Finite Group Databasebeta

← #13141065

Subgroups — #13141065

Order 1728.

583 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#41×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721105C211yesmaximal, characteristic, complemented
3#11721053C211yesmaximal, characteristic, complemented
4#11720586C211yesmaximal, characteristic, complemented
5#63664⋊C3C211yesmaximal, characteristic, complemented
6#11720322C211yesmaximal, characteristic, complemented
7#11720305C211yesmaximal, characteristic, complemented
8#10591672—31nomaximal
9#10591670—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#88722C411yescharacteristic, complemented
12Q8×(C3⋊D3)⋊C3C411yescharacteristic, complemented
13#3830⋊C3—21no—
14#3830⋊C3C411yescharacteristic, complemented
15D4×(C3⋊D3)⋊C3—21no—
16D4×(C3⋊D3)⋊C3C411yescharacteristic, complemented
17#3828⋊C3—21no—
18#3828⋊C3C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3—21no—
20C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
21C4×#1084C2211yescharacteristic, complemented
22#88538C2211yescharacteristic, complemented
23C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
24#3748⋊C3C2211yescharacteristic, complemented
25#88450—21no—
26#88448—21no—
27#88430—21no—
28((C3⋊D3)⋊C3)⋊C8—21no—
29#41×C3⋊D3D311yescharacteristic
30C3×D3×#41—31no—
31#63874—31no—
32#63855—31no—
33#63818—31no—
34D3×#208—31no—
35#63664D311yescharacteristic, complemented
36C3×#837—31no—
37C3×#837—31no—
38C3×#837—31no—
39C3×#837—61no—
40#63570—31no—
41#63570—31no—
42#63569—31no—
43#63568—31no—
44C22×(C3⋊D3)⋊C3—21no—
45C22×(C3⋊D3)⋊C3D411yescharacteristic
46Q8×H(3,𝔽3)C2×C411yescharacteristic
47D4×H(3,𝔽3)—21no—
48D4×H(3,𝔽3)C2×C411yescharacteristic
49C2×C4×H(3,𝔽3)—21no—
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×#1084—21no—
52#638⋊C3—21no—
53#638⋊C3—21no—
54#638⋊C3—21no—
55C2×(C3⋊Dic3)⋊C3—21no—
56C2×(C3⋊Dic3)⋊C3—21no—
57C2×(C3⋊Dic3)⋊C3D411yescharacteristic
58C4⋊((C3⋊D3)⋊C3)—21no—
59C4⋊((C3⋊D3)⋊C3)C2×C411yescharacteristic
60C4×(C3⋊D3)⋊C3—21no—
61C4×(C3⋊D3)⋊C3—21no—
62C4×(C3⋊D3)⋊C3—21no—
63C4×(C3⋊D3)⋊C3C2×C411yescharacteristic
64C4×(C3⋊D3)⋊C3C2×C411yescharacteristic
65C4×(C3⋊D3)⋊C3D411yescharacteristic
66C4×(C3⋊D3)⋊C3D411yescharacteristic
67#634⋊C3—21no—
68#634⋊C3C2×C411yescharacteristic
69#6180—21no—
70(C3⋊(C3⋊C8))⋊C3—21no—
71#5236—91no—
72#4821—91no—
73#4821—91no—
74C32×#41—21no—
75C32×#41C2×D311yescharacteristic
76C32×#41C2×D311yescharacteristic
77#3832Dic311yescharacteristic, complemented
78Q8×C3⋊D3Dic311yescharacteristic, complemented
79#3830—21no—
80#3830Dic311yescharacteristic, complemented
81D4×C3⋊D3—21no—
82D4×C3⋊D3Dic311yescharacteristic, complemented
83#3828—21no—
84#3828C2×D311yescharacteristic
85C2×C4×C3⋊D3—21no—
86C2×C4×C3⋊D3C2×D311yescharacteristic
87C3×#239—31no—
88C3×D3×Q8—31no—
89C3×#237—31no—
90C3×#237—61no—
91C3×D3×D4—31no—
92C3×D3×D4—61no—
93C3×#235—31no—
94C3×#235—61no—
95C2×C3×C4×D3—31no—
96C2×C3×C4×D3—61no—
97C4×C3⋊Dic3C2×D311yescharacteristic, complemented
98#3748C2×D311yescharacteristic, complemented
99C3×C4×Dic3—31no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—61no—
103C3×#208—31no—
104C3×#208—31no—
105C3×#208—31no—
106C3×#208—61no—
107#3724—61no—
108Dic3⋊Dic3—61no—
