Finite Group Databasebeta

← #13160093

Subgroups — #13160093

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×#89014C211yesmaximal, characteristic, complemented
2C2×C32×D24C211yesmaximal, characteristic, complemented
3#11723522C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic, complemented
5#11722186C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592857—31nomaximal
9#10590962—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C2×#6290—21no—
16C2×D3×C3⋊Dic3—21no—
17#89014—22no—
18#89014—22no—
19C2×C33×C8C2211yescharacteristic
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×D24—22no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88794—22no—
25C2×#3830—31no—
26C2×#3795—34no—
27C2×C32×D8—31no—
28#64147—31no—
29C2×#3754—31no—
30C2×C3×D24D314yescomplemented
31#63908—34no—
32C2×#3716—34no—
33#63659—31no—
34#63647D311yescharacteristic, complemented
35#63643—31no—
36#63643—34no—
37#63643—64no—
38C3×(C2×D3)⋊C8—34no—
39#63563—34no—
40C22×C3×C3⋊D3D411yescharacteristic
41C22×C32×D3—21no—
42C22×C32×D3—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)—21no—
46#6308—21no—
47#6308D412yescomplemented
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C32×D12—21no—
52C32×D12—21no—
53C32×D12—21no—
54C32×D12D412yescomplemented
55#6290—24no—
56#6290—42no—
57D3×C3⋊Dic3—24no—
58C33×C8—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5158—94no—
61#4734—91no—
62C2×C32×D4—31no—
63C2×C32×D4—31no—
64C2×#638—61no—
65C2×#638—61no—
66C22×C3⋊Dic3—61no—
67#3830—62no—
68#3830—62no—
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×#634—31no—
71C2×#634—34no—
72C2×#634—64no—
73C2×C3×D12C2×D314yes—
74C2×C3×D12C2×D314yes—
75C2×C3×C4×D3—34no—
76C2×#625—64no—
77C2×#625—64no—
78C2×D3×Dic3—64no—
79#3795—68no—
80#3795—68no—
81C32×D8—62no—
82C2×C32×C8—24no—
83C2×C32×C8C2×D314yes—
84C2×C32×C8C2×D311yescharacteristic
85#3757—61no—
86#3754—62no—
87C4⋊(C3⋊Dic3)—31no—
88C4⋊(C3⋊Dic3)—34no—
89C4⋊(C3⋊Dic3)—64no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3×C3⋊C8—34no—
92C3×D24—28no—
93(C2×D3)⋊Dic3—64no—
94#3716—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3—21no—
99C2×C32×D3—22no—
100C2×C32×D3—41no—
101C2×C32×D3—41no—
102C2×C32×D3—41no—
103C2×C32×D3—41no—
104C2×C32×D3—41no—
105C33×C4C2×D411yescharacteristic
106C33×C4C2×D411yescharacteristic
107C3⋊(C3⋊Dic3)—22no—
108C3⋊(C3⋊Dic3)—41no—
109C3×C3⋊Dic3—21no—
110C2×#237—94no—
111C2×#235—91no—
112C2×C3×D8—34no—
113#938—94no—
114C2×#214—94no—
115#930—91no—
116C2×D24C3⋊D311yescharacteristic, complemented
117C2×#204—91no—
118C3×C22⋊C8—91no—
119#832—94no—
120#821—91no—
121(C2×D3)⋊C8—34no—
122#816—94no—
123#816—91no—
124#816—184no—
125C23×C32—61no—
126C23×C32—61no—
127C22×C3⋊D3D1211yescharacteristic
128C22×C3×D3—24no—
129C22×C3×D3—24no—
130C22×C3×D3—34no—
131C32×D4—32no—
132C32×D4—61no—
133C32×D4—61no—
134C32×D4—61no—
135C2×C32×C4—24no—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C22×D311yescharacteristic
138#638—64no—
139#638—122no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—64no—
