Finite Group Databasebeta

← #13175507

Subgroups — #13175507

Order 1728.

1924 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6293C211yesmaximal, characteristic, complemented
2C2×#89029C211yesmaximal, characteristic
3#11723918C211yesmaximal, characteristic, complemented
4C3×#64095C211yesmaximal, characteristic, complemented
5C32×#886C211yesmaximal, characteristic, complemented
6#11723605C211yesmaximal, characteristic, complemented
7#11723602C211yesmaximal, characteristic, complemented
8#11723536C212yesmaximal, complemented
9#11723530C212yesmaximal, complemented
10#11723502C212yesmaximal, complemented
11#11723500C212yesmaximal, complemented
12#10596119—31nomaximal
13#10594084—34nomaximal
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×#638C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×#67C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C2×#6293—24no—
20C2×#6293C2212yes—
21C2×#6293C2211yescharacteristic
22C2×#6293C2212yes—
23C2×#6290C2211yescharacteristic
24C2×Dic3×C3⋊D3C2211yescharacteristic
25#89029—24no—
26C2×D3×C3⋊Dic3C2211yescharacteristic
27C33×C22⋊C4C2211yescharacteristic
28#88864C2211yescharacteristic
29#88857C2212yes—
30C4×C3⋊(C3⋊Dic3)C2212yes—
31C3×#3752C2212yes—
32C3×C4⋊(C3⋊Dic3)C2212yes—
33C32×(C2×D3)⋊C4C2212yes—
34C32×C4⋊Dic3C2212yes—
35#88806C2212yes—
36#88806C2212yes—
37#88803C2212yes—
38(C2×D3)⋊(C3⋊Dic3)C2212yes—
39C2×#3830—31no—
40C22×#634—31no—
41C22×#627—34no—
42C2×#3805—34no—
43C32×#123—31no—
44#64144—31no—
45#64138—31no—
46#64105—32no—
47#64097—32no—
48#64095D311yescharacteristic, complemented
49#64094—32no—
50#64087—32no—
51#64086—31no—
52#64086—34no—
53#64086—64no—
54C3×#886—34no—
55C3×#886D314yescomplemented
56#63961—34no—
57#63961—34no—
58#63919—38no—
59#63915—38no—
60#63899—38no—
61#63899—38no—
62C23×C33C2311yescharacteristic
63C22×C3×C3⋊D3C2311yescharacteristic
64C22×C32×D3C2311yescharacteristic
65C2×C33×C4C2312yes—
66C2×C3⋊(C3⋊Dic3)D414yes—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C3×#638—22no—
71C2×C3×C3⋊Dic3—22no—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2312yes—
74C32×#67—22no—
75C2×C32×Dic3—22no—
76C2×C32×Dic3C2311yescharacteristic
77C2×C32×Dic3C2312yes—
78#6293—22no—
79#6293—24no—
80#6293—24no—
81#6290—22no—
82Dic3×C3⋊D3—22no—
83D3×C3⋊Dic3—22no—
84#5648—94no—
85#5648—91no—
86C2×C32×D4—31no—
87C22×C32×C4—31no—
88C2×#638—31no—
89C2×#638C2×D311yescharacteristic
90C22×C3⋊Dic3—31no—
91C22×C3⋊Dic3—34no—
92C22×C3⋊Dic3—31no—
93C22×C3⋊Dic3—64no—
94C22×C3⋊Dic3C2×D311yescharacteristic
95#3830—64no—
96C2×C4×C3⋊D3—31no—
97C2×#634—32no—
98C2×#634—32no—
99C2×#634—31no—
100C2×#634—64no—
101C2×C3×#67—34no—
102C2×C3×#67C2×D314yes—
103C22×C3×Dic3—34no—
104C22×C3×Dic3C2×D314yes—
105C2×#627—34no—
106C2×#627—38no—
107C2×#627—38no—
108C2×#627—616no—
109C2×#625—34no—
110#3805—616no—
111C2×D3×Dic3—34no—
