3×D411yescharacteristic, complemented
192C2×C4×D3—21no—
193C2×C4×D3—21no—
194C3×Q16—41no—
195C3×Q16—41no—
196C3×Q16—42no—
197C3×SD16—41no—
198C3×SD16—41no—
199C3×SD16—42no—
200C3×M16—22no—
201C3×M16—21no—
202C3×M16—21no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C4⋊C4—61no—
207C3×C4⋊C4—121no—
208C3×C22⋊C4—21no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—21no—
211C3×C22⋊C4—21no—
212C3×C22⋊C4—31no—
213C3×C22⋊C4—31no—
214C3×C22⋊C4—41no—
215C3×C22⋊C4—41no—
216C3×C22⋊C4—61no—
217C3×C22⋊C4—61no—
218C3×C22⋊C4—61no—
219C3×C22⋊C4—61no—
220C3×C22⋊C4—61no—
221C3×C22⋊C4—61no—
222C3×C22⋊C4—121no—
223C3×C22⋊C4—121no—
224C3×C42—21no—
225C3×C42—61no—
226C3×C42#6711yescharacteristic, complemented
227C3×C42C3×D411yescharacteristic, complemented
228#217—21no—
229#217—21no—
230#217—21no—
231#217—41no—
232#217C3×D411yescharacteristic
233#217C3×D411yescharacteristic, complemented
234(C2×D3)⋊C4—21no—
235(C2×D3)⋊C4—21no—
236(C2×D3)⋊C4—21no—
237(C2×D3)⋊C4—41no—
238C4⋊Dic3—21no—
239Dic3⋊C4—21no—
240Dic3⋊C4—21no—
241Dic3⋊C4—21no—
242Dic3⋊C4—41no—
243C4×Dic3—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32—41no—
249C22×C32—81no—
250C22×C32—81no—
251C22×C32#12011yescharacteristic
252C2×C3×D3—22no—
253C2×C3×D3—41no—
254C2×C3×D3—41no—
255C2×C3×D3—42no—
256C2×C3×D3—81no—
257C32×C4—21no—
258C32×C4—41no—
259C32×C4—41no—
260C32×C4—41no—
261C3×Dic3—41no—
262C3×Dic3—41no—
263C3×Dic3—41no—
264C3×Dic3—81no—
265C3×Dic3—81no—
266H(5,𝔽2)C2×C3×D311yescharacteristic
267#137—21no—
268#126—61no—
269#124C2×C3×D311yescharacteristic
270#123—31no—
271#123—61no—
272#123—61no—
273C22⋊Q8—61no—
274#120—31no—
275#120—61no—
276C4×D4—61no—
277#117—61no—
278C2×C22⋊C4—61no—
279#104—21no—
280#100C2×C3×D311yescharacteristic
281#99—31no—
282#99—31no—
283#99—61no—
284#99—61no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—61no—
294C23×C3—121no—
295C23×C3—122no—
296C23×C3D3×D411yescharacteristic
297C23×C3D3×D411yescharacteristic
298C23×C3C2×C3×D411yescharacteristic
299C23×C3C2×C3×D411yescharacteristic
300C22×D3—21no—
301C22×D3—42no—
302C22×D3—41no—
303C3×Q8—22no—
304C3×Q8—21no—
305C3×Q8—21no—
306C3×Q8—41no—
307C3×Q8—41no—
308C3×Q8—42no—
309C3×D4—22no—
310C3×D4—21no—
311C3×D4—21no—
312C3×D4—41no—
313C3×D4—41no—
314C3×D4—42no—
315C3×D4—41no—
316C3×D4—41no—
317C3×D4—41no—
318C3×D4—41no—
319C3×D4—42no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—41no—
323C3×D4—41no—
324C3×D4—122no—
325C3×D4—242no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—122no—
346C2×C3×C4—121no—
347C2×C3×C4—121no—
348C2×C3×C4—121no—
349C2×C3×C4—241no—
350C2×C3×C4C2×#6711yescharacteristic
351C2×C3×C4C2×C3×D411yescharacteristic
352#67—42no—
353#67—82no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C4×D3—42no—
361C4×D3—81no—
362C3×C8—42no—
363C3×C8—41no—
364C3×C8—41no—
365C2×C32—21no—
366C2×C32—41no—
367C2×C32—41no—
368C2×C32—81no—
369C2×C32B(4,𝔽2)11yescharacteristic
370C3×D3—42no—
371C24—31no—
372#41—21no—
373#41—41no—
374C2×Q8C3×#6711yescharacteristic, complemented
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—61no—
379C2×D4—121no—
380C2×D4C22×C3×D311yescharacteristic
381C22×C4—31no—
382C22×C4—61no—
383C22×C4—61no—
384Q16—41no—
385SD16—41no—
386M16—21no—
387C4⋊C4—61no—
388C4⋊C4—61no—
389C4⋊C4—61no—
390C4⋊C4—61no—
391C4⋊C4—121no—
392C22⋊C4—21no—
393C22⋊C4—21no—
394C22⋊C4—31no—
395C22⋊C4—31no—
396C22⋊C4—61no—
397C22⋊C4—61no—
398C22⋊C4—61no—
399C22⋊C4—61no—
400C22⋊C4—61no—
401C22⋊C4—61no—
402C22⋊C4—121no—
403C22⋊C4—121no—
404C42—61no—
405C42C3×#6711yescharacteristic, complemented
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—42no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—62no—
421C22×C3—81no—
422C22×C3—81no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3—81no—
427C22×C3—121no—
428C22×C3—122no—
429C22×C3—121no—
430C22×C3—161no—
431C22×C3—161no—
432C22×C3—241no—
433C22×C3#93711yescharacteristic
434C22×C3C3×#12011yescharacteristic
435C2×D3—22no—
436C2×D3—41no—
437C2×D3—41no—
438C2×D3—42no—
439C2×D3—81no—
440C3×C4—22no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—42no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—81no—
451C3×C4—81no—
452C3×C4—121no—
453C3×C4—121no—
454C3×C4—121no—
455C3×C4—241no—
456C3×C4—241no—
457Dic3—41no—
458Dic3—41no—
459Dic3—41no—
460Dic3—81no—
461Dic3—81no—
462C32#219711yescharacteristic, complemented, Sylow
463C23—21no—
464C23—21no—
465C23—61no—
466C23—121no—
467C23—122no—
468C23C3×D3×D411yescharacteristic
469C23C3×D3×D411yescharacteristic
470Q8—21no—
471Q8—41no—
472D4—21no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—122no—
478D4—242no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—122no—
488C2×C4—121no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—241no—
492C2×C4C2×C3×#6711yescharacteristic
493C8—41no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—41no—
498C2×C3—41no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—81no—
503C2×C3—81no—
504C2×C3—82no—
505C2×C3—81no—
506C2×C3—81no—
507C2×C3—122no—
508C2×C3#503211yescharacteristic
509C2×C3C3×B(4,𝔽2)11yescharacteristic
510D3—42no—
511C22—21no—
512C22—21no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—62no—
517C22—81no—
518C22—81no—
519C22—121no—
520C22—122no—
521C22—121no—
522C22—241no—
523C22C3×#93711yescharacteristic
524C4—21no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—121no—
529C4—121no—
530C4—121no—
531C4—241no—
532C4—241no—
533C3—21no—
534C3#7099011yescharacteristic, direct product factor
535C3C3×#219711yescharacteristic, complemented
536C2—21no—
537C2—41no—
538C2—41no—
539C2—81no—
540C2—122no—
541C2C3×#503211yescharacteristic
subgroupG/H
centerC2×C3#5032
commutator, cosocleC2×C3×D4C23×C3
FrattiniC2×D4C22×C3×D3
FittingC32×#455C2
socleC2×C32B(4,𝔽2)
prime radicalC3×#5244C2
hypercenterC3×#455D3
hypocenterC3C3×#2197
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#455

