Finite Group Databasebeta

← C3×#69153 #11798564

Subgroups — C3×#69153 #11798564

Order 1152.

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

normal subgroups only

This group has 551 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
2C3×#5061C211yesmaximal, characteristic, complemented
3C3×#4742C211yesmaximal, characteristic, complemented
4C3×#4741C211yesmaximal, characteristic
5C32×#354C211yesmaximal, characteristic, complemented
6C3×#4472C211yesmaximal, characteristic, complemented
7C3×#4471C211yesmaximal, characteristic, complemented
8C3×#2197—31nomaximal
9#69153C311yesmaximal, characteristic, direct product factor
10C3×#1004C2211yescharacteristic
11C32×#124C2211yescharacteristic
12C3×#946—21no—
13C3×#942—21no—
14C3×#937C2211yescharacteristic
15C3×#935C2211yescharacteristic
16C3×#886—21no—
17C3×#883—21no—
18C3×#882—21no—
19C3×#67⋊C4—21no—
20C3×D3×C22⋊C4—21no—
21C3×#879—21no—
22C3×#872—21no—
23C32×#99C2211yescharacteristic
24C32×#99C2211yescharacteristic
25C3×#833C2211yescharacteristic
26C3×#806—21no—
27C3×#806—21no—
28C3×#805—21no—
29C3×#551—31no—
30#5679C2×C311yescharacteristic, complemented
31C3×#458—31no—
32C3×B(4,𝔽2)—31no—
33C3×#455—31no—
34#5061C2×C311yescharacteristic, complemented
35#4742C2×C311yescharacteristic, complemented
36#4741C2×C311yescharacteristic
37C3×#354—21no—
38C3×#354D311yescharacteristic, complemented
39C3×#354C2×C311yescharacteristic, complemented
40C3×#352—31no—
41C3×AΣL(1,𝔽16)—31no—
42#4472C2×C311yescharacteristic, complemented
43#4471C2×C311yescharacteristic, complemented
44C23×C3×D3D411yescharacteristic
45C2×C32×Q8D411yescharacteristic, complemented
46C2×C32×D4C2311yescharacteristic
47C2×C3×#67—21no—
48C2×C3×#67—21no—
49C2×C3×#67—21no—
50C22×C3×Dic3D411yescharacteristic, complemented
51C3×#237—41no—
52C3×D3×D4—41no—
53C3×#235—21no—
54C2×C3×D12—41no—
55C2×C3×C4×D3—21no—
56C2×C3×C4×D3—21no—
57C32×C22⋊C4—21no—
58C32×C22⋊C4—21no—
59C32×C22⋊C4—21no—
60C32×C22⋊C4—21no—
61C32×C42D411yescharacteristic, complemented
62C3×#217—21no—
63C3×#217D411yescharacteristic
64C3×#217D411yescharacteristic, complemented
65C3×#216—41no—
66C3×#215—41no—
67C3×(C2×D3)⋊C4—21no—
68C3×(C2×D3)⋊C4—21no—
69C3×(C2×D3)⋊C4—21no—
70C3×(C2×D3)⋊C4—41no—
71C3×(C2×D3)⋊C4—41no—
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3—41no—
74C3×Dic3⋊C4—21no—
75C3×Dic3⋊C4—21no—
76C3×Dic3⋊C4—21no—
77C3×C4×Dic3—21no—
78C3×#208—21no—
79#2197—31noSylow
80C3×H(5,𝔽2)—31no—
81#1004C22×C311yescharacteristic
82C3×#137—61no—
83C3×#126—61no—
84C3×#124—21no—
85C3×#124C2×D311yescharacteristic
86C3×#124C22×C311yescharacteristic
87C3×#123—31no—
88C3×#123—61no—
89C3×#123—61no—
90C3×C22⋊Q8—61no—
91C3×#120—31no—
92C3×#120—61no—
93C3×C4×D4—61no—
94C3×#117—61no—
95C2×C3×C22⋊C4—61no—
96#946—21no—
97#942—21no—
98#937C22×C311yescharacteristic
99#935C22×C311yescharacteristic
100#886—21no—
101#883—21no—
102#882—21no—
103#67⋊C4—21no—
104D3×C22⋊C4—21no—
105#879—21no—
106#872—21no—
107C3×#104—61no—
108C3×#100—31no—
109C3×#99—21no—
110C3×#99—21no—
111C3×#99—61no—
112C3×#99—61no—
113C3×#99C2×D311yescharacteristic
114C3×#99C2×D311yescharacteristic
115C3×#99C22×C311yescharacteristic
116C3×#99C22×C311yescharacteristic
117#833C22×C311yescharacteristic
118#806—21no—
119#806—21no—
120#805—21no—
121C23×C32C2×D411yescharacteristic
122C23×C32C2×D411yescharacteristic
123C22×C3×D3—21no—
124C22×C3×D3—21no—
125C22×C3×D3—21no—
126C22×C3×D3—42no—
127C22×C3×D3—41no—
128C32×Q8—41no—
129C32×D4—41no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
