Finite Group Databasebeta

← #13156455

Subgroups — #13156455

Order 1728.

566 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723674C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic
3#11722053C211yesmaximal, characteristic, complemented
4#11722039C211yesmaximal, characteristic, complemented
5#11722023C211yesmaximal, characteristic
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63567C211yesmaximal, characteristic, complemented
8#10590686—31nomaximal
9#10590516—31nomaximal
10#10590426—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C3×#624—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88815—21no—
20#88814—21no—
21C3×#3724—21no—
22C3×D3×C4⋊C4—31no—
23C3×D3×C4⋊C4—31no—
24#63900D311yescharacteristic
25#63871—31no—
26#63857—31no—
27#63856—31no—
28#63647—31no—
29#63637—31no—
30#63635—31no—
31C3×#817—31no—
32C3×#816—31no—
33C3×#810—31no—
34C3×#808—31no—
35#63574—31no—
36#63574—31no—
37#63574—61no—
38#63573D311yescharacteristic, complemented
39#63567D311yescharacteristic, complemented
40#63565—31no—
41#63565—31no—
42#63565—61no—
43(C2×D3)⋊(C3⋊C8)—31no—
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—61no—
46C22×C3×C3⋊D3D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C3×#634—21no—
54C3×#634—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C3×#624—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#4971—91no—
61#4862—91no—
62#4805—91no—
63C2×C4×C3⋊D3C2×D311yescharacteristic
64C2×#634C2×D311yescharacteristic
65C2×C3×C4×D3—31no—
66C2×C3×C4×D3—31no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3—61no—
69C2×C3×C4×D3—61no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—61no—
73C2×#624—21no—
74C2×C32×C8—31no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—31no—
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—61no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—61no—
84C2×C3⋊(C3⋊C8)—121no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99#3724—21no—
100#3724—61no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103#3722—61no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—21no—
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3SD1612yes—
110C33×C4#4111yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—21no—
113C3×C3⋊Dic3—41no—
114C3×C3⋊Dic3—41no—
115C32×Dic3—41no—
116C32×Dic3—41no—
117C2×C3×C4⋊C4—91no—
118#946—91no—
119#924—91no—
120#897—91no—
121D3×C4⋊C4—31no—
122D3×C4⋊C4—31no—
123C3×#106—91no—
124C3×#103—91no—
125#835—91no—
126#830—91no—
127(C2×D3)⋊C8—91no—
128(C2×D3)⋊C8—91no—
129(C2×D3)⋊C8—181no—
130#817—31no—
131#816—31no—
132#810—31no—
133#810—91no—
134#810—91no—
135#810—181no—
136#808—31no—
137#808—91no—
138#808—91no—
139#808—91no—
140#808—181no—
141C22×C3⋊D3#6711yescharacteristic
142C22×C3×D3—31no—
143C22×C3×D3—31no—
144C22×C3×D3—61no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—41no—
149C2×C32×C4—61no—
150C2×C32×C4—61no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3#6711yescharacteristic
157C4×C3⋊D3—21no—
158C4×C3⋊D3—21no—
159#634—21no—
160#634—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—41no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—121no—
175C2×C3×Dic3—121no—
176C3×C4×D3—61no—
177C3×C4×D3—61no—
178C3×C4×D3—61no—
179C3×C4×D3—61no—
180C3×C4×D3—121no—
181C3×C4×D3—121no—
182C3×C4×D3—122no—
183C3×C4×D3—122no—
184C3×Dic6—62no—
185C3×Dic6—61no—
