Finite Group Databasebeta

← D3×#5419 #11922216

Subgroups — D3×#5419 #11922216

Order 1152.

723 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 723 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22×D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×D3×#226C211yesmaximal, characteristic, complemented
3#10593653C211yesmaximal, characteristic, complemented
4C3×#5419C211yesmaximal, characteristic, complemented
5D3×#859C211yesmaximal, characteristic, complemented
6#10593211C211yesmaximal, characteristic, complemented
7#10593210C211yesmaximal, characteristic, complemented
8C2×#5419—31nomaximal
9D3×#451—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C2×D3×SL(2,𝔽3)—22no—
12C2×D3×SL(2,𝔽3)C2211yescharacteristic
13C2×D3×SL(2,𝔽3)C2211yescharacteristic
14C2×C3⋊#226C2211yescharacteristic, complemented
15C2×C3×#226C2211yescharacteristic, complemented
16D3×#226—22no—
17D3×#217—41nomaximal
18C3⋊#859C2211yescharacteristic, complemented
19C3×#859C2211yescharacteristic, complemented
20C22×D3×Q8D311yescharacteristic
21C23×SL(2,𝔽3)—31no—
22C22×#226—31no—
23C2×D3×Q16—31no—
24C2×D3×Q16—31no—
25D3×C22⋊Q8—31no—
26#5419—31no—
27#5419—31no—
28#5419—31no—
29#5419—61no—
30#5419D311yescharacteristic, direct product factor
31C2×#859—31no—
32C3×#451—31no—
33#5228—31no—
34#5186—31no—
35#5186—31no—
36#5050—31no—
37#4809—31no—
38#4809—31no—
39D3×#103—31no—
40D3×#103—31no—
41#4739—31no—
42D3×C22⋊C8—31no—
43C23×C3×D3—41no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)C2311yescharacteristic
46C2×D3×Dic3—41no—
47C2×D3×Dic3—41no—
48D3×SL(2,𝔽3)—21no—
49D3×SL(2,𝔽3)D412yes—
50C3⋊#226—21no—
51C3×#226—21no—
52#3757—41no—
53C3×#217—41no—
54(C2×D3)⋊Dic3—41no—
55(C2×D3)⋊Dic3—41no—
56C2×#451—91noSylow
57C22×C3×Q8C2×D311yescharacteristic
58C2×D3×Q8—22no—
59C2×D3×Q8—31no—
60C2×D3×Q8—31no—
61C2×D3×Q8—31no—
62C2×D3×Q8—62no—
63C2×D3×Q8C2×D311yescharacteristic
64C2×D3×Q8C2×D311yescharacteristic
65C22×C4×D3—31no—
66C22×Dic6—31no—
67C22×SL(2,𝔽3)—21no—
68C22×SL(2,𝔽3)—31no—
69C22×SL(2,𝔽3)—31no—
70C22×SL(2,𝔽3)—62no—
71C22×SL(2,𝔽3)C2×D311yescharacteristic
72C2×#226—31no—
73C2×#226—61no—
74C2×#226—62no—
75C2×#226C2×D311yescharacteristic, complemented
76C2×C3×Q16—31no—
77C2×C3×Q16—31no—
78C3×C22⋊Q8—31no—
79C2×#217—121no—
80#946—31no—
81C2×#216—31no—
82C2×#216—31no—
83C2×#216—31no—
84C2×#216—31no—
85#924—31no—
86D3×Q16—62no—
87D3×Q16—64no—
88D3×Q16—62no—
89C2×Dic12—31no—
90C2×Dic12—31no—
91C2×D3×C8—31no—
92C2×D3×C8—31no—
93#897—31no—
94#896—61no—
95D3×C4⋊C4—31no—
96D3×C4⋊C4—61no—
97D3×C22⋊C4—61no—
98#878—61no—
99#859—31no—
100#859—61no—
101#859C2×D311yescharacteristic, complemented
102C3×#103—31no—
103C3×#103—31no—
104C3×C22⋊C8—31no—
105#835—31no—
106#835—31no—
107#830—31no—
