Finite Group Databasebeta

← #13161114

Subgroups — #13161114

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×#88861C211yesmaximal, characteristic, complemented
3C3×#64147C211yesmaximal, characteristic, complemented
4C2×C32×#213C211yesmaximal, characteristic, complemented
5#11722042C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592857—31nomaximal
9#10591519—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6290—21no—
16C2×D3×C3⋊Dic3—21no—
17#89014—22no—
18#89014—22no—
19#88861—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#3757—21no—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C32×#213—22no—
24C2×C32×C3⋊C8C2211yescharacteristic
25C2×#3830—31no—
26C2×#3795—34no—
27C2×C32×D8—31no—
28#64147D311yescharacteristic, complemented
29C2×#3754—31no—
30C2×#3754—34no—
31C2×#3754—64no—
32C3×#938—34no—
33C2×C3×#213D314yescomplemented
34#63659—31no—
35#63647—31no—
36#63643—31no—
37#63567—34no—
38#63562—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C33×D4—21no—
44C33×D4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47#6308—21no—
48#6308D412yescomplemented
49C3×#638—42no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C32×D12—21no—
55C32×D12D412yescomplemented
56#6290—24no—
57D3×C3⋊Dic3—24no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5158—94no—
61#5037—91no—
62C2×C32×D4—24no—
63C2×C32×D4—31no—
64C2×C32×D4C2×D314yes—
65C2×C32×D4C2×D311yescharacteristic
66C2×#638—21no—
67C2×#638—61no—
68C22×C3⋊Dic3—61no—
69#3830—62no—
70#3830—62no—
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C2×#634—31no—
73C2×#634—34no—
74C2×#634—64no—
75C2×C3×#67—64no—
76C2×C3×D12C2×D314yes—
77C2×C3×C4×D3—34no—
78C2×#625—64no—
79C2×D3×Dic3—64no—
80#3795—68no—
81#3795—68no—
82C32×D8—62no—
83C2×C32×C8—31no—
84#3757—21no—
85#3754—62no—
86#3754—68no—
87#3754—616no—
88C4⋊(C3⋊Dic3)C2×D311yescharacteristic
89C2×C3⋊(C3⋊C8)—31no—
90C2×C3⋊(C3⋊C8)—34no—
91C2×C3⋊(C3⋊C8)—64no—
92C3×#217—64no—
93C3×#213—28no—
94C3×C4⋊Dic3—34no—
95C2×C3×C3⋊C8C2×D314yes—
96C22×C33—41no—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33C2×D411yescharacteristic
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—
110C3×C3⋊Dic3—41no—
111C2×#237—94no—
112C2×#235—91no—
113C2×C3×D8—34no—
114C3×#121—91no—
115#938—34no—
116C2×#214—94no—
117C2×#214—91no—
118C2×#214—184no—
119C2×#213C3⋊D311yescharacteristic, complemented
120#832—94no—
121#830—91no—
122(C2×D3)⋊C8—94no—
123#816—94no—
124#810—91no—
125#809—91no—
126C23×C32—21no—
127C23×C32—24no—
128C23×C32—44no—
129C23×C32—61no—
130C22×C3⋊D3#6711yescharacteristic
131C22×C3×D3—24no—
132C22×C3×D3—34no—
133C32×D4—28no—
134C32×D4—24no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—24no—
138C32×D4—32no—
139C32×D4—44no—
140C32×D4—61no—
141C2×C32×C4—24no—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C22×D311yescharacteristic
144#638—42no—
145#638—64no—
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3—64no—
148C2×C3⋊Dic3—61no—
149C2×C3⋊Dic3—64no—
