Finite Group Databasebeta

← #13161770

Subgroups — #13161770

Order 1728.

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

normal subgroups only

This group has 856 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×M16×C3⋊D3C211yesmaximal, characteristic, complemented
3C4×#6294C211yesmaximal, characteristic, complemented
4C3×#63652C212yesmaximal, complemented
5#11722077C212yesmaximal, complemented
6#10591008—31nomaximal
7D3×#825—32nomaximal
8C3×#3828C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10#89049—21no—
11#89046—21no—
12C2×#6294—21no—
13C4⋊#1113C411yescharacteristic, complemented
14C4×#1113—21no—
15#89042—21no—
16#89041C412yescomplemented
17#89040C411yescharacteristic, complemented
18C33×M16C2211yescharacteristic, complemented
19C3×C4×C3⋊Dic3C2212yescomplemented
20C3×#3748C2211yescharacteristic, complemented
21C3×#3743—21no—
22C3×C8×C3⋊D3—21no—
23#88813—21no—
24#64306—31no—
25D3×#235—32no—
26M16×C3⋊D3D311yescharacteristic, complemented
27C3×D3×M16—32no—
28C3×D3×M16—61no—
29C4×#624—31no—
30C4×D3×Dic3—32no—
31#63652D312yescomplemented
32C3×#825—32no—
33C3×#825—32no—
34C3×#825—62no—
35#63570—32no—
36#63569—32no—
37#63568—32no—
38C22×C3×C3⋊D3D411yescharacteristic
39C2×#1113—21no—
40C2×C33×C4C2311yescharacteristic
41C3×#638—22no—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4⋊(C3⋊D3)C2×C412yescomplemented
45C3×C4×C3⋊D3—22no—
46C3×C4×C3⋊D3C2×C411yescharacteristic
47C3×C4×C3⋊D3C2×C411yescharacteristic
48C3×C4×C3⋊D3D411yescharacteristic
49C3×C4×C3⋊D3D411yescharacteristic
50C3×#634C2×C412yescomplemented
51#6296—21no—
52#6295—22no—
53#6295—21no—
54#6294—24no—
55#6294—21no—
56#6294—22no—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C2×#825—91no—
60D3×#104—92no—
61#3828C2×D312yes—
62C2×C4×C3⋊D3C2×D311yescharacteristic
63C3×#235—32no—
64C3×#235—32no—
65C3×#235—62no—
66C2×C3×C4×D3—32no—
67C2×C3×C4×D3—61no—
68#3811—61no—
69D3×#67—62no—
70C2×#624—61no—
71#3805—61no—
72#3804—62no—
73C2×D3×Dic3—62no—
74#3802—31no—
75D3×D12—32no—
76C4×D32—62no—
77C4×D32—61no—
78#3799—32no—
79#3798—62no—
80#3798—61no—
81#3797—31no—
82#3796—32no—
83#3795—32no—
84D3×Dic6—32no—
85C32×M16—21no—
86C32×M16—22no—
87C32×M16—41no—
88C32×M16C2×D312yescomplemented
89C32×M16C2×D311yescharacteristic, complemented
90C4×C3⋊Dic3C2×D312yescomplemented
91#3748C2×D311yescharacteristic, complemented
92#3743—21no—
93C8×C3⋊D3—21no—
94C3×C4×Dic3—32no—
95C3×C4×Dic3—32no—
96C3×C4×Dic3—62no—
97C3×#208—32no—
98C3×#208—61no—
99C3×ΓL(1,𝔽25)—62no—
100C3×ΓL(1,𝔽25)—62no—
101C3×D3×C8—62no—
102C3×D3×C8—62no—
103Dic32—62no—
104Dic32—61no—
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×C4C22×C411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3×C3⋊Dic3—24no—
114C3×C3⋊Dic3—22no—
115C3×C3⋊Dic3C22⋊C411yescharacteristic
116C3×C3⋊Dic3C22⋊C411yescharacteristic
