Finite Group Databasebeta

← #85873⋊C3 #11923213

Subgroups — #85873⋊C3 #11923213

Order 1152.

769 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 769 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×#1007⋊C3C211yesmaximal, characteristic
2C3×#519⋊C3C211yesmaximal, characteristic, complemented
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#5572⋊C3C211yesmaximal, characteristic, complemented
5C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#5574⋊C3C211yesmaximal, characteristic, complemented
7C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
8#85873C311yesmaximal, characteristic, complemented
9C4×2−1+4⋊C3—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11#1007⋊C3C414yescomplemented
12C2×D3×SL(2,𝔽3)C2211yescharacteristic
13C2×C3⋊(#41⋊C3)C2211yescharacteristic
14C3×#67⋊C4—41nomaximal
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16C3×C4×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18Dic3×SL(2,𝔽3)C2211yescharacteristic
19C2×#1007C2×C311yescharacteristic
20C2×2−1+4⋊C3—31no—
21C3×#519C2×C311yescharacteristic, complemented
22C2×Q8×Dic3C2×C311yescharacteristic, complemented
23#5574—31no—
24#5574C2×C311yescharacteristic, complemented
25C4×#239—31no—
26C4×#239C2×C311yescharacteristic, complemented
27#5572—31no—
28#5572C2×C311yescharacteristic, complemented
29C4×D3×Q8—31no—
30C4×D3×Q8C2×C311yescharacteristic, complemented
31#5523—31no—
32#5523—31no—
33#5522—31no—
34#5522—31no—
35#520⋊C3—31no—
36#519⋊C3—21no—
37#519⋊C3—31no—
38#519⋊C3D311yescharacteristic, complemented
39C4×#41⋊C3—31no—
40C2×C4×SL(2,𝔽3)—31no—
41C2×C4×SL(2,𝔽3)—31no—
42C2×C3×#67—41no—
43C22×C3×Dic3—41no—
44C2×C3×C4×D3—41no—
45C2×C3×SL(2,𝔽3)C2×C412yescomplemented
46C2×C3×SL(2,𝔽3)C2311yescharacteristic
47D3×SL(2,𝔽3)C2×C412yescomplemented
48C3⋊(#41⋊C3)C2×C412yescomplemented
49C32×C22⋊C4—41no—
50C3×(C2×D3)⋊C4—41no—
51C3×Dic3⋊C4—41no—
52C3×C4×Dic3—41no—
53C4×2−1+4—31noSylow
54C22×C3×Q8C22×C311yescharacteristic
55#1007—34no—
56#1007C3×C414yescomplemented
57C2×#239—31no—
58C2×#239C22×C311yescharacteristic
59C2×D3×Q8—31no—
60C2×D3×Q8C22×C311yescharacteristic
61C2×#235—31no—
62C2×#235—31no—
632−1+4⋊C3—34no—
64C2×#41⋊C3—31no—
65C22×SL(2,𝔽3)—21no—
66C22×SL(2,𝔽3)—31no—
67C22×SL(2,𝔽3)C2×D311yescharacteristic
68C3×C4×Q8—31no—
69C3×C4×Q8—31no—
70C3×C4×Q8C22×C311yescharacteristic
71C3×C4×Q8C22×C311yescharacteristic
72C3×C4×D4—121no—
73C3×#117—31no—
74C3×#117—31no—
75Q8×Dic3—31no—
76Q8×Dic3—31no—
77Q8×Dic3C22×C311yescharacteristic
78Q8×Dic3C22×C311yescharacteristic
79C2×C4⋊Dic3—31no—
80C2×C4×Dic3—31no—
81#893—31no—
82#893—31no—
83#893—31no—
84#893—31no—
85#892—32no—
86#892—31no—
87#892—31no—
88D3×C4⋊C4—31no—
89D3×C4⋊C4—32no—
90D3×C4⋊C4—31no—
91#887—31no—
92#887—31no—
93#887—31no—
94#887—31no—
95#67⋊C4—32no—
96#67⋊C4—31no—
97#67⋊C4—31no—
98#67⋊C4—31no—
99#67⋊C4A411yescharacteristic
100C4×D12—32no—
101C4×D12—31no—
102C4×D12—31no—
103#872—32no—
104#872—31no—
105#872—31no—
106C42×D3—32no—
107C42×D3—31no—