134C2×C32×C4C2×D411yescharacteristic
135C3×#67—42no—
136C3×#67—41no—
137C3×#67—41no—
138C3×#67—41no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—41no—
145C3×D12—21no—
146C3×D12—82no—
147C3×C4×D3—41no—
148C3×C4×D3—42no—
149C3×C4×D3—81no—
150C3×Dic6—21no—
151C3×C3⋊C8—41no—
152#551—31no—
153#458—31no—
154B(4,𝔽2)—31no—
155#455—31no—
156#354C3×D311yescharacteristic, complemented
157#352—31no—
158AΣL(1,𝔽16)—31no—
159C24×C3—31no—
160C23×D3C3×D411yescharacteristic
161C3×#41—61no—
162C3×#41—121no—
163C2×C3×Q8—21no—
164C2×C3×Q8D1211yescharacteristic, complemented
165C2×C3×Q8C3×D411yescharacteristic, complemented
166C2×C3×D4—21no—
167C2×C3×D4—61no—
168C2×C3×D4—61no—
169C2×C3×D4—61no—
170C2×C3×D4—121no—
171C2×C3×D4—121no—
172C2×C3×D4C22×D311yescharacteristic
173C2×C3×D4C23×C311yescharacteristic
174C22×C3×C4—31no—
175C22×C3×C4—61no—
176C22×C3×C4—61no—
177C2×#67—21no—
178C2×#67—21no—
179C2×#67—21no—
180C22×Dic3C3×D411yescharacteristic, complemented
181#237—41no—
182D3×D4—41no—
183#235—21no—
184C2×D12—41no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C3×Q16—121no—
188C3×SD16—121no—
189C3×M16—61no—
190C3×C4⋊C4—61no—
191C3×C4⋊C4—61no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—121no—
195C3×C22⋊C4—21no—
196C3×C22⋊C4—21no—
197C3×C22⋊C4—21no—
198C3×C22⋊C4—21no—
199C3×C22⋊C4—21no—
200C3×C22⋊C4—21no—
201C3×C22⋊C4—21no—
202C3×C22⋊C4—21no—
203C3×C22⋊C4—31no—
204C3×C22⋊C4—31no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4—41no—
207C3×C22⋊C4—41no—
208C3×C22⋊C4—41no—
209C3×C22⋊C4—61no—
210C3×C22⋊C4—61no—
211C3×C22⋊C4—61no—
212C3×C22⋊C4—61no—
213C3×C22⋊C4—121no—
214C3×C22⋊C4—121no—
215C3×C42—21no—
216C3×C42—61no—
217C3×C42D1211yescharacteristic, complemented
218C3×C42C3×D411yescharacteristic, complemented
219#217—21no—
220#217C3×D411yescharacteristic
221#217C3×D411yescharacteristic, complemented
222#216—41no—
223#215—41no—
224(C2×D3)⋊C4—21no—
225(C2×D3)⋊C4—21no—
226(C2×D3)⋊C4—21no—
227(C2×D3)⋊C4—41no—
228(C2×D3)⋊C4—41no—
229C4⋊Dic3—21no—
230C4⋊Dic3—41no—
231Dic3⋊C4—21no—
232Dic3⋊C4—21no—
233Dic3⋊C4—21no—
234C4×Dic3—21no—
235#208—21no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—41no—
239C22×C32—41no—
240C22×C32#12011yescharacteristic
241C2×C3×D3—22no—
242C2×C3×D3—41no—
243C2×C3×D3—41no—
244C2×C3×D3—42no—
245C2×C3×D3—41no—
246C2×C3×D3—81no—
247C2×C3×D3—81no—
248C2×C3×D3—81no—
249C32×C4—21no—
250C32×C4—41no—
251C32×C4—41no—
252C32×C4—81no—
253C32×C4—81no—
254C3×Dic3—41no—
255C3×Dic3—41no—
256C3×Dic3—41no—
257C3×Dic3—41no—
258H(5,𝔽2)—31no—
259#137—61no—
260#126—61no—
261#124C2×C3×D311yescharacteristic
262#123—31no—
263#123—61no—
264#123—61no—
265C22⋊Q8—61no—
266#120—31no—
267#120—61no—
268C4×D4—61no—
269#117—61no—
270C2×C22⋊C4—61no—
271#104—61no—
272#100—31no—
273#99—61no—
274#99—61no—
275#99C2×C3×D311yescharacteristic
276#99C2×C3×D311yescharacteristic
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—61no—
280C23×C3—61no—
281C23×C3—61no—
282C23×C3—121no—
283C23×C3—122no—
284C23×C3D3×D411yescharacteristic
285C23×C3D3×D411yescharacteristic
286C23×C3C2×C3×D411yescharacteristic
287C23×C3C2×C3×D411yescharacteristic
288C22×D3—21no—
289C22×D3—21no—
290C22×D3—21no—
291C22×D3—42no—
292C22×D3—41no—
293C3×Q8—41no—
294C3×Q8—41no—
295C3×Q8—61no—
296C3×Q8—81no—
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—61no—
300C3×D4—81no—
301C3×D4—121no—
302C3×D4—121no—
303C3×D4—122no—
304C3×D4—121no—
305C3×D4—242no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