150C2×C3⋊Dic3—124no—
151C2×C3⋊Dic3#6711yescharacteristic
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—
160C3×#67—128no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—64no—
163C3×D12—24no—
164C3×D12#6718yescomplemented
165C3×C4×D3—64no—
166C3×C4×D3—64no—
167#625—616no—
168D3×Dic3—616no—
169C32×C8—61no—
170C3⋊(C3⋊C8)—68no—
171C3⋊(C3⋊C8)—64no—
172C3⋊(C3⋊C8)—61no—
173C3×C3⋊C8—24no—
174#449—271noSylow
175C2×C33—81no—
176C2×C33#12011yescharacteristic
177C2×C33#13511yescharacteristic
178C2×C33AGL(1,ℤ/8)11yescharacteristic
179C3×C3⋊D3—41no—
180C32×D3—42no—
181C2×C3×D4—24no—
182C2×C3×D4—34no—
183C2×C3×D4—181no—
184C2×C3×D4D3214yes—
185C2×C3×D4C2×C3⋊D311yescharacteristic
186C22×C3×C4—91no—
187C2×#67—64no—
188C2×#67—181no—
189C2×#67—184no—
190C22×Dic3—184no—
191#237—188no—
192#237—188no—
193#235—182no—
194#235—182no—
195C2×D12C2×C3⋊D311yescharacteristic
196C2×C4×D3—34no—
197C2×C4×D3—181no—
198C2×Dic6—94no—
199C2×Dic6—91no—
200C2×Dic6—184no—
201C3×D8—68no—
202C2×C3×C8—34no—
203C3×C4⋊C4—91no—
204C3×C22⋊C4—181no—
205#217—64no—
206#214—188no—
207#214—182no—
208#214—1816no—
209#213—22no—
210C4⋊Dic3—34no—
211C2×C3⋊C8—94no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—184no—
214C2×C3⋊C8C2×C3⋊D311yescharacteristic
215C22×C32—24no—
216C22×C32—48no—
217C22×C32—41no—
218C22×C32—44no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—44no—
222C22×C32—44no—
223C22×C32—62no—
224C22×C32—84no—
225C22×C32—84no—
226C22×C32—121no—
227C22×C32—121no—
228C22×C32D3×D414yes—
229C22×C32C2×#6711yescharacteristic
230C2×C3⋊D3—21no—
231C2×C3⋊D3—21no—
232C2×C3⋊D3—21no—
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×C4D3×D414yes—
242C32×C4D3×D411yescharacteristic
243C32×C4D3×D411yescharacteristic
244C32×C4C2×#6714yes—
245C3⋊Dic3—21no—
246C3⋊Dic3—41no—
247C3⋊Dic3—62no—
248C3⋊Dic3—68no—
249C3⋊Dic3—128no—
250C3×Dic3—64no—
251C3×Dic3—124no—
252C2×#41—271no—
253C2×SD16—271no—
254C2×D8—31no—
255#121—91no—
256#103—271no—
257#102—271no—
258C22⋊C8—271no—
259C33#44911yescharacteristic, complemented, Sylow
260C23×C3—24no—
261C23×C3—21no—
262C23×C3—44no—
263C23×C3—64no—
264C23×C3—91no—
265C22×D3—21no—
266C22×D3—34no—
267C3×D4—28no—
268C3×D4—24no—
269C3×D4—24no—
270C3×D4—21no—
271C3×D4—21no—
272C3×D4—38no—
273C3×D4—44no—
274C3×D4—64no—
275C3×D4—362no—
276C2×C3×C4—24no—
277C2×C3×C4—91no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4—181no—
281C2×C3×C4C2×D3214yes—
282C2×C3×C4C22×C3⋊D311yescharacteristic
283#67—128no—
284#67—184no—
285#67—1816no—
286C2×Dic3—34no—
287C2×Dic3—64no—
288C2×Dic3—1816no—
289C2×Dic3—184no—
290C2×Dic3—181no—
291C2×Dic3—364no—
292D12—21no—
293D12#63812yescomplemented
294C4×D3—64no—
295C4×D3—64no—
296C4×D3—184no—
297Dic6—98no—
298Dic6—92no—
299Dic6—188no—
300Dic6—184no—
301Dic6—181no—
302Dic6—364no—
303C3×C8—64no—
304C3⋊C8—21no—
305C3⋊C8—188no—
306C3⋊C8—184no—
307C3⋊C8—181no—
308C2×C32—24no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32—88no—
312C2×C32—84no—
313C2×C32—81no—
314C2×C32—122no—
