Finite Group Databasebeta

← #13156276

Subgroups — #13156276

Order 1728.

578 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723691C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic, complemented
3#11722056C211yesmaximal, characteristic, complemented
4#11722051C211yesmaximal, characteristic, complemented
5#11722033C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8#10590778—31nomaximal
9#10590464—31nomaximal
10#10590463—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×#6294—21no—
13C2×C3×#627—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88814—21no—
21C3×#3724—21no—
22C3×#888—31no—
23C3×#888—31no—
24#63919D311yescharacteristic, complemented
25#63898—31no—
26#63894—31no—
27Dic3×D12—31no—
28#63641—31no—
29#63636—31no—
30#63634—31no—
31C3×#821—31no—
32C3×#818—31no—
33C3×#809—31no—
34C3×#807—31no—
35#63576—31no—
36#63576—31no—
37#63576—61no—
38#63575D311yescharacteristic, complemented
39Dic3⋊(C3⋊C8)—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—61no—
42#63564D311yescharacteristic
43#63562—31no—
44#63562—31no—
45#63562—61no—
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)Q812yescomplemented
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55#6294—42no—
56C3×#627—42no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5087—91no—
60#4890—91no—
61#4759—91no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—61no—
66C2×C3×Dic6—61no—
67C2×C3×Dic6—61no—
68C2×#627—21no—
69C2×#624—61no—
70C2×D3×Dic3—61no—
71C2×D3×Dic3—61no—
72C2×C32×C8—31no—
73C32×C4⋊C4—31no—
74C32×C4⋊C4—31no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76C4×C3⋊Dic3C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—61no—
81C2×C3⋊(C3⋊C8)—61no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—121no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—61no—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101#3724—21no—
102#3722—61no—
103(C2×D3)⋊Dic3—61no—
104(C2×D3)⋊Dic3—61no—
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3—41no—
108C2×C3×C3⋊D3—41no—
109C33×C4C2×Q811yescharacteristic
110C33×C4#4111yescharacteristic
111C3×C3⋊Dic3—21no—
112C3×C3⋊Dic3—41no—
113C3×C3⋊Dic3D812yes—
114C32×Dic3—41no—
115C32×Dic3—41no—
116C3×C4⋊Q8—91no—
117D4×Dic3—91no—
118#893—91no—
119#888—31no—
120#888—31no—
121C4×D12—91no—
122C3×#107—91no—
123C3×#102—91no—
124#832—91no—
125#821—31no—
126#818—31no—
127Dic3⋊C8—91no—
128Dic3⋊C8—91no—
129Dic3⋊C8—181no—
130#809—31no—
131#809—91no—
132#809—91no—
133#809—181no—
134#807—31no—
135#807—91no—
136#807—91no—
137#807—91no—
138#807—181no—
139C4⋊(C3⋊C8)—91no—
140C22×C3⋊D3—21no—
141C22×C3×D3—61no—
142C22×C3×D3—61no—
143C22×C3×D3—121no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—41no—
148C2×C32×C4—61no—
149C2×C32×C4—61no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3#6711yescharacteristic
155C2×C3⋊Dic3#6711yescharacteristic
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)Dic612yescomplemented
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—41no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—121no—
173C3×D12—32no—
174C3×D12—32no—
175C3×D12—62no—
176C3×D12—61no—
177C3×D12—61no—
178C3×D12—121no—
179C3×Dic6—122no—
180C3×Dic6—122no—
181#627—42no—
182#624—122no—
183D3×Dic3—122no—
184D3×Dic3—122no—
185C32×C8—61no—
186C3⋊(C3⋊C8)—61no—
187C3⋊(C3⋊C8)—62no—
188C3⋊(C3⋊C8)—61no—