109#3722—61no—
110(C2×D3)⋊Dic3—61no—
111#3712—61no—
112#3711—61no—
113#3710—61no—
114D3×C3⋊C8—61no—
115C22×H(3,𝔽3)—21no—
116C22×H(3,𝔽3)C2×D411yescharacteristic
117C2×(C3⋊D3)⋊C3—21no—
118C2×(C3⋊D3)⋊C3—41no—
119C2×(C3⋊D3)⋊C3—42no—
120C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
121C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
122C4×H(3,𝔽3)—21no—
123C4×H(3,𝔽3)C22×C411yescharacteristic
124C4×H(3,𝔽3)C2×D411yescharacteristic
125#1084—41no—
126(C3⋊Dic3)⋊C3—21no—
127(C3⋊Dic3)⋊C3—41no—
128(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
129(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
130C2×C3×#41—91no—
131D3×#41—31no—
132D3×#41—91no—
133#926—91no—
134C2×#208—91no—
135D3×M16—91no—
136D3×M16—91no—
137#872—91no—
138#872—91no—
139C3×#104—91no—
140C3×#104—91no—
141#837—31no—
142#837—31no—
143#837—31no—
144#837—61no—
145#837—91no—
146#837—91no—
147#837—91no—
148#825—91no—
149#825—91no—
150#805—91no—
151#805—91no—
152C22×C3⋊D3—21no—
153C22×C3⋊D3#6711yescharacteristic
154C22×C3×D3—31no—
155C22×C3×D3—61no—
156C32×Q8—21no—
157C32×Q8C4×D311yescharacteristic
158C32×Q8C2×Dic311yescharacteristic
159C32×D4—21no—
160C32×D4—21no—
161C32×D4—21no—
162C32×D4—41no—
163C32×D4C4×D311yescharacteristic
164C32×D4C2×Dic311yescharacteristic
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—41no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171#638—21no—
172#638—21no—
173#638—21no—
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3#6711yescharacteristic
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)C2×Dic311yescharacteristic
179C4×C3⋊D3—21no—
180C4×C3⋊D3—21no—
181C4×C3⋊D3—21no—
182C4×C3⋊D3C2×Dic311yescharacteristic
183C4×C3⋊D3C2×Dic311yescharacteristic
184C4×C3⋊D3#6711yescharacteristic
185C4×C3⋊D3#6711yescharacteristic
186#634—21no—
187#634C2×Dic311yescharacteristic
188C3×#67—61no—
189C3×#67—61no—
190C3×#67—61no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—121no—
197C3×D12—31no—
198C3×D12—61no—
199C3×C4×D3—31no—
200C3×C4×D3—31no—
201C3×C4×D3—31no—
202C3×C4×D3—31no—
203C3×C4×D3—61no—
204C3×C4×D3—61no—
205C3×C4×D3—61no—
206C3×Dic6—31no—
207C3×Dic6—61no—
208C3⋊(C3⋊C8)—21no—
209C3×C3⋊C8—61no—
210C3×C3⋊C8—61no—
211C3×C3⋊C8—61no—
212C3×C3⋊C8—121no—
213#421—271noSylow
214C2×H(3,𝔽3)—21no—
215C2×H(3,𝔽3)—41no—
216C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
217(C3⋊D3)⋊C3—21no—
218(C3⋊D3)⋊C3—21no—
219(C3⋊D3)⋊C3—41no—
220C3×#41—31no—
221C3×#41—31no—
222C3×#41—61no—
223C3×#41—91no—
224C3×#41—91no—
225C3×#41—91no—
226C3×#41—181no—
227C3×#41—181no—
228C3×#41D3211yescharacteristic
229C2×C3×Q8—91no—
230C2×C3×D4—91no—
231C2×C3×D4—181no—
232C22×C3×C4—91no—
233C22×C3×C4—181no—
234#239—31no—
235#239—91no—
236D3×Q8—31no—
237D3×Q8—91no—
238#237—31no—
239#237—61no—
240#237—91no—
241#237—181no—
242D3×D4—31no—
243D3×D4—61no—
244D3×D4—91no—
245D3×D4—181no—
246#235—31no—
247#235—61no—
248#235—91no—
249#235—181no—
250C2×C4×D3—31no—
251C2×C4×D3—61no—
252C2×C4×D3—91no—
253C2×C4×D3—181no—
254C3×M16—91no—
255C3×M16—91no—
256C3×C42—91no—
257C3×C42—91no—
258#217—181no—
259(C2×D3)⋊C4—181no—
260(C2×D3)⋊C4—181no—
261C4⋊Dic3—181no—
262Dic3⋊C4—181no—
263Dic3⋊C4—181no—
264C4×Dic3—31no—
265C4×Dic3—31no—
266C4×Dic3—31no—
267C4×Dic3—61no—
