Finite Group Databasebeta

← #13155732

Subgroups — #13155732

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#11723657C211yesmaximal, characteristic, complemented
2#11723349C211yesmaximal, characteristic, complemented
3C3×#63829C211yesmaximal, characteristic
4#11722058C211yesmaximal, characteristic, complemented
5#11722046C211yesmaximal, characteristic, complemented
6C3×#63569C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10591675—31nomaximal
9#10590915—31nomaximal
10#10590341—31nomaximal
11C3×#3828C2211yescharacteristic
12C32×#235C2211yescharacteristic
13#88855C2211yescharacteristic
14C3×#3748C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88809—21no—
19#88808—21no—
20#88786—21no—
21D3×C3⋊(C3⋊C8)—21no—
22C3×#3719C411yescharacteristic, complemented
23C3×#3717C411yescharacteristic, complemented
24C3×#3716C411yescharacteristic, complemented
25C3×#3714C411yescharacteristic, complemented
26#64157—31no—
27C3×#950—31no—
28C3×#904—31no—
29#63829D311yescharacteristic
30#63817—31no—
31#63815—31no—
32#63814—31no—
33#63814—31no—
34#63814—31no—
35#63814—61no—
36#63646—31no—
37#63632—31no—
38C3×#837—31no—
39C3×C8×Dic3—31no—
40C3×#805—31no—
41C3×C4×C3⋊C8—31no—
42#63577—31no—
43#63577—31no—
44#63577—61no—
45#63569—31no—
46#63569—31no—
47#63569—61no—
48#63569D311yescharacteristic, complemented
49Dic3×C3⋊C8D311yescharacteristic, complemented
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))—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)D411yescharacteristic
63C3×C3⋊(C3⋊C8)D411yescharacteristic
64C32×C3⋊C8C2×C411yescharacteristic
65C32×C3⋊C8C2×C411yescharacteristic
66#5196—91no—
67#4824—91no—
68#4700—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
77C2×C32×C8—31no—
78C32×C42—31no—
79#3748—31no—
80#3748—31no—
81#3748—31no—
82#3748—61no—
83#3748—61no—
84#3748—61no—
85#3748—61no—
86#3748—121no—
87#3748C2×D311yescharacteristic
88C2×C3⋊(C3⋊C8)—61no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×#216—31no—
91C3×#215—31no—
92C3×#214—31no—
93C3×#213—31no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3C2×D311yescharacteristic
96C3×C4×Dic3C2×D311yescharacteristic
97C3×#208—31no—
98C3×#208—31no—
99C3×#208—61no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C3×Dic12—31no—
107C3×D24—31no—
108C3×#204—31no—
109C3×#204—31no—
110#3719Dic311yescharacteristic, complemented
111#3717Dic311yescharacteristic, complemented
112#3716Dic311yescharacteristic, complemented
113#3714Dic311yescharacteristic, complemented
114#3712—61no—
115#3711—61no—
116#3711—61no—
117#3711—61no—
118#3711—62no—
119#3711—61no—
120#3710—61no—
121D3×C3⋊C8—61no—
122D3×C3⋊C8—61no—
123D3×C3⋊C8—61no—
124D3×C3⋊C8—62no—
125D3×C3⋊C8—61no—
126C22×C33#4111yescharacteristic
127C2×C3×C3⋊D3—21no—
128C2×C32×D3—21no—
129C33×C4C22×C411yescharacteristic
130C33×C4C2×D411yescharacteristic
131C3×C3⋊Dic3—21no—
132C32×Dic3—22no—
133C32×Dic3—21no—
134C3×#135—91no—
135#950—31no—
136#948—91no—
137#948—91no—
138#948—91no—
139#948—181no—
140#907—91no—
141#904—31no—
142#901—91no—
143#901—91no—
144C3×#104—91no—
145C3×C4×C8—91no—
146#837—31no—
147#837—91no—
148#822—91no—
149#819—91no—
150#819—91no—
151#819—181no—
152C8×Dic3—31no—
153#805—31no—
154#805—91no—
155#805—91no—
156#805—181no—
157C4×C3⋊C8—31no—
158C32×Q8—31no—
159C32×D4—31no—
160C32×D4—61no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—41no—
165C2×C32×C4—61no—
166C2×C32×C4—61no—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170#638—21no—
171C4⋊(C3⋊D3)C2×Dic311yescharacteristic
172C4×C3⋊D3—21no—
173#634C2×Dic311yescharacteristic
174C3×#67—22no—
175C3×#67—21no—
176C3×#67—21no—
177C3×#67—61no—
