Finite Group Databasebeta

← #13161988

Subgroups — #13161988

Order 1728.

934 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 143 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×C3×#3832C211yesmaximal, characteristic
2#11723689C211yesmaximal, characteristic, complemented
3C2×#88809C211yesmaximal, characteristic, complemented
4C3×#63662C212yesmaximal, complemented
5#11722075C212yesmaximal, complemented
6#10591554—31nomaximal
7#10591550—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C3×#3832—22no—
10C3×#3832C414yescomplemented
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×#6294—21no—
16C3×C4⋊(C3⋊Dic3)C2212yes—
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18#88815—21no—
19#88809—22no—
20C2×#3832D311yescharacteristic
21C2×C3×#239—32no—
22C2×C3×#239—61no—
23#63900—31no—
24D3×C4⋊Dic3—32no—
25C2×#3712—31no—
26C2×#3711—32no—
27#63662D312yescomplemented
28C3×#835—32no—
29C3×#835—32no—
30C3×#835—62no—
31#63564—32no—
32#63563—32no—
33#63562—32no—
34C22×C3×C3⋊D3—21no—
35C22×C3×C3⋊D3D411yescharacteristic
36C33×Q8—21no—
37C33×Q8C2×C412yescomplemented
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4⋊(C3⋊D3)—24no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)C2×C412yescomplemented
45C3×C4×C3⋊D3—24no—
46C3×C4×C3⋊D3C2×C412yes—
47C3×C4×C3⋊D3D412yes—
48#6294—42no—
49C3×C3⋊(C3⋊C8)—22no—
50#5232—91no—
51#4803—92no—
52C2×C32×Q8—21no—
53C2×C32×Q8—22no—
54C2×C32×Q8—41no—
55C2×C32×Q8C2×D312yes—
56C2×C32×Q8C2×D311yescharacteristic
57#3832—22no—
58#3832Dic314yescomplemented
59C2×C4⋊(C3⋊D3)—21no—
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C4×C3⋊D3—21no—
62C2×C4×C3⋊D3C2×D311yescharacteristic
63C3×#239—38no—
64C3×#239—64no—
65C3×#239—64no—
66C3×#239—64no—
67C2×C3×D12—32no—
68C2×C3×D12—62no—
69C2×C3×D12—61no—
70C2×C3×D12—62no—
71C2×C3×C4×D3—32no—
72C2×C3×C4×D3—62no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3—62no—
75C2×#624—61no—
76C2×D3×Dic3—62no—
77C4⋊(C3⋊Dic3)C2×D312yes—
78C2×C3⋊(C3⋊C8)C2×D312yes—
79C3×C4⋊Dic3—32no—
80C3×C4⋊Dic3—32no—
81C3×C4⋊Dic3—62no—
82C2×C3×C3⋊C8—32no—
83C2×C3×C3⋊C8—32no—
84C2×C3×C3⋊C8—62no—
85#3724—61no—
86Dic3⋊Dic3—62no—
87#3712—62no—
88#3711—64no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—22no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3C22⋊C412yes—
96C33×C4—22no—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3×C3⋊Dic3—42no—
100C3×C3⋊Dic3C22⋊C412yes—
101C2×C3×#41—91no—
102C2×#239—32no—
103C2×#239—61no—
104C2×C4⋊Dic3—91no—
105C2×#208—91no—
106C2×ΓL(1,𝔽25)—92no—
107D3×C4⋊C4—92no—
108C3×#103—92no—
109#835—32no—
110#835—32no—
111#835—62no—
112#832—92no—
113#821—92no—
114#809—92no—
115C22×C3⋊D3—21no—
116C22×C3⋊D3#6711yescharacteristic
117C22×C3×D3—32no—
118C22×C3×D3—61no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C32×Q8—21no—
122C32×Q8—22no—
123C32×Q8—22no—
124C32×Q8—24no—
125C32×Q8—22no—
126C32×Q8—42no—
127C32×Q8—42no—
128C32×Q8—42no—
129C32×Q8C4×D314yescomplemented
130C32×Q8C2×Dic312yescomplemented
131C2×C32×C4—21no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—21no—
136C2×C32×C4—41no—
137C2×C32×C4—42no—
138C2×C32×C4—42no—
139C2×C32×C4C22×D312yes—
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3#6711yescharacteristic
143C4⋊(C3⋊D3)—24no—
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)C2×Dic312yescomplemented
146C4×C3⋊D3—24no—
