Finite Group Databasebeta

← #13161530

Subgroups — #13161530

Order 1728.

906 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724618C211yesmaximal, characteristic
2C3×C42×C3⋊D3C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4C3×#63632C212yesmaximal, complemented
5#11722077C212yesmaximal, complemented
6#10590393—31nomaximal
7D3×#805—32nomaximal
8C3×#3828C2212yes—
9C2×C3×C4×C3⋊D3—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11#89049—21no—
12#89046—21no—
13C4⋊#1113C411yescharacteristic, complemented
14C4×#1113—21no—
15#89042—21no—
16#89041C412yescomplemented
17#89040C411yescharacteristic, complemented
18C33×C42C2211yescharacteristic, complemented
19C3×C4×C3⋊Dic3—21no—
20C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
21C3×#3748C2212yescomplemented
22#88809—21no—
23#88808—21no—
24#64306—31no—
25D3×#235—32no—
26C42×C3⋊D3D311yescharacteristic, complemented
27C3×C42×D3—32no—
28C3×C42×D3—61no—
29#63818—31no—
30D3×#208—32no—
31#63632D312yescomplemented
32C3×#805—32no—
33C3×#805—32no—
34C3×#805—62no—
35#63570—32no—
36#63569—32no—
37#63568—32no—
38C22×C3×C3⋊D3D411yescharacteristic
39C2×#1113—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C3×#638—22no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4⋊(C3⋊D3)C2×C412yescomplemented
46C3×C4×C3⋊D3—24no—
47C3×C4×C3⋊D3—22no—
48C3×C4×C3⋊D3C2×C411yescharacteristic
49C3×C4×C3⋊D3C2×C411yescharacteristic
50C3×C4×C3⋊D3D411yescharacteristic
51C3×C4×C3⋊D3D411yescharacteristic
52C3×#634C2×C412yescomplemented
53#6296—21no—
54#6295—22no—
55#6295—21no—
56#6294—21no—
57#6294—22no—
58C3×C3⋊(C3⋊C8)—22no—
59C2×#805—91no—
60D3×#104—92no—
61#3828C2×D312yes—
62C2×C4×C3⋊D3—21no—
63C2×C4×C3⋊D3C2×D311yescharacteristic
64C3×#235—32no—
65C3×#235—32no—
66C3×#235—62no—
67C2×C3×C4×D3—32no—
68C2×C3×C4×D3—62no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—62no—
71#3811—61no—
72D3×#67—62no—
73#3805—61no—
74#3804—62no—
75#3802—31no—
76D3×D12—32no—
77C4×D32—62no—
78C4×D32—61no—
79#3799—32no—
80#3798—62no—
81#3798—61no—
82#3797—31no—
83#3796—32no—
84#3795—32no—
85D3×Dic6—32no—
86C32×C42—21no—
87C32×C42—22no—
88C32×C42—41no—
89C32×C42C2×D311yescharacteristic, complemented
90C32×C42C2×D312yescomplemented
91C4×C3⋊Dic3—21no—
92C4×C3⋊Dic3C2×D311yescharacteristic, complemented
93#3748C2×D312yescomplemented
94C3×C4×Dic3—32no—
95C3×C4×Dic3—62no—
96C3×C4×Dic3—61no—
97C3×C4×Dic3—62no—
98C3×#208—32no—
99C3×#208—32no—
100C3×#208—62no—
101#3712—61no—
102#3711—62no—
103#3710—61no—
104D3×C3⋊C8—62no—
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3C22⋊C411yescharacteristic
109C2×C3×C3⋊D3C22⋊C411yescharacteristic
110#1113—42no—
111C33×C4—22no—
112C33×C4C22×C411yescharacteristic
113C33×C4C2×D411yescharacteristic
114C3×C3⋊Dic3—22no—
115C3×C3⋊Dic3—22no—
116C3×C3⋊Dic3C22⋊C411yescharacteristic
117C3×C3⋊Dic3C22⋊C411yescharacteristic
