Finite Group Databasebeta

← #13155756

Subgroups — #13155756

Order 1728.

570 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723657C211yesmaximal, characteristic, complemented
2#11723363C211yesmaximal, characteristic
3C3×#63815C211yesmaximal, characteristic, complemented
4#11722059C211yesmaximal, characteristic, complemented
5#11722019C211yesmaximal, characteristic, complemented
6C3×#63570C211yesmaximal, characteristic, complemented
7C3×#63569C211yesmaximal, characteristic, complemented
8#10591663—31nomaximal
9#10590976—31nomaximal
10#10590390—31nomaximal
11C3×#3828C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×#3748C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C32×#208C2211yescharacteristic
18#88809—21no—
19#88808—21no—
20#88798C411yescharacteristic, complemented
21#88793C411yescharacteristic, complemented
22#88792C411yescharacteristic, complemented
23#88789C411yescharacteristic, complemented
24C3×#3711—21no—
25C3×D3×C3⋊C8—21no—
26#64159—31no—
27C3×#948—31no—
28C3×#907—31no—
29#63822—31no—
30#63822—31no—
31#63822—61no—
32#63817—31no—
33#63815—31no—
34#63815D311yescharacteristic, complemented
35#63814—31no—
36#63649—31no—
37#63630—31no—
38C3×#837—31no—
39C3×#825—31no—
40C3×#805—31no—
41C3×#805—31no—
42#63578—31no—
43#63578—31no—
44#63578—61no—
45#63570D311yescharacteristic, complemented
46#63569D311yescharacteristic, complemented
47#63554—31no—
48#63554—31no—
49#63554—61no—
50C2×C33×C4C2311yescharacteristic
51C3×#638—21no—
52C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
53C3×C4×C3⋊D3—21no—
54C3×#634C2×C411yescharacteristic
55C32×#67—21no—
56C2×C32×Dic3—21no—
57C32×D12C2×C411yescharacteristic
58C32×C4×D3—21no—
59C32×Dic6C2×C411yescharacteristic
60C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
61C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)D411yescharacteristic
64C3×C3⋊(C3⋊C8)D411yescharacteristic
65C32×C3⋊C8—21no—
66#5145—91no—
67#4823—91no—
68#4823—91no—
69C32×#41—31no—
70#3828C2×D311yescharacteristic
71C3×#235—21no—
72C3×#235—31no—
73C3×#235—31no—
74C3×#235—61no—
75C3×#235C2×D311yescharacteristic
76C3×#235C2×D311yescharacteristic
77C32×M16—31no—
78C32×C42—31no—
79#3756—31no—
80#3755—31no—
81#3754—31no—
82#3753—31no—
83#3748C2×D311yescharacteristic
84C2×C3⋊(C3⋊C8)—31no—
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—61no—
88C2×C3⋊(C3⋊C8)—61no—
89C2×C3⋊(C3⋊C8)—61no—
90C2×C3⋊(C3⋊C8)—121no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×C4×Dic3—21no—
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D311yescharacteristic
95C3×#208—21no—
96C3×#208—31no—
97C3×#208—31no—
98C3×#208—61no—
99C3×#208—61no—
100C3×#208C2×D311yescharacteristic
101C3×#208C2×D311yescharacteristic
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8—61no—
106C3×ΓL(1,𝔽25)—61no—
107C3×D3×C8—61no—
108#3718—31no—
109#3718—31no—
110#3718—61no—
111#3715—31no—
112#3715—31no—
113#3715—31no—
