Finite Group Databasebeta

← #13160084

Subgroups — #13160084

Order 1728.

1140 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1140 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×C3×#3745C211yesmaximal, characteristic, complemented
3#11723515C211yesmaximal, characteristic, complemented
4C2×#88796C211yesmaximal, characteristic, complemented
5#11722186C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592433—31nomaximal
9#10590962—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6290—21no—
15C2×#6290—21no—
16C2×Dic3×C3⋊D3—21no—
17#89021—22no—
18#89021—22no—
19C2×C33×C8C2211yescharacteristic
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C3×#3745—22no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88803—21no—
24#88796—22no—
25C2×#3828—31no—
26C2×#3795—34no—
27#64139—31no—
28C2×#3745D311yescharacteristic, complemented
29C2×#3744—31no—
30C2×C3×D24—34no—
31#63908—34no—
32C2×#3716—34no—
33C32×C22⋊C8—31no—
34#63648—31no—
35#63643—31no—
36#63643—34no—
37#63643—64no—
38C3×(C2×D3)⋊C8D314yescomplemented
39#63563—34no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—21no—
42C22×C32×D3D411yescharacteristic
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)—21no—
46#6308—21no—
47#6308D412yescomplemented
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)D412yescomplemented
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55#6290—24no—
56#6290—42no—
57Dic3×C3⋊D3—24no—
58C33×C8—21no—
59C32×C3⋊C8—21no—
60#5158—91no—
61#4734—94no—
62C22×C32×C4—31no—
63C2×#638—61no—
64C2×#638—61no—
65#3828—62no—
66#3828—62no—
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3—61no—
70C2×#634—31no—
71C2×#634—34no—
72C2×#634—64no—
73C2×C3×D12—34no—
74C2×C3×D12—34no—
75C2×C3×C4×D3C2×D314yes—
76C2×#625—64no—
77C2×#625—64no—
78C2×D3×Dic3—64no—
79#3795—68no—
80#3795—68no—
81C2×C32×C8—24no—
82C2×C32×C8—31no—
83C2×C32×C8C2×D314yes—
84C2×C32×C8C2×D311yescharacteristic
85#3752—61no—
86C4⋊(C3⋊Dic3)—31no—
87C4⋊(C3⋊Dic3)—34no—
88C4⋊(C3⋊Dic3)—64no—
89#3745—22no—
90#3744—62no—
91C2×C3×C3⋊C8C2×D314yes—
92C3×D24—68no—
93(C2×D3)⋊Dic3—64no—
94#3716—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—22no—
97C2×C3×C3⋊D3—41no—
98C2×C3×C3⋊D3—41no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3—41no—
102C2×C32×D3—21no—
103C2×C32×D3—21no—
104C2×C32×D3—21no—
105C33×C4C2×D411yescharacteristic
106C33×C4C2×D411yescharacteristic
107C3⋊(C3⋊Dic3)—22no—
108C3⋊(C3⋊Dic3)—41no—
109C32×Dic3—21no—
110C2×#237—91no—
111C2×#235—94no—
112C2×C3×D8—91no—
113#938—91no—
114C2×#214—91no—
115#930—94no—
116C2×D24—34no—
117C2×#204—94no—
118C3×C22⋊C8—34no—
119#832—91no—
120#821—94no—
121(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
122#816—94no—
123#816—91no—
124#816—184no—
125C23×C32—31no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3—21no—
128C22×C3×D3—64no—
129C22×C3×D3—64no—
130C22×C3×D3D1214yes—
131C2×C32×C4—24no—
132C2×C32×C4—31no—
133C2×C32×C4—61no—
134C2×C32×C4—61no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C22×D311yescharacteristic
137#638—64no—
138#638—122no—
139C2×C3⋊Dic3—61no—
