Finite Group Databasebeta

← #13161763

Subgroups — #13161763

Order 1728.

580 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 99 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×#64116C211yesmaximal, characteristic, complemented
2#11723660C211yesmaximal, characteristic
3#11723657C211yesmaximal, characteristic, complemented
4C3×#63652C211yesmaximal, characteristic, complemented
5C3×#63649C211yesmaximal, characteristic, complemented
6#11722077C211yesmaximal, characteristic, complemented
7#11722070C211yesmaximal, characteristic, complemented
8#10590995—31nomaximal
9#10590977—31nomaximal
10#10590975—31nomaximal
11C3×#3828C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C33×M16C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#3743—21no—
19C3×C8×C3⋊D3—21no—
20#88812C411yescharacteristic, complemented
21#88811C411yescharacteristic, complemented
22#88811C411yescharacteristic, complemented
23#88810C411yescharacteristic, complemented
24#88809—21no—
25#88808—21no—
26#64116D311yescharacteristic, complemented
27C3×#907—31no—
28C3×#907—31no—
29C3×#907—61no—
30#63829—31no—
31#63829—31no—
32#63822—31no—
33#63817—31no—
34#63815—31no—
35#63814—31no—
36#63652D311yescharacteristic, complemented
37#63649D311yescharacteristic, complemented
38C3×#825—31no—
39C3×#825—31no—
40C3×#825—61no—
41C3×#822—31no—
42C3×#822—31no—
43C3×#822—61no—
44#63570—31no—
45#63569—31no—
46#63568—31no—
47#63553—31no—
48Dic3×C3⋊C8—31no—
49Dic3×C3⋊C8—31no—
50C2×C33×C4C2311yescharacteristic
51C3×#638—21no—
52C3×#638—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
55C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
56C3×C4×C3⋊D3—21no—
57C3×C4×C3⋊D3—21no—
58C3×#634C2×C411yescharacteristic
59C3×#634C2×C411yescharacteristic
60C33×C8—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
63C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
64C3×C3⋊(C3⋊C8)D411yescharacteristic
65C3×C3⋊(C3⋊C8)D411yescharacteristic
66#5144—91no—
67#4895—91no—
68#4824—91no—
69#3828C2×D311yescharacteristic
70#3828C2×D311yescharacteristic
71C3×#235—31no—
72C3×#235—31no—
73C3×#235—31no—
74C3×#235—31no—
75C3×#235—61no—
76C3×#235—61no—
77C32×M16—21no—
78C32×M16—21no—
79C32×M16—21no—
80C32×M16—41no—
81C32×M16C2×D311yescharacteristic
82C32×M16C2×D311yescharacteristic
83C32×M16C2×D311yescharacteristic
84C4×C3⋊Dic3C2×D311yescharacteristic
85#3748C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88#3743—21no—
89C8×C3⋊D3—21no—
90C3×C4×Dic3—31no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—61no—
93C3×#208—31no—
94C3×#208—31no—
95C3×#208—61no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8—61no—
102C3×ΓL(1,𝔽25)—61no—
103C3×ΓL(1,𝔽25)—61no—
104C3×ΓL(1,𝔽25)—62no—
105C3×D3×C8—61no—
106C3×D3×C8—61no—
107C3×D3×C8—62no—
108#3719—31no—
109#3719—31no—
110#3718—31no—
111#3717—31no—
112#3717—31no—
113#3716—31no—
114#3716—31no—
115#3715—31no—
116#3715—31no—
117#3714—31no—
118#3714—31no—
119#3713—31no—
120#3712—61no—
121#3711—61no—
122#3711—61no—
123#3710—61no—
124D3×C3⋊C8—61no—
125D3×C3⋊C8—61no—
126C22×C33#4111yescharacteristic
127C2×C3×C3⋊D3—21no—
128C2×C3×C3⋊D3—21no—
129C33×C4C22×C411yescharacteristic
130C33×C4C2×D411yescharacteristic
131C3×C3⋊Dic3—22no—
132C3×C3⋊Dic3—21no—
133C3×C3⋊Dic3—21no—
134C3×#131—91no—
135#950—91no—
136#950—91no—
137#948—91no—
138#907—31no—
139#907—31no—
140#907—61no—
141#907—91no—
142#904—91no—
143#901—91no—
144C3×#108—91no—
145C3×#104—91no—
146#837—91no—
147#825—31no—
148#825—31no—
149#825—61no—
150#825—91no—
151#822—31no—
152#822—31no—
153#822—61no—
154C8×Dic3—91no—
155C8×Dic3—91no—
156#805—91no—
157C4×C3⋊C8—91no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—41no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165#638—21no—
166#638—21no—
167C2×C3⋊Dic3—21no—
168C4⋊(C3⋊D3)C4×D311yescharacteristic