178C3×#67—61no—
179C3×#67—121no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—41no—
183C3×D12—21no—
184C3×D12—31no—
185C3×D12—31no—
186C3×D12—61no—
187C3×D12C4×D311yescharacteristic
188C3×D12C2×Dic311yescharacteristic
189C3×C4×D3—22no—
190C3×C4×D3—21no—
191C3×C4×D3—21no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—121no—
195C3×Dic6—21no—
196C3×Dic6—31no—
197C3×Dic6—31no—
198C3×Dic6—61no—
199C3×Dic6C4×D311yescharacteristic
200C3×Dic6C2×Dic311yescharacteristic
201C32×C8—31no—
202C32×C8—31no—
203C3⋊(C3⋊C8)—21no—
204C3⋊(C3⋊C8)—61no—
205C3⋊(C3⋊C8)—62no—
206C3⋊(C3⋊C8)—61no—
207C3⋊(C3⋊C8)—61no—
208C3⋊(C3⋊C8)—121no—
209C3⋊(C3⋊C8)—121no—
210C3⋊(C3⋊C8)—122no—
211C3⋊(C3⋊C8)D1211yescharacteristic
212C3⋊(C3⋊C8)D1211yescharacteristic
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—61no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—121no—
224C3×C3⋊C8C4×D311yescharacteristic
225C3×C3⋊C8C4×D311yescharacteristic
226C3×C3⋊C8C2×Dic311yescharacteristic
227C3×C3⋊C8C2×Dic311yescharacteristic
228#443—271noSylow
229C2×C33—21no—
230C2×C33C4×D411yescharacteristic
231C3×C3⋊D3—41no—
232C32×D3—41no—
233C3×#41—31no—
234C3×#41—31no—
235C3×#41—61no—
236C3×#41—91no—
237#235—31no—
238#235—31no—
239#235—61no—
240#235D3211yescharacteristic
241C3×Q16—91no—
242C3×SD16—91no—
243C3×SD16—91no—
244C3×D8—91no—
245C3×M16—91no—
246C2×C3×C8—31no—
247C2×C3×C8—31no—
248C2×C3×C8—61no—
249C2×C3×C8—91no—
250C3×C42—31no—
251C3×C42—31no—
252C3×C42—61no—
253#216—31no—
254#215—31no—
255#214—31no—
256#213—31no—
257C4×Dic3D3211yescharacteristic
258#208—31no—
259#208—31no—
260#208—61no—
261#208—91no—
262#208—91no—
263#208—91no—
264#208—181no—
265#208—181no—
266#208—181no—
267#208—181no—
268#208—181no—
269#208—181no—
270#208—182no—
271#208—361no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—61no—
275C2×C3⋊C8—181no—
276C2×C3⋊C8—181no—
277C2×C3⋊C8—181no—
278C2×C3⋊C8—182no—
279C2×C3⋊C8D3211yescharacteristic
280Dic12—31no—
281D24—31no—
282#204—31no—
283#204—31no—
284ΓL(1,𝔽25)—181no—
285ΓL(1,𝔽25)—181no—
286ΓL(1,𝔽25)—181no—
287D3×C8—181no—
288D3×C8—181no—
289D3×C8—181no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32—61no—
295C22×C32#23511yescharacteristic
296C22×C32#23711yescharacteristic
297C22×C32#23711yescharacteristic
298C2×C3⋊D3—21no—
299C2×C3×D3—22no—
300C2×C3×D3—21no—
301C2×C3×D3—21no—
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×C4—61no—
314C32×C4—62no—
315C32×C4C2×C4×D311yescharacteristic
316C32×C4C2×C4×D311yescharacteristic
317C32×C4C2×D1211yescharacteristic
318C32×C4D3×D411yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4C22×Dic311yescharacteristic
321C3⋊Dic3—21no—
322C3×Dic3—22no—
323C3×Dic3—22no—
324C3×Dic3—22no—
325C3×Dic3—21no—
326C3×Dic3—21no—
327C3×Dic3—42no—
328C3×Dic3—61no—
329C3×Dic3—61no—
330C3×Dic3—121no—
331#135—91no—
332#131—271no—
333#131—271no—
334#108—271no—
335#104—91no—
336#104—271no—
337C4×C8—91no—
338C33#44311yescharacteristic, complemented, Sylow
339C3×Q8—31no—
340C3×Q8—31no—
341C3×Q8—61no—
342C3×Q8—91no—
343C3×D4—31no—
344C3×D4—31no—
345C3×D4—61no—
346C3×D4—62no—
347C3×D4—61no—
348C3×D4—61no—
349C3×D4—91no—
350C3×D4—181no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—41no—
355C2×C3×C4—62no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—121no—
361C2×C3×C4—181no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365#67—21no—
366#67—61no—
367#67—61no—
368#67—121no—
369C2×Dic3—21no—
370D12—31no—
371D12—31no—
372D12—61no—
373D12D3×Dic311yescharacteristic
374C4×D3—21no—
375C4×D3—61no—
376C4×D3—61no—
377C4×D3—121no—
378Dic6—31no—
379Dic6—31no—
380Dic6—61no—
381Dic6D3×Dic311yescharacteristic
382C3×C8—31no—