140C2×C3⋊Dic3—61no—
141C2×C3⋊Dic3—64no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—124no—
144C2×C3⋊Dic3—124no—
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)—21no—
148C4⋊(C3⋊D3)#6712yescomplemented
149C4×C3⋊D3—64no—
150#634—32no—
151#634—38no—
152#634—68no—
153#634—61no—
154#634—64no—
155#634—124no—
156C2×C3×Dic3D1214yes—
157C3×D12—38no—
158C3×D12—64no—
159C3×D12—64no—
160C3×D12—64no—
161C3×C4×D3—24no—
162C3×C4×D3—24no—
163#625—616no—
164#625—128no—
165D3×Dic3—616no—
166C32×C8—21no—
167C32×C8—24no—
168C32×C8—44no—
169C32×C8—61no—
170C3×C3⋊C8—24no—
171#449—271noSylow
172C2×C33#12011yescharacteristic
173C2×C33#13511yescharacteristic
174C2×C33AGL(1,ℤ/8)11yescharacteristic
175C3×C3⋊D3—42no—
176C3×C3⋊D3—81no—
177C32×D3—41no—
178C2×C3×D4—91no—
179C2×C3×D4—91no—
180C22×C3×C4—34no—
181C2×#67—184no—
182C2×#67—181no—
183C2×#67—184no—
184C2×#67—181no—
185C22×Dic3—181no—
186#237—182no—
187#237—182no—
188#235—188no—
189#235—188no—
190C2×D12—34no—
191C2×D12—34no—
192C2×C4×D3—184no—
193C2×C4×D3C2×C3⋊D311yescharacteristic
194C2×Dic6—94no—
195C2×Dic6—91no—
196C2×Dic6—184no—
197C3×D8—182no—
198C2×C3×C8—24no—
199C2×C3×C8—34no—
200C2×C3×C8D3214yes—
201C2×C3×C8C2×C3⋊D311yescharacteristic
202#217—181no—
203#214—182no—
204(C2×D3)⋊C4—184no—
205C4⋊Dic3—94no—
206C4⋊Dic3—91no—
207C4⋊Dic3—184no—
208C2×C3⋊C8C2×C3⋊D311yescharacteristic
209D24—68no—
210#204—188no—
211C22×C32—24no—
212C22×C32—61no—
213C22×C32—61no—
214C22×C32—61no—
215C22×C32C2×D1214yes—
216C22×C32D3×D411yescharacteristic
217C2×C3⋊D3—22no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3—41no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3×D3—24no—
224C2×C3×D3—24no—
225C2×C3×D3—24no—
226C2×C3×D3—68no—
227C2×C3×D3—124no—
228C2×C3×D3—124no—
229C2×C3×D3—124no—
230C2×C3×D3—124no—
231C2×C3×D3—124no—
232C32×C4—24no—
233C32×C4—24no—
234C32×C4—61no—
235C32×C4D3×D414yes—
236C32×C4D3×D414yes—
237C32×C4D3×D411yescharacteristic
238C32×C4C2×#6711yescharacteristic
239C3⋊Dic3—62no—
240C3⋊Dic3—68no—
241C3⋊Dic3—128no—
242C3⋊Dic3—121no—
243C3⋊Dic3—124no—
244C3⋊Dic3—244no—
245C3×Dic3—24no—
246C2×#41—271no—
247C2×SD16—271no—
248C2×D8—91no—
249#121—271no—
250#103—271no—
251#102—271no—
252C22⋊C8—31no—
253C33#44911yescharacteristic, complemented, Sylow
254C23×C3—34no—
255C23×C3—181no—
256C23×C3—181no—
257C22×D3—64no—
258C22×D3—64no—
259C22×D3C4⋊(C3⋊D3)11yescharacteristic
260C3×D4—92no—
261C3×D4—181no—
262C3×D4—181no—
263C3×D4—181no—
264C2×C3×C4—24no—
265C2×C3×C4—34no—
266C2×C3×C4—64no—
267C2×C3×C4—64no—
268C2×C3×C4C2×D3214yes—
269C2×C3×C4C22×C3⋊D311yescharacteristic
270#67—184no—
271#67—1816no—
272#67—362no—
273#67—368no—
274C2×Dic3—184no—
275C2×Dic3—184no—
276C2×Dic3—181no—
277C2×Dic3—184no—
278C2×Dic3—181no—
279C2×Dic3—364no—
280C2×Dic3—364no—
281C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
282D12—38no—
283D12—64no—
284D12—64no—
285D12—64no—
286C4×D3—21no—
287C4×D3—21no—
288C4×D3—1816no—
289Dic6—98no—
290Dic6—92no—
291Dic6—188no—
292Dic6—184no—
293Dic6—181no—
294Dic6—364no—
295C3×C8—24no—
296C3×C8—21no—
297C3×C8—44no—
298C3×C8—64no—