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—61no—
321C2×C3×C4—61no—
322C2×C3×C4—81no—
323C2×C3×C4—81no—
324C2×C3×C4—121no—
325C2×C3×C4—122no—
326C2×C3×C4—121no—
327C2×C3×C4—241no—
328C2×C3×C4C2×D1211yescharacteristic
329C2×C3×C4C2×C3×D411yescharacteristic
330#67—42no—
331#67—41no—
332#67—41no—
333#67—41no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—41no—
340D12—21no—
341D12—82no—
342C4×D3—41no—
343C4×D3—42no—
344C4×D3—81no—
345Dic6—21no—
346C3×C8—121no—
347C3⋊C8—41no—
348C2×C32—21no—
349C2×C32—41no—
350C2×C32—41no—
351C2×C32B(4,𝔽2)11yescharacteristic
352C3×D3—42no—
353C3×D3—81no—
354C24—31no—
355#41—61no—
356#41—121no—
357C2×Q8C3×D1211yescharacteristic, complemented
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—121no—
362C2×D4—121no—
363C2×D4C22×C3×D311yescharacteristic
364C22×C4—31no—
365C22×C4—61no—
366C22×C4—61no—
367Q16—121no—
368SD16—121no—
369M16—61no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—61no—
373C4⋊C4—61no—
374C4⋊C4—121no—
375C22⋊C4—21no—
376C22⋊C4—21no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—31no—
380C22⋊C4—31no—
381C22⋊C4—61no—
382C22⋊C4—61no—
383C22⋊C4—61no—
384C22⋊C4—61no—
385C22⋊C4—121no—
386C22⋊C4—121no—
387C42—61no—
388C42C3×D1211yescharacteristic, complemented
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—62no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—121no—
404C22×C3—122no—
405C22×C3—121no—
406C22×C3—121no—
407C22×C3—241no—
408C22×C3—241no—
409C22×C3—241no—
410C22×C3#88211yescharacteristic
411C22×C3C3×#12011yescharacteristic
412C2×D3—22no—
413C2×D3—41no—
414C2×D3—41no—
415C2×D3—42no—
416C2×D3—41no—
417C2×D3—81no—
418C2×D3—81no—
419C2×D3—81no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—41no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—81no—
428C3×C4—81no—
429C3×C4—81no—
430C3×C4—81no—
431C3×C4—81no—
432C3×C4—81no—
433C3×C4—121no—
434C3×C4—121no—
435C3×C4—121no—
436C3×C4—121no—
437C3×C4—161no—
438C3×C4—161no—
439Dic3—41no—
440Dic3—41no—
441Dic3—41no—
442Dic3—41no—
443C32#219711yescharacteristic, complemented, Sylow
444C23—61no—
445C23—61no—
446C23—61no—
447C23—121no—
448C23—122no—
449C23C3×D3×D411yescharacteristic
450C23C3×D3×D411yescharacteristic
451Q8—41no—
452Q8—61no—
453D4—41no—
454D4—61no—
455D4—121no—
456D4—121no—
457D4—122no—
458D4—121no—
459D4—242no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—121no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4—241no—
473C2×C4C2×C3×D1211yescharacteristic
474C8—121no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—81no—
484C2×C3—81no—
485C2×C3—122no—
486C2×C3—241no—
487C2×C3#474111yescharacteristic
488C2×C3C3×B(4,𝔽2)11yescharacteristic
489D3—42no—
490D3—81no—
491C22—21no—
492C22—21no—
493C22—41no—
494C22—41no—
495C22—62no—
496C22—121no—
497C22—122no—
498C22—121no—
499C22—121no—
500C22—241no—
501C22—241no—
502C22—241no—
503C22C3×#88211yescharacteristic
504C4—21no—
505C4—41no—
506C4—41no—
507C4—81no—
508C4—81no—
509C4—121no—
510C4—121no—
511C4—121no—
512C4—121no—
513C3—21no—
514C3#6915311yescharacteristic, direct product factor
515C3C3×#219711yescharacteristic, complemented
516C2—21no—
517C2—41no—
518C2—41no—
519C2—122no—
520C2—241no—
521C2C3×#474111yescharacteristic
subgroupG/H
centerC2×C3#4741
commutator, cosocleC2×C3×D4C23×C3
FrattiniC2×D4C22×C3×D3
FittingC32×#354C2
socleC2×C32B(4,𝔽2)
prime radicalC3×#4741C2
hypercenterC3×#354D3
hypocenterC3C3×#2197
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#354

