Finite Group Databasebeta

← #13155728

Subgroups — #13155728

Order 1728.

580 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723349C211yesmaximal, characteristic, complemented
2C3×#63822C211yesmaximal, characteristic
3C3×#63815C211yesmaximal, characteristic, complemented
4#11722070C211yesmaximal, characteristic, complemented
5#11722060C211yesmaximal, characteristic, complemented
6#11722044C211yesmaximal, characteristic, complemented
7C3×#63568C211yesmaximal, characteristic, complemented
8#10591675—31nomaximal
9#10591673—31nomaximal
10#10590975—31nomaximal
11C32×#235C2211yescharacteristic
12C32×#235C2211yescharacteristic
13#88855C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#208C2211yescharacteristic
18#88786—21no—
19D3×C3⋊(C3⋊C8)—21no—
20C3×#3718C411yescharacteristic, complemented
21C3×#3715C411yescharacteristic, complemented
22C3×#3715C411yescharacteristic, complemented
23C3×#3713C411yescharacteristic, complemented
24C3×#3711—21no—
25C3×D3×C3⋊C8—21no—
26#64157—31no—
27C3×#950—31no—
28C3×#950—31no—
29C3×#948—31no—
30C3×#907—31no—
31#63822D311yescharacteristic
32#63815D311yescharacteristic, complemented
33#63814—31no—
34#63814—31no—
35#63814—61no—
36#63664—31no—
37#63649—31no—
38C3×#837—31no—
39C3×#825—31no—
40#63577—31no—
41#63577—31no—
42#63577—61no—
43#63569—31no—
44#63569—31no—
45#63569—61no—
46#63568D311yescharacteristic, complemented
47#63553—31no—
48Dic3×C3⋊C8—31no—
49Dic3×C3⋊C8—31no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C32×#67—21no—
53C32×#67—21no—
54C32×D12C2×C411yescharacteristic
55C32×D12C2×C411yescharacteristic
56C32×C4×D3—21no—
57C32×C4×D3—21no—
58C32×Dic6C2×C411yescharacteristic
59C32×Dic6C2×C411yescharacteristic
60C3⋊(C3⋊(C3⋊C8))—21no—
61C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
62C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
63C3×C3⋊(C3⋊C8)D411yescharacteristic
64C3×C3⋊(C3⋊C8)D411yescharacteristic
65C32×C3⋊C8—21no—
66#5196—91no—
67#5144—91no—
68#4824—91no—
69C32×#41—31no—
70C32×#41—31no—
71C3×#235—21no—
72C3×#235—21no—
73C3×#235C2×D311yescharacteristic
74C3×#235C2×D311yescharacteristic
75C3×#235C2×D311yescharacteristic
76C3×#235C2×D311yescharacteristic
77C32×M16—31no—
78C4×C3⋊Dic3C2×D311yescharacteristic
79#3748—31no—
80#3748—31no—
81#3748—31no—
82#3748—61no—
83#3748—61no—
84#3748—61no—
85#3748—61no—
86#3748—121no—
87C2×C3⋊(C3⋊C8)—61no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×#216—31no—
91C3×#216—31no—
92C3×#215—31no—
93C3×#215—31no—
94C3×#214—31no—
95C3×#214—31no—
96C3×#213—31no—
97C3×#213—31no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—61no—
101C3×#208—21no—
102C3×#208—61no—
103C3×#208C2×D311yescharacteristic
104C3×#208C2×D311yescharacteristic
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—61no—
110C2×C3×C3⋊C8—61no—
111C2×C3×C3⋊C8—61no—
112C3×ΓL(1,𝔽25)—61no—
113C3×D3×C8—61no—
114#3718Dic311yescharacteristic, complemented
115#3715Dic311yescharacteristic, complemented
116#3715Dic311yescharacteristic, complemented
117#3713Dic311yescharacteristic, complemented
118#3711—21no—
119#3711—61no—
120#3711—61no—
121#3711—62no—
122D3×C3⋊C8—21no—
123D3×C3⋊C8—61no—
124D3×C3⋊C8—61no—
125D3×C3⋊C8—62no—
126C22×C33#4111yescharacteristic
127C2×C32×D3—21no—
128C2×C32×D3—21no—
129C33×C4C22×C411yescharacteristic
130C33×C4C2×D411yescharacteristic
131C3×C3⋊Dic3—22no—
132C32×Dic3—21no—
133C32×Dic3—21no—
134C3×#135—91no—
135C3×#131—91no—
136#950—31no—
137#950—31no—
138#948—31no—
139#948—91no—
140#948—91no—
141#948—181no—
142#907—31no—
143#901—91no—
144C3×#104—91no—
145#837—31no—
146#837—91no—
147#837—91no—
148#837—181no—
149#825—31no—
150#822—91no—
151#822—91no—
152#822—181no—
153#819—91no—
154C8×Dic3—91no—
155C8×Dic3—91no—
156#805—91no—
157C4×C3⋊C8—91no—
158C32×Q8—31no—
159C32×Q8—31no—
160C32×D4—31no—
161C32×D4—31no—
162C32×D4—61no—