315C2×C32#91114yes—
316C2×C32#91214yes—
317C2×C32#93211yescharacteristic
318C2×C32#93714yes—
319C2×C32#93711yescharacteristic
320C2×C32#95011yescharacteristic
321C3⋊D3—41no—
322C3×D3—48no—
323C3×D3—124no—
324#41—542no—
325#41—542no—
326C2×Q8—271no—
327C2×D4—31no—
328C2×D4—181no—
329C2×D4—541no—
330C2×D4D3×C3⋊D311yescharacteristic
331C22×C4—91no—
332C22×C4—541no—
333SD16—542no—
334D8—62no—
335C2×C8—31no—
336C2×C8—271no—
337C4⋊C4—91no—
338C22⋊C4—181no—
339C22×C3—24no—
340C22×C3—48no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—44no—
345C22×C3—44no—
346C22×C3—44no—
347C22×C3—68no—
348C22×C3—84no—
349C22×C3—84no—
350C22×C3—124no—
351C22×C3—124no—
352C22×C3—181no—
353C22×C3—181no—
354C22×C3—181no—
355C22×C3D3×#6714yes—
356C22×C3D4×C3⋊D311yescharacteristic
357C2×D3—22no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—64no—
361C2×D3—64no—
362C2×D3—64no—
363C3×C4—24no—
364C3×C4—24no—
365C3×C4—181no—
366C3×C4—361no—
367C3×C4D3×#6714yes—
368C3×C4#381114yes—
369C3×C4D4×C3⋊D311yescharacteristic
370C3×C4C2×#63811yescharacteristic
371Dic3—64no—
372Dic3—124no—
373Dic3—182no—
374Dic3—188no—
375Dic3—368no—
376C32—24no—
377C32#503711yescharacteristic, complemented
378C32#515814yescomplemented
379C23—21no—
380C23—61no—
381C23—91no—
382Q8—272no—
383Q8—541no—
384D4—21no—
385D4—21no—
386D4—32no—
387D4—61no—
388D4—362no—
389D4—544no—
390C2×C4—91no—
391C2×C4—181no—
392C2×C4—181no—
393C2×C4—181no—
394C2×C4—544no—
395C2×C4—541no—
396C2×C4C2×D3×C3⋊D311yescharacteristic
397C8—61no—
398C8—541no—
399C2×C3—24no—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—88no—
403C2×C3—84no—
404C2×C3—81no—
405C2×C3—128no—
406C2×C3—361no—
407C2×C3#6393214yes—
408C2×C3#6393314yes—
409C2×C3#6398314yes—
410C2×C3#6412011yescharacteristic
411C2×C3#6412111yescharacteristic
412C2×C3#6414611yescharacteristic
413D3—42no—
414D3—124no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—62no—
419C22—121no—
420C22—121no—
421C22—181no—
422C22—181no—
423C22—181no—
424C22C3⋊D3×#6711yescharacteristic
425C4—181no—
426C4—361no—
427C4—542no—
428C4D3×#63811yescharacteristic
429C4#8903911yescharacteristic
430C3—24no—
431C3#1059151914yescomplemented
432C3#1059285711yescharacteristic, complemented
433C2—81no—
434C2—122no—
435C2—361no—
436C2#1172355911yescharacteristic
437C2#1172356011yescharacteristic
438C2#1172364311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89014C2semidirect
2. C2×#88861C2semidirect
3. C3×#64147C2semidirect
4. C2×C32×#213C2semidirect
5. #11722042C2semidirect
6. #11722031C2semidirect
7. #11722023C2semidirect
28. #64147D3semidirect
33. C2×C3×#213D3semidirect
48. #6308D4semidirect
55. C32×D12D4semidirect
119. C2×#213C3⋊D3semidirect
164. C3×D12#67semidirect
259. C33#449semidirect
293. D12#638semidirect
377. C32#5037semidirect
378. C32#5158semidirect
431. C3#10591519semidirect
432. 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×#888612
C3×#641472
C2×C32×#2132
#117220422
#117220312
#117220232
#105928573
#105915193