169C4⋊(C3⋊D3)C4×D311yescharacteristic
170C4×C3⋊D3—21no—
171C4×C3⋊D3—21no—
172#634C4×D311yescharacteristic
173#634C4×D311yescharacteristic
174C3×#67—62no—
175C3×#67—61no—
176C3×#67—61no—
177C3×#67—61no—
178C3×#67—61no—
179C3×#67—121no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—121no—
183C3×D12—31no—
184C3×D12—31no—
185C3×D12—31no—
186C3×D12—31no—
187C3×D12—61no—
188C3×D12—61no—
189C3×C4×D3—62no—
190C3×C4×D3—61no—
191C3×C4×D3—61no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—121no—
195C3×Dic6—31no—
196C3×Dic6—31no—
197C3×Dic6—31no—
198C3×Dic6—31no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C32×C8—21no—
202C32×C8—22no—
203C32×C8—21no—
204C32×C8—21no—
205C32×C8—41no—
206C32×C8—41no—
207C32×C8—42no—
208C3⋊(C3⋊C8)—21no—
209C3⋊(C3⋊C8)C4×D311yescharacteristic
210C3⋊(C3⋊C8)C4×D311yescharacteristic
211C3⋊(C3⋊C8)#6711yescharacteristic
212C3⋊(C3⋊C8)#6711yescharacteristic
213C3×C3⋊C8—31no—
214C3×C3⋊C8—31no—
215C3×C3⋊C8—31no—
216C3×C3⋊C8—31no—
217C3×C3⋊C8—31no—
218C3×C3⋊C8—31no—
219C3×C3⋊C8—31no—
220C3×C3⋊C8—31no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224C3×C3⋊C8—61no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—121no—
228#443—271noSylow
229C2×C33—21no—
230C2×C33C4×D411yescharacteristic
231C3×C3⋊D3—41no—
232C3×C3⋊D3—41no—
233C3×#41—91no—
234C3×#41—91no—
235#235—31no—
236#235—31no—
237#235—31no—
238#235—31no—
239#235—61no—
240#235—61no—
241C3×M16—21no—
242C3×M16—21no—
243C3×M16—21no—
244C3×M16—41no—
245C3×M16—91no—
246C3×M16—181no—
247C3×M16D3211yescharacteristic
248C3×M16D3211yescharacteristic
249C3×M16D3211yescharacteristic
250C2×C3×C8—91no—
251C2×C3×C8—91no—
252C2×C3×C8—181no—
253C3×C42—91no—
254#216—91no—
255#216—91no—
256#215—91no—
257#215—91no—
258#214—91no—
259#214—91no—
260#213—91no—
261#213—91no—
262C4×Dic3—31no—
263C4×Dic3—31no—
264C4×Dic3—61no—
265#208—31no—
266#208—31no—
267#208—61no—
268#208—181no—
269C2×C3⋊C8—31no—
270C2×C3⋊C8—31no—
271C2×C3⋊C8—31no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—61no—
274C2×C3⋊C8—61no—
275C2×C3⋊C8—181no—
276Dic12—91no—
277D24—91no—
278#204—91no—
279#204—91no—
280ΓL(1,𝔽25)—61no—
281ΓL(1,𝔽25)—61no—
282ΓL(1,𝔽25)—62no—
283ΓL(1,𝔽25)—181no—
284ΓL(1,𝔽25)—181no—
285D3×C8—61no—
286D3×C8—61no—
287D3×C8—62no—
288D3×C8—181no—
289D3×C8—181no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32#23511yescharacteristic
295C22×C32#23711yescharacteristic
296C22×C32#23911yescharacteristic
297C2×C3⋊D3—21no—
298C2×C3⋊D3—21no—
299C2×C3×D3—62no—
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—121no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4C2×C4×D311yescharacteristic
314C32×C4C2×C4×D311yescharacteristic
315C32×C4C2×C4×D311yescharacteristic
316C32×C4D3×D411yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4C2×#6711yescharacteristic
319C3⋊Dic3—21no—
320C3⋊Dic3—22no—
321C3⋊Dic3—21no—
322C3×Dic3—62no—
323C3×Dic3—62no—
324C3×Dic3—62no—
325C3×Dic3—61no—
326C3×Dic3—61no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—121no—
330C3×Dic3—122no—
331#135—271no—
332#131—91no—
333#131—271no—
334#108—91no—
335#104—91no—
336#104—271no—
337C4×C8—271no—
338C33#44311yescharacteristic, complemented, Sylow
339C3×Q8—91no—
340C3×Q8—91no—
341C3×D4—91no—
342C3×D4—91no—
343C3×D4—181no—
344C3×D4—181no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—41no—
349C2×C3×C4—181no—
350C2×C3×C4—181no—
351C2×C3×C4—181no—
352C2×C3×C4C2×D3211yescharacteristic
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355#67—62no—
356#67—61no—
357#67—61no—
358#67—61no—
359#67—61no—
360#67—121no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—121no—
364D12—31no—
365D12—31no—
366D12—31no—
367D12—31no—
368D12—61no—
369D12—61no—
370C4×D3—62no—
371C4×D3—61no—
372C4×D3—61no—
373C4×D3—61no—
374C4×D3—61no—
375C4×D3—121no—
376Dic6—31no—