147C4×C3⋊D3C2×Dic312yes—
148C4×C3⋊D3#6712yes—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—122no—
154C3×D12—34no—
155C3×D12—62no—
156C3×D12—68no—
157C3×D12—68no—
158C3×D12—62no—
159C3×D12—62no—
160C3×C4×D3—34no—
161C3×C4×D3—34no—
162C3×C4×D3—62no—
163C3×C4×D3—62no—
164C3×C4×D3—68no—
165C3×C4×D3—68no—
166#624—122no—
167D3×Dic3—124no—
168C3⋊(C3⋊C8)—22no—
169C3×C3⋊C8—62no—
170C3×C3⋊C8—62no—
171C3×C3⋊C8—122no—
172#417—271noSylow
173C2×C33C2×C22⋊C411yescharacteristic
174C2×C33AGL(1,ℤ/8)11yescharacteristic
175C2×C33#13711yescharacteristic
176C3×C3⋊D3—22no—
177C3×C3⋊D3—42no—
178C3×#41—94no—
179C3×#41—182no—
180C2×C3×Q8—21no—
181C2×C3×Q8—22no—
182C2×C3×Q8—41no—
183C2×C3×Q8D3212yes—
184C2×C3×Q8D3211yescharacteristic
185C2×C3×D4—91no—
186C2×C3×D4—181no—
187C22×C3×C4—91no—
188C22×C3×C4—181no—
189C22×Dic3—181no—
190#239—38no—
191#239—64no—
192#239—64no—
193#239—64no—
194C2×D12—32no—
195C2×D12—61no—
196C2×D12—62no—
197C2×D12—62no—
198C2×C4×D3—32no—
199C2×C4×D3—61no—
200C2×C4×D3—62no—
201C2×C4×D3—62no—
202C2×C4×D3—182no—
203C2×C3×C8—92no—
204C3×C4⋊C4—92no—
205C4⋊Dic3—32no—
206C4⋊Dic3—32no—
207C4⋊Dic3—62no—
208C4⋊Dic3—181no—
209Dic3⋊C4—182no—
210#208—182no—
211C2×C3⋊C8—32no—
212C2×C3⋊C8—32no—
213C2×C3⋊C8—62no—
214ΓL(1,𝔽25)—184no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32D3×D412yes—
219C22×C32C2×#6711yescharacteristic
220C2×C3⋊D3—22no—
221C2×C3⋊D3—21no—
222C2×C3⋊D3—21no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3⋊D3#21712yes—
226C2×C3×D3—34no—
227C2×C3×D3—62no—
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—64no—
231C2×C3×D3—64no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C2×C3×D3—122no—
235C2×C3×D3—122no—
236C2×C3×D3—122no—
237C2×C3×D3—122no—
238C32×C4—22no—
239C32×C4—24no—
240C32×C4—24no—
241C32×C4—22no—
242C32×C4—21no—
243C32×C4—22no—
244C32×C4—21no—
245C32×C4—41no—
246C32×C4—41no—
247C32×C4—44no—
248C32×C4—44no—
249C32×C4C2×C4×D312yes—
250C32×C4D3×D412yes—
251C32×C4C22×Dic311yescharacteristic
252C32×C4C2×#6711yescharacteristic
253C3⋊Dic3—42no—
254C3⋊Dic3#21712yes—
255C3×Dic3—34no—
256C3×Dic3—62no—
257C3×Dic3—122no—
258C3×Dic3—122no—
259C3×Dic3—242no—
260C2×#41—91no—
261C2×M16—271no—
262C2×C4⋊C4—271no—
263#103—92no—
264#102—272no—
265C33#41711yescharacteristic, complemented, Sylow
266C23×C3—91no—
267C23×C3—181no—
268C22×D3—32no—
269C22×D3—61no—
270C22×D3—62no—
271C22×D3—62no—
272C3×Q8—22no—
273C3×Q8—22no—
274C3×Q8—21no—
275C3×Q8—22no—
276C3×Q8—24no—
277C3×Q8—42no—
278C3×Q8—42no—
279C3×Q8—42no—
280C3×Q8D3×Dic314yescomplemented
281C3×Q8#62412yescomplemented
282C3×D4—92no—
283C3×D4—184no—
284C3×D4—181no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—21no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—91no—
294C2×C3×C4—92no—
295C2×C3×C4—92no—
296C2×C3×C4—184no—
297C2×C3×C4—182no—
298C2×C3×C4C2×D3212yes—
299C2×C3×C4C2×D3211yescharacteristic
300C2×Dic3—32no—
301C2×Dic3—61no—
302C2×Dic3—62no—
303C2×Dic3—62no—
304C2×Dic3—122no—
305C2×Dic3—362no—
306D12—34no—
307D12—62no—
308D12—68no—
309D12—68no—
310D12—62no—
311D12—62no—
312C4×D3—34no—
313C4×D3—34no—
314C4×D3—62no—
315C4×D3—62no—
316C4×D3—68no—
317C4×D3—68no—
318C4×D3—364no—
319C3×C8—182no—
320C3⋊C8—62no—
321C3⋊C8—62no—
322C3⋊C8—122no—
323C2×C32—22no—
324C2×C32—21no—
325C2×C32—22no—
326C2×C32—21no—
327C2×C32—22no—
328C2×C32—21no—
329C2×C32—41no—
330C2×C32—41no—
331C2×C32—41no—
