Finite Group Databasebeta

← #13160100

Subgroups — #13160100

Order 1728.

1244 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 158 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×#89014C211yesmaximal, characteristic, complemented
2C2×#88851C211yesmaximal, characteristic, complemented
3#11723526C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722041C211yesmaximal, characteristic, complemented
8#10592480—31nomaximal
9#10590968—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6292—21no—
15C2×#6290—21no—
16C2×D3×C3⋊Dic3—21no—
17#89014—22no—
18#89014—22no—
19C2×C33×C8C2211yescharacteristic
20#88851—22no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×C4⋊Dic3C2211yescharacteristic
23#88804—21no—
24#88790—22no—
25C2×#3830—31no—
26C2×#3795—34no—
27C2×#3753—31no—
28C2×#3744—31no—
29C2×#3744—34no—
30C2×#3744—64no—
31#64104—31no—
32#63909—34no—
33C2×#3714—34no—
34C32×#102—31no—
35#63647D311yescharacteristic, complemented
36#63637—31no—
37C3×#821D314yescomplemented
38C3×(C2×D3)⋊C8—34no—
39#63567—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47#6308—21no—
48#6308D412yescomplemented
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3—21no—
51C3×C4×C3⋊D3—21no—
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12D412yescomplemented
55#6292—42no—
56#6290—24no—
57D3×C3⋊Dic3—24no—
58C33×C8—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#4781—94no—
61#4734—91no—
62C2×C32×D4—31no—
63C2×#638—61no—
64C22×C3⋊Dic3—61no—
65#3830—62no—
66#3830—62no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—34no—
69C2×C4⋊(C3⋊D3)—64no—
70C2×C4⋊(C3⋊D3)—61no—
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C2×#634—31no—
73C2×#634—34no—
74C2×#634—64no—
75C2×C3×D12C2×D314yes—
76C2×C3×C4×D3—34no—
77C2×#626—64no—
78C2×#625—64no—
79C2×D3×Dic3—64no—
80#3795—68no—
81#3795—68no—
82C2×C32×C8—24no—
83C2×C32×C8C2×D314yes—
84C2×C32×C8C2×D311yescharacteristic
85C32×C4⋊C4—31no—
86#3753—62no—
87#3752—61no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89#3744—62no—
90#3744—68no—
91#3744—616no—
92C3×C4⋊Dic3C2×D314yes—
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×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—21no—
103C2×C32×D3—22no—
104C2×C32×D3—41no—
105C2×C32×D3—41no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3⋊(C3⋊Dic3)—22no—
109C3×C3⋊Dic3—21no—
110C32×Dic3—41no—
111C2×#237—94no—
112C2×#235—91no—
113C2×#213—94no—
114#930—91no—
115C2×D24—91no—
116C2×#204—94no—
117C2×#204—91no—
118C2×#204—184no—
119#895—94no—
120C3×#102—34no—
121C3×C22⋊C8—91no—
122#821C3⋊D311yescharacteristic, complemented
123(C2×D3)⋊C8—34no—
124#816—91no—
125#810—94no—
126C23×C32—61no—
127C22×C3⋊D3—61no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—124no—
130C22×C3⋊D3D1211yescharacteristic
131C22×C3×D3—24no—
132C22×C3×D3—34no—
133C32×D4—32no—
134C32×D4—61no—
135C2×C32×C4—24no—
136C2×C32×C4—61no—
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D311yescharacteristic
139#638—64no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—124no—
144C2×C3⋊Dic3D1211yescharacteristic
145C4⋊(C3⋊D3)—68no—
146C4⋊(C3⋊D3)—64no—