189C3⋊(C3⋊C8)—61no—
190C3⋊(C3⋊C8)—121no—
191C3⋊(C3⋊C8)—121no—
192C3⋊(C3⋊C8)—122no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—41no—
196#474—271noSylow
197C2×C33C22⋊Q811yescharacteristic
198C2×C33C2×D811yescharacteristic
199C2×C33#13711yescharacteristic
200C3×C3⋊D3—42no—
201C2×C3×Q8—181no—
202C2×C3×D4—91no—
203C22×Dic3—181no—
204C2×D12—31no—
205C2×D12—31no—
206C2×D12—61no—
207C2×C4×D3—181no—
208C2×C4×D3—181no—
209C2×Dic6—61no—
210C2×Dic6—61no—
211C2×C3×C8—31no—
212C2×C3×C8—31no—
213C2×C3×C8—61no—
214C3×C4⋊C4—31no—
215C3×C4⋊C4—31no—
216C3×C4⋊C4—31no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—61no—
219C3×C4⋊C4—61no—
220C3×C4⋊C4—91no—
221C3×C4⋊C4—181no—
222C3×C42—91no—
223#217—181no—
224(C2×D3)⋊C4—181no—
225(C2×D3)⋊C4—181no—
226C4⋊Dic3—31no—
227C4⋊Dic3—31no—
228C4⋊Dic3—61no—
229C4⋊Dic3D3211yescharacteristic
230C4⋊Dic3D3211yescharacteristic
231Dic3⋊C4—61no—
232Dic3⋊C4—61no—
233C4×Dic3—31no—
234C4×Dic3—31no—
235C4×Dic3—61no—
236C2×C3⋊C8—91no—
237C2×C3⋊C8—91no—
238C2×C3⋊C8—91no—
239C2×C3⋊C8—181no—
240C2×C3⋊C8—181no—
241C2×C3⋊C8—181no—
242C2×C3⋊C8—361no—
243C2×C3⋊C8D3211yescharacteristic
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—41no—
248C22×C32D3×D411yescharacteristic
249C22×C32D3×D411yescharacteristic
250C22×C32C2×#6711yescharacteristic
251C2×C3⋊D3—22no—
252C2×C3⋊D3—41no—
253C2×C3⋊D3—41no—
254C2×C3×D3—62no—
255C2×C3×D3—62no—
256C2×C3×D3—122no—
257C2×C3×D3—121no—
258C2×C3×D3—121no—
259C2×C3×D3—121no—
260C2×C3×D3—121no—
261C2×C3×D3—241no—
262C2×C3×D3—241no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—41no—
270C32×C4—41no—
271C32×C4—121no—
272C32×C4—121no—
273C32×C4C2×Dic611yescharacteristic
274C32×C4#23511yescharacteristic
275C32×C4#23711yescharacteristic
276C32×C4D3×Q811yescharacteristic
277C32×C4D3×Q811yescharacteristic
278C32×C4#23911yescharacteristic
279C3⋊Dic3—21no—
280C3⋊Dic3—41no—
281C3⋊Dic3#21312yes—
282C3×Dic3—32no—
283C3×Dic3—32no—
284C3×Dic3—41no—
285C3×Dic3—41no—
286C3×Dic3—41no—
287C3×Dic3—41no—
288C3×Dic3—62no—
289C3×Dic3—61no—
290C3×Dic3—61no—
291C3×Dic3—81no—
292C3×Dic3—81no—
293C3×Dic3—121no—
294C3×Dic3—121no—
295C3×Dic3—121no—
296C3×Dic3—241no—
297C4⋊Q8—91no—
298C4×D4—271no—
299#107—91no—
300#107—271no—
301C4⋊C8—271no—
302#102—91no—
303#102—271no—
304C33#47411yescharacteristic, complemented, Sylow
305C23×C3—181no—
306C22×D3—61no—
307C22×D3—61no—
308C22×D3—121no—
309C3×Q8—362no—
310C3×D4—92no—
311C3×D4—181no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—41no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—91no—
321C2×C3×C4—91no—
322C2×C3×C4—121no—
323C2×C3×C4—121no—
324C2×C3×C4—181no—
325C2×C3×C4C2×D3211yescharacteristic
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—31no—
331C2×Dic3—31no—
332C2×Dic3—31no—
333C2×Dic3—31no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—121no—
339C2×Dic3—362no—
340D12—32no—
341D12—32no—
342D12—62no—
343D12—61no—
344D12—61no—
345D12—121no—
346C4×D3—362no—
347C4×D3—362no—
348Dic6—122no—
349Dic6—122no—
350C3×C8—61no—
351C3×C8—61no—
352C3×C8—121no—
353C3⋊C8—21no—
354C3⋊C8—182no—
355C3⋊C8—181no—
356C3⋊C8—181no—
357C3⋊C8—181no—
358C3⋊C8—361no—
359C3⋊C8—361no—
360C3⋊C8—362no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32#87811yescharacteristic
374C2×C32#89611yescharacteristic
375C2×C32D3×D811yescharacteristic
376C2×C32D3×D811yescharacteristic