117D3×#41—92no—
118C2×#235—91no—
119C2×C3×M16—91no—
120C2×C4×Dic3—91no—
121D3×M16—32no—
122D3×M16—61no—
123C42×D3—92no—
124C3×#104—92no—
125#837—92no—
126#825—32no—
127#825—32no—
128#825—62no—
129#825—92no—
130#805—92no—
131C22×C3⋊D3D1211yescharacteristic
132C22×C3×D3—32no—
133C22×C3×D3—61no—
134C2×D32—62no—
135C2×D32—61no—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—41no—
139C2×C32×C4C22×D312yes—
140C2×C32×C4C22×D311yescharacteristic
141#638—22no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3D1211yescharacteristic
144C4⋊(C3⋊D3)C4×D312yescomplemented
145C4×C3⋊D3—22no—
146C4×C3⋊D3C4×D311yescharacteristic
147C4×C3⋊D3C4×D311yescharacteristic
148C4×C3⋊D3#6711yescharacteristic
149C4×C3⋊D3#6711yescharacteristic
150#634C4×D312yescomplemented
151C3×#67—62no—
152C3×#67—62no—
153C3×#67—122no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—122no—
159C3×D12—32no—
160C3×D12—32no—
161C3×D12—62no—
162C3×C4×D3—32no—
163C3×C4×D3—32no—
164C3×C4×D3—32no—
165C3×C4×D3—32no—
166C3×C4×D3—61no—
167C3×C4×D3—61no—
168C3×C4×D3—61no—
169C3×C4×D3—61no—
170C3×C4×D3—62no—
171C3×C4×D3—62no—
172C3×C4×D3—122no—
173C3×Dic6—32no—
174C3×Dic6—32no—
175C3×Dic6—62no—
176#627—62no—
177#627—61no—
178#626—62no—
179#626—62no—
180#626—62no—
181#625—62no—
182#625—61no—
183#624—62no—
184#624—64no—
185#624—61no—
186D3×Dic3—62no—
187D3×Dic3—62no—
188D3×Dic3—68no—
189D3×Dic3—62no—
190C32×C8—21no—
191C32×C8—22no—
192C32×C8—22no—
193C32×C8—42no—
194C32×C8—42no—
195C3⋊(C3⋊C8)—21no—
196C3×C3⋊C8—62no—
197C3×C3⋊C8—62no—
198C2×#104—271noSylow
199C2×C33—21no—
200C2×C33C2×C22⋊C411yescharacteristic
201C3×C3⋊D3—21no—
202C3×C3⋊D3—42no—
203C3×C3⋊D3#10412yescomplemented
204C3×#41—92no—
205C22×C3×C4—91no—
206C2×#67—181no—
207C22×Dic3—181no—
208#239—92no—
209D3×Q8—92no—
210#237—92no—
211#237—182no—
212D3×D4—92no—
213D3×D4—182no—
214#235—32no—
215#235—32no—
216#235—62no—
217#235—92no—
218#235—182no—
219#235—181no—
220#235—181no—
221C2×D12—91no—
222C2×C4×D3—32no—
223C2×C4×D3—61no—
224C2×C4×D3—182no—
225C2×C4×D3—181no—
226C2×C4×D3—182no—
227C2×Dic6—91no—
228C3×M16—21no—
229C3×M16—22no—
230C3×M16—41no—
231C3×M16—91no—
232C3×M16—181no—
233C3×M16D3211yescharacteristic, complemented
234C3×M16D3212yescomplemented
235C2×C3×C8—181no—
236C3×C42—92no—
237C4×Dic3—32no—
238C4×Dic3—32no—
239C4×Dic3—62no—
240C4×Dic3—182no—
241C4×Dic3—181no—
242#208—32no—
243#208—61no—
244ΓL(1,𝔽25)—62no—
245ΓL(1,𝔽25)—62no—
246D3×C8—62no—
247D3×C8—62no—
248C22×C32—22no—
249C22×C32—21no—
250C22×C32—41no—
251C22×C32C2×D1211yescharacteristic
252C22×C32D3×D412yes—
253C2×C3⋊D3—22no—
254C2×C3⋊D3—22no—
255C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
256C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
257C2×C3×D3—32no—
258C2×C3×D3—32no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—64no—