114#3715—31no—
115#3715—61no—
116#3715—61no—
117#3713—31no—
118#3713—31no—
119#3713—61no—
120#3712—61no—
121#3711—21no—
122#3711—61no—
123#3711—61no—
124#3710—61no—
125D3×C3⋊C8—21no—
126D3×C3⋊C8—61no—
127D3×C3⋊C8—61no—
128C22×C33#4111yescharacteristic
129C2×C3×C3⋊D3—21no—
130C2×C32×D3—21no—
131C33×C4C22×C411yescharacteristic
132C33×C4C2×D411yescharacteristic
133C3×C3⋊Dic3—21no—
134C32×Dic3—21no—
135C32×Dic3—41no—
136C3×#131—91no—
137#950—91no—
138#950—91no—
139#950—91no—
140#950—181no—
141#948—31no—
142#948—91no—
143#907—31no—
144#907—91no—
145#901—91no—
146C3×#104—91no—
147C3×#104—91no—
148#837—31no—
149#825—31no—
150#822—91no—
151#822—91no—
152#822—91no—
153#822—181no—
154#815—91no—
155#805—31no—
156#805—31no—
157#803—91no—
158#803—91no—
159#803—181no—
160C32×Q8—31no—
161C32×D4—31no—
162C32×D4—61no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—41no—
167C2×C32×C4—61no—
168C2×C32×C4—61no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172#638—21no—
173C4⋊(C3⋊D3)C4×D311yescharacteristic
174C4×C3⋊D3—21no—
175#634C4×D311yescharacteristic
176C3×#67—21no—
177C3×#67—21no—
178C3×#67—41no—
179C3×#67—61no—
180C3×#67—61no—
181C3×#67—121no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C3×D12—21no—
186C3×D12—31no—
187C3×D12—31no—
188C3×D12—61no—
189C3×D12C4×D311yescharacteristic
190C3×D12C4×D311yescharacteristic
191C3×C4×D3—21no—
192C3×C4×D3—21no—
193C3×C4×D3—41no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—121no—
197C3×Dic6—21no—
198C3×Dic6—31no—
199C3×Dic6—31no—
200C3×Dic6—61no—
201C3×Dic6C4×D311yescharacteristic
202C3×Dic6C4×D311yescharacteristic
203C32×C8—61no—
204C3⋊(C3⋊C8)—21no—
205C3⋊(C3⋊C8)—31no—
206C3⋊(C3⋊C8)—31no—
207C3⋊(C3⋊C8)—31no—
208C3⋊(C3⋊C8)—31no—
209C3⋊(C3⋊C8)—31no—
210C3⋊(C3⋊C8)—31no—
211C3⋊(C3⋊C8)—61no—
212C3⋊(C3⋊C8)—61no—
213C3⋊(C3⋊C8)—61no—
214C3⋊(C3⋊C8)—61no—
215C3⋊(C3⋊C8)—61no—
216C3⋊(C3⋊C8)—61no—
217C3⋊(C3⋊C8)—121no—
218C3⋊(C3⋊C8)—121no—
219C3⋊(C3⋊C8)#6711yescharacteristic
220C3⋊(C3⋊C8)#6711yescharacteristic
221C3×C3⋊C8—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—31no—
224C3×C3⋊C8—31no—
225C3×C3⋊C8—31no—
226C3×C3⋊C8—31no—
227C3×C3⋊C8—41no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—121no—
233#444—271noSylow
234C2×C33—21no—
235C2×C33C4×D411yescharacteristic
236C3×C3⋊D3—41no—
237C32×D3—41no—
238C3×#41—31no—
239C3×#41—31no—
240C3×#41—61no—
241C3×#41—91no—
242#235—31no—
243#235—31no—
244#235—61no—
245#235D3211yescharacteristic
246C3×M16—31no—
247C3×M16—31no—
248C3×M16—61no—
249C3×M16—91no—
250C3×M16—181no—
251C2×C3×C8—91no—
252C2×C3×C8—181no—
253C3×C42—31no—
254C3×C42—31no—
255C3×C42—61no—
256#216—91no—
257#216—91no—
258#216—91no—
259#216—181no—
260#215—91no—
261#215—91no—
262#215—91no—
263#215—181no—
264#214—91no—
265#214—91no—
266#214—91no—
267#214—181no—
268#213—91no—
269#213—91no—
270#213—91no—
271#213—181no—
272C4×Dic3D3211yescharacteristic
273#208—31no—
274#208—31no—