108(C2×D3)⋊C8—31no—
109(C2×D3)⋊C8—31no—
110#816—31no—
111#816—31no—
112#810—31no—
113#810—31no—
114C23×C32—41no—
115C22×C3×D3—44no—
116C22×C3×D3—41no—
117C22×C3×D3—41no—
118C22×C3×D3—82no—
119C22×C3×D3—82no—
120C2×C3⋊Dic3—41no—
121C2×C3⋊Dic3—41no—
122C2×C3×Dic3—41no—
123C2×C3×Dic3—41no—
124C3×SL(2,𝔽3)C2×D411yescharacteristic
125D3×Dic3—82no—
126D3×Dic3—82no—
127C23×Q8—31no—
128C22×Q16—91no—
129C22×Q16—91no—
130C2×C22⋊Q8—91no—
131#451—31no—
132#451—91no—
133#451—91no—
134#451—91no—
135#451—91no—
136#451—91no—
137#451—91no—
138#451—91no—
139C2×#103—91no—
140C2×#103—91no—
141C2×C22⋊C8—91no—
142C24×C3—121no—
143C23×D3S411yescharacteristic
144C2×C3×Q8—21no—
145C2×C3×Q8—31no—
146C2×C3×Q8—31no—
147C2×C3×Q8—61no—
148C2×C3×Q8C22×D311yescharacteristic
149C22×C3×C4—31no—
150C22×Dic3—31no—
151C22×Dic3—121no—
152C22×Dic3—121no—
153D3×Q8—21no—
154D3×Q8—32no—
155D3×Q8—34no—
156D3×Q8—61no—
157D3×Q8—62no—
158D3×Q8—62no—
159D3×Q8—62no—
160D3×Q8—62no—
161D3×Q8—62no—
162D3×Q8#6712yes—
163C2×C4×D3—31no—
164C2×C4×D3—31no—
165C2×C4×D3—31no—
166C2×C4×D3—31no—
167C2×C4×D3—62no—
168C2×C4×D3—62no—
169C2×C4×D3—61no—
170C2×C4×D3—61no—
171C2×Dic6—31no—
172C2×Dic6—31no—
173C2×Dic6—31no—
174C2×Dic6—62no—
175C2×Dic6—62no—
176C2×Dic6—61no—
177C2×Dic6—61no—
178C2×SL(2,𝔽3)—21no—
179C2×SL(2,𝔽3)—21no—
180C2×SL(2,𝔽3)—32no—
181C2×SL(2,𝔽3)—41no—
182C2×SL(2,𝔽3)—61no—
183C2×SL(2,𝔽3)C22×D311yescharacteristic
184#226—21no—
185#226—61no—
186#226—121no—
187C3×Q16—61no—
188C3×Q16—62no—
189C3×Q16—61no—
190C2×C3×C8—31no—
191C2×C3×C8—31no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—61no—
194C3×C22⋊C4—61no—
195#217—41no—
196#217—61no—
197#217—121no—
198#217—121no—
199#217—121no—
200#217—241no—
201#216—62no—
202#216—62no—
203#216—61no—
204#216—62no—
205#216—61no—
206(C2×D3)⋊C4—61no—
207(C2×D3)⋊C4—61no—
208C4⋊Dic3—31no—
209C4⋊Dic3—61no—
210Dic3⋊C4—61no—
211Dic3⋊C4—61no—
212Dic3⋊C4—61no—
213C2×C3⋊C8—31no—
214C2×C3⋊C8—31no—
215Dic12—62no—
216Dic12—62no—
217D3×C8—62no—
218D3×C8—62no—
219C22×C32—42no—
220C22×C32—41no—
221C22×C32—81no—
222C22×C32—81no—
223C2×C3×D3—42no—
224C2×C3×D3—42no—
225C2×C3×D3—42no—
226C2×C3×D3—42no—
227C2×C3×D3—88no—
228C2×C3×D3—81no—
229C2×C3×D3—81no—
230C3⋊Dic3—81no—
231C3⋊Dic3—81no—
232C3×Dic3—81no—
233C3×Dic3—81no—
234C22×Q8—31no—
235C22×Q8—31no—
236C22×Q8—62no—
237C22×Q8—91no—
238C22×Q8—91no—
239C22×Q8—91no—
240C22×Q8—91no—
241C22×Q8—182no—
242C22×Q8D3211yescharacteristic
243C23×C4—91no—
244C2×Q16—31no—
245C2×Q16—31no—
246C2×Q16—91no—
247C2×Q16—91no—
248C2×Q16—91no—
249C2×Q16—91no—
250C2×Q16—91no—
251C2×Q16—91no—
252C2×Q16—182no—
253C2×Q16—182no—
254C2×Q16—184no—
255C22×C8—91no—
256C22×C8—91no—
257C22⋊Q8—31no—
258C22⋊Q8—91no—
259C22⋊Q8—91no—
260C22⋊Q8—91no—
