Finite Group Databasebeta

← #13160099

Subgroups — #13160099

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×#89021C211yesmaximal, characteristic, complemented
2C2×#88851C211yesmaximal, characteristic, complemented
3#11723516C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11722042C211yesmaximal, characteristic, complemented
8#10592433—31nomaximal
9#10590968—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6291—21no—
15C2×#6290—21no—
16C2×Dic3×C3⋊D3—21no—
17#89021—22no—
18#89021—22no—
19C2×C33×C8C2211yescharacteristic
20#88851—22no—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88804—21no—
24#88791—22no—
25C2×#3828—31no—
26C2×#3795—34no—
27#64139—31no—
28C2×#3745—31no—
29C2×#3744—31no—
30C2×#3744—34no—
31C2×#3744—64no—
32#63909—34no—
33C2×#3714—34no—
34C32×C22⋊C8—31no—
35#63648D311yescharacteristic, complemented
36#63643—31no—
37C3×#821—34no—
38C3×(C2×D3)⋊C8D314yescomplemented
39#63567—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3—21no—
42C22×C32×D3D411yescharacteristic
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⋊Dic3—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)D412yescomplemented
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55#6291—42no—
56#6290—24no—
57Dic3×C3⋊D3—24no—
58C33×C8—21no—
59C32×C3⋊C8—21no—
60#4781—91no—
61#4734—94no—
62C22×C32×C4—31no—
63C2×#638—61no—
64C2×#638—61no—
65#3828—62no—
66#3828—62no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—34no—
69C2×C4⋊(C3⋊D3)—64no—
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C4×C3⋊D3—61no—
72C2×#634—31no—
73C2×#634—34no—
74C2×#634—64no—
75C2×C3×D12—34no—
76C2×C3×C4×D3C2×D314yes—
77C2×#626—64no—
78C2×#625—64no—
79C2×D3×Dic3—64no—
80#3795—68no—
81#3795—68no—
82C2×C32×C8—24no—
83C2×C32×C8—31no—
84C2×C32×C8C2×D314yes—
85C2×C32×C8C2×D311yescharacteristic
86#3752—61no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88#3745—62no—
89#3744—62no—
90#3744—68no—
91#3744—616no—
92C3×C4⋊Dic3—34no—
93C2×C3×C3⋊C8C2×D314yes—
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—22no—
101C2×C3×C3⋊D3—41no—
102C2×C3×C3⋊D3—41no—
103C2×C32×D3—21no—
104C2×C32×D3—21no—
105C2×C32×D3—21no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3⋊(C3⋊Dic3)—22no—
109C3×C3⋊Dic3—41no—
110C32×Dic3—21no—
111C2×#237—91no—
112C2×#235—94no—
113C2×#213—91no—
114#930—94no—
115C2×D24—94no—
116C2×#204—94no—
117C2×#204—91no—
118C2×#204—184no—
119#895—91no—
120C3×#102—91no—
121C3×C22⋊C8—34no—
122#821—34no—
123(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
124#816—94no—
125#810—91no—
126C23×C32—31no—
127C22×C3⋊D3—21no—
128C22×C3⋊D3—61no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—124no—
131C22×C3×D3—64no—
132C22×C3×D3D1214yes—
133C2×C32×C4—24no—
134C2×C32×C4—31no—
135C2×C32×C4—61no—
136C2×C32×C4—61no—
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D311yescharacteristic
139#638—64no—
140#638—122no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—64no—
144C2×C3⋊Dic3—124no—
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)—68no—
147C4⋊(C3⋊D3)—61no—