112C2×D3×Dic3—34no—
113C32×C4⋊C4—32no—
114C32×C22⋊C4—24no—
115C32×C22⋊C4—32no—
116C32×C22⋊C4C2×D314yes—
117C32×C22⋊C4C2×D311yescharacteristic
118#3757—31no—
119#3757—34no—
120#3757—32no—
121#3757—64no—
122#3752—32no—
123#3752C2×D312yes—
124C4⋊(C3⋊Dic3)C2×D312yes—
125#3750—32no—
126#3750—38no—
127#3750—32no—
128#3750—32no—
129#3750—68no—
130C4×C3⋊Dic3—32no—
131C4×C3⋊Dic3—38no—
132C4×C3⋊Dic3—68no—
133C3×(C2×D3)⋊C4—38no—
134C3×(C2×D3)⋊C4C2×D318yes—
135C3×C4⋊Dic3—38no—
136C3×C4⋊Dic3C2×D318yes—
137#3724—38no—
138#3724—38no—
139(C2×D3)⋊Dic3—38no—
140(C2×D3)⋊Dic3—38no—
141C22×C33—22no—
142C22×C33C2×D412yes—
143C22×C33C2411yescharacteristic
144C2×C3×C3⋊D3—21no—
145C2×C3×C3⋊D3—22no—
146C2×C32×D3—21no—
147C2×C32×D3—22no—
148C33×C4—22no—
149C3⋊(C3⋊Dic3)—22no—
150C3⋊(C3⋊Dic3)C2×D412yes—
151C3⋊(C3⋊Dic3)C2×D412yes—
152C3×C3⋊Dic3—21no—
153C3×C3⋊Dic3—22no—
154C32×Dic3—21no—
155C32×Dic3—22no—
156C2×#237—94no—
157C2×#237—91no—
158C22×Dic6—94no—
159C22×Dic6—91no—
160C3×#123—34no—
161C3×#123—91no—
162#935—94no—
163#935—91no—
164#929—94no—
165#929—91no—
166#896—98no—
167#896—92no—
168#888—98no—
169#888—92no—
170#886—34no—
171#886C3⋊D311yescharacteristic, complemented
172#885—98no—
173#885—92no—
174#878—98no—
175#878—92no—
176#877—94no—
177#877—91no—
178#877—184no—
179C23×C32—24no—
180C23×C32—31no—
181C23×C32C22×D314yes—
182C23×C32C22×D311yescharacteristic
183C22×C3⋊D3C22×D311yescharacteristic
184C22×C3×D3—34no—
185C22×C3×D3C22×D314yes—
186C32×D4—62no—
187C2×C32×C4—28no—
188C2×C32×C4—31no—
189C2×C32×C4—32no—
190C2×C32×C4—62no—
191C2×C32×C4C22×D318yes—
192C2×C32×C4C22×D312yes—
193#638—22no—
194#638—62no—
195C2×C3⋊Dic3—22no—
196C2×C3⋊Dic3—34no—
197C2×C3⋊Dic3—31no—
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—32no—
200C2×C3⋊Dic3—316no—
201C2×C3⋊Dic3—34no—
202C2×C3⋊Dic3—34no—
203C2×C3⋊Dic3—38no—
204C2×C3⋊Dic3—64no—
205C2×C3⋊Dic3—64no—
206C2×C3⋊Dic3—616no—
207C2×C3⋊Dic3—68no—
208C2×C3⋊Dic3—62no—
209C2×C3⋊Dic3C22×D311yescharacteristic
210C2×C3⋊Dic3C22×D312yes—
211C4×C3⋊D3—62no—
212#634—64no—
213#634—64no—
214#634—62no—
215C3×#67—28no—
216C3×#67—68no—
217C2×C3×Dic3—28no—
218C2×C3×Dic3—38no—
219C2×C3×Dic3—34no—
220C2×C3×Dic3—68no—
221C2×C3×Dic3C22×D318yes—
222C2×C3×Dic3C22×D314yes—
223#627—68no—
224#627—616no—
225#627—616no—
226#625—68no—
227D3×Dic3—68no—
228D3×Dic3—68no—
229#536—271noSylow
230C2×C33—22no—
231C2×C33C22×D411yescharacteristic
232C2×C332−1+412yes—
233C3×C3⋊D3—41no—
234C32×D3—41no—
235C2×C3×D4—34no—
236C2×C3×D4—91no—
237C22×C3×C4—34no—
238C22×C3×C4—91no—
239C2×#67—34no—
240C2×#67—94no—
241C2×#67—91no—
242C2×#67C2×C3⋊D311yescharacteristic
243C22×Dic3—34no—
244C22×Dic3—94no—
245C22×Dic3—94no—
246C22×Dic3—91no—
247C22×Dic3—91no—
248C22×Dic3—184no—
249C22×Dic3C2×C3⋊D311yescharacteristic
250#237—184no—
251#237—1816no—
252C2×C4×D3—94no—
253C2×C4×D3—91no—
254C2×Dic6—98no—