163C32×D4—61no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—41no—
168C2×C32×C4—61no—
169C2×C32×C4—61no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—21no—
174C3×#67—22no—
175C3×#67—21no—
176C3×#67—21no—
177C3×#67—21no—
178C3×#67—21no—
179C3×#67—41no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—121no—
183C3×D12—21no—
184C3×D12—21no—
185C3×D12C4×D311yescharacteristic
186C3×D12C4×D311yescharacteristic
187C3×D12C2×Dic311yescharacteristic
188C3×D12C2×Dic311yescharacteristic
189C3×C4×D3—22no—
190C3×C4×D3—21no—
191C3×C4×D3—21no—
192C3×C4×D3—21no—
193C3×C4×D3—21no—
194C3×C4×D3—41no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6C4×D311yescharacteristic
198C3×Dic6C4×D311yescharacteristic
199C3×Dic6C2×Dic311yescharacteristic
200C3×Dic6C2×Dic311yescharacteristic
201C32×C8—61no—
202C3⋊(C3⋊C8)—61no—
203C3⋊(C3⋊C8)—62no—
204C3⋊(C3⋊C8)—61no—
205C3⋊(C3⋊C8)—61no—
206C3⋊(C3⋊C8)—121no—
207C3⋊(C3⋊C8)—121no—
208C3⋊(C3⋊C8)—122no—
209C3⋊(C3⋊C8)C2×Dic311yescharacteristic
210C3⋊(C3⋊C8)C2×Dic311yescharacteristic
211C3⋊(C3⋊C8)#6711yescharacteristic
212C3⋊(C3⋊C8)#6711yescharacteristic
213C3×C3⋊C8—21no—
214C3×C3⋊C8—21no—
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—31no—
222C3×C3⋊C8—31no—
223C3×C3⋊C8—41no—
224C3×C3⋊C8—61no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—61no—
228#443—271noSylow
229C2×C33—21no—
230C2×C33C4×D411yescharacteristic
231C32×D3—41no—
232C32×D3—41no—
233C3×#41—31no—
234C3×#41—31no—
235C3×#41—31no—
236C3×#41—31no—
237C3×#41—61no—
238C3×#41—61no—
239#235D3211yescharacteristic
240#235D3211yescharacteristic
241C3×Q16—91no—
242C3×SD16—91no—
243C3×SD16—91no—
244C3×D8—91no—
245C3×M16—31no—
246C3×M16—31no—
247C3×M16—61no—
248C3×M16—181no—
249C2×C3×C8—91no—
250C2×C3×C8—91no—
251C2×C3×C8—181no—
252C3×C42—91no—
253#216—31no—
254#216—31no—
255#215—31no—
256#215—31no—
257#214—31no—
258#214—31no—
259#213—31no—
260#213—31no—
261C4×Dic3—31no—
262C4×Dic3—31no—
263C4×Dic3—61no—
264#208—61no—
265#208—91no—
266#208—91no—
267#208—91no—
268#208—181no—
269#208—181no—
270#208—181no—
271#208—181no—
272#208—181no—
273#208—182no—
274#208—361no—
275#208D3211yescharacteristic
276C2×C3⋊C8—31no—
277C2×C3⋊C8—31no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—61no—
281C2×C3⋊C8—61no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8—181no—
284C2×C3⋊C8—181no—
285C2×C3⋊C8—182no—
286ΓL(1,𝔽25)—61no—
287ΓL(1,𝔽25)—181no—
288D3×C8—61no—
289D3×C8—181no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32—61no—
295C22×C32—61no—
296C22×C32#23511yescharacteristic
297C22×C32#23711yescharacteristic
298C22×C32#23711yescharacteristic
299C2×C3×D3—22no—
300C2×C3×D3—21no—
301C2×C3×D3—21no—
302C2×C3×D3—21no—
303C2×C3×D3—21no—
304C2×C3×D3—41no—
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×C4—61no—
314C32×C4—61no—
315C32×C4C2×C4×D311yescharacteristic
316C32×C4C2×C4×D311yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4D3×D411yescharacteristic
319C32×C4C22×Dic311yescharacteristic
320C32×C4C2×#6711yescharacteristic
321C3⋊Dic3—22no—
322C3×Dic3—22no—
323C3×Dic3—21no—
324C3×Dic3—21no—
325C3×Dic3—21no—
326C3×Dic3—21no—
327C3×Dic3—41no—
328C3×Dic3—62no—
329C3×Dic3—62no—
330C3×Dic3—122no—
331#135—91no—
332#131—91no—
333#131—271no—
334#108—271no—
335#104—91no—
336#104—271no—
337C4×C8—271no—
338C33#44311yescharacteristic, complemented, Sylow
339C3×Q8—31no—
340C3×Q8—31no—
341C3×Q8—31no—
342C3×Q8—31no—
343C3×Q8—61no—
344C3×Q8—61no—
345C3×D4—31no—
346C3×D4—31no—
347C3×D4—31no—
348C3×D4—31no—
349C3×D4—61no—
350C3×D4—61no—
351C3×D4—62no—
352C3×D4—61no—
353C3×D4—61no—
354C3×D4—61no—
355C3×D4—61no—
356C3×D4—121no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—62no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—121no—
367C2×C3×C4—181no—
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371#67—21no—