148C4⋊(C3⋊D3)—64no—
149C4⋊(C3⋊D3)—61no—
150C4⋊(C3⋊D3)—64no—
151C4⋊(C3⋊D3)—124no—
152C4⋊(C3⋊D3)D1212yescomplemented
153C4×C3⋊D3—64no—
154#634—32no—
155#634—38no—
156#634—68no—
157#634—61no—
158#634—64no—
159#634—124no—
160C2×C3×Dic3—64no—
161C2×C3×Dic3D1214yes—
162C3×D12—38no—
163C3×D12—64no—
164C3×C4×D3—24no—
165C3×C4×D3—24no—
166#626—128no—
167#625—616no—
168D3×Dic3—616no—
169C32×C8—28no—
170C32×C8—24no—
171C32×C8—21no—
172C32×C8—61no—
173C3×C3⋊C8—24no—
174#449—271noSylow
175C2×C33#12011yescharacteristic
176C2×C33#13511yescharacteristic
177C2×C33AGL(1,ℤ/8)11yescharacteristic
178C3⋊(C3⋊D3)—81no—
179C3×C3⋊D3—42no—
180C32×D3—41no—
181C2×C3×D4—91no—
182C22×C3×C4—34no—
183C2×#67—184no—
184C2×#67—184no—
185C2×#67—181no—
186C22×Dic3—181no—
187#237—182no—
188#237—182no—
189#235—188no—
190#235—188no—
191C2×D12—34no—
192C2×D12—94no—
193C2×D12—91no—
194C2×D12—184no—
195C2×D12—181no—
196C2×C4×D3—184no—
197C2×C4×D3C2×C3⋊D311yescharacteristic
198C2×Dic6—94no—
199C2×Dic6—91no—
200C2×Dic6—184no—
201C2×C3×C8—24no—
202C2×C3×C8—34no—
203C2×C3×C8D3214yes—
204C2×C3×C8C2×C3⋊D311yescharacteristic
205C3×C4⋊C4—91no—
206#213—182no—
207(C2×D3)⋊C4—184no—
208(C2×D3)⋊C4—181no—
209C4⋊Dic3—34no—
210C2×C3⋊C8C2×C3⋊D311yescharacteristic
211D24—188no—
212#204—188no—
213#204—182no—
214#204—1816no—
215C22×C32—24no—
216C22×C32—61no—
217C22×C32—61no—
218C22×C32—61no—
219C22×C32C2×D1214yes—
220C22×C32D3×D411yescharacteristic
221C2×C3⋊D3—22no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
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—24no—
234C2×C3×D3—24no—
235C2×C3×D3—24no—
236C2×C3×D3—68no—
237C2×C3×D3—124no—
238C2×C3×D3—124no—
239C32×C4—24no—
240C32×C4—24no—
241C32×C4—61no—
242C32×C4C2×D1211yescharacteristic
243C32×C4D3×D414yes—
244C32×C4D3×D414yes—
245C32×C4D3×D411yescharacteristic
246C3⋊Dic3—41no—
247C3⋊Dic3—62no—
248C3⋊Dic3—68no—
249C3⋊Dic3—128no—
250C3×Dic3—24no—
251C3×Dic3—124no—
252C2×#41—271no—
253C2×SD16—271no—
254C2×D8—271no—
255#121—271no—
256#103—271no—
257#102—91no—
258C22⋊C8—31no—
259C33#44911yescharacteristic, complemented, Sylow
260C23×C3—34no—
261C23×C3—181no—
262C22×D3—64no—
263C22×D3—184no—
264C22×D3—181no—
265C22×D3—364no—
266C22×D3C4⋊(C3⋊D3)11yescharacteristic
267C3×D4—92no—
268C3×D4—181no—
269C2×C3×C4—24no—
270C2×C3×C4—34no—
271C2×C3×C4—64no—
272C2×C3×C4—64no—
273C2×C3×C4—181no—
274C2×C3×C4C2×D3214yes—
275C2×C3×C4C22×C3⋊D311yescharacteristic
276#67—184no—
277#67—1816no—
278#67—368no—
279C2×Dic3—64no—
280C2×Dic3—184no—
281C2×Dic3—184no—
282C2×Dic3—181no—
283C2×Dic3—364no—
284C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
285D12—38no—
286D12—64no—
287D12—188no—
288D12—184no—
289D12—184no—
290D12—181no—
291D12—181no—
292D12—364no—
293D12—362no—
294C4×D3—21no—
295C4×D3—21no—
296C4×D3—1816no—
297Dic6—98no—
298Dic6—92no—
299Dic6—188no—
300Dic6—184no—
301Dic6—181no—
302Dic6—364no—
303C3×C8—28no—
304C3×C8—24no—
305C3×C8—21no—
306C3×C8—64no—
307C3⋊C8—21no—
308C2×C32—24no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32—121no—
312C2×C32#88214yes—