144C2×C3⋊Dic3—64no—
145C2×C3⋊Dic3—124no—
146C2×C3⋊Dic3—124no—
147C2×C3⋊Dic3D1211yescharacteristic
148C4×C3⋊D3—21no—
149C4×C3⋊D3—21no—
150#634—32no—
151#634—38no—
152#634—68no—
153#634—61no—
154#634—64no—
155#634—124no—
156C2×C3×Dic3—34no—
157C3×D12—24no—
158C3×D12—24no—
159C3×D12—24no—
160C3×D12#6718yescomplemented
161C3×C4×D3—64no—
162C3×C4×D3—64no—
163#625—616no—
164#625—128no—
165D3×Dic3—616no—
166C32×C8—21no—
167C32×C8—24no—
168C32×C8—44no—
169C3⋊(C3⋊C8)—21no—
170C3×C3⋊C8—64no—
171#449—271noSylow
172C2×C33#12011yescharacteristic
173C2×C33#13511yescharacteristic
174C2×C33AGL(1,ℤ/8)11yescharacteristic
175C3×C3⋊D3—41no—
176C32×D3—42no—
177C32×D3—81no—
178C2×C3×D4—34no—
179C2×C3×D4—34no—
180C22×C3×C4—91no—
181C2×#67—181no—
182C2×#67—184no—
183C2×#67—181no—
184C2×#67—184no—
185C22×Dic3—184no—
186#237—188no—
187#237—188no—
188#235—182no—
189#235—182no—
190C2×D12C2×C3⋊D311yescharacteristic
191C2×D12C2×C3⋊D311yescharacteristic
192C2×C4×D3—34no—
193C2×C4×D3—181no—
194C2×Dic6—94no—
195C2×Dic6—91no—
196C2×Dic6—184no—
197C3×D8—68no—
198C2×C3×C8—24no—
199C2×C3×C8—91no—
200C2×C3×C8D3214yes—
201C2×C3×C8C2×C3⋊D311yescharacteristic
202#217—184no—
203#214—188no—
204(C2×D3)⋊C4—181no—
205C4⋊Dic3—94no—
206C4⋊Dic3—91no—
207C4⋊Dic3—184no—
208C2×C3⋊C8—34no—
209D24—22no—
210#204—182no—
211C22×C32—24no—
212C22×C32—62no—
213C22×C32—121no—
214C22×C32—121no—
215C22×C32—121no—
216C22×C32—121no—
217C22×C32—121no—
218C22×C32C2×D1211yescharacteristic
219C22×C32D3×D414yes—
220C2×C3⋊D3—21no—
221C2×C3⋊D3—21no—
222C2×C3⋊D3—21no—
223C2×C3×D3—28no—
224C2×C3×D3—44no—
225C2×C3×D3—44no—
226C2×C3×D3—44no—
227C2×C3×D3—44no—
228C2×C3×D3—44no—
229C2×C3×D3—64no—
230C2×C3×D3—64no—
231C2×C3×D3—64no—
232C32×C4—24no—
233C32×C4—24no—
234C32×C4D3×D414yes—
235C32×C4D3×D411yescharacteristic
236C32×C4D3×D411yescharacteristic
237C32×C4C2×#6714yes—
238C3⋊Dic3—21no—
239C3⋊Dic3—62no—
240C3⋊Dic3—68no—
241C3⋊Dic3—128no—
242C3⋊Dic3—121no—
243C3⋊Dic3—124no—
244C3⋊Dic3—244no—
245C3×Dic3—64no—
246C2×#41—271no—
247C2×SD16—271no—
248C2×D8—31no—
249#121—271no—
250#103—271no—
251#102—271no—
252C22⋊C8—91no—
253C33#44911yescharacteristic, complemented, Sylow
254C23×C3—64no—
255C23×C3—64no—
256C23×C3—91no—
257C22×D3—21no—
258C22×D3—21no—
259C22×D3—34no—
260C3×D4—38no—
261C3×D4—64no—
262C3×D4—64no—
263C3×D4—64no—
264C2×C3×C4—24no—
265C2×C3×C4—91no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4C2×D3214yes—
269C2×C3×C4C22×C3⋊D311yescharacteristic
270#67—184no—
271#67—1816no—
272#67—362no—
273#67—368no—
274C2×Dic3—34no—
275C2×Dic3—1816no—
276C2×Dic3—184no—
277C2×Dic3—181no—
278C2×Dic3—184no—
279C2×Dic3—181no—
280C2×Dic3—364no—
281C2×Dic3—364no—
282D12—21no—
283D12—21no—
284D12—21no—
285D12#63812yescomplemented
286C4×D3—64no—
287C4×D3—64no—
288C4×D3—184no—
289Dic6—98no—
290Dic6—92no—
291Dic6—188no—
292Dic6—184no—
293Dic6—181no—
294Dic6—364no—
295C3×C8—24no—
296C3×C8—21no—
297C3×C8—44no—
298C3×C8—181no—
299C3⋊C8—64no—