377C2×C32#91511yescharacteristic
378C2×C32#91811yescharacteristic
379C2×C32C2×#21311yescharacteristic
380C2×C32#94211yescharacteristic
381C2×C32#94611yescharacteristic
382C3⋊D3—42no—
383C3×D3—122no—
384C3×D3—122no—
385C3×D3—242no—
386C2×Q8—181no—
387C2×D4—91no—
388C22×C4—541no—
389C2×C8—31no—
390C2×C8—271no—
391C4⋊C4—31no—
392C4⋊C4—31no—
393C4⋊C4—91no—
394C4⋊C4—181no—
395C22⋊C4—541no—
396C42—91no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—41no—
401C22×C3—182no—
402C22×C3—361no—
403C22×C3—361no—
404C22×C3#380211yescharacteristic
405C22×C3D3×#6711yescharacteristic
406C22×C3D3×#6711yescharacteristic
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—122no—
410C2×D3—121no—
411C2×D3—121no—
412C2×D3—121no—
413C2×D3—121no—
414C2×D3—241no—
415C2×D3—241no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—92no—
425C3×C4—121no—
426C3×C4—121no—
427C3×C4—121no—
428C3×C4—121no—
429C3×C4—181no—
430C3×C4—241no—
431C3×C4—241no—
432C3×C4—361no—
433C3×C4D3×Dic611yescharacteristic
434C3×C4D3×Dic611yescharacteristic
435C3×C4#379611yescharacteristic
436C3×C4#379711yescharacteristic
437C3×C4#379911yescharacteristic
438C3×C4#380411yescharacteristic
439Dic3—32no—
440Dic3—32no—
441Dic3—41no—
442Dic3—41no—
443Dic3—62no—
444Dic3—61no—
445Dic3—61no—
446Dic3—121no—
447Dic3—121no—
448Dic3—121no—
449Dic3—241no—
450C32—21no—
451C32—21no—
452C32—21no—
453C32—41no—
454C32#475911yescharacteristic, complemented
455C32#489011yescharacteristic, complemented
456C32#508711yescharacteristic, complemented
457C23—181no—
458Q8—362no—
459D4—92no—
460D4—181no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—91no—
464C2×C4—91no—
465C2×C4—181no—
466C2×C4—1082no—
467C2×C4D3311yescharacteristic
468C8—61no—
469C8—541no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—362no—
483C2×C3#6379711yescharacteristic
484C2×C3#6381111yescharacteristic
485C2×C3#6388711yescharacteristic
486C2×C3#6388811yescharacteristic
487C2×C3#6389911yescharacteristic
488C2×C3D3×#21311yescharacteristic
489C2×C3D3×#21311yescharacteristic
490C2×C3#6394611yescharacteristic
491C2×C3#6394811yescharacteristic
492D3—122no—
493D3—122no—
494D3—242no—
495C22—182no—
496C22—361no—
497C22—361no—
498C22#8895311yescharacteristic
499C4—92no—
500C4—121no—
501C4—121no—
502C4—181no—
503C4—361no—
504C4#8895711yescharacteristic
505C4#8896211yescharacteristic
506C3—21no—
507C3—21no—
508C3—21no—
509C3—41no—
510C3#1059046311yescharacteristic, complemented
511C3#1059046411yescharacteristic, complemented
512C3#1059077811yescharacteristic, complemented
513C2—362no—
514C2#1172283111yescharacteristic
515C2#1172289211yescharacteristic
516C2#1172297911yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#474
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63919C2semidirect
3. #11722056C2semidirect
4. #11722051C2semidirect
6. C3×#63575C2semidirect
24. #63919D3semidirect
38. #63575D3semidirect
52. C3×C4⋊(C3⋊D3)Q8semidirect
157. C4⋊(C3⋊D3)Dic6semidirect
304. C33#474semidirect
454. C32#4759semidirect
455. C32#4890semidirect
456. C32#5087semidirect
510. C3#10590463semidirect
511. C3#10590464semidirect
512. C3#10590778semidirect

Maximal subgroups

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

subgroupindex
#117236912
C3×#639192
#117220562
#117220512
#117220332
C3×#635752
C3×#635642
#105907783
#105904643
#105904633