268C4×Dic3—91no—
269#208—31no—
270#208—31no—
271#208—31no—
272#208—61no—
273#208—91no—
274#208—181no—
275C2×C3⋊C8—181no—
276ΓL(1,𝔽25)—181no—
277ΓL(1,𝔽25)—181no—
278D3×C8—181no—
279D3×C8—181no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—41no—
284C22×C32D3×D411yescharacteristic
285C22×C32C2×#6711yescharacteristic
286C2×C3⋊D3—21no—
287C2×C3⋊D3—41no—
288C2×C3⋊D3—42no—
289C2×C3⋊D3#21711yescharacteristic
290C2×C3⋊D3#21711yescharacteristic
291C2×C3×D3—31no—
292C2×C3×D3—31no—
293C2×C3×D3—61no—
294C2×C3×D3—121no—
295C2×C3×D3—122no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—41no—
301C32×C4C2×C4×D311yescharacteristic
302C32×C4D3×D411yescharacteristic
303C32×C4C22×Dic311yescharacteristic
304C32×C4C2×#6711yescharacteristic
305C3⋊Dic3—21no—
306C3⋊Dic3—41no—
307C3⋊Dic3#21711yescharacteristic
308C3⋊Dic3#21711yescharacteristic
309C3×Dic3—31no—
310C3×Dic3—31no—
311C3×Dic3—61no—
312C3×Dic3—121no—
313C3×Dic3—121no—
314C3×Dic3—121no—
315C3×Dic3—122no—
316C2×#41—91no—
317C2×M16—271no—
318#117—271no—
319#104—91no—
320#104—91no—
321#104—271no—
322#104—271no—
323H(3,𝔽3)#42111yescharacteristic, complemented, Sylow
324C23×C3—91no—
325C23×C3—181no—
326C22×D3—31no—
327C22×D3—61no—
328C22×D3—91no—
329C22×D3—181no—
330C3×Q8—31no—
331C3×Q8—31no—
332C3×Q8—61no—
333C3×Q8—91no—
334C3×Q8—181no—
335C3×Q8D3×Dic311yescharacteristic
336C3×D4—21no—
337C3×D4—31no—
338C3×D4—31no—
339C3×D4—61no—
340C3×D4—61no—
341C3×D4—61no—
342C3×D4—91no—
343C3×D4—121no—
344C3×D4—181no—
345C3×D4—181no—
346C3×D4—181no—
347C3×D4—181no—
348C3×D4D3×Dic311yescharacteristic
349C2×C3×C4—21no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—91no—
356C2×C3×C4—91no—
357C2×C3×C4—91no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—121no—
361C2×C3×C4—181no—
362C2×C3×C4—181no—
363C2×C3×C4—181no—
364C2×C3×C4—181no—
365C2×C3×C4—181no—
366C2×C3×C4—181no—
367C2×C3×C4C2×D3211yescharacteristic
368#67—61no—
369#67—61no—
370#67—61no—
371#67—181no—
372#67—181no—
373#67—181no—
374C2×Dic3—31no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—61no—
378C2×Dic3—61no—
379C2×Dic3—91no—
380C2×Dic3—121no—
381C2×Dic3—181no—
382C2×Dic3—181no—
383D12—31no—
384D12—61no—
385D12—91no—
386D12—181no—
387C4×D3—31no—
388C4×D3—31no—
389C4×D3—31no—
390C4×D3—31no—
391C4×D3—61no—
392C4×D3—61no—
393C4×D3—61no—
394C4×D3—91no—
395C4×D3—91no—
396C4×D3—91no—
397C4×D3—91no—
398C4×D3—181no—
399C4×D3—181no—
400C4×D3—181no—
401Dic6—31no—
402Dic6—61no—
403Dic6—91no—
404Dic6—181no—
405C3×C8—181no—
406C3×C8—181no—
407C3⋊C8—61no—
408C3⋊C8—61no—
409C3⋊C8—61no—
410C3⋊C8—121no—
411C3⋊C8—181no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—81no—
419C2×C32D3×C22⋊C411yescharacteristic
420C2×C32C2×#21711yescharacteristic
421C3⋊D3—21no—
422C3⋊D3—21no—
423C3⋊D3—41no—
424C3×D3—61no—
425C3×D3—61no—
426C3×D3—121no—
427#41—91no—
428#41—91no—
429#41—91no—
430#41—181no—
431#41—181no—
432#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
433C2×Q8—91no—
434C2×D4—91no—
435C2×D4—181no—
436C22×C4—91no—
437C22×C4—181no—
438M16—91no—
439M16—91no—
440M16—541no—
441C2×C8—541no—
442C4⋊C4—541no—
443C22⋊C4—541no—
444C42—91no—
445C42—91no—
446C22×C3—21no—
447C22×C3—31no—
448C22×C3—31no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—91no—