186C3×Dic6—61no—
187C3×Dic6—61no—
188C3×Dic6—61no—
189C3×Dic6—121no—
190#624—42no—
191C32×C8—61no—
192C3⋊(C3⋊C8)—61no—
193C3⋊(C3⋊C8)—62no—
194C3⋊(C3⋊C8)—61no—
195C3⋊(C3⋊C8)—61no—
196C3⋊(C3⋊C8)—121no—
197C3⋊(C3⋊C8)—121no—
198C3⋊(C3⋊C8)—122no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—41no—
202#483—271noSylow
203C2×C33#12311yescharacteristic
204C2×C33C2×SD1611yescharacteristic
205C2×C33#13711yescharacteristic
206C3×C3⋊D3—22no—
207C2×C3×Q8—91no—
208C22×C3×C4—91no—
209C22×C3×C4—181no—
210C2×C4×D3—31no—
211C2×C4×D3—31no—
212C2×C4×D3—61no—
213C2×C4×D3—61no—
214C2×C4×D3—61no—
215C2×Dic6—31no—
216C2×Dic6—31no—
217C2×Dic6—61no—
218C2×C3×C8—31no—
219C2×C3×C8—31no—
220C2×C3×C8—61no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—181no—
229#217—181no—
230(C2×D3)⋊C4—181no—
231(C2×D3)⋊C4—181no—
232C4⋊Dic3—31no—
233C4⋊Dic3—31no—
234C4⋊Dic3—61no—
235C4⋊Dic3—181no—
236C4⋊Dic3D3211yescharacteristic
237C4⋊Dic3D3211yescharacteristic
238Dic3⋊C4—61no—
239Dic3⋊C4—61no—
240Dic3⋊C4—181no—
241Dic3⋊C4—181no—
242C2×C3⋊C8—91no—
243C2×C3⋊C8—91no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—181no—
246C2×C3⋊C8—181no—
247C2×C3⋊C8—181no—
248C2×C3⋊C8—361no—
249C2×C3⋊C8D3211yescharacteristic
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32D3×D411yescharacteristic
255C22×C32D3×D411yescharacteristic
256C22×C32C2×#6711yescharacteristic
257C2×C3⋊D3—21no—
258C2×C3⋊D3—21no—
259C2×C3⋊D3#21412yes—
260C2×C3×D3—32no—
261C2×C3×D3—32no—
262C2×C3×D3—62no—
263C2×C3×D3—61no—
264C2×C3×D3—61no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—121no—
268C2×C3×D3—121no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—121no—
278C32×C4—121no—
279C32×C4#23511yescharacteristic
280C32×C4#23511yescharacteristic
281C32×C4#23511yescharacteristic
282C32×C4#23511yescharacteristic
283C32×C4#23711yescharacteristic
284C32×C4#23911yescharacteristic
285C3⋊Dic3—21no—
286C3⋊Dic3—41no—
287C3⋊Dic3—41no—
288C3×Dic3—41no—
289C3×Dic3—41no—
290C3×Dic3—41no—
291C3×Dic3—41no—
292C3×Dic3—61no—
293C3×Dic3—61no—
294C3×Dic3—81no—
295C3×Dic3—81no—
296C3×Dic3—121no—
297C3×Dic3—122no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—121no—
301C3×Dic3—121no—
302C3×Dic3—241no—
303C22⋊Q8—271no—
304C2×C4⋊C4—91no—
305#106—91no—
306#106—271no—
307#103—91no—
308#103—271no—
309C22⋊C8—271no—
310C33#48311yescharacteristic, complemented, Sylow
311C23×C3—91no—
312C22×D3—31no—
313C22×D3—31no—
314C22×D3—61no—
315C3×Q8—181no—
316C3×Q8—181no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—41no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—91no—
326C2×C3×C4—121no—
327C2×C3×C4—121no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4—181no—
332C2×C3×C4—362no—
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—31no—
339C2×Dic3—31no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—121no—
346C2×Dic3—121no—
347C4×D3—61no—
348C4×D3—61no—
349C4×D3—61no—
350C4×D3—61no—
351C4×D3—121no—
352C4×D3—121no—
353C4×D3—122no—
354C4×D3—122no—
355Dic6—62no—
356Dic6—61no—
357Dic6—61no—
358Dic6—61no—
359Dic6—61no—
360Dic6—121no—
361C3×C8—61no—
362C3×C8—61no—
363C3×C8—121no—
364C3⋊C8—21no—
365C3⋊C8—182no—
366C3⋊C8—181no—
367C3⋊C8—181no—
368C3⋊C8—181no—
369C3⋊C8—361no—
370C3⋊C8—361no—
371C3⋊C8—362no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32#88311yescharacteristic
385C2×C32#89411yescharacteristic
