Finite Group Databasebeta

← #13161768

Subgroups — #13161768

Order 1728.

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

normal subgroups only

This group has 924 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
1C2×C4⋊#1113C211yesmaximal, characteristic
2C3×M16×C3⋊D3C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4C3×#63650C212yesmaximal, complemented
5#11722073C212yesmaximal, complemented
6#10591001—31nomaximal
7D3×#823—32nomaximal
8C22×#1113C412yes—
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×#6295C412yes—
12C4⋊#1113—21no—
13C4⋊#1113—21no—
14C4⋊#1113—21no—
15C4⋊#1113—21no—
16C33×M16C2211yescharacteristic, complemented
17C3×#3748C2211yescharacteristic, complemented
18C3×#3748C2212yescomplemented
19C3×#3743—21no—
20C3×C8×C3⋊D3—21no—
21#88809—21no—
22#88808—21no—
23C2×#3802—31no—
24C2×D3×D12—32no—
25M16×C3⋊D3D311yescharacteristic, complemented
26C3×D3×M16—32no—
27C3×D3×M16—61no—
28#63818—31no—
29D3×#208—32no—
30#63650D312yescomplemented
31C3×#823—32no—
32C3×#823—32no—
33C3×#823—62no—
34#63559—32no—
35#63558—32no—
36#63558—32no—
37C22×C3×C3⋊D3C2×C411yescharacteristic
38C22×C3×C3⋊D3C2×C414yes—
39C2×#1113—22no—
40C2×#1113—22no—
41C2×#1113—44no—
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C3×C4⋊(C3⋊D3)—22no—
45C3×C4⋊(C3⋊D3)—22no—
46C3×C4×C3⋊D3D411yescharacteristic
47C3×C4×C3⋊D3D411yescharacteristic
48C3×C4×C3⋊D3D411yescharacteristic
49C3×C4×C3⋊D3D411yescharacteristic
50#6295—22no—
51#6295—22no—
52C33×C8—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C3×C3⋊(C3⋊C8)—22no—
55C2×#823—91no—
56D3×#100—92no—
57C22×D32—34no—
58C22×D32—32no—
59C2×C4⋊(C3⋊D3)C2×D312yes—
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×C3×D12—32no—
62C2×C3×D12—32no—
63C2×C3×D12—62no—
64C2×C3×C4×D3—32no—
65C2×C3×C4×D3—61no—
66C2×#626—34no—
67C2×#625—32no—
68#3802—61no—
69#3802—61no—
70#3802—61no—
71#3802—61no—
72D3×D12—62no—
73D3×D12—62no—
74D3×D12—62no—
75D3×D12—62no—
76C32×M16—21no—
77C32×M16—22no—
78C32×M16—41no—
79C32×M16C2×D312yescomplemented
80C32×M16C2×D311yescharacteristic, complemented
81#3748C2×D312yescomplemented
82#3748C2×D311yescharacteristic, complemented
83#3743—21no—
84C8×C3⋊D3—21no—
85C3×#208—32no—
86C3×#208—32no—
87C3×#208—32no—
88C3×#208—61no—
89C3×#208—62no—
90C3×ΓL(1,𝔽25)—62no—
91C3×ΓL(1,𝔽25)—62no—
92C3×D3×C8—62no—
93C3×D3×C8—62no—
94#3712—61no—
95#3711—62no—
96#3710—61no—
97D3×C3⋊C8—62no—
98C22×C33C22×C411yescharacteristic
99C2×C3×C3⋊D3—22no—
100C2×C3×C3⋊D3—24no—
101C2×C3×C3⋊D3—44no—
102C2×C3×C3⋊D3C22⋊C411yescharacteristic
103C2×C3×C3⋊D3C22⋊C411yescharacteristic
104#1113—44no—