262C2×C3×D3—62no—
263C2×C3×D3—62no—
264C2×C3×D3—122no—
265C2×C3×D3—122no—
266D32—124no—
267D32—122no—
268C32×C4—22no—
269C32×C4—21no—
270C32×C4—22no—
271C32×C4—21no—
272C32×C4—41no—
273C32×C4—41no—
274C32×C4C2×C4×D312yes—
275C32×C4C2×C4×D311yescharacteristic
276C32×C4D3×D412yes—
277C32×C4C2×#6711yescharacteristic
278C3⋊Dic3—24no—
279C3⋊Dic3—22no—
280C3⋊Dic3(C2×D3)⋊C411yescharacteristic
281C3⋊Dic3(C2×D3)⋊C411yescharacteristic
282C3×Dic3—32no—
283C3×Dic3—32no—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C3×Dic3—64no—
287C3×Dic3—64no—
288C3×Dic3—62no—
289C3×Dic3—62no—
290C3×Dic3—122no—
291C3×Dic3—124no—
292C2×#41—271no—
293C2×M16—91no—
294C2×C42—271no—
295#104—92no—
296#104—272no—
297C33C2×#10411yescharacteristic, complemented, Sylow
298C23×C3—91no—
299C22×D3—32no—
300C22×D3—61no—
301C22×D3—182no—
302C22×D3—181no—
303C3×Q8—92no—
304C3×D4—92no—
305C3×D4—182no—
306C2×C3×C4—21no—
307C2×C3×C4—22no—
308C2×C3×C4—41no—
309C2×C3×C4—91no—
310C2×C3×C4—91no—
311C2×C3×C4—91no—
312C2×C3×C4—91no—
313C2×C3×C4—91no—
314C2×C3×C4—182no—
315C2×C3×C4—182no—
316C2×C3×C4C2×D3212yes—
317C2×C3×C4C2×D3211yescharacteristic
318#67—62no—
319#67—62no—
320#67—122no—
321#67—182no—
322#67—182no—
323#67—182no—
324C2×Dic3—32no—
325C2×Dic3—61no—
326C2×Dic3—62no—
327C2×Dic3—62no—
328C2×Dic3—122no—
329C2×Dic3—182no—
330C2×Dic3—181no—
331C2×Dic3—184no—
332D12—32no—
333D12—32no—
334D12—62no—
335D12—181no—
336D12—182no—
337C4×D3—32no—
338C4×D3—32no—
339C4×D3—32no—
340C4×D3—32no—
341C4×D3—61no—
342C4×D3—61no—
343C4×D3—61no—
344C4×D3—61no—
345C4×D3—62no—
346C4×D3—62no—
347C4×D3—122no—
348C4×D3—182no—
349C4×D3—182no—
350C4×D3—182no—
351C4×D3—188no—
352Dic6—32no—
353Dic6—32no—
354Dic6—62no—
355Dic6—182no—
356Dic6—181no—
357C3×C8—22no—
358C3×C8—22no—
359C3×C8—21no—
360C3×C8—42no—
361C3×C8—42no—
362C3×C8—181no—
363C3⋊C8—62no—
364C3⋊C8—62no—
365C2×C32—21no—
366C2×C32—22no—
367C2×C32—22no—
368C2×C32—21no—
369C2×C32—41no—
370C2×C32—42no—
371C2×C32—41no—
372C2×C32—81no—
373C2×C32D3×C22⋊C412yes—
374C2×C32C2×(C2×D3)⋊C411yescharacteristic
375C3⋊D3—21no—
376C3⋊D3—42no—
377C3⋊D3#82512yescomplemented
378C3×D3—34no—
379C3×D3—62no—
380C3×D3—62no—
381C3×D3—121no—
382C3×D3—122no—
383C3×D3—122no—
384C3×D3—242no—
385#41—92no—
386#41—272no—
387#41—541no—
388#41—541no—
389C2×Q8—271no—
390C2×D4—271no—
391C2×D4—541no—
392C22×C4—91no—
393C22×C4—541no—
394C22×C4—541no—
395M16—91no—
396M16—181no—
397M16#111311yescharacteristic, complemented
398C2×C8—181no—
399C42—92no—
400C42—541no—
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—91no—
405C22×C3—91no—
406C22×C3—182no—
407C22×C3—182no—
408C22×C3D3×D1212yes—
409C22×C3#380211yescharacteristic
410C2×D3—32no—
411C2×D3—32no—
412C2×D3—61no—
413C2×D3—61no—
414C2×D3—64no—
415C2×D3—62no—
416C2×D3—62no—
417C2×D3—122no—
418C2×D3—122no—
419C2×D3—364no—