386C2×C32D3×SD1611yescharacteristic
387C2×C32D3×SD1611yescharacteristic
388C2×C32#91511yescharacteristic
389C2×C32#91811yescharacteristic
390C2×C32#92911yescharacteristic
391C2×C32C2×#21411yescharacteristic
392C2×C32#94211yescharacteristic
393C3⋊D3—22no—
394C3×D3—62no—
395C3×D3—62no—
396C3×D3—122no—
397C2×Q8—91no—
398C22×C4—91no—
399C22×C4—181no—
400C2×C8—31no—
401C2×C8—271no—
402C4⋊C4—31no—
403C4⋊C4—31no—
404C4⋊C4—91no—
405C4⋊C4—181no—
406C4⋊C4—541no—
407C22⋊C4—541no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—41no—
412C22×C3—92no—
413C22×C3—181no—
414C22×C3—181no—
415C22×C3#380211yescharacteristic
416C22×C3D3×#6711yescharacteristic
417C22×C3D3×#6711yescharacteristic
418C2×D3—32no—
419C2×D3—32no—
420C2×D3—62no—
421C2×D3—61no—
422C2×D3—61no—
423C2×D3—61no—
424C2×D3—61no—
425C2×D3—121no—
426C2×D3—121no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—121no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—121no—
439C3×C4—181no—
440C3×C4—241no—
441C3×C4—241no—
442C3×C4—361no—
443C3×C4—361no—
444C3×C4#379511yescharacteristic
445C3×C4#379811yescharacteristic
446C3×C4#379811yescharacteristic
447C3×C4#379911yescharacteristic
448C3×C4#379911yescharacteristic
449C3×C4#380411yescharacteristic
450Dic3—41no—
451Dic3—41no—
452Dic3—61no—
453Dic3—61no—
454Dic3—121no—
455Dic3—122no—
456Dic3—121no—
457Dic3—121no—
458Dic3—121no—
459Dic3—121no—
460Dic3—241no—
461C32—21no—
462C32—21no—
463C32—21no—
464C32—41no—
465C32#480511yescharacteristic, complemented
466C32#486211yescharacteristic, complemented
467C32#497111yescharacteristic, complemented
468C23—91no—
469Q8—181no—
470Q8—181no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—91no—
474C2×C4—181no—
475C2×C4—181no—
476C2×C4—181no—
477C2×C4—181no—
478C2×C4—362no—
479C2×C4D3311yescharacteristic
480C8—61no—
481C8—541no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—182no—
495C2×C3#6379611yescharacteristic
496C2×C3#6381111yescharacteristic
497C2×C3#6385011yescharacteristic
498C2×C3#6385411yescharacteristic
499C2×C3#6390511yescharacteristic
500C2×C3D3×#21411yescharacteristic
501C2×C3D3×#21411yescharacteristic
502C2×C3#6394611yescharacteristic
503C2×C3#6394811yescharacteristic
504D3—62no—
505D3—62no—
506D3—122no—
507C22—92no—
508C22—181no—
509C22—181no—
510C22#8895311yescharacteristic
511C4—121no—
512C4—121no—
513C4—181no—
514C4—361no—
515C4—361no—
516C4#8896011yescharacteristic
517C4#8896511yescharacteristic
518C3—21no—
519C3—21no—
520C3—21no—
521C3—41no—
522C3#1059042611yescharacteristic, complemented
523C3#1059051611yescharacteristic, complemented
524C3#1059068611yescharacteristic, complemented
525C2—182no—
526C2#1172283911yescharacteristic
527C2#1172289211yescharacteristic
528C2#1172299511yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#483
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
3. #11722053C2semidirect
4. #11722039C2semidirect
6. C3×#63573C2semidirect
7. C3×#63567C2semidirect
38. #63573D3semidirect
39. #63567D3semidirect
310. C33#483semidirect
465. C32#4805semidirect
466. C32#4862semidirect
467. C32#4971semidirect
522. C3#10590426semidirect
523. C3#10590516semidirect
524. C3#10590686semidirect

Maximal subgroups

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

subgroupindex
#117236742
C3×#639002
#117220532
#117220392
#117220232
C3×#635732
C3×#635672
#105906863
#105905163
#105904263