Complements

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

NH ≅ G/Ntype
1. C3×#5679C2semidirect
2. C32×#455C2semidirect
4. C3×#5243C2semidirect
5. C3×#4541C2semidirect
6. C3×#4476C2semidirect
7. C3×#4475C2semidirect
9. #70990C3direct (N × H)
30. #5679C2×C3semidirect
34. C3×#455D3semidirect
35. C3×#455C2×C3semidirect
37. #5243C2×C3semidirect
41. #4541C2×C3semidirect
42. #4476C2×C3semidirect
43. #4475C2×C3semidirect
47. C2×C32×Q8D4semidirect
54. C22×C3×Dic3D4semidirect
62. C32×C42D4semidirect
68. C3×#217D4semidirect
157. #455C3×D3semidirect
170. C2×C3×Q8#67semidirect
171. C2×C3×Q8C3×D4semidirect
191. C22×Dic3C3×D4semidirect
226. C3×C42#67semidirect
227. C3×C42C3×D4semidirect
233. #217C3×D4semidirect
374. C2×Q8C3×#67semidirect
405. C42C3×#67semidirect
462. C32#2197semidirect
534. C3#70990direct (N × H)
535. C3C3×#2197semidirect

Maximal subgroups

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

subgroupindex
C3×#56792
C32×#4552
C3×#52442
C3×#52432
C3×#45412
C3×#44762
C3×#44752
C3×#21973
#709903