420C2×D3—362no—
421C3×C4—22no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—22no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—91no—
428C3×C4—91no—
429C3×C4—184no—
430C3×C4—182no—
431C3×C4C4×D3212yes—
432C3×C4C4×D3211yescharacteristic
433C3×C4D3×#6712yes—
434C3×C4#381111yescharacteristic
435Dic3—32no—
436Dic3—32no—
437Dic3—61no—
438Dic3—61no—
439Dic3—64no—
440Dic3—64no—
441Dic3—62no—
442Dic3—62no—
443Dic3—122no—
444Dic3—124no—
445C32—22no—
446C32—21no—
447C32—41no—
448C32D3×#10412yescomplemented
449C32C2×#82511yescharacteristic, complemented
450C23—91no—
451C23—541no—
452Q8—92no—
453Q8—541no—
454D4—92no—
455D4—182no—
456D4—541no—
457D4—542no—
458C2×C4—91no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—182no—
464C2×C4—182no—
465C2×C4—542no—
466C2×C4—541no—
467C2×C4—544no—
468C2×C4C2×#111311yescharacteristic
469C8—21no—
470C8—181no—
471C2×C3—22no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—22no—
475C2×C3—42no—
476C2×C3—41no—
477C2×C3—41no—
478C2×C3—81no—
479C2×C3—92no—
480C2×C3—181no—
481C2×C3—362no—
482C2×C3D3×(C2×D3)⋊C412yes—
483C2×C3#6392111yescharacteristic
484D3—34no—
485D3—62no—
486D3—62no—
487D3—121no—
488D3—122no—
489D3—122no—
490D3—242no—
491C22—91no—
492C22—91no—
493C22—182no—
494C22—182no—
495C22—1082no—
496C22C4⋊#111311yescharacteristic
497C4—91no—
498C4—91no—
499C4—184no—
500C4—182no—
501C4C4×#111311yescharacteristic
502C4#8904911yescharacteristic
503C3—21no—
504C3—22no—
505C3—41no—
506C3D3×#82512yescomplemented
507C3#1059100811yescharacteristic, complemented
508C2—21no—
509C2—92no—
510C2—181no—
511C2—362no—
512C2#1172370011yescharacteristic
subgroupG/H
centerC4#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC33×M16C22
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
cosocleC2×C33×C4C23
hypercenterM16#1113
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C3×M16×C3⋊D3C2semidirect
3. C4×#6294C2semidirect
4. C3×#63652C2semidirect
5. #11722077C2semidirect
13. C4⋊#1113C4semidirect
16. #89041C4semidirect
17. #89040C4semidirect
18. C33×M16C22semidirect
19. C3×C4×C3⋊Dic3C22semidirect
20. C3×#3748C22semidirect
26. M16×C3⋊D3D3semidirect
31. #63652D3semidirect
44. C3×C4⋊(C3⋊D3)C2×C4semidirect
50. C3×#634C2×C4semidirect
88. C32×M16C2×D3semidirect
89. C32×M16C2×D3semidirect
90. C4×C3⋊Dic3C2×D3semidirect
91. #3748C2×D3semidirect
144. C4⋊(C3⋊D3)C4×D3semidirect
150. #634C4×D3semidirect
203. C3×C3⋊D3#104semidirect
233. C3×M16D32semidirect
234. C3×M16D32semidirect
297. C33C2×#104semidirect
377. C3⋊D3#825semidirect
397. M16#1113semidirect
448. C32D3×#104semidirect
449. C32C2×#825semidirect
506. C3D3×#825semidirect
507. C3#10591008semidirect

Maximal subgroups

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

subgroupindex
#117246182
C3×M16×C3⋊D32
C4×#62942
C3×#636522
#117220772
#105910083
D3×#8253