Finite Group Databasebeta

← #13161194

Subgroups — #13161194

Order 1728.

1244 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 188 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×C32×#239C211yesmaximal, characteristic, complemented
2C2×#88862C211yesmaximal, characteristic, complemented
3#11723516C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592937—31nomaximal
9#10591588—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C32×#239—22no—
13C32×#239—22no—
14C2×C32×D12—21no—
15C2×C32×D12C2211yescharacteristic
16C2×C32×C4×D3—21no—
17C2×C32×C4×D3C2211yescharacteristic
18C2×#6291—21no—
19#88862—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23#88804—21no—
24#88790—22no—
25C2×C32×#41—31no—
26C2×C3×#239D314yescomplemented
27C2×#3755—31no—
28C2×#3755—34no—
29C2×#3755—64no—
30C2×#3753—31no—
31#64139—31no—
32#63909—34no—
33C2×#3714—34no—
34#63662D311yescharacteristic, complemented
35#63659—31no—
36#63657—31no—
37C3×#835—34no—
38#63562—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C32×D3—21no—
42C22×C32×D3D411yescharacteristic
43C33×Q8—21no—
44C33×Q8D412yescomplemented
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C32×Dic3D411yescharacteristic
51C32×D12—24no—
52C32×D12—21no—
53C32×D12D412yescomplemented
54C32×C4×D3—24no—
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57#6291—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5238—94no—
61#4810—91no—
62C32×#41—62no—
63C32×#41—62no—
64C2×C32×Q8—24no—
65C2×C32×Q8C2×D311yescharacteristic
66C2×C32×Q8C2×D314yes—
67C2×C32×D4—31no—
68C2×C32×D4—61no—
69C22×C32×C4—31no—
70C22×C32×C4—61no—
71C2×#638—61no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—34no—
74C2×C4⋊(C3⋊D3)—64no—
75C3×#239—28no—
76C3×#239—28no—
77C2×C3×D12—24no—
78C2×C3×D12C2×D314yes—
79C2×C3×C4×D3—24no—
80C2×C3×C4×D3C2×D314yes—
81C2×#626—64no—
82#3755—616no—
83#3755—62no—
84#3755—68no—
85#3753—62no—
86#3752—61no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×C4⋊Dic3—34no—
93C2×C3×C3⋊C8—34no—
94#3722—64no—
95#3714—68no—
96C22×C33C2×D411yescharacteristic
97C2×C3⋊(C3⋊D3)—41no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C32×D3—21no—
101C2×C32×D3—22no—
102C2×C32×D3—21no—
103C2×C32×D3—21no—
104C2×C32×D3—41no—
105C2×C32×D3—41no—
106C33×C4—22no—
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3×C3⋊Dic3—41no—
110C32×Dic3—21no—
111C2×C3×#41—34no—
112C2×#239C3⋊D311yescharacteristic, complemented
113C2×#215—94no—
114C2×#215—91no—
115C2×#215—184no—
116C2×#213—94no—
117#930—94no—
118C2×D24—91no—
119#895—91no—
120C3×#103—91no—
121#835—34no—
122#832—94no—
123#830—94no—
124(C2×D3)⋊C8—91no—
125#809—91no—
126C23×C32—31no—
127C23×C32—61no—
128C22×C3⋊D3—61no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—124no—
131C22×C3×D3—24no—
132C22×C3×D3#6714yes—
133C32×Q8—21no—
134C32×Q8—28no—
135C32×Q8—24no—
136C32×Q8—44no—
137C32×Q8D1212yescomplemented
138C32×Q8#6718yescomplemented
139C32×D4—32no—
140C32×D4—64no—
141C32×D4—61no—
142C2×C32×C4—21no—
143C2×C32×C4—24no—
144C2×C32×C4—24no—
145C2×C32×C4—31no—
146C2×C32×C4—44no—
147C2×C32×C4—64no—
148C2×C32×C4—61no—
149C2×C32×C4—61no—
150C2×C32×C4C22×D314yes—
151C2×C32×C4C22×D311yescharacteristic
152#638—122no—
153C2×C3⋊Dic3—21no—