261C22⋊Q8—181no—
262C22⋊Q8—181no—
263C2×C4⋊C4—91no—
264C2×C4⋊C4—181no—
265C2×C22⋊C4—181no—
266#103—31no—
267#103—31no—
268#103—91no—
269#103—91no—
270#103—91no—
271#103—91no—
272#103—91no—
273#103—91no—
274C22⋊C8—31no—
275C22⋊C8—91no—
276C22⋊C8—91no—
277C22⋊C8—91no—
278C23×C3—41no—
279C23×C3—81no—
280C23×C3—121no—
281C23×C3—121no—
282C23×C3—124no—
283C23×C3—242no—
284C23×C3—242no—
285C23×C3C2×S411yescharacteristic
286C22×D3—22no—
287C22×D3—22no—
288C22×D3#22614yescomplemented
289C22×D3C2×S411yescharacteristic
290C22×D3C2×S411yescharacteristic
291C3×Q8—31no—
292C3×Q8—32no—
293C3×Q8—61no—
294C3×Q8—61no—
295C3×Q8—61no—
296C3×Q8C2×#6711yescharacteristic
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×Dic3—31no—
304C2×Dic3—31no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—62no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—121no—
311C2×Dic3—121no—
312C2×Dic3—241no—
313C2×Dic3—241no—
314C2×Dic3—242no—
315C2×Dic3—242no—
316C4×D3—32no—
317C4×D3—32no—
318C4×D3—62no—
319C4×D3—61no—
320C4×D3—61no—
321C4×D3—62no—
322C4×D3—122no—
323C4×D3—122no—
324Dic6—32no—
325Dic6—32no—
326Dic6—61no—
327Dic6—64no—
328Dic6—61no—
329Dic6—61no—
330Dic6—61no—
331Dic6—122no—
332SL(2,𝔽3)—21no—
333SL(2,𝔽3)D3×D411yescharacteristic
334C3×C8—61no—
335C3×C8—61no—
336C3⋊C8—61no—
337C3⋊C8—61no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—41no—
341C2×C32—82no—
342C3×D3—44no—
343C3×D3—82no—
344C24—31no—
345C2×Q8—21no—
346C2×Q8—31no—
347C2×Q8—31no—
348C2×Q8—32no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—91no—
352C2×Q8—91no—
353C2×Q8—91no—
354C2×Q8—92no—
355C2×Q8—94no—
356C2×Q8—181no—
357C2×Q8—181no—
358C2×Q8—182no—
359C2×Q8—182no—
360C2×Q8—182no—
361C2×Q8—182no—
362C2×Q8—182no—
363C2×Q8—182no—
364C2×Q8—181no—
365C2×Q8—182no—
366C2×Q8C2×D3211yescharacteristic
367C22×C4—31no—
368C22×C4—91no—
369C22×C4—91no—
370C22×C4—91no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—182no—
374C22×C4—182no—
375C22×C4—181no—
376C22×C4—181no—
377Q16—61no—
378Q16—62no—
379Q16—61no—
380Q16—182no—
381Q16—182no—
382Q16—181no—
383Q16—182no—
384Q16—182no—
385Q16—182no—
386Q16—181no—
387C2×C8—31no—
388C2×C8—31no—
389C2×C8—91no—
390C2×C8—91no—
391C2×C8—182no—
392C2×C8—182no—
393C4⋊C4—31no—
394C4⋊C4—61no—
395C4⋊C4—91no—
396C4⋊C4—181no—
397C4⋊C4—181no—
398C4⋊C4—181no—
399C4⋊C4—181no—
400C22⋊C4—61no—
401C22⋊C4—181no—
402C22⋊C4—181no—
403C22⋊C4—181no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—41no—
407C22×C3—42no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3—82no—
412C22×C3—122no—
413C22×C3—122no—
414C22×C3—122no—
415C22×C3—122no—
416C22×C3—161no—
417C22×C3—161no—
418C22×C3—241no—
419C22×C3—241no—
420C22×C3—248no—
421C22×C3C2×#22612yescomplemented