377Dic6—31no—
378Dic6—31no—
379Dic6—31no—
380Dic6—61no—
381Dic6—61no—
382C3×C8—22no—
383C3×C8—21no—
384C3×C8—21no—
385C3×C8—21no—
386C3×C8—41no—
387C3×C8—41no—
388C3×C8—42no—
389C3×C8—91no—
390C3×C8—91no—
391C3×C8—91no—
392C3×C8—91no—
393C3×C8—181no—
394C3⋊C8—31no—
395C3⋊C8—31no—
396C3⋊C8—31no—
397C3⋊C8—31no—
398C3⋊C8—31no—
399C3⋊C8—31no—
400C3⋊C8—31no—
401C3⋊C8—31no—
402C3⋊C8—61no—
403C3⋊C8—61no—
404C3⋊C8—61no—
405C3⋊C8—61no—
406C3⋊C8—61no—
407C3⋊C8—61no—
408C3⋊C8—121no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32—81no—
420C2×C32#67⋊C411yescharacteristic
421C2×C32#89311yescharacteristic
422C2×C32C4×#6711yescharacteristic
423C3⋊D3—41no—
424C3⋊D3—41no—
425C3×D3—122no—
426C3×D3—121no—
427C3×D3—121no—
428C3×D3—121no—
429C3×D3—121no—
430C3×D3—241no—
431#41—91no—
432#41—91no—
433Q16—271no—
434SD16—271no—
435SD16—271no—
436D8—271no—
437M16—91no—
438M16—181no—
439M16—541no—
440M16#111311yescharacteristic
441C2×C8—91no—
442C2×C8—91no—
443C2×C8—181no—
444C2×C8—541no—
445C42—91no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—41no—
450C22×C3—181no—
451C22×C3—181no—
452C22×C3#379511yescharacteristic
453C22×C3#379611yescharacteristic
454C22×C3#379911yescharacteristic
455C2×D3—62no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—61no—
460C2×D3—121no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—182no—
470C3×C4—181no—
471C3×C4—181no—
472C3×C4C4×D3211yescharacteristic
473C3×C4C4×D3211yescharacteristic
474C3×C4C4×D3211yescharacteristic
475C3×C4D3×#6711yescharacteristic
476C3×C4D3×#6711yescharacteristic
477C3×C4#381111yescharacteristic
478Dic3—62no—
479Dic3—62no—
480Dic3—62no—
481Dic3—61no—
482Dic3—61no—
483Dic3—61no—
484Dic3—61no—
485Dic3—121no—
486Dic3—122no—
487C32—21no—
488C32—21no—
489C32—21no—
490C32—41no—
491C32#482411yescharacteristic, complemented
492C32#489511yescharacteristic, complemented
493C32#514411yescharacteristic, complemented
494Q8—91no—
495Q8—91no—
496D4—91no—
497D4—91no—
498D4—181no—
499D4—181no—
500C2×C4—181no—
501C2×C4—181no—
502C2×C4—181no—
503C2×C4C2×#111311yescharacteristic
504C8—21no—
505C8—91no—
506C8—91no—
507C8—91no—
508C8—91no—
509C8—181no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—41no—
519C2×C3—41no—
520C2×C3—81no—
521C2×C3—361no—
522C2×C3—361no—
523C2×C3#6387911yescharacteristic
524C2×C3#6388111yescharacteristic
525C2×C3#6390411yescharacteristic
526D3—122no—
527D3—121no—
528D3—121no—
529D3—121no—
530D3—121no—
531D3—241no—
532C22—181no—
533C22—181no—
534C22#8904111yescharacteristic
535C4—182no—
536C4—181no—
537C4—181no—
538C4C4×#111311yescharacteristic
539C4#8904911yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059097511yescharacteristic, complemented
545C3#1059097711yescharacteristic, complemented
546C3#1059099511yescharacteristic, complemented
547C2—21no—
548C2—361no—
549C2—361no—
550C2#1172368411yescharacteristic
subgroupG/H
centerC4#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC33×M16C22
socleC2×C33C4×D4
prime radical#11723660C2
cosocleC2×C33×C4C23
hypercenterM16#1113
hypocenterC33#443
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. C3×#64116C2semidirect
3. #11723657C2semidirect
4. C3×#63652C2semidirect
5. C3×#63649C2semidirect
6. #11722077C2semidirect
7. #11722070C2semidirect
20. #88812C4semidirect
21. #88811C4semidirect
22. #88811C4semidirect
23. #88810C4semidirect
26. #64116D3semidirect
36. #63652D3semidirect
37. #63649D3semidirect
338. C33#443semidirect
491. C32#4824semidirect
492. C32#4895semidirect
493. C32#5144semidirect
544. C3#10590975semidirect
545. C3#10590977semidirect
546. C3#10590995semidirect

Maximal subgroups

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

subgroupindex
C3×#641162
#117236602
#117236572
C3×#636522
C3×#636492
#117220772
#117220702
#105909953
#105909773
#105909753