299C3⋊C8—21no—
300C2×C32—24no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—121no—
304C2×C32#88214yes—
305C2×C32#90414yes—
306C2×C32#90814yes—
307C2×C32#91111yescharacteristic
308C2×C32#91211yescharacteristic
309C2×C32#93711yescharacteristic
310C3⋊D3—42no—
311C3⋊D3—81no—
312C3×D3—44no—
313C3×D3—128no—
314C3×D3—244no—
315#41—542no—
316#41—542no—
317C2×Q8—271no—
318C2×D4—91no—
319C2×D4—91no—
320C2×D4—541no—
321C2×D4—541no—
322C22×C4—31no—
323C22×C4—541no—
324SD16—542no—
325D8—182no—
326C2×C8—31no—
327C2×C8D3×C3⋊D311yescharacteristic
328C4⋊C4—271no—
329C22⋊C4—541no—
330C22×C3—24no—
331C22×C3—64no—
332C22×C3—64no—
333C22×C3—64no—
334C22×C3—182no—
335C22×C3—361no—
336C22×C3—361no—
337C22×C3—361no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3D3×D1214yes—
341C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
342C2×D3—21no—
343C2×D3—21no—
344C2×D3—21no—
345C2×D3—68no—
346C2×D3—124no—
347C2×D3—124no—
348C2×D3—124no—
349C2×D3—124no—
350C2×D3—124no—
351C3×C4—24no—
352C3×C4—24no—
353C3×C4—64no—
354C3×C4#380214yes—
355C3×C4D3×#6714yes—
356C3×C4D4×C3⋊D311yescharacteristic
357C3×C4D4×C3⋊D311yescharacteristic
358Dic3—21no—
359Dic3—182no—
360Dic3—188no—
361Dic3—368no—
362Dic3—364no—
363Dic3—361no—
364Dic3—724no—
365C32—24no—
366C32#473414yescomplemented
367C32#515811yescharacteristic, complemented
368C23—31no—
369C23—181no—
370C23—181no—
371Q8—272no—
372Q8—541no—
373D4—92no—
374D4—181no—
375D4—181no—
376D4—181no—
377D4—544no—
378D4—1082no—
379C2×C4—31no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—544no—
383C2×C4—541no—
384C2×C4—541no—
385C2×C4C2×D3×C3⋊D311yescharacteristic
386C8—21no—
387C8—61no—
388C2×C3—24no—
389C2×C3—24no—
390C2×C3—24no—
391C2×C3—124no—
392C2×C3—362no—
393C2×C3—721no—
394C2×C3#6379514yes—
395C2×C3#6380714yes—
396C2×C3#6392214yes—
397C2×C3#6409111yescharacteristic
398C2×C3#6411311yescharacteristic
399C2×C3#6411711yescharacteristic
400D3—41no—
401D3—128no—
402D3—244no—
403C22—61no—
404C22—61no—
405C22—61no—
406C22—182no—
407C22—361no—
408C22—361no—
409C22—361no—
410C22—361no—
411C22—361no—
412C22D3×C4⋊(C3⋊D3)11yescharacteristic
413C4—61no—
414C4—542no—
415C4—1081no—
416C4#8902611yescharacteristic
417C4C3⋊D3×#6711yescharacteristic
418C3—24no—
419C3#1059096214yescomplemented
420C3#1059243311yescharacteristic, complemented
421C2—121no—
422C2—362no—
423C2—721no—
424C2#1172331611yescharacteristic
425C2#1172333611yescharacteristic
426C2#1172354111yescharacteristic
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×C3×#3745C2semidirect
3. #11723515C2semidirect
4. C2×#88796C2semidirect
5. #11722186C2semidirect
6. C32×(C2×D3)⋊C8C2semidirect
7. #11722034C2semidirect
28. C2×#3745D3semidirect
38. C3×(C2×D3)⋊C8D3semidirect
47. #6308D4semidirect
51. C3×C4⋊(C3⋊D3)D4semidirect
121. (C2×D3)⋊C8C3⋊D3semidirect
148. C4⋊(C3⋊D3)#67semidirect
253. C33#449semidirect
366. C32#4734semidirect
367. C32#5158semidirect
419. C3#10590962semidirect
420. 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×C3×#37452
#117235152
C2×#887962
#117221862
C32×(C2×D3)⋊C82
#117220342
#105924333
#105909623