118D3×#41—92no—
119C2×#235—91no—
120C2×C3×C42—91no—
121C2×#208—91no—
122D3×M16—92no—
123C42×D3—32no—
124C42×D3—61no—
125C3×#104—92no—
126#837—92no—
127#825—92no—
128#805—32no—
129#805—32no—
130#805—62no—
131#805—92no—
132C22×C3⋊D3#6711yescharacteristic
133C22×C3×D3—32no—
134C22×C3×D3—61no—
135C2×D32—62no—
136C2×D32—61no—
137C2×C32×C4—21no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—21no—
142C2×C32×C4—41no—
143C2×C32×C4—42no—
144C2×C32×C4—42no—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D312yes—
147#638—22no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3#6711yescharacteristic
150C4⋊(C3⋊D3)C4×D312yescomplemented
151C4×C3⋊D3—24no—
152C4×C3⋊D3—22no—
153C4×C3⋊D3C4×D311yescharacteristic
154C4×C3⋊D3C4×D311yescharacteristic
155C4×C3⋊D3D1211yescharacteristic
156C4×C3⋊D3D1211yescharacteristic
157#634C4×D312yescomplemented
158C3×#67—62no—
159C3×#67—62no—
160C3×#67—122no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—62no—
165C3×D12—32no—
166C3×D12—32no—
167C3×D12—62no—
168C3×C4×D3—32no—
169C3×C4×D3—32no—
170C3×C4×D3—32no—
171C3×C4×D3—32no—
172C3×C4×D3—61no—
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—68no—
177C3×C4×D3—68no—
178C3×C4×D3—62no—
179C3×C4×D3—62no—
180C3×C4×D3—122no—
181C3×Dic6—32no—
182C3×Dic6—32no—
183C3×Dic6—62no—
184#627—62no—
185#627—61no—
186#626—62no—
187#626—62no—
188#626—62no—
189#625—62no—
190#625—61no—
191#624—62no—
192#624—61no—
193D3×Dic3—62no—
194D3×Dic3—62no—
195D3×Dic3—62no—
196C3⋊(C3⋊C8)—22no—
197C3×C3⋊C8—62no—
198C3×C3⋊C8—62no—
199C3×C3⋊C8—122no—
200C2×#104—271noSylow
201C2×C33—21no—
202C2×C33C2×C22⋊C411yescharacteristic
203C3×C3⋊D3—21no—
204C3×C3⋊D3—42no—
205C3×C3⋊D3#10412yescomplemented
206C3×#41—92no—
207C22×C3×C4—91no—
208C22×C3×C4—181no—
209C2×#67—181no—
210#239—92no—
211D3×Q8—92no—
212#237—92no—
213#237—182no—
214D3×D4—92no—
215D3×D4—182no—
216#235—32no—
217#235—32no—
218#235—62no—
219#235—92no—
220#235—182no—
221#235—181no—
222#235—181no—
223C2×D12—91no—
224C2×C4×D3—32no—
225C2×C4×D3—61no—
226C2×C4×D3—62no—
227C2×C4×D3—62no—
228C2×C4×D3—182no—
229C2×C4×D3—181no—
230C2×Dic6—91no—
231C3×M16—92no—
232C3×C42—21no—
233C3×C42—22no—
234C3×C42—41no—
235C3×C42—91no—
236C3×C42—181no—
237C3×C42D3211yescharacteristic, complemented
238C3×C42D3212yescomplemented
239C4×Dic3—32no—
240C4×Dic3—61no—
241C4×Dic3—62no—
242C4×Dic3—62no—
243#208—32no—
244#208—32no—
245#208—62no—
246#208—181no—
247C2×C3⋊C8—181no—
248ΓL(1,𝔽25)—182no—
249D3×C8—182no—
250C22×C32—22no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32D3×D412yes—
254C22×C32C2×#6711yescharacteristic
255C2×C3⋊D3—22no—
256C2×C3⋊D3—22no—
257C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
258C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
259C2×C3×D3—32no—
260C2×C3×D3—32no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—64no—