372#67—21no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—121no—
376D12D3×Dic311yescharacteristic
377D12D3×Dic311yescharacteristic
378C4×D3—21no—
379C4×D3—21no—
380Dic6D3×Dic311yescharacteristic
381Dic6D3×Dic311yescharacteristic
382C3×C8—61no—
383C3×C8—61no—
384C3×C8—91no—
385C3×C8—91no—
386C3×C8—91no—
387C3×C8—91no—
388C3×C8—121no—
389C3⋊C8—21no—
390C3⋊C8—31no—
391C3⋊C8—31no—
392C3⋊C8—31no—
393C3⋊C8—31no—
394C3⋊C8—31no—
395C3⋊C8—31no—
396C3⋊C8—31no—
397C3⋊C8—31no—
398C3⋊C8—61no—
399C3⋊C8—61no—
400C3⋊C8—61no—
401C3⋊C8—61no—
402C3⋊C8—182no—
403C3⋊C8—181no—
404C3⋊C8—181no—
405C3⋊C8—181no—
406C3⋊C8—361no—
407C3⋊C8—361no—
408C3⋊C8—362no—
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—121no—
421C2×C32—121no—
422C2×C32#67⋊C411yescharacteristic
423C2×C32C4×#6711yescharacteristic
424C2×C32D4×Dic311yescharacteristic
425C3×D3—42no—
426C3×D3—41no—
427C3×D3—41no—
428C3×D3—41no—
429C3×D3—41no—
430C3×D3—81no—
431#41—31no—
432#41—31no—
433Q16—91no—
434SD16—91no—
435SD16—91no—
436D8—91no—
437M16—31no—
438M16—181no—
439M16—271no—
440M16—541no—
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—62no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—61no—
454C22×C3—61no—
455C22×C3—121no—
456C22×C3#379511yescharacteristic
457C22×C3#380411yescharacteristic
458C22×C3#380511yescharacteristic
459C2×D3—21no—
460C2×D3—21no—
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—62no—
470C3×C4—61no—
471C3×C4—61no—
472C3×C4—61no—
473C3×C4—61no—
474C3×C4—121no—
475C3×C4—182no—
476C3×C4C4×D3211yescharacteristic
477C3×C4C2×D3×Dic311yescharacteristic
478C3×C4C2×D3×Dic311yescharacteristic
479C3×C4D3×#6711yescharacteristic
480C3×C4D3×#6711yescharacteristic
481C3×C4#381111yescharacteristic
482Dic3—21no—
483Dic3—21no—
484Dic3—62no—
485Dic3—62no—
486Dic3—122no—
487C32—21no—
488C32—21no—
489C32—21no—
490C32—41no—
491C32#482411yescharacteristic, complemented
492C32#514411yescharacteristic, complemented
493C32#519611yescharacteristic, complemented
494Q8—31no—
495Q8—31no—
496D4—31no—
497D4—31no—
498D4—61no—
499D4—61no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—181no—
503C2×C4D3311yescharacteristic
504C8—61no—
505C8—91no—
506C8—91no—
507C8—91no—
508C8—91no—
509C8—541no—
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—122no—
522C2×C3—121no—
523C2×C3—121no—
524C2×C3—121no—
525C2×C3—121no—
526C2×C3—241no—
527C2×C3#6387911yescharacteristic
528C2×C3Dic3×#6711yescharacteristic
529C2×C3#67⋊Dic311yescharacteristic
530D3—41no—
531D3—41no—
532C22—61no—
533C22—61no—
534C22#8896111yescharacteristic
535C4—61no—
536C4—61no—
537C4—182no—
538C4D32×Dic311yescharacteristic
539C4#8895211yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059097511yescharacteristic, complemented
545C3#1059167311yescharacteristic, complemented
546C3#1059167511yescharacteristic, complemented
547C2—21no—
548C2—121no—
549C2—121no—
550C2#1172292411yescharacteristic
subgroupG/H
centerC4#88952
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radicalC3×#63822C2
hypocenterC33#443
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. #11723349C2semidirect
3. C3×#63815C2semidirect
4. #11722070C2semidirect
5. #11722060C2semidirect
6. #11722044C2semidirect
7. C3×#63568C2semidirect
20. C3×#3718C4semidirect
21. C3×#3715C4semidirect
22. C3×#3715C4semidirect
23. C3×#3713C4semidirect
32. #63815D3semidirect
46. #63568D3semidirect
114. #3718Dic3semidirect
115. #3715Dic3semidirect
116. #3715Dic3semidirect
117. #3713Dic3semidirect
338. C33#443semidirect
491. C32#4824semidirect
492. C32#5144semidirect
493. C32#5196semidirect
544. C3#10590975semidirect
545. C3#10591673semidirect
546. C3#10591675semidirect

Maximal subgroups

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

subgroupindex
#117233492
C3×#638222
C3×#638152
#117220702
#117220602
#117220442
C3×#635682
#105916753
#105916733
#105909753