105#1113—44no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3C22⋊C411yescharacteristic
109C3×C3⋊Dic3C22⋊C411yescharacteristic
110C2×D3×D4—92no—
111C22×D12—91no—
112C2×C3×M16—91no—
113C2×#208—91no—
114D3×M16—32no—
115D3×M16—61no—
116D3×M16—92no—
117C3×#100—92no—
118#833—92no—
119#823—32no—
120#823—32no—
121#823—62no—
122#823—92no—
123#823—92no—
124C22×C3⋊D3C4×D311yescharacteristic
125C22×C3⋊D3C4×D314yes—
126C22×C3×D3—32no—
127C22×C3×D3—34no—
128C22×C3×D3—34no—
129C22×C3×D3—61no—
130C22×C3×D3—64no—
131C2×D32—64no—
132C2×D32—64no—
133C2×D32—62no—
134C2×D32—62no—
135C2×D32—128no—
136C2×D32—124no—
137C2×C32×C4—21no—
138C2×C32×C4—22no—
139C2×C32×C4—41no—
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D312yes—
142C2×C3⋊Dic3C4×D311yescharacteristic
143C4⋊(C3⋊D3)—22no—
144C4⋊(C3⋊D3)—22no—
145C4×C3⋊D3D1211yescharacteristic
146C4×C3⋊D3D1211yescharacteristic
147C4×C3⋊D3#6711yescharacteristic
148C4×C3⋊D3#6711yescharacteristic
149C2×C3×Dic3—32no—
150C2×C3×Dic3—61no—
151C3×D12—62no—
152C3×D12—62no—
153C3×D12—62no—
154C3×D12—62no—
155C3×D12—122no—
156C3×D12—122no—
157C3×C4×D3—32no—
158C3×C4×D3—32no—
159C3×C4×D3—32no—
160C3×C4×D3—32no—
161C3×C4×D3—61no—
162C3×C4×D3—61no—
163C3×C4×D3—61no—
164C3×C4×D3—61no—
165#626—64no—
166#626—64no—
167#625—62no—
168#625—62no—
169C32×C8—21no—
170C32×C8—22no—
171C32×C8—22no—
172C32×C8—42no—
173C32×C8—42no—
174C3⋊(C3⋊C8)—21no—
175C3⋊(C3⋊C8)—22no—
176C3×C3⋊C8—62no—
177C3×C3⋊C8—62no—
178C3×C3⋊C8—62no—
179C3×C3⋊C8—62no—
180C3×C3⋊C8—122no—
181C2×#100—271noSylow
182C2×C33—21no—
183C2×C33C2×C22⋊C411yescharacteristic
184C3×C3⋊D3—21no—
185C3×C3⋊D3—44no—
186C3×C3⋊D3#10012yescomplemented
187C23×D3—92no—
188C23×D3—94no—
189C2×C3×D4—92no—
190C22×C3×C4—91no—
191C2×#67—94no—
192D3×D4—182no—
193D3×D4—182no—
194D3×D4—182no—
195D3×D4—182no—
196C2×D12—32no—
197C2×D12—32no—
198C2×D12—62no—
199C2×D12—92no—
200C2×D12—181no—
201C2×D12—181no—
202C2×D12—181no—
203C2×D12—181no—
204C2×C4×D3—32no—
205C2×C4×D3—61no—
206C3×M16—21no—
207C3×M16—22no—
208C3×M16—41no—
209C3×M16—91no—
210C3×M16—92no—
211C3×M16—181no—
212C3×M16D3211yescharacteristic, complemented
213C3×M16D3212yescomplemented
214C2×C3×C8—181no—
215#208—32no—
216#208—32no—
217#208—32no—
218#208—61no—
219#208—62no—
220#208—181no—
221C2×C3⋊C8—181no—
222ΓL(1,𝔽25)—62no—
223ΓL(1,𝔽25)—62no—
224ΓL(1,𝔽25)—182no—
225D3×C8—62no—
226D3×C8—62no—
227D3×C8—182no—
228C22×C32—21no—
229C22×C32—22no—
230C22×C32—41no—
231C22×C32C2×C4×D312yes—
232C22×C32C2×C4×D311yescharacteristic
233C2×C3⋊D3—22no—
234C2×C3⋊D3—24no—
235C2×C3⋊D3—44no—
236C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