313C2×C32#88211yescharacteristic
314C2×C32#90414yes—
315C2×C32#90814yes—
316C2×C32#91111yescharacteristic
317C2×C32#91611yescharacteristic
318C3⋊D3—42no—
319C3⋊D3—248no—
320C3⋊D3—244no—
321C3⋊D3—241no—
322C3×D3—44no—
323C3×D3—128no—
324#41—542no—
325#41—542no—
326C2×Q8—271no—
327C2×D4—91no—
328C2×D4—271no—
329C2×D4—541no—
330C2×D4—541no—
331C22×C4—31no—
332C22×C4—541no—
333SD16—542no—
334D8—542no—
335C2×C8—31no—
336C2×C8D3×C3⋊D311yescharacteristic
337C4⋊C4—91no—
338C22⋊C4—541no—
339C22×C3—24no—
340C22×C3—64no—
341C22×C3—64no—
342C22×C3—64no—
343C22×C3—182no—
344C22×C3—361no—
345C22×C3—361no—
346C22×C3D3×D1214yes—
347C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
348C2×D3—21no—
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—68no—
352C2×D3—124no—
353C2×D3—124no—
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—64no—
366C3×C4—361no—
367C3×C4D3×D1214yes—
368C3×C4#381114yes—
369C3×C4D4×C3⋊D311yescharacteristic
370C3×C4D4×C3⋊D311yescharacteristic
371Dic3—21no—
372Dic3—124no—
373Dic3—182no—
374Dic3—188no—
375Dic3—368no—
376C32—24no—
377C32#473414yescomplemented
378C32#478111yescharacteristic, complemented
379C23—31no—
380C23—181no—
381C23—541no—
382Q8—272no—
383Q8—541no—
384D4—92no—
385D4—181no—
386D4—544no—
387D4—541no—
388D4—541no—
389D4—1082no—
390C2×C4—31no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—181no—
394C2×C4—544no—
395C2×C4—541no—
396C2×C4C2×D3×C3⋊D311yescharacteristic
397C8—21no—
398C8—61no—
399C2×C3—24no—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—124no—
403C2×C3—362no—
404C2×C3#6379514yes—
405C2×C3#6380614yes—
406C2×C3#6392314yes—
407C2×C3#6409111yescharacteristic
408C2×C3#6411311yescharacteristic
409C2×C3#6411711yescharacteristic
410D3—41no—
411D3—128no—
412D3—728no—
413D3—721no—
414D3—724no—
415C22—61no—
416C22—61no—
417C22—61no—
418C22—182no—
419C22—361no—
420C22—361no—
421C22—1081no—
422C22—1081no—
423C22—1081no—
424C22D3×C4⋊(C3⋊D3)11yescharacteristic
425C4—61no—
426C4—361no—
427C4—542no—
428C4C3⋊D3×D1211yescharacteristic
429C4#8903911yescharacteristic
430C3—24no—
431C3#1059096814yescomplemented
432C3#1059243311yescharacteristic, complemented
433C2—121no—
434C2—362no—
435C2—2161no—
436C2#1172331611yescharacteristic
437C2#1172333511yescharacteristic
438C2#1172354211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
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×#89021C2semidirect
2. C2×#88851C2semidirect
3. #11723516C2semidirect
4. C2×#88791C2semidirect
5. C3×#63648C2semidirect
6. C32×(C2×D3)⋊C8C2semidirect
7. #11722042C2semidirect
35. #63648D3semidirect
38. C3×(C2×D3)⋊C8D3semidirect
48. #6308D4semidirect
51. C3×C4⋊(C3⋊D3)D4semidirect
123. (C2×D3)⋊C8C3⋊D3semidirect
152. C4⋊(C3⋊D3)D12semidirect
259. C33#449semidirect
377. C32#4734semidirect
378. C32#4781semidirect
431. C3#10590968semidirect
432. C3#10592433semidirect

Maximal subgroups

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

subgroupindex
C2×#890212
C2×#888512
#117235162
C2×#887912
C3×#636482
C32×(C2×D3)⋊C82
#117220422
#105924333
#105909683