108C42×D3—31no—
109C4×Dic6—32no—
110C4×Dic6—31no—
111C4×Dic6—31no—
112C4×SL(2,𝔽3)—21no—
113C4×SL(2,𝔽3)—21no—
114C4×SL(2,𝔽3)—31no—
115C4×SL(2,𝔽3)—31no—
116C4×SL(2,𝔽3)C2×D311yescharacteristic
117C4×SL(2,𝔽3)C2×D311yescharacteristic
118C23×C32—41no—
119C22×C3×D3—41no—
120C2×C32×C4—41no—
121C2×C32×C4—41no—
122C3×#67—44no—
123C2×C3×Dic3—41no—
124C2×C3×Dic3—41no—
125C2×C3×Dic3—41no—
126C2×C3×Dic3—82no—
127C3×C4×D3—82no—
128C3×SL(2,𝔽3)C22×C411yescharacteristic
129C2×2−1+4—31no—
130#520—31no—
131#520—91no—
132#520—91no—
133#520—91no—
134#519—31no—
135#519—91no—
136#519C3×D311yescharacteristic, complemented
137C4×#41—31no—
138C4×#41—91no—
139C4×#41—91no—
140C4×#41—91no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×Q8—91no—
144C2×C3×Q8—31no—
145C2×C3×Q8—32no—
146C2×C3×Q8C2×C3×C412yescomplemented
147C2×C3×Q8C23×C311yescharacteristic
148C2×C3×D4—121no—
149C22×C3×C4—31no—
150C22×C3×C4—121no—
151C22×C3×C4—121no—
152C2×#67—31no—
153C2×#67C2×A411yescharacteristic
154C22×Dic3—31no—
155C22×Dic3C2×A411yescharacteristic
156#239—34no—
157#239—34no—
158#239—32no—
159#239C2×C3×C412yescomplemented
160D3×Q8—34no—
161D3×Q8—32no—
162D3×Q8—34no—
163D3×Q8C2×C3×C412yescomplemented
164#235—34no—
165#235—34no—
166#235—34no—
167#235—34no—
168C2×D12—31no—
169C2×D12—31no—
170C2×C4×D3—32no—
171C2×C4×D3—31no—
172C2×C4×D3—31no—
173C2×C4×D3—31no—
174C2×C4×D3—32no—
175C2×C4×D3—31no—
176C2×C4×D3—31no—
177C2×C4×D3C2×A411yescharacteristic
178C2×Dic6—31no—
179C2×Dic6—31no—
180#41⋊C3—32no—
181C2×SL(2,𝔽3)—22no—
182C2×SL(2,𝔽3)—21no—
183C2×SL(2,𝔽3)—32no—
184C2×SL(2,𝔽3)C4×D312yescomplemented
185C2×SL(2,𝔽3)C22×D311yescharacteristic
186C3×C4⋊C4—31no—
187C3×C4⋊C4—31no—
188C3×C4⋊C4—31no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—121no—
191C3×C22⋊C4—31no—
192C3×C22⋊C4—41no—
193C3×C22⋊C4—81no—
194C3×C22⋊C4—121no—
195C3×C22⋊C4C2×A411yescharacteristic
196C3×C42—32no—
197C3×C42—31no—
198C3×C42—31no—
199C3×C42—121no—
200(C2×D3)⋊C4—32no—
201(C2×D3)⋊C4—31no—
202(C2×D3)⋊C4—31no—
203(C2×D3)⋊C4—31no—
204(C2×D3)⋊C4C2×A411yescharacteristic
205C4⋊Dic3—32no—
206C4⋊Dic3—31no—
207C4⋊Dic3—31no—
208Dic3⋊C4—32no—
209Dic3⋊C4—31no—
210Dic3⋊C4—31no—
211Dic3⋊C4—31no—
212Dic3⋊C4C2×A411yescharacteristic
213C4×Dic3—32no—
214C4×Dic3—31no—
215C4×Dic3—31no—
216C4×Dic3—31no—
217C4×Dic3—31no—
218C4×Dic3—31no—
219C4×Dic3—31no—
220C4×Dic3—31no—
221C4×Dic3C2×A411yescharacteristic
222C22×C32—42no—
223C22×C32—41no—
224C22×C32—81no—
225C22×C32—81no—
226C2×C3×D3—42no—
227C2×C3×D3—81no—
228C2×C3×D3—81no—
229C32×C4—81no—
230C32×C4—81no—
231C3×Dic3—42no—
232C3×Dic3—42no—
233C3×Dic3—81no—
2342−1+4—34no—
2352−1+4—94no—
236C2×#41—31no—
237C2×#41—91no—
238C2×#41—91no—
239C2×#41—91no—
240C22×Q8—31no—
241C22×Q8—91no—
242C22×Q8C2×C3×D311yescharacteristic
243C4×Q8—31no—
244C4×Q8—31no—
245C4×Q8—31no—
246C4×Q8—31no—
247C4×Q8—91no—
248C4×Q8—92no—
249C4×Q8—91no—
250C4×Q8—91no—
251C4×Q8—91no—
252C4×Q8—91no—
253C4×Q8—91no—
254C4×Q8—91no—
255C4×Q8—91no—
256C4×Q8C2×C3×D311yescharacteristic