237C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
238C2×C3×D3—32no—
239C2×C3×D3—32no—
240C2×C3×D3—61no—
241C2×C3×D3—61no—
242C2×C3×D3—64no—
243C2×C3×D3—64no—
244C2×C3×D3—64no—
245C2×C3×D3—122no—
246C2×C3×D3—124no—
247C2×C3×D3—124no—
248C2×C3×D3—124no—
249C2×C3×D3—244no—
250D32—128no—
251D32—128no—
252D32—124no—
253D32—124no—
254C32×C4—22no—
255C32×C4—21no—
256C32×C4—22no—
257C32×C4—21no—
258C32×C4—41no—
259C32×C4—41no—
260C32×C4C2×D1211yescharacteristic
261C32×C4D3×D412yes—
262C32×C4D3×D412yes—
263C32×C4C2×#6711yescharacteristic
264C3⋊Dic3(C2×D3)⋊C411yescharacteristic
265C3⋊Dic3(C2×D3)⋊C411yescharacteristic
266C3×Dic3—32no—
267C3×Dic3—32no—
268C3×Dic3—61no—
269C3×Dic3—61no—
270C22×D4—271no—
271C2×M16—91no—
272C2×M16—271no—
273#100—92no—
274#100—272no—
275C33C2×#10011yescharacteristic, complemented, Sylow
276C23×C3—91no—
277C23×C3—94no—
278C22×D3—32no—
279C22×D3—34no—
280C22×D3—34no—
281C22×D3—61no—
282C22×D3—64no—
283C22×D3—184no—
284C22×D3—184no—
285C22×D3—182no—
286C22×D3—182no—
287C22×D3—368no—
288C22×D3—364no—
289C3×D4—182no—
290C3×D4—182no—
291C2×C3×C4—21no—
292C2×C3×C4—22no—
293C2×C3×C4—41no—
294C2×C3×C4—91no—
295C2×C3×C4—91no—
296C2×C3×C4—91no—
297C2×C3×C4—91no—
298C2×C3×C4—91no—
299C2×C3×C4C2×D3212yes—
300C2×C3×C4C2×D3211yescharacteristic
301#67—184no—
302#67—184no—
303C2×Dic3—32no—
304C2×Dic3—61no—
305D12—62no—
306D12—62no—
307D12—62no—
308D12—62no—
309D12—122no—
310D12—122no—
311D12—182no—
312D12—182no—
313C4×D3—32no—
314C4×D3—32no—
315C4×D3—32no—
316C4×D3—32no—
317C4×D3—61no—
318C4×D3—61no—
319C4×D3—61no—
320C4×D3—61no—
321C3×C8—22no—
322C3×C8—22no—
323C3×C8—21no—
324C3×C8—42no—
325C3×C8—42no—
326C3×C8—182no—
327C3×C8—181no—
328C3⋊C8—62no—
329C3⋊C8—62no—
330C3⋊C8—62no—
331C3⋊C8—62no—
332C3⋊C8—122no—
333C2×C32—21no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32—42no—
340C2×C32—81no—
341C2×C32D3×C22⋊C412yes—
342C2×C32C2×(C2×D3)⋊C411yescharacteristic
343C3⋊D3—21no—
344C3⋊D3—44no—
345C3⋊D3#82312yescomplemented
346C3×D3—34no—
347C3×D3—62no—
348C3×D3—62no—
349C3×D3—121no—
350C3×D3—124no—
351C3×D3—124no—
352C3×D3—244no—
353C24—272no—
354C2×D4—92no—
355C2×D4—272no—
356C2×D4—541no—
357C2×D4—541no—
358C2×D4—541no—
359C2×D4—541no—
360C22×C4—91no—
361M16—92no—
362M16—91no—
363M16—181no—
364M16—541no—
365M16#111311yescharacteristic, complemented
366C2×C8—181no—
367C2×C8—541no—
368C22×C3—21no—
369C22×C3—22no—
370C22×C3—41no—
371C22×C3—91no—
372C22×C3—91no—
373C22×C3—182no—
374C22×C3—184no—
375C22×C3—364no—
376C22×C3C4×D3212yes—
377C22×C3C4×D3211yescharacteristic
378C2×D3—32no—
379C2×D3—32no—
380C2×D3—61no—
381C2×D3—61no—
382C2×D3—64no—