255C2×Dic6—98no—
256C2×Dic6—94no—
257C2×Dic6—92no—
258C2×Dic6—92no—
259C2×Dic6—91no—
260C2×Dic6—184no—
261C2×Dic6—1816no—
262C3×C4⋊C4—38no—
263C3×C4⋊C4—92no—
264C3×C22⋊C4—24no—
265C3×C22⋊C4—38no—
266C3×C22⋊C4—92no—
267C3×C22⋊C4D3214yes—
268C3×C22⋊C4C2×C3⋊D311yescharacteristic
269#217—98no—
270#217—94no—
271#217—91no—
272#217—92no—
273#217—184no—
274(C2×D3)⋊C4—38no—
275(C2×D3)⋊C4—98no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4C2×C3⋊D312yes—
278C4⋊Dic3—38no—
279C4⋊Dic3C2×C3⋊D312yes—
280Dic3⋊C4—98no—
281Dic3⋊C4—98no—
282Dic3⋊C4—98no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—188no—
287C4×Dic3—98no—
288C4×Dic3—92no—
289C4×Dic3—188no—
290C22×C32—22no—
291C22×C32—24no—
292C22×C32—28no—
293C22×C32—28no—
294C22×C32—48no—
295C22×C32—61no—
296C22×C32—62no—
297C22×C32D3×D418yes—
298C22×C32D3×D412yes—
299C22×C32C23×D314yes—
300C22×C32C23×D311yescharacteristic
301C2×C3⋊D3—21no—
302C2×C3⋊D3—22no—
303C2×C3×D3—24no—
304C2×C3×D3—28no—
305C2×C3×D3—64no—
306C2×C3×D3—68no—
307C32×C4—22no—
308C32×C4—28no—
309C32×C4—48no—
310C32×C4—62no—
311C32×C4—61no—
312C3⋊Dic3—22no—
313C3⋊Dic3—21no—
314C3⋊Dic3—32no—
315C3⋊Dic3—32no—
316C3⋊Dic3—38no—
317C3⋊Dic3—38no—
318C3⋊Dic3—68no—
319C3⋊Dic3—68no—
320C3⋊Dic3—62no—
321C3⋊Dic3—68no—
322C3⋊Dic3—128no—
323C3×Dic3—24no—
324C3×Dic3—28no—
325C3×Dic3—64no—
326C3×Dic3—68no—
327C2×#41—271no—
328C22×Q8—271no—
329C4⋊Q8—272no—
330#124—272no—
331#123—31no—
332#123—91no—
333#123—271no—
334#123—271no—
335C22⋊Q8—272no—
336C22⋊Q8—272no—
337#117—271no—
338C33#53611yescharacteristic, complemented, Sylow
339C23×C3—24no—
340C23×C3—34no—
341C23×C3—91no—
342C23×C3C2×D3214yes—
343C23×C3C22×C3⋊D311yescharacteristic
344C22×D3—34no—
345C22×D3C22×C3⋊D311yescharacteristic
346C3×D4—68no—
347C3×D4—182no—
348C2×C3×C4—28no—
349C2×C3×C4—34no—
350C2×C3×C4—38no—
351C2×C3×C4—68no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—182no—
355C2×C3×C4C2×D3218yes—
356C2×C3×C4C22×C3⋊D312yes—
357#67—22no—
358#67—68no—
359#67—182no—
360#67—188no—
361C2×Dic3—22no—
362C2×Dic3—34no—
363C2×Dic3—38no—
364C2×Dic3—68no—
365C2×Dic3—94no—
366C2×Dic3—94no—
367C2×Dic3—91no—
368C2×Dic3—91no—
369C2×Dic3—916no—
370C2×Dic3—94no—
371C2×Dic3—98no—
372C2×Dic3—92no—
373C2×Dic3—184no—
374C2×Dic3—184no—
375C2×Dic3—1816no—
376C2×Dic3—188no—
377C2×Dic3—182no—
378C2×Dic3—188no—
379C2×Dic3C22×C3⋊D311yescharacteristic
380C2×Dic3C22×C3⋊D312yes—
381C4×D3—188no—
382C4×D3—182no—
383Dic6—184no—
384Dic6—184no—
385Dic6—182no—
386Dic6—1816no—
387Dic6—1816no—
388Dic6—188no—
389C2×C32—22no—
390C2×C32—24no—
391C2×C32—28no—
392C2×C32—28no—
393C2×C32—48no—
394C2×C32—121no—
395C2×C32C2×D3×D414yes—
396C2×C32C2×D3×D411yescharacteristic
397C2×C32#101018yes—
398C2×C32#101012yes—
399C3⋊D3—41no—
400C3×D3—44no—
401C3×D3—124no—
402#41—544no—
403C2×Q8—272no—
404C2×Q8—272no—
405C2×Q8—271no—
406C2×Q8—544no—