332C2×C32D3×C22⋊C412yes—
333C2×C32#91412yes—
334C2×C32#91812yes—
335C2×C32#94911yescharacteristic
336C2×C32#95111yescharacteristic
337C2×C32C2×#21711yescharacteristic
338C3⋊D3—22no—
339C3⋊D3—42no—
340C3×D3—64no—
341C3×D3—122no—
342C3×D3—124no—
343C3×D3—124no—
344#41—94no—
345#41—182no—
346C2×Q8#111311yescharacteristic
347C2×D4—91no—
348C2×D4—181no—
349C22×C4—91no—
350C22×C4—181no—
351C22×C4—541no—
352M16—542no—
353C2×C8—92no—
354C4⋊C4—92no—
355C4⋊C4—541no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—41no—
359C22×C3—92no—
360C22×C3—181no—
361C22×C3—181no—
362C22×C3—182no—
363C22×C3—361no—
364C22×C3—361no—
365C22×C3D3×#6712yes—
366C22×C3#381111yescharacteristic
367C2×D3—34no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—64no—
372C2×D3—64no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—122no—
376C2×D3—122no—
377C2×D3—122no—
378C2×D3—122no—
379C3×C4—24no—
380C3×C4—24no—
381C3×C4—22no—
382C3×C4—21no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—21no—
386C3×C4—44no—
387C3×C4—44no—
388C3×C4—41no—
389C3×C4—41no—
390C3×C4—92no—
391C3×C4—362no—
392C3×C4C2×D3×Dic312yes—
393C3×C4C2×#62411yescharacteristic
394C3×C4D3×#6712yes—
395C3×C4#381111yescharacteristic
396Dic3—34no—
397Dic3—62no—
398Dic3—122no—
399Dic3—122no—
400Dic3—242no—
401C32—22no—
402C32—21no—
403C32—41no—
404C32#480312yescomplemented
405C32#523211yescharacteristic, complemented
406C23—91no—
407C23—181no—
408Q8—21no—
409Q8#629412yescomplemented
410D4—92no—
411D4—184no—
412D4—181no—
413C2×C4—21no—
414C2×C4—91no—
415C2×C4—92no—
416C2×C4—92no—
417C2×C4—184no—
418C2×C4—182no—
419C2×C4—1082no—
420C2×C4C2×#111311yescharacteristic
421C8—182no—
422C2×C3—22no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—22no—
426C2×C3—21no—
427C2×C3—22no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3—182no—
432C2×C3—362no—
433C2×C3#6393912yes—
434C2×C3#6394111yescharacteristic
435C2×C3#6394312yes—
436C2×C3#6394511yescharacteristic
437C2×C3D3×#21712yes—
438C2×C3#6397411yescharacteristic
439D3—64no—
440D3—122no—
441D3—124no—
442D3—124no—
443C22—92no—
444C22—181no—
445C22—181no—
446C22—182no—
447C22—361no—
448C22—361no—
449C22#8904911yescharacteristic
450C4—22no—
451C4—92no—
452C4—362no—
453C4C2×#629411yescharacteristic
454C4#8904911yescharacteristic
455C3—21no—
456C3—22no—
457C3—41no—
458C3#1059155012yescomplemented
459C3#1059155411yescharacteristic, complemented
460C2—182no—
461C2—362no—
462C2#1172370911yescharacteristic
463C2#1172371111yescharacteristic
464C2#1172372311yescharacteristic
subgroupG/H
centerC22#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3832C2
cosocleC2×C33×C4C23
hypercenterC2×Q8#1113
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11723689C2semidirect
3. C2×#88809C2semidirect
4. C3×#63662C2semidirect
5. #11722075C2semidirect
10. C3×#3832C4semidirect
27. #63662D3semidirect
37. C33×Q8C2×C4semidirect
44. C3×C4⋊(C3⋊D3)C2×C4semidirect
58. #3832Dic3semidirect
129. C32×Q8C4×D3semidirect
130. C32×Q8C2×Dic3semidirect
145. C4⋊(C3⋊D3)C2×Dic3semidirect
265. C33#417semidirect
280. C3×Q8D3×Dic3semidirect
281. C3×Q8#624semidirect
404. C32#4803semidirect
405. C32#5232semidirect
409. Q8#6294semidirect
458. C3#10591550semidirect
459. C3#10591554semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
#117236892
C2×#888092
C3×#636622
#117220752
#105915543
#105915503