275#208—61no—
276#208—61no—
277#208—181no—
278#208D3211yescharacteristic
279C2×C3⋊C8—31no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—61no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8—91no—
284C2×C3⋊C8—91no—
285C2×C3⋊C8—91no—
286C2×C3⋊C8—181no—
287C2×C3⋊C8—181no—
288C2×C3⋊C8—181no—
289C2×C3⋊C8—181no—
290C2×C3⋊C8—361no—
291ΓL(1,𝔽25)—61no—
292ΓL(1,𝔽25)—181no—
293ΓL(1,𝔽25)—181no—
294D3×C8—61no—
295D3×C8—181no—
296D3×C8—181no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—41no—
301C22×C32—61no—
302C22×C32#23511yescharacteristic
303C22×C32#23711yescharacteristic
304C22×C32#23711yescharacteristic
305C2×C3⋊D3—21no—
306C2×C3×D3—21no—
307C2×C3×D3—21no—
308C2×C3×D3—41no—
309C2×C3×D3—61no—
310C2×C3×D3—61no—
311C2×C3×D3—121no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—61no—
321C32×C4—121no—
322C32×C4C2×C4×D311yescharacteristic
323C32×C4C2×C4×D311yescharacteristic
324C32×C4C2×C4×D311yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4D3×D411yescharacteristic
327C32×C4C2×#6711yescharacteristic
328C3⋊Dic3—21no—
329C3×Dic3—21no—
330C3×Dic3—21no—
331C3×Dic3—41no—
332C3×Dic3—42no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—61no—
336C3×Dic3—61no—
337C3×Dic3—121no—
338#135—271no—
339#131—91no—
340#131—271no—
341#108—271no—
342#104—91no—
343#104—91no—
344#97—271no—
345C33#44411yescharacteristic, complemented, Sylow
346C3×Q8—31no—
347C3×Q8—31no—
348C3×Q8—61no—
349C3×Q8—91no—
350C3×D4—31no—
351C3×D4—31no—
352C3×D4—61no—
353C3×D4—61no—
354C3×D4—61no—
355C3×D4—91no—
356C3×D4—121no—
357C3×D4—181no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—62no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—121no—
368C2×C3×C4—181no—
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372#67—21no—
373#67—61no—
374#67—61no—
375#67—121no—
376C2×Dic3—21no—
377D12—31no—
378D12—31no—
379D12—61no—
380D12#62411yescharacteristic
381C4×D3—21no—
382C4×D3—61no—
383C4×D3—61no—
384C4×D3—121no—
385Dic6—31no—
386Dic6—31no—
387Dic6—61no—
388Dic6#62411yescharacteristic
389C3×C8—61no—
390C3×C8—61no—
391C3×C8—91no—
392C3×C8—91no—
393C3×C8—121no—
394C3×C8—181no—
395C3⋊C8—21no—
396C3⋊C8—31no—
397C3⋊C8—31no—
398C3⋊C8—31no—
399C3⋊C8—31no—
400C3⋊C8—61no—
401C3⋊C8—61no—
402C3⋊C8—61no—
403C3⋊C8—61no—
404C3⋊C8—91no—
405C3⋊C8—91no—
406C3⋊C8—91no—
407C3⋊C8—91no—
408C3⋊C8—91no—
409C3⋊C8—91no—
410C3⋊C8—121no—
411C3⋊C8—181no—
412C3⋊C8—181no—
413C3⋊C8—181no—
414C3⋊C8—181no—
415C3⋊C8—181no—
416C3⋊C8—181no—
417C3⋊C8—361no—
418C3⋊C8—361no—
419C2×C32—21no—
420C2×C32—22no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—41no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—42no—
430C2×C32—121no—
431C2×C32#67⋊C411yescharacteristic
432C2×C32#67⋊C411yescharacteristic
433C2×C32C4×#6711yescharacteristic
434C3⋊D3—41no—
435C3×D3—41no—
436C3×D3—41no—
437C3×D3—81no—
438C3×D3—121no—
439C3×D3—121no—
440C3×D3—241no—
441#41—31no—
442#41—91no—