154C4⋊(C3⋊D3)—68no—
155C4⋊(C3⋊D3)—61no—
156C4⋊(C3⋊D3)—61no—
157C4⋊(C3⋊D3)—64no—
158C4⋊(C3⋊D3)—64no—
159C4⋊(C3⋊D3)—124no—
160C2×C3×Dic3—64no—
161C2×C3×Dic3#6714yes—
162C3×D12—216no—
163C3×D12—24no—
164C3×D12#6718yescomplemented
165C3×C4×D3—216no—
166C3×C4×D3—24no—
167C3×C4×D3—24no—
168#626—128no—
169C3⋊(C3⋊C8)—21no—
170C3⋊(C3⋊C8)—68no—
171C3⋊(C3⋊C8)—61no—
172C3⋊(C3⋊C8)—64no—
173C3×C3⋊C8—64no—
174#449—271noSylow
175C2×C33#12011yescharacteristic
176C2×C33#13511yescharacteristic
177C2×C33AGL(1,ℤ/8)11yescharacteristic
178C3⋊(C3⋊D3)—81no—
179C32×D3—41no—
180C32×D3—42no—
181C3×#41—68no—
182C3×#41—68no—
183C2×C3×Q8—24no—
184C2×C3×Q8D3214yes—
185C2×C3×Q8C2×C3⋊D311yescharacteristic
186C2×C3×D4—34no—
187C2×C3×D4—64no—
188C22×C3×C4—34no—
189C22×C3×C4—64no—
190C2×#67—184no—
191#239—22no—
192#239—22no—
193C2×D12—21no—
194C2×D12—94no—
195C2×D12—91no—
196C2×D12—184no—
197C2×D12—181no—
198C2×D12C2×C3⋊D311yescharacteristic
199C2×C4×D3—21no—
200C2×C4×D3C2×C3⋊D311yescharacteristic
201C2×C3×C8—91no—
202C3×C4⋊C4—91no—
203#215—1816no—
204#215—188no—
205#215—182no—
206#213—188no—
207(C2×D3)⋊C4—184no—
208(C2×D3)⋊C4—181no—
209C4⋊Dic3—34no—
210C2×C3⋊C8—34no—
211C2×C3⋊C8—94no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—184no—
214D24—182no—
215C22×C32—24no—
216C22×C32—62no—
217C22×C32—61no—
218C22×C32—61no—
219C22×C32—61no—
220C22×C32—121no—
221C22×C32—121no—
222C22×C32D3×D411yescharacteristic
223C22×C32C2×#6714yes—
224C2×C3⋊D3—128no—
225C2×C3⋊D3—121no—
226C2×C3⋊D3—121no—
227C2×C3⋊D3—121no—
228C2×C3⋊D3—124no—
229C2×C3⋊D3—124no—
230C2×C3⋊D3—124no—
231C2×C3⋊D3—244no—
232C2×C3⋊D3—244no—
233C2×C3×D3—24no—
234C2×C3×D3—24no—
235C2×C3×D3—24no—
236C2×C3×D3—28no—
237C2×C3×D3—44no—
238C2×C3×D3—44no—
239C32×C4—22no—
240C32×C4—24no—
241C32×C4—24no—
242C32×C4—28no—
243C32×C4—48no—
244C32×C4—61no—
245C32×C4C2×D1211yescharacteristic
246C32×C4D3×D411yescharacteristic
247C32×C4C2×#6714yes—
248C32×C4C2×#6714yes—
249C3⋊Dic3—41no—
250C3×Dic3—24no—
251C3×Dic3—124no—
252C2×#41—31no—
253C2×SD16—271no—
254C2×D8—271no—
255#121—271no—
256#103—91no—
257#102—271no—
258C22⋊C8—271no—
259C33#44911yescharacteristic, complemented, Sylow
260C23×C3—34no—
261C23×C3—64no—
262C22×D3—21no—
263C22×D3—184no—
264C22×D3—181no—
265C22×D3—364no—
266C22×D3#63811yescharacteristic
267C3×Q8—24no—
268C3×Q8—21no—
269C3×Q8—28no—
270C3×Q8—44no—
271C3×Q8#62618yescomplemented
272C3×Q8#63812yescomplemented
273C3×D4—38no—
274C3×D4—616no—
275C3×D4—64no—
276C2×C3×C4—24no—
277C2×C3×C4—21no—
278C2×C3×C4—24no—
279C2×C3×C4—34no—
280C2×C3×C4—44no—
281C2×C3×C4—616no—
282C2×C3×C4—64no—
283C2×C3×C4—64no—
284C2×C3×C4—181no—
285C2×C3×C4C2×D3214yes—
286C2×C3×C4C22×C3⋊D311yescharacteristic
287#67—368no—
288C2×Dic3—64no—
289C2×Dic3#63811yescharacteristic
290D12—24no—
291D12—21no—
292D12—188no—
293D12—184no—
294D12—184no—
295D12—181no—
296D12—181no—
297D12—364no—
298D12—362no—
299D12#63812yescomplemented
300C4×D3—24no—
301C4×D3—21no—
302C4×D3—21no—
303C3×C8—181no—
304C3⋊C8—64no—
305C3⋊C8—188no—
306C3⋊C8—184no—
307C3⋊C8—181no—
308C2×C32—24no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32—121no—
312C2×C32—122no—
313C2×C32#88211yescharacteristic
314C2×C32#91411yescharacteristic
315C2×C32#91911yescharacteristic