407C2×D4—31no—
408C2×D4—91no—
409C2×D4—271no—
410C22×C4—31no—
411C22×C4—91no—
412C22×C4—271no—
413C22×C4—271no—
414C22×C4—271no—
415C4⋊C4—32no—
416C4⋊C4—92no—
417C4⋊C4—272no—
418C4⋊C4—272no—
419C4⋊C4—272no—
420C22⋊C4—32no—
421C22⋊C4—92no—
422C22⋊C4—272no—
423C22⋊C4—272no—
424C22⋊C4—271no—
425C22⋊C4D3×C3⋊D311yescharacteristic
426C42—272no—
427C22×C3—28no—
428C22×C3—22no—
429C22×C3—24no—
430C22×C3—28no—
431C22×C3—48no—
432C22×C3—64no—
433C22×C3—68no—
434C22×C3—181no—
435C22×C3—182no—
436C22×C3#380218yes—
437C22×C3D4×C3⋊D312yes—
438C22×C3C22×D3214yes—
439C22×C3C23×C3⋊D311yescharacteristic
440C2×D3—21no—
441C2×D3—22no—
442C2×D3—64no—
443C2×D3—68no—
444C3×C4—28no—
445C3×C4—22no—
446C3×C4—48no—
447C3×C4—68no—
448C3×C4—64no—
449C3×C4—182no—
450C3×C4—181no—
451Dic3—22no—
452Dic3—21no—
453Dic3—68no—
454Dic3—64no—
455Dic3—98no—
456Dic3—98no—
457Dic3—92no—
458Dic3—92no—
459Dic3—188no—
460Dic3—188no—
461Dic3—188no—
462Dic3—182no—
463Dic3—368no—
464C32—24no—
465C32#564814yescomplemented
466C32#564811yescharacteristic, complemented
467C23—31no—
468C23—91no—
469C23C2×D3×C3⋊D311yescharacteristic
470Q8—544no—
471Q8—544no—
472Q8—542no—
473D4—62no—
474D4—182no—
475D4—542no—
476C2×C4—32no—
477C2×C4—31no—
478C2×C4—62no—
479C2×C4—92no—
480C2×C4—91no—
481C2×C4—182no—
482C2×C4—271no—
483C2×C4—271no—
484C2×C4—274no—
485C2×C4—272no—
486C2×C4—542no—
487C2×C4—542no—
488C2×C4C2×D3×C3⋊D312yes—
489C2×C3—22no—
490C2×C3—28no—
491C2×C3—24no—
492C2×C3—28no—
493C2×C3—48no—
494C2×C3—124no—
495C2×C3—361no—
496C2×C3C2×#380214yes—
497C2×C3#6430918yes—
498C2×C3C2×D4×C3⋊D311yescharacteristic
499C2×C3#6436712yes—
500D3—41no—
501D3—124no—
502C22—22no—
503C22—61no—
504C22—62no—
505C22—181no—
506C22—182no—
507C22#8902612yes—
508C22C22×D3×C3⋊D311yescharacteristic
509C4—22no—
510C4—62no—
511C4—61no—
512C4—182no—
513C4—181no—
514C4—272no—
515C4—272no—
516C4—542no—
517C3—24no—
518C3#1059408414yescomplemented
519C3#1059611911yescharacteristic, complemented
520C2—22no—
521C2—121no—
522C2—361no—
523C2C2×#8902611yescharacteristic
524C2#1172458212yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×#89029C2
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6293C2semidirect
3. #11723918C2semidirect
4. C3×#64095C2semidirect
5. C32×#886C2semidirect
6. #11723605C2semidirect
7. #11723602C2semidirect
8. #11723536C2semidirect
9. #11723530C2semidirect
10. #11723502C2semidirect
11. #11723500C2semidirect
48. #64095D3semidirect
55. C3×#886D3semidirect
171. #886C3⋊D3semidirect
338. C33#536semidirect
465. C32#5648semidirect
466. C32#5648semidirect
518. C3#10594084semidirect
519. C3#10596119semidirect

Maximal subgroups

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

subgroupindex
C22×#62932
C2×#890292
#117239182
C3×#640952
C32×#8862
#117236052
#117236022
#117235362
#117235302
#117235022
#117235002
#105961193
#105940843