383C3×C8—31no—
384C3×C8—31no—
385C3×C8—31no—
386C3×C8—61no—
387C3×C8—61no—
388C3×C8—91no—
389C3×C8—91no—
390C3×C8—181no—
391C3⋊C8—31no—
392C3⋊C8—31no—
393C3⋊C8—31no—
394C3⋊C8—31no—
395C3⋊C8—61no—
396C3⋊C8—61no—
397C3⋊C8—61no—
398C3⋊C8—61no—
399C3⋊C8—121no—
400C3⋊C8—182no—
401C3⋊C8—181no—
402C3⋊C8—181no—
403C3⋊C8—181no—
404C3⋊C8—361no—
405C3⋊C8—361no—
406C3⋊C8—362no—
407C3⋊C8D3×Dic311yescharacteristic
408C3⋊C8D3×Dic311yescharacteristic
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×C32C4×D1211yescharacteristic
422C2×C32#67⋊C411yescharacteristic
423C2×C32D4×Dic311yescharacteristic
424C3⋊D3—41no—
425C3×D3—42no—
426C3×D3—41no—
427C3×D3—41no—
428C3×D3—121no—
429C3×D3—121no—
430C3×D3—241no—
431#41—31no—
432#41—91no—
433Q16—91no—
434SD16—91no—
435SD16—91no—
436D8—91no—
437M16—91no—
438M16—271no—
439M16—541no—
440M16—541no—
441C2×C8—31no—
442C2×C8—91no—
443C2×C8—541no—
444C2×C8—541no—
445C42—31no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—41no—
450C22×C3—62no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—181no—
454C22×C3#379511yescharacteristic
455C22×C3#380411yescharacteristic
456C22×C3#380511yescharacteristic
457C2×D3—21no—
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—62no—
470C3×C4—62no—
471C3×C4—62no—
472C3×C4—61no—
473C3×C4—61no—
474C3×C4—122no—
475C3×C4—181no—
476C3×C4C4×D3211yescharacteristic
477C3×C4D3×D1211yescharacteristic
478C3×C4D3×D1211yescharacteristic
479C3×C4C2×D3×Dic311yescharacteristic
480C3×C4C2×D3×Dic311yescharacteristic
481C3×C4#381111yescharacteristic
482Dic3—22no—
483Dic3—21no—
484Dic3—61no—
485Dic3—61no—
486Dic3—121no—
487C32—21no—
488C32—21no—
489C32—21no—
490C32—41no—
491C32#470011yescharacteristic, complemented
492C32#482411yescharacteristic, complemented
493C32#519611yescharacteristic, complemented
494Q8—31no—
495Q8—91no—
496D4—31no—
497D4—61no—
498D4—91no—
499D4—181no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—181no—
503C2×C4D3311yescharacteristic
504C8—31no—
505C8—31no—
506C8—91no—
507C8—91no—
508C8—181no—
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—361no—
525C2×C3#6388011yescharacteristic
526C2×C3Dic3×D1211yescharacteristic
527C2×C3#67⋊Dic311yescharacteristic
528D3—41no—
529D3—121no—
530D3—121no—
531D3—241no—
532C22—61no—
533C22—181no—
534C22#8896111yescharacteristic
535C4—62no—
536C4—61no—
537C4—181no—
538C4D32×Dic311yescharacteristic
539C4#8895511yescharacteristic
540C3—21no—
541C3—21no—
542C3—21no—
543C3—41no—
544C3#1059034111yescharacteristic, complemented
545C3#1059091511yescharacteristic, complemented
546C3#1059167511yescharacteristic, complemented
547C2—21no—
548C2—121no—
549C2—361no—
550C2#1172292811yescharacteristic
subgroupG/H
centerC4#88955
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radicalC3×#63829C2
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. #11723657C2semidirect
2. #11723349C2semidirect
4. #11722058C2semidirect
5. #11722046C2semidirect
6. C3×#63569C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
22. C3×#3719C4semidirect
23. C3×#3717C4semidirect
24. C3×#3716C4semidirect
25. C3×#3714C4semidirect
48. #63569D3semidirect
49. Dic3×C3⋊C8D3semidirect
110. #3719Dic3semidirect
111. #3717Dic3semidirect
112. #3716Dic3semidirect
113. #3714Dic3semidirect
338. C33#443semidirect
491. C32#4700semidirect
492. C32#4824semidirect
493. C32#5196semidirect
544. C3#10590341semidirect
545. C3#10590915semidirect
546. C3#10591675semidirect

Maximal subgroups

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

subgroupindex
#117236572
#117233492
C3×#638292
#117220582
#117220462
C3×#635692
C3×Dic3×C3⋊C82
#105916753
#105909153
#105903413