453C22×C3—91no—
454C22×C3—121no—
455C22×C3—181no—
456C22×C3—361no—
457C22×C3—362no—
458C22×C3D3×#6711yescharacteristic
459C2×D3—31no—
460C2×D3—31no—
461C2×D3—61no—
462C2×D3—91no—
463C2×D3—91no—
464C2×D3—121no—
465C2×D3—122no—
466C2×D3—181no—
467C2×D3—361no—
468C2×D3—362no—
469C3×C4—21no—
470C3×C4—31no—
471C3×C4—31no—
472C3×C4—31no—
473C3×C4—31no—
474C3×C4—61no—
475C3×C4—61no—
476C3×C4—61no—
477C3×C4—61no—
478C3×C4—91no—
479C3×C4—91no—
480C3×C4—121no—
481C3×C4—181no—
482C3×C4—361no—
483C3×C4—361no—
484C3×C4C2×D3×Dic311yescharacteristic
485C3×C4D3×#6711yescharacteristic
486Dic3—31no—
487Dic3—31no—
488Dic3—61no—
489Dic3—91no—
490Dic3—91no—
491Dic3—121no—
492Dic3—121no—
493Dic3—121no—
494Dic3—122no—
495Dic3—181no—
496Dic3—361no—
497C32—21no—
498C32#482111yescharacteristic, complemented
499C32#523611yescharacteristic, complemented
500C23—91no—
501C23—181no—
502Q8—91no—
503Q8—181no—
504Q8((C3⋊D3)⋊C3)⋊C411yescharacteristic
505D4—21no—
506D4—91no—
507D4—181no—
508D4—181no—
509D4—181no—
510D4—181no—
511D4((C3⋊D3)⋊C3)⋊C411yescharacteristic
512C2×C4—21no—
513C2×C4—91no—
514C2×C4—91no—
515C2×C4—91no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—181no—
519C2×C4—181no—
520C2×C4—181no—
521C2×C4—181no—
522C2×C4—181no—
523C2×C4—181no—
524C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
525C8—181no—
526C8—181no—
527C2×C3—21no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—41no—
531C2×C3—61no—
532C2×C3—61no—
533C2×C3—61no—
534C2×C3—121no—
535C2×C3—121no—
536C2×C3—121no—
537C2×C3—181no—
538C2×C3—181no—
539C2×C3—241no—
540C2×C3—361no—
541C2×C3D3×#21711yescharacteristic
542D3—61no—
543D3—61no—
544D3—121no—
545D3—181no—
546D3—181no—
547D3—361no—
548C22—21no—
549C22—91no—
550C22—91no—
551C22—181no—
552C22—361no—
553C22—362no—
554C22#8867611yescharacteristic
555C4—21no—
556C4—91no—
557C4—91no—
558C4—181no—
559C4—361no—
560C4—361no—
561C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
562C4#8867611yescharacteristic
563C3—31no—
564C3—31no—
565C3—61no—
566C3#1059167011yescharacteristic
567C2—21no—
568C2—41no—
569C2—181no—
570C2—181no—
571C2—361no—
572C2#1172131011yescharacteristic
subgroupG/H
centerC4#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3D3×#217
prime radical#41×(C3⋊D3)⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#421
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #11721105C2semidirect
3. #11721053C2semidirect
4. #11720586C2semidirect
5. #63664⋊C3C2semidirect
6. #11720322C2semidirect
7. #11720305C2semidirect
11. #88722C4semidirect
12. Q8×(C3⋊D3)⋊C3C4semidirect
14. #3830⋊C3C4semidirect
16. D4×(C3⋊D3)⋊C3C4semidirect
21. C4×#1084C22semidirect
22. #88538C22semidirect
23. C4×(C3⋊Dic3)⋊C3C22semidirect
24. #3748⋊C3C22semidirect
35. #63664D3semidirect
77. #3832Dic3semidirect
78. Q8×C3⋊D3Dic3semidirect
80. #3830Dic3semidirect
82. D4×C3⋊D3Dic3semidirect
97. C4×C3⋊Dic3C2×D3semidirect
98. #3748C2×D3semidirect
323. H(3,𝔽3)#421semidirect
498. C32#4821semidirect
499. C32#5236semidirect

Maximal subgroups

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

subgroupindex
#41×(C3⋊D3)⋊C32
#117211052
#117210532
#117205862
#63664⋊C32
#117203222
#117203052
#105916723
#105916703