147C4⋊(C3⋊D3)—61no—
148C4⋊(C3⋊D3)—61no—
149C4⋊(C3⋊D3)—64no—
150C4⋊(C3⋊D3)—124no—
151C4⋊(C3⋊D3)—122no—
152C4×C3⋊D3—21no—
153C4×C3⋊D3—21no—
154#634—32no—
155#634—38no—
156#634—68no—
157#634—61no—
158#634—64no—
159#634—124no—
160C2×C3×Dic3—24no—
161C2×C3×Dic3—34no—
162C3×D12—24no—
163C3×D12D1218yescomplemented
164C3×C4×D3—64no—
165C3×C4×D3—64no—
166#626—128no—
167#625—616no—
168D3×Dic3—616no—
169C32×C8—28no—
170C32×C8—24no—
171C32×C8—21no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—64no—
174#449—271noSylow
175C2×C33#12011yescharacteristic
176C2×C33#13511yescharacteristic
177C2×C33AGL(1,ℤ/8)11yescharacteristic
178C3⋊(C3⋊D3)—81no—
179C3×C3⋊D3—41no—
180C32×D3—42no—
181C2×C3×D4—34no—
182C22×C3×C4—91no—
183C2×#67—181no—
184C2×#67—181no—
185C2×#67—184no—
186C22×Dic3—184no—
187#237—188no—
188#237—188no—
189#235—182no—
190#235—182no—
191C2×D12—94no—
192C2×D12—91no—
193C2×D12—184no—
194C2×D12—184no—
195C2×D12C2×C3⋊D311yescharacteristic
196C2×C4×D3—34no—
197C2×C4×D3—181no—
198C2×Dic6—94no—
199C2×Dic6—91no—
200C2×Dic6—184no—
201C2×C3×C8—24no—
202C2×C3×C8—91no—
203C2×C3×C8D3214yes—
204C2×C3×C8C2×C3⋊D311yescharacteristic
205C3×C4⋊C4—34no—
206#213—188no—
207(C2×D3)⋊C4—181no—
208(C2×D3)⋊C4—184no—
209C4⋊Dic3C2×C3⋊D311yescharacteristic
210C2×C3⋊C8—34no—
211D24—182no—
212#204—188no—
213#204—182no—
214#204—1816no—
215C22×C32—24no—
216C22×C32—62no—
217C22×C32—121no—
218C22×C32—121no—
219C22×C32C2×D1211yescharacteristic
220C22×C32D3×D414yes—
221C2×C3⋊D3—21no—
222C2×C3⋊D3—21no—
223C2×C3⋊D3—21no—
224C2×C3⋊D3—128no—
225C2×C3⋊D3—124no—
226C2×C3⋊D3—121no—
227C2×C3⋊D3—121no—
228C2×C3⋊D3—121no—
229C2×C3⋊D3—124no—
230C2×C3⋊D3—124no—
231C2×C3⋊D3—244no—
232C2×C3⋊D3—244no—
233C2×C3×D3—28no—
234C2×C3×D3—44no—
235C2×C3×D3—44no—
236C2×C3×D3—64no—
237C2×C3×D3—64no—
238C2×C3×D3—64no—
239C32×C4—24no—
240C32×C4—24no—
241C32×C4—121no—
242C32×C4C2×D1214yes—
243C32×C4D3×D414yes—
244C32×C4D3×D411yescharacteristic
245C32×C4D3×D411yescharacteristic
246C3⋊Dic3—21no—
247C3⋊Dic3—62no—
248C3⋊Dic3—68no—
249C3⋊Dic3—128no—
250C3×Dic3—44no—
251C3×Dic3—64no—
252C2×#41—271no—
253C2×SD16—271no—
254C2×D8—271no—
255#121—271no—
256#103—271no—
257#102—31no—
258C22⋊C8—91no—
259C33#44911yescharacteristic, complemented, Sylow
260C23×C3—64no—
261C23×C3—91no—
262C22×D3—21no—
263C22×D3—34no—
264C22×D3—184no—
265C22×D3—181no—
266C22×D3—364no—
267C3×D4—38no—
268C3×D4—64no—
269C2×C3×C4—24no—
270C2×C3×C4—64no—
271C2×C3×C4—91no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4C2×D3214yes—
275C2×C3×C4C22×C3⋊D311yescharacteristic
276#67—184no—
277#67—1816no—
278#67—362no—
279C2×Dic3—21no—
280C2×Dic3—34no—
281C2×Dic3—1816no—
282C2×Dic3—184no—
283C2×Dic3—181no—
284C2×Dic3—364no—
285D12—21no—
286D12—188no—
287D12—184no—
288D12—184no—
289D12—181no—
290D12—181no—
291D12—364no—
292D12—368no—
293D12C4⋊(C3⋊D3)12yescomplemented
294C4×D3—64no—
295C4×D3—64no—
296C4×D3—184no—
297Dic6—98no—
298Dic6—92no—
299Dic6—188no—
300Dic6—184no—
301Dic6—181no—
302Dic6—364no—
303C3×C8—28no—
304C3×C8—24no—
305C3×C8—21no—
306C3×C8—181no—
307C3⋊C8—64no—
308C2×C32—24no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32—122no—