316C2×C32#94914yes—
317C2×C32#95014yes—
318C2×C32#95314yes—
319C3⋊D3—248no—
320C3⋊D3—244no—
321C3⋊D3—241no—
322C3×D3—44no—
323C3×D3—48no—
324#41—62no—
325#41—62no—
326C2×Q8D3×C3⋊D311yescharacteristic
327C2×D4—31no—
328C2×D4—61no—
329C2×D4—271no—
330C2×D4—541no—
331C22×C4—31no—
332C22×C4—61no—
333SD16—542no—
334D8—542no—
335C2×C8—91no—
336C2×C8—271no—
337C4⋊C4—91no—
338C22⋊C4—541no—
339C22×C3—24no—
340C22×C3—64no—
341C22×C3—64no—
342C22×C3—68no—
343C22×C3—64no—
344C22×C3—124no—
345C22×C3—124no—
346C22×C3D3×#6714yes—
347C22×C3C2×#63811yescharacteristic
348C2×D3—21no—
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—22no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—368no—
355C2×D3—361no—
356C2×D3—361no—
357C2×D3—361no—
358C2×D3—364no—
359C2×D3—364no—
360C2×D3—364no—
361C2×D3—724no—
362C2×D3—724no—
363C3×C4—22no—
364C3×C4—28no—
365C3×C4—24no—
366C3×C4—24no—
367C3×C4—48no—
368C3×C4—64no—
369C3×C4—361no—
370C3×C4C2×#62614yes—
371C3×C4D3×#6714yes—
372C3×C4C2×#63811yescharacteristic
373C3×C4C2×#63811yescharacteristic
374Dic3—21no—
375Dic3—124no—
376C32—24no—
377C32#481011yescharacteristic, complemented
378C32#523814yescomplemented
379C23—31no—
380C23—61no—
381C23—541no—
382Q8—21no—
383Q8#629112yescomplemented
384D4—32no—
385D4—64no—
386D4—61no—
387D4—541no—
388D4—541no—
389D4—1082no—
390C2×C4—21no—
391C2×C4—31no—
392C2×C4—64no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—181no—
396C2×C4C2×D3×C3⋊D311yescharacteristic
397C8—181no—
398C8—541no—
399C2×C3—24no—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—124no—
403C2×C3—128no—
404C2×C3#6393914yes—
405C2×C3#6394914yes—
406C2×C3#6397714yes—
407C2×C3#6415811yescharacteristic
408C2×C3#6415911yescharacteristic
409C2×C3#6416211yescharacteristic
410D3—41no—
411D3—42no—
412D3—728no—
413D3—724no—
414D3—721no—
415C22—61no—
416C22—61no—
417C22—62no—
418C22—61no—
419C22—121no—
420C22—121no—
421C22—1081no—
422C22—1081no—
423C22—1081no—
424C22D3×#63811yescharacteristic
425C4—22no—
426C4—61no—
427C4—361no—
428C4C2×#629111yescharacteristic
429C4D3×#63811yescharacteristic
430C3—24no—
431C3#1059158814yescomplemented
432C3#1059293711yescharacteristic, complemented
433C2—121no—
434C2—122no—
435C2—2161no—
436C2#1172356811yescharacteristic
437C2#1172358211yescharacteristic
438C2#1172363011yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#449
LCSG ▷ C2×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
1. C2×C32×#239C2semidirect
2. C2×#88862C2semidirect
3. #11723516C2semidirect
4. C2×#88790C2semidirect
5. C3×#63662C2semidirect
6. #11722031C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
26. C2×C3×#239D3semidirect
34. #63662D3semidirect
44. C33×Q8D4semidirect
53. C32×D12D4semidirect
112. C2×#239C3⋊D3semidirect
137. C32×Q8D12semidirect
138. C32×Q8#67semidirect
164. C3×D12#67semidirect
259. C33#449semidirect
271. C3×Q8#626semidirect
272. C3×Q8#638semidirect
299. D12#638semidirect
377. C32#4810semidirect
378. C32#5238semidirect
383. Q8#6291semidirect
431. C3#10591588semidirect
432. C3#10592937semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
C2×#888622
#117235162
C2×#887902
C3×#636622
#117220312
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105929373
#105915883