257C4×Q8C2×C3×D311yescharacteristic
258C4×D4—31no—
259C4×D4—92no—
260C4×D4—91no—
261C4×D4—91no—
262C4×D4—91no—
263C4×D4—91no—
264C4×D4—91no—
265C4×D4—92no—
266C4×D4—91no—
267C4×D4—91no—
268C4×D4—91no—
269C4×D4—91no—
270#117—31no—
271#117—31no—
272#117—91no—
273#117—91no—
274#117—92no—
275#117—92no—
276#117—91no—
277#117—91no—
278C2×C4⋊C4—91no—
279C2×C4⋊C4—92no—
280C2×C4⋊C4—91no—
281C2×C4⋊C4—91no—
282C2×C42—91no—
283C2×C42—91no—
284C2×C42—92no—
285C2×C42—91no—
286C23×C3—41no—
287C23×C3—81no—
288C23×C3—121no—
289C23×C3C22×A411yescharacteristic
290C22×D3—31no—
291C22×D3C22×A411yescharacteristic
292C3×Q8—32no—
293C3×Q8—31no—
294C3×Q8—32no—
295C3×Q8C22×C3×C411yescharacteristic
296C3×D4—124no—
297C2×C3×C4—32no—
298C2×C3×C4—32no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—81no—
306C2×C3×C4—81no—
307C2×C3×C4—121no—
308C2×C3×C4—121no—
309C2×C3×C4—121no—
310C2×C3×C4—242no—
311C2×C3×C4—242no—
312C2×C3×C4C22×A411yescharacteristic
313C2×C3×C4C22×A411yescharacteristic
314#67—34no—
315#67C4×A414yes—
316C2×Dic3—22no—
317C2×Dic3—31no—
318C2×Dic3—31no—
319C2×Dic3—31no—
320C2×Dic3—121no—
321C2×Dic3C22×A411yescharacteristic
322C2×Dic3C22×A411yescharacteristic
323C2×Dic3C22×A411yescharacteristic
324D12—32no—
325D12—32no—
326D12—34no—
327C4×D3—22no—
328C4×D3—38no—
329C4×D3—32no—
330C4×D3—32no—
331C4×D3—32no—
332C4×D3—32no—
333C4×D3—62no—
334C4×D3—121no—
335C4×D3—121no—
336C4×D3—122no—
337Dic6—32no—
338Dic6—34no—
339Dic6—32no—
340SL(2,𝔽3)—21no—
341SL(2,𝔽3)C2×C4×D311yescharacteristic
342C2×C32—41no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—82no—
346C3×D3—82no—
347#41—32no—
348#41—94no—
349#41—94no—
350#41—94no—
351#41—94no—
352#41—94no—
353#41—94no—
354#41—92no—
355C2×Q8—31no—
356C2×Q8—32no—
357C2×Q8—32no—
358C2×Q8—91no—
359C2×Q8—91no—
360C2×Q8—94no—
361C2×Q8—94no—
362C2×Q8—92no—
363C2×Q8C3×C4×D312yescomplemented
364C2×Q8C22×C3×D311yescharacteristic
365C2×D4—31no—
366C2×D4—91no—
367C2×D4—91no—
368C2×D4—91no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—91no—
375C22×C4—92no—
376C22×C4—91no—
377C22×C4—92no—
378C22×C4—91no—
379C22×C4—91no—
380C4⋊C4—31no—
381C4⋊C4—31no—
382C4⋊C4—31no—
383C4⋊C4—31no—
384C4⋊C4—31no—
385C4⋊C4—91no—
386C4⋊C4—91no—
387C4⋊C4—91no—
388C4⋊C4—91no—
389C4⋊C4—92no—
390C4⋊C4—92no—
391C4⋊C4—91no—
392C22⋊C4—31no—
393C22⋊C4—31no—
394C22⋊C4—91no—
395C22⋊C4—91no—
396C22⋊C4—92no—
397C22⋊C4—91no—
398C22⋊C4D3×A411yescharacteristic
399C42—32no—
400C42—31no—
401C42—31no—
402C42—31no—
403C42—91no—
404C42—91no—
405C42—91no—
406C42—91no—
407C42—91no—
408C42—91no—
409C42—92no—
410C42—91no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—42no—
414C22×C3—41no—
415C22×C3—81no—
416C22×C3—81no—
417C22×C3—82no—
418C22×C3—81no—
419C22×C3—122no—
420C22×C3—161no—
421C22×C3—161no—
422C22×C3—241no—
423C22×C3—241no—
424C22×C3C2×C4×A412yes—
425C22×C3C23×A411yescharacteristic
426C2×D3—21no—
427C2×D3—21no—
428C2×D3—32no—
429C2×D3—61no—
430C2×D3—61no—