312C2×C32#88214yes—
313C2×C32#88211yescharacteristic
314C2×C32#90411yescharacteristic
315C2×C32#90811yescharacteristic
316C2×C32#91114yes—
317C2×C32#91614yes—
318C3⋊D3—41no—
319C3⋊D3—248no—
320C3⋊D3—241no—
321C3⋊D3—244no—
322C3×D3—48no—
323C3×D3—124no—
324#41—542no—
325#41—542no—
326C2×Q8—271no—
327C2×D4—31no—
328C2×D4—271no—
329C2×D4—541no—
330C2×D4—541no—
331C22×C4—91no—
332C22×C4—541no—
333SD16—542no—
334D8—542no—
335C2×C8—91no—
336C2×C8D3×C3⋊D311yescharacteristic
337C4⋊C4—31no—
338C22⋊C4—541no—
339C22×C3—24no—
340C22×C3—68no—
341C22×C3—124no—
342C22×C3—124no—
343C22×C3—181no—
344C22×C3—181no—
345C22×C3—181no—
346C22×C3D3×D1214yes—
347C22×C3D4×C3⋊D311yescharacteristic
348C2×D3—22no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—64no—
352C2×D3—64no—
353C2×D3—64no—
354C2×D3—368no—
355C2×D3—364no—
356C2×D3—364no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—361no—
360C2×D3—364no—
361C2×D3—724no—
362C2×D3—724no—
363C3×C4—24no—
364C3×C4—24no—
365C3×C4—124no—
366C3×C4—181no—
367C3×C4D3×D1214yes—
368C3×C4#381114yes—
369C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
370C3×C4D4×C3⋊D311yescharacteristic
371Dic3—41no—
372Dic3—64no—
373Dic3—182no—
374Dic3—188no—
375Dic3—368no—
376C32—24no—
377C32#473411yescharacteristic, complemented
378C32#478114yescomplemented
379C23—61no—
380C23—91no—
381C23—541no—
382Q8—272no—
383Q8—541no—
384D4—32no—
385D4—61no—
386D4—544no—
387D4—541no—
388D4—541no—
389D4—1082no—
390C2×C4—61no—
391C2×C4—91no—
392C2×C4—181no—
393C2×C4—181no—
394C2×C4—544no—
395C2×C4—541no—
396C2×C4C2×D3×C3⋊D311yescharacteristic
397C8—21no—
398C8—181no—
399C2×C3—24no—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—128no—
403C2×C3—361no—
404C2×C3#6379514yes—
405C2×C3#6380614yes—
406C2×C3#6392314yes—
407C2×C3#6409111yescharacteristic
408C2×C3#6412011yescharacteristic
409C2×C3#6412511yescharacteristic
410D3—42no—
411D3—124no—
412D3—728no—
413D3—721no—
414D3—724no—
415C22—62no—
416C22—121no—
417C22—121no—
418C22—181no—
419C22—181no—
420C22—181no—
421C22—1081no—
422C22—1081no—
423C22—1081no—
424C22C3⋊D3×D1211yescharacteristic
425C4—121no—
426C4—181no—
427C4—542no—
428C4D3×C4⋊(C3⋊D3)11yescharacteristic
429C4#8903911yescharacteristic
430C3—24no—
431C3#1059096814yescomplemented
432C3#1059248011yescharacteristic, complemented
433C2—122no—
434C2—361no—
435C2—2161no—
436C2#1172332011yescharacteristic
437C2#1172333811yescharacteristic
438C2#1172354211yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#89014C2semidirect
2. C2×#88851C2semidirect
3. #11723526C2semidirect
4. C2×#88790C2semidirect
5. C3×#63647C2semidirect
6. C32×#821C2semidirect
7. #11722041C2semidirect
35. #63647D3semidirect
37. C3×#821D3semidirect
48. #6308D4semidirect
54. C32×D12D4semidirect
122. #821C3⋊D3semidirect
163. C3×D12D12semidirect
259. C33#449semidirect
293. D12C4⋊(C3⋊D3)semidirect
377. C32#4734semidirect
378. C32#4781semidirect
431. C3#10590968semidirect
432. C3#10592480semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
C2×#888512
#117235262
C2×#887902
C3×#636472
C32×#8212
#117220412
#105924803
#105909683