383C2×D3—64no—
384C2×D3—64no—
385C2×D3—122no—
386C2×D3—124no—
387C2×D3—124no—
388C2×D3—124no—
389C2×D3—244no—
390C2×D3—368no—
391C2×D3—368no—
392C2×D3—364no—
393C2×D3—364no—
394C3×C4—22no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—22no—
398C3×C4—41no—
399C3×C4—41no—
400C3×C4—91no—
401C3×C4—91no—
402C3×C4D3×D1212yes—
403C3×C4#380211yescharacteristic
404C3×C4D3×#6712yes—
405C3×C4#381111yescharacteristic
406Dic3—32no—
407Dic3—32no—
408Dic3—61no—
409Dic3—61no—
410C32—22no—
411C32—21no—
412C32—41no—
413C32D3×#10012yescomplemented
414C32C2×#82311yescharacteristic, complemented
415C23—91no—
416C23—94no—
417C23—542no—
418C23—542no—
419C23—1084no—
420D4—182no—
421D4—182no—
422D4—542no—
423D4—542no—
424C2×C4—91no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—91no—
429C2×C4C2×#111311yescharacteristic
430C8—21no—
431C8—182no—
432C8—181no—
433C2×C3—22no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—22no—
437C2×C3—41no—
438C2×C3—42no—
439C2×C3—41no—
440C2×C3—81no—
441C2×C3—92no—
442C2×C3—181no—
443C2×C3—364no—
444C2×C3D3×(C2×D3)⋊C412yes—
445C2×C3#6392111yescharacteristic
446D3—34no—
447D3—62no—
448D3—62no—
449D3—121no—
450D3—124no—
451D3—124no—
452D3—244no—
453C22—91no—
454C22—91no—
455C22—182no—
456C22—184no—
457C22—364no—
458C22—1084no—
459C22—1084no—
460C22C4×#111311yescharacteristic
461C4—91no—
462C4—91no—
463C4C4⋊#111311yescharacteristic
464C4#8904911yescharacteristic
465C3—21no—
466C3—22no—
467C3—41no—
468C3D3×#82312yescomplemented
469C3#1059100111yescharacteristic, complemented
470C2—21no—
471C2—92no—
472C2—181no—
473C2—364no—
474C2#1172370011yescharacteristic
subgroupG/H
centerC2#11723700
commutatorC22×C33C22×C4
FrattiniC2×C4C2×#1113
FittingC33×M16C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C4⋊#1113C2
cosocleC2×C33×C4C23
hypercenterM16#1113
hypocenterC33C2×#100
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C3×M16×C3⋊D3C2semidirect
3. #11723656C2semidirect
4. C3×#63650C2semidirect
5. #11722073C2semidirect
16. C33×M16C22semidirect
17. C3×#3748C22semidirect
18. C3×#3748C22semidirect
25. M16×C3⋊D3D3semidirect
30. #63650D3semidirect
79. C32×M16C2×D3semidirect
80. C32×M16C2×D3semidirect
81. #3748C2×D3semidirect
82. #3748C2×D3semidirect
186. C3×C3⋊D3#100semidirect
212. C3×M16D32semidirect
213. C3×M16D32semidirect
275. C33C2×#100semidirect
345. C3⋊D3#823semidirect
365. M16#1113semidirect
413. C32D3×#100semidirect
414. C32C2×#823semidirect
468. C3D3×#823semidirect
469. C3#10591001semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C3×M16×C3⋊D32
#117236562
C3×#636502
#117220732
#105910013
D3×#8233