431C2×D3C2×C4×A412yes—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—31no—
435C3×C4—32no—
436C3×C4—31no—
437C3×C4—61no—
438C3×C4—61no—
439C3×C4—81no—
440C3×C4—81no—
441C3×C4—122no—
442C3×C4—122no—
443C3×C4—161no—
444C3×C4—161no—
445C3×C4—241no—
446Dic3—21no—
447Dic3—32no—
448Dic3—61no—
449Dic3—61no—
450Dic3C2×C4×A412yes—
451Dic32−1+4⋊C312yescomplemented
452C32—41noSylow
453C23—31no—
454C23—91no—
455C23C2×D3×A411yescharacteristic
456Q8—32no—
457Q8—31no—
458Q8—32no—
459Q8—94no—
460Q8—92no—
461Q8—92no—
462Q8C2×C3×C4×D311yescharacteristic
463D4—34no—
464D4—94no—
465D4—92no—
466D4—92no—
467D4—94no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—31no—
471C2×C4—31no—
472C2×C4—31no—
473C2×C4—31no—
474C2×C4—31no—
475C2×C4—31no—
476C2×C4—31no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—92no—
480C2×C4—92no—
481C2×C4—92no—
482C2×C4—92no—
483C2×C4—98no—
484C2×C4—91no—
485C2×C4—91no—
486C2×C4—91no—
487C2×C4—182no—
488C2×C4—361no—
489C2×C4—361no—
490C2×C4—362no—
491C2×C4—361no—
492C2×C4C2×D3×A411yescharacteristic
493C2×C4C2×D3×A411yescharacteristic
494C2×C3—22no—
495C2×C3—41no—
496C2×C3—41no—
497C2×C3—41no—
498C2×C3—82no—
499C2×C3—81no—
500C2×C3—81no—
501C2×C3—81no—
502C2×C3—162no—
503C2×C3—242no—
504C2×C3C22×C4×A411yescharacteristic
505C2×C3C2×2−1+4⋊C311yescharacteristic
506C2×C3#594511yescharacteristic
507D3—22no—
508D3—62no—
509C22—21no—
510C22—21no—
511C22—32no—
512C22—61no—
513C22—61no—
514C22—92no—
515C22—181no—
516C22—181no—
517C22C4×D3×A412yes—
518C22C22×D3×A411yescharacteristic
519C4—21no—
520C4—21no—
521C4—31no—
522C4—32no—
523C4—31no—
524C4—32no—
525C4—32no—
526C4—61no—
527C4—61no—
528C4—61no—
529C4—92no—
530C4—181no—
531C4—181no—
532C3—41no—
533C3—81no—
534C3C4×2−1+4⋊C311yescharacteristic, complemented
535C2—22no—
536C2—62no—
537C2—182no—
538C2C2×C4×D3×A411yescharacteristic
539C2D3×2−1+4⋊C311yescharacteristic
540C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FittingC3×#519C2×C3
socleC22×C3C23×A4
prime radicalC2×#1007⋊C3C2
cosocleC2×C3×Q8C23×C3
hypercenterC22⋊C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#519⋊C3C2semidirect
3. C2×Dic3×SL(2,𝔽3)C2semidirect
4. #5572⋊C3C2semidirect
5. C4×D3×SL(2,𝔽3)C2semidirect
6. #5574⋊C3C2semidirect
7. C4×C3⋊(#41⋊C3)C2semidirect
8. #85873C3semidirect
11. #1007⋊C3C4semidirect
21. C3×#519C2×C3semidirect
22. C2×Q8×Dic3C2×C3semidirect
24. #5574C2×C3semidirect
26. C4×#239C2×C3semidirect
28. #5572C2×C3semidirect
30. C4×D3×Q8C2×C3semidirect
38. #519⋊C3D3semidirect
45. C2×C3×SL(2,𝔽3)C2×C4semidirect
47. D3×SL(2,𝔽3)C2×C4semidirect
48. C3⋊(#41⋊C3)C2×C4semidirect
56. #1007C3×C4semidirect
136. #519C3×D3semidirect
146. C2×C3×Q8C2×C3×C4semidirect
159. #239C2×C3×C4semidirect
163. D3×Q8C2×C3×C4semidirect
184. C2×SL(2,𝔽3)C4×D3semidirect
363. C2×Q8C3×C4×D3semidirect
451. Dic32−1+4⋊C3semidirect
534. C3C4×2−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1007⋊C32
C3×#519⋊C32
C2×Dic3×SL(2,𝔽3)2
#5572⋊C32
C4×D3×SL(2,𝔽3)2
#5574⋊C32
C4×C3⋊(#41⋊C3)2
#858733
C4×2−1+4⋊C33
C3×#67⋊C44