300C2×C32—24no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—122no—
304C2×C32—241no—
305C2×C32#88211yescharacteristic
306C2×C32#90411yescharacteristic
307C2×C32#90811yescharacteristic
308C2×C32#91114yes—
309C2×C32#91214yes—
310C2×C32#93714yes—
311C3⋊D3—41no—
312C3×D3—48no—
313C3×D3—84no—
314C3×D3—124no—
315#41—542no—
316#41—542no—
317C2×Q8—271no—
318C2×D4—31no—
319C2×D4—31no—
320C2×D4—541no—
321C2×D4—541no—
322C22×C4—91no—
323C22×C4—541no—
324SD16—542no—
325D8—62no—
326C2×C8—91no—
327C2×C8D3×C3⋊D311yescharacteristic
328C4⋊C4—271no—
329C22⋊C4—541no—
330C22×C3—24no—
331C22×C3—68no—
332C22×C3—124no—
333C22×C3—124no—
334C22×C3—124no—
335C22×C3—124no—
336C22×C3—124no—
337C22×C3—181no—
338C22×C3—181no—
339C22×C3—181no—
340C22×C3D3×D1214yes—
341C22×C3D4×C3⋊D311yescharacteristic
342C2×D3—22no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—41no—
347C2×D3—41no—
348C2×D3—64no—
349C2×D3—64no—
350C2×D3—64no—
351C3×C4—24no—
352C3×C4—24no—
353C3×C4—181no—
354C3×C4#380214yes—
355C3×C4D3×#6714yes—
356C3×C4D4×C3⋊D311yescharacteristic
357C3×C4C2×#63811yescharacteristic
358Dic3—64no—
359Dic3—182no—
360Dic3—188no—
361Dic3—368no—
362Dic3—364no—
363Dic3—361no—
364Dic3—724no—
365C32—24no—
366C32#473411yescharacteristic, complemented
367C32#515814yescomplemented
368C23—61no—
369C23—61no—
370C23—91no—
371Q8—272no—
372Q8—541no—
373D4—32no—
374D4—61no—
375D4—61no—
376D4—61no—
377D4—544no—
378D4—1082no—
379C2×C4—91no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4—544no—
383C2×C4—541no—
384C2×C4—541no—
385C2×C4C2×D3×C3⋊D311yescharacteristic
386C8—21no—
387C8—181no—
388C2×C3—24no—
389C2×C3—24no—
390C2×C3—24no—
391C2×C3—128no—
392C2×C3—244no—
393C2×C3—361no—
394C2×C3#6379514yes—
395C2×C3#6380714yes—
396C2×C3#6392214yes—
397C2×C3#6412011yescharacteristic
398C2×C3#6412111yescharacteristic
399C2×C3#6414611yescharacteristic
400D3—42no—
401D3—81no—
402D3—124no—
403C22—62no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22—121no—
409C22—181no—
410C22—181no—
411C22—181no—
412C22C3⋊D3×D1211yescharacteristic
413C4—181no—
414C4—542no—
415C4—1081no—
416C4#8902611yescharacteristic
417C4D3×#63811yescharacteristic
418C3—24no—
419C3#1059096214yescomplemented
420C3#1059285711yescharacteristic, complemented
421C2—122no—
422C2—241no—
423C2—361no—
424C2#1172332011yescharacteristic
425C2#1172333911yescharacteristic
426C2#1172354011yescharacteristic
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×C32×D24C2semidirect
3. #11723522C2semidirect
4. C2×#88794C2semidirect
5. #11722186C2semidirect
6. C3×#63647C2semidirect
7. #11722032C2semidirect
30. C2×C3×D24D3semidirect
34. #63647D3semidirect
47. #6308D4semidirect
54. C32×D12D4semidirect
116. C2×D24C3⋊D3semidirect
160. C3×D12#67semidirect
253. C33#449semidirect
285. D12#638semidirect
366. C32#4734semidirect
367. C32#5158semidirect
419. C3#10590962semidirect
420. C3#10592857semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
C2×C32×D242
#117235222
C2×#887942
#117221862
C3×#636472
#117220322
#105928573
#105909623