264C2×C3×D3—64no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—122no—
268D32—124no—
269D32—122no—
270C32×C4—22no—
271C32×C4—24no—
272C32×C4—24no—
273C32×C4—22no—
274C32×C4—21no—
275C32×C4—22no—
276C32×C4—21no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—44no—
280C32×C4—44no—
281C32×C4C2×C4×D312yes—
282C32×C4C2×C4×D311yescharacteristic
283C32×C4C2×D1211yescharacteristic
284C32×C4D3×D412yes—
285C3⋊Dic3—22no—
286C3⋊Dic3—22no—
287C3⋊Dic3(C2×D3)⋊C411yescharacteristic
288C3⋊Dic3(C2×D3)⋊C411yescharacteristic
289C3×Dic3—32no—
290C3×Dic3—32no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3—64no—
294C3×Dic3—64no—
295C3×Dic3—62no—
296C3×Dic3—62no—
297C3×Dic3—122no—
298C2×#41—271no—
299C2×M16—271no—
300C2×C42—91no—
301#104—92no—
302#104—272no—
303C33C2×#10411yescharacteristic, complemented, Sylow
304C23×C3—91no—
305C22×D3—32no—
306C22×D3—61no—
307C22×D3—182no—
308C22×D3—181no—
309C3×Q8—92no—
310C3×D4—92no—
311C3×D4—182no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—91no—
321C2×C3×C4—91no—
322C2×C3×C4—91no—
323C2×C3×C4—91no—
324C2×C3×C4—91no—
325C2×C3×C4—184no—
326C2×C3×C4—181no—
327C2×C3×C4—182no—
328C2×C3×C4C2×D3212yes—
329C2×C3×C4C2×D3211yescharacteristic
330#67—62no—
331#67—62no—
332#67—122no—
333#67—182no—
334#67—182no—
335#67—182no—
336C2×Dic3—32no—
337C2×Dic3—61no—
338C2×Dic3—62no—
339C2×Dic3—62no—
340C2×Dic3—182no—
341C2×Dic3—181no—
342D12—32no—
343D12—32no—
344D12—62no—
345D12—181no—
346D12—182no—
347C4×D3—32no—
348C4×D3—32no—
349C4×D3—32no—
350C4×D3—32no—
351C4×D3—61no—
352C4×D3—61no—
353C4×D3—61no—
354C4×D3—61no—
355C4×D3—68no—
356C4×D3—68no—
357C4×D3—62no—
358C4×D3—62no—
359C4×D3—122no—
360C4×D3—182no—
361C4×D3—182no—
362C4×D3—182no—
363Dic6—32no—
364Dic6—32no—
365Dic6—62no—
366Dic6—182no—
367Dic6—181no—
368C3×C8—182no—
369C3⋊C8—62no—
370C3⋊C8—62no—
371C3⋊C8—122no—
372C2×C32—21no—
373C2×C32—22no—
374C2×C32—22no—
375C2×C32—22no—
376C2×C32—21no—
377C2×C32—41no—
378C2×C32—42no—
379C2×C32—42no—
380C2×C32D3×C22⋊C412yes—
381C2×C32C2×(C2×D3)⋊C411yescharacteristic
382C3⋊D3—21no—
383C3⋊D3—42no—
384C3⋊D3#80512yescomplemented
385C3×D3—34no—
386C3×D3—62no—
387C3×D3—62no—
388C3×D3—62no—
389C3×D3—122no—
390C3×D3—122no—
391C3×D3—242no—
392#41—92no—
393#41—272no—
394#41—541no—
395#41—541no—
396C2×Q8—271no—
397C2×D4—271no—
398C2×D4—541no—
399C22×C4—91no—
400C22×C4—181no—
401C22×C4—541no—
402M16—92no—
403M16—541no—
404C2×C8—541no—
405C42—91no—
406C42—181no—
407C42#111311yescharacteristic, complemented
408C22×C3—21no—
409C22×C3—22no—
410C22×C3—41no—
411C22×C3—91no—
412C22×C3—91no—
413C22×C3—182no—
414C22×C3—182no—
415C22×C3D3×#6712yes—
416C22×C3#381111yescharacteristic
417C2×D3—32no—
418C2×D3—32no—
419C2×D3—61no—
420C2×D3—61no—
421C2×D3—64no—
422C2×D3—64no—
423C2×D3—62no—
424C2×D3—62no—
425C2×D3—122no—
426C2×D3—364no—
427C2×D3—362no—
428C3×C4—24no—
429C3×C4—24no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—21no—