Complements

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

NH ≅ G/Ntype
1. C3×#5679C2semidirect
2. C3×#5061C2semidirect
3. C3×#4742C2semidirect
5. C32×#354C2semidirect
6. C3×#4472C2semidirect
7. C3×#4471C2semidirect
9. #69153C3direct (N × H)
30. #5679C2×C3semidirect
34. #5061C2×C3semidirect
35. #4742C2×C3semidirect
38. C3×#354D3semidirect
39. C3×#354C2×C3semidirect
42. #4472C2×C3semidirect
43. #4471C2×C3semidirect
45. C2×C32×Q8D4semidirect
50. C22×C3×Dic3D4semidirect
61. C32×C42D4semidirect
64. C3×#217D4semidirect
156. #354C3×D3semidirect
164. C2×C3×Q8D12semidirect
165. C2×C3×Q8C3×D4semidirect
180. C22×Dic3C3×D4semidirect
217. C3×C42D12semidirect
218. C3×C42C3×D4semidirect
221. #217C3×D4semidirect
357. C2×Q8C3×D12semidirect
388. C42C3×D12semidirect
443. C32#2197semidirect
514. C3#69153direct (N × H)
515. C3C3×#2197semidirect

Maximal subgroups

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

subgroupindex
C3×#56792
C3×#50612
C3×#47422
C3×#47412
C32×#3542
C3×#44722
C3×#44712
C3×#21973
#691533