422C22×C3C22×S411yescharacteristic
423C2×D3—21no—
424C2×D3—21no—
425C2×D3—28no—
426C2×D3C2×#22612yes—
427C2×D3C2×#22612yescomplemented
428C2×D3#98312yes—
429C2×D3GL(2,ℤ/4)12yes—
430C3×C4—31no—
431C3×C4—31no—
432C3×C4—61no—
433C3×C4—121no—
434C3×C4—121no—
435Dic3—32no—
436Dic3—61no—
437Dic3—81no—
438Dic3—81no—
439Dic3—121no—
440Dic3—121no—
441Dic3—241no—
442Dic3—241no—
443Dic3—481no—
444Dic3—481no—
445C32—41noSylow
446C23—31no—
447C23—31no—
448C23—34no—
449C23—62no—
450C23—62no—
451C23D3×S411yescharacteristic
452Q8—31no—
453Q8—32no—
454Q8—61no—
455Q8—61no—
456Q8—61no—
457Q8—92no—
458Q8—92no—
459Q8—181no—
460Q8—181no—
461Q8—181no—
462Q8—181no—
463Q8—184no—
464Q8—362no—
465Q8D3×#6711yescharacteristic
466C2×C4—31no—
467C2×C4—31no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—91no—
475C2×C4—91no—
476C2×C4—181no—
477C2×C4—181no—
478C2×C4—182no—
479C2×C4—182no—
480C2×C4—182no—
481C2×C4—181no—
482C2×C4—181no—
483C2×C4—362no—
484C2×C4—362no—
485C8—61no—
486C8—61no—
487C8—181no—
488C8—181no—
489C2×C3—22no—
490C2×C3—41no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—82no—
494C2×C3—81no—
495C2×C3—81no—
496C2×C3—81no—
497C2×C3—124no—
498C2×C3—162no—
499C2×C3—242no—
500C2×C3C22×#22611yescharacteristic
501C2×C3C2×#98311yescharacteristic
502C2×C3C2×GL(2,ℤ/4)11yescharacteristic
503D3—22no—
504D3#541914yesdirect product factor
505C22—21no—
506C22—21no—
507C22—32no—
508C22—32no—
509C22—32no—
510C22—32no—
511C22—68no—
512C22—61no—
513C22—61no—
514C22D3×#22612yescomplemented
515C22C2×D3×S411yescharacteristic
516C4—31no—
517C4—31no—
518C4—61no—
519C4—92no—
520C4—121no—
521C4—121no—
522C4—181no—
523C4—361no—
524C4—361no—
525C3—41no—
526C3—81no—
527C3C2×#541911yescharacteristic, complemented
528C2—22no—
529C2—34no—
530C2—62no—
531C2C2×D3×#22611yescharacteristic
532C2D3×#98311yescharacteristic
533C2D3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC22×C3×Q8C2×D3
socleC22×C3C22×S4
prime radicalC22×D3×SL(2,𝔽3)C2
hypercenterC23D3×S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×D3×#226C2semidirect
3. #10593653C2semidirect
4. C3×#5419C2semidirect
5. D3×#859C2semidirect
6. #10593211C2semidirect
7. #10593210C2semidirect
14. C2×C3⋊#226C22semidirect
15. C2×C3×#226C22semidirect
18. C3⋊#859C22semidirect
19. C3×#859C22semidirect
30. #5419D3direct (N × H)
75. C2×#226C2×D3semidirect
101. #859C2×D3semidirect
288. C22×D3#226semidirect
421. C22×C3C2×#226semidirect
427. C2×D3C2×#226semidirect
504. D3#5419direct (N × H)
514. C22D3×#226semidirect
527. C3C2×#5419semidirect

Maximal subgroups

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

subgroupindex
C22×D3×SL(2,𝔽3)2
C2×D3×#2262
#105936532
C3×#54192
D3×#8592
#105932112
#105932102
C2×#54193
D3×#4513
D3×#2174