433C3×C4—22no—
434C3×C4—21no—
435C3×C4—44no—
436C3×C4—44no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—91no—
440C3×C4—91no—
441C3×C4—182no—
442C3×C4—182no—
443C3×C4C4×D3212yes—
444C3×C4C4×D3211yescharacteristic
445C3×C4D3×D1212yes—
446C3×C4#380211yescharacteristic
447Dic3—32no—
448Dic3—32no—
449Dic3—61no—
450Dic3—61no—
451Dic3—64no—
452Dic3—64no—
453Dic3—62no—
454Dic3—62no—
455Dic3—122no—
456C32—22no—
457C32—21no—
458C32—41no—
459C32D3×#10412yescomplemented
460C32C2×#80511yescharacteristic, complemented
461C23—91no—
462C23—541no—
463Q8—92no—
464Q8—541no—
465D4—92no—
466D4—182no—
467D4—541no—
468D4—542no—
469C2×C4—21no—
470C2×C4—91no—
471C2×C4—91no—
472C2×C4—91no—
473C2×C4—91no—
474C2×C4—91no—
475C2×C4—184no—
476C2×C4—181no—
477C2×C4—182no—
478C2×C4—542no—
479C2×C4—541no—
480C2×C4C2×#111311yescharacteristic
481C8—182no—
482C2×C3—22no—
483C2×C3—22no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—22no—
487C2×C3—42no—
488C2×C3—42no—
489C2×C3—41no—
490C2×C3—92no—
491C2×C3—181no—
492C2×C3—362no—
493C2×C3D3×(C2×D3)⋊C412yes—
494C2×C3#6392111yescharacteristic
495D3—34no—
496D3—62no—
497D3—62no—
498D3—62no—
499D3—122no—
500D3—122no—
501D3—242no—
502C22—91no—
503C22—91no—
504C22—182no—
505C22—182no—
506C22—1082no—
507C22#8904911yescharacteristic
508C4—22no—
509C4—91no—
510C4—91no—
511C4—182no—
512C4—182no—
513C4C4×#111311yescharacteristic
514C4C4⋊#111311yescharacteristic
515C3—21no—
516C3—22no—
517C3—41no—
518C3D3×#80512yescomplemented
519C3#1059039311yescharacteristic, complemented
520C2—21no—
521C2—92no—
522C2—181no—
523C2—362no—
524C2#1172370011yescharacteristic
subgroupG/H
centerC4C4⋊#1113
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC33×C42C22
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
cosocleC2×C33×C4C23
hypercenterC42#1113
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C42×C3⋊D3C2semidirect
3. #11723656C2semidirect
4. C3×#63632C2semidirect
5. #11722077C2semidirect
13. C4⋊#1113C4semidirect
16. #89041C4semidirect
17. #89040C4semidirect
18. C33×C42C22semidirect
20. C3×C4×C3⋊Dic3C22semidirect
21. C3×#3748C22semidirect
26. C42×C3⋊D3D3semidirect
31. #63632D3semidirect
45. C3×C4⋊(C3⋊D3)C2×C4semidirect
52. C3×#634C2×C4semidirect
89. C32×C42C2×D3semidirect
90. C32×C42C2×D3semidirect
92. C4×C3⋊Dic3C2×D3semidirect
93. #3748C2×D3semidirect
150. C4⋊(C3⋊D3)C4×D3semidirect
157. #634C4×D3semidirect
205. C3×C3⋊D3#104semidirect
237. C3×C42D32semidirect
238. C3×C42D32semidirect
303. C33C2×#104semidirect
384. C3⋊D3#805semidirect
407. C42#1113semidirect
459. C32D3×#104semidirect
460. C32C2×#805semidirect
518. C3D3×#805semidirect
519. C3#10590393semidirect

Maximal subgroups

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

subgroupindex
#117246182
C3×C42×C3⋊D32
#117236562
C3×#636322
#117220772
#105903933
D3×#8053