443Q16—271no—
444SD16—271no—
445SD16—271no—
446D8—271no—
447M16—31no—
448M16—91no—
449M16—181no—
450M16—541no—
451C2×C8—91no—
452C2×C8—181no—
453C2×C8—271no—
454C2×C8—541no—
455C42—31no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—41no—
460C22×C3—61no—
461C22×C3—61no—
462C22×C3—121no—
463C22×C3—181no—
464C22×C3#379511yescharacteristic
465C22×C3#380411yescharacteristic
466C22×C3#380511yescharacteristic
467C2×D3—21no—
468C2×D3—61no—
469C2×D3—61no—
470C2×D3—121no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—61no—
480C3×C4—61no—
481C3×C4—121no—
482C3×C4—122no—
483C3×C4—121no—
484C3×C4—121no—
485C3×C4—181no—
486C3×C4C4×D3211yescharacteristic
487C3×C4C4×D3211yescharacteristic
488C3×C4#380211yescharacteristic
489C3×C4C2×#62411yescharacteristic
490C3×C4D3×#6711yescharacteristic
491C3×C4D3×#6711yescharacteristic
492Dic3—21no—
493Dic3—41no—
494Dic3—61no—
495Dic3—61no—
496Dic3—121no—
497C32—21no—
498C32—21no—
499C32—21no—
500C32—41no—
501C32#482311yescharacteristic, complemented
502C32#482311yescharacteristic, complemented
503C32#514511yescharacteristic, complemented
504Q8—31no—
505Q8—91no—
506D4—31no—
507D4—61no—
508D4—91no—
509D4—181no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—181no—
513C2×C4D3311yescharacteristic
514C8—61no—
515C8—91no—
516C8—91no—
517C8—181no—
518C8—271no—
519C8—271no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—41no—
528C2×C3—41no—
529C2×C3—42no—
530C2×C3—41no—
531C2×C3—121no—
532C2×C3—121no—
533C2×C3—241no—
534C2×C3—361no—
535C2×C3#6387911yescharacteristic
536C2×C3#6390211yescharacteristic
537C2×C3#6395211yescharacteristic
538D3—41no—
539D3—121no—
540D3—121no—
541D3—241no—
542C22—61no—
543C22—181no—
544C22#8896111yescharacteristic
545C4—61no—
546C4—121no—
547C4—181no—
548C4D3×#62411yescharacteristic
549C4#8895311yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059039011yescharacteristic, complemented
555C3#1059097611yescharacteristic, complemented
556C3#1059166311yescharacteristic, complemented
557C2—21no—
558C2—121no—
559C2—361no—
560C2#1172293611yescharacteristic
subgroupG/H
centerC4#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radical#11723363C2
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723657C2semidirect
3. C3×#63815C2semidirect
4. #11722059C2semidirect
5. #11722019C2semidirect
6. C3×#63570C2semidirect
7. C3×#63569C2semidirect
20. #88798C4semidirect
21. #88793C4semidirect
22. #88792C4semidirect
23. #88789C4semidirect
34. #63815D3semidirect
45. #63570D3semidirect
46. #63569D3semidirect
345. C33#444semidirect
501. C32#4823semidirect
502. C32#4823semidirect
503. C32#5145semidirect
554. C3#10590390semidirect
555. C3#10590976semidirect
556. C3#10591663semidirect

Maximal subgroups

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

subgroupindex
#117236572
#117233632
C3×#638152
#117220592
#117220192
C3×#635702
C3×#635692
#105916633
#105909763
#105903903