Finite Group Databasebeta

← #13160040

Subgroups — #13160040

Order 1728.

1328 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1328 conjugacy classes of subgroups — too many to draw. Only its 162 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×C3⋊D3×D12C211yesmaximal, characteristic
2C2×C32×#204C211yesmaximal, characteristic, complemented
3#11723435C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592868—31nomaximal
9#10590942—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic, complemented
15C2×#6291—21no—
16C3⋊D3×D12—22no—
17C3⋊D3×D12—22no—
18C2×C33×C8C2211yescharacteristic, complemented
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
21C32×#204—22no—
22#88806—21no—
23#88803—21no—
24#88795—22no—
25C2×D4×C3⋊D3—31no—
26C2×D3×D12—34no—
27C2×C32×SD16—31no—
28#64155—31no—
29C2×#3755—31no—
30C2×C3×#204D314yescomplemented
31#63856—34no—
32C2×#3717—34no—
33#63659—31no—
34#63648—31no—
35#63648—34no—
36#63648—64no—
37#63647D311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—34no—
39#63563—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C2×D3×C3⋊D3—24no—
44C2×D3×C3⋊D3—42no—
45C2×D3×C3⋊D3—42no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊(C3⋊D3))—21no—
49C4⋊(C3⋊(C3⋊D3))D412yes—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C2×C32×Dic3—21no—
54C32×D12—21no—
55C32×D12D412yes—
56C32×Dic6—21no—
57C32×Dic6—21no—
58#6291—24no—
59C33×C8—21no—
60C3×C3⋊(C3⋊C8)—21no—
61#5169—94no—
62#4731—91no—
63C23×C3⋊D3—61no—
64C22×D32—64no—
65C2×C32×Q8—31no—
66C2×C32×D4—31no—
67C2×#638—61no—
68D4×C3⋊D3—62no—
69D4×C3⋊D3—62no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—34no—
72C2×C4⋊(C3⋊D3)—64no—
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C2×C3×D12C2×D314yes—
75C2×C3×C4×D3—34no—
76C2×C3×Dic6C2×D314yescomplemented
77C2×#626—64no—
78D3×D12—68no—
79D3×D12—68no—
80C32×SD16—62no—
81C2×C32×C8—24no—
82C2×C32×C8C2×D314yescomplemented
83C2×C32×C8C2×D311yescharacteristic, complemented
84#3755—62no—
85#3752—61no—
86C4⋊(C3⋊Dic3)—31no—
87C4⋊(C3⋊Dic3)—34no—
88C4⋊(C3⋊Dic3)—64no—
89#3750—61no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
91C2×C3×C3⋊C8—34no—
92C3×#204—28no—
93#3724—64no—
94(C2×D3)⋊Dic3—64no—
95#3717—68no—
96C22×C33C2×D411yescharacteristic
97C2×C3⋊(C3⋊D3)—22no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3SD1612yescomplemented
103C2×C32×D3—22no—
104C2×C32×D3—41no—
105C2×C32×D3—41no—
106D3×C3⋊D3—48no—
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3⋊(C3⋊Dic3)—41no—
110C3×C3⋊Dic3—21no—
111C32×Dic3—41no—
112C2×D3×D4—94no—
113C22×D12—91no—
114C2×C3×SD16—34no—
115#946—94no—
116C2×#215—94no—
117#924—91no—
118C2×#204—91no—
119C2×#204C3⋊D311yescharacteristic, complemented
120C3×C22⋊C8—91no—
121#832—94no—
122#821—91no—
123#821—94no—
124#821—91no—
125#821—184no—
126(C2×D3)⋊C8—34no—
127C23×C32—61no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—61no—
130C22×C3⋊D3—64no—
131C22×C3⋊D3—124no—
132C22×C3⋊D3—122no—
133C22×C3⋊D3—122no—
134C22×C3⋊D3D1211yescharacteristic
135C22×C3×D3—24no—
136C22×C3×D3—34no—
137C2×D32—616no—
138C2×D32—128no—
139C2×D32—128no—
140C32×Q8—61no—
141C32×Q8—61no—
142C32×D4—32no—
143C32×D4—61no—
144C2×C32×C4—24no—
145C2×C32×C4—61no—
146C2×C32×C4C22×D314yes—
147C2×C32×C4C22×D311yescharacteristic
148#638—64no—
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—64no—
151C2×C3⋊Dic3—124no—
152C2×C3⋊Dic3D1211yescharacteristic
153C4⋊(C3⋊D3)—32no—
154C4⋊(C3⋊D3)—38no—
155C4⋊(C3⋊D3)—68no—
156C4⋊(C3⋊D3)—61no—
157C4⋊(C3⋊D3)—64no—
158C4⋊(C3⋊D3)—124no—
159C4×C3⋊D3—21no—
160C4×C3⋊D3—21no—
161C2×C3×Dic3—24no—
162C2×C3×Dic3—34no—
163C3×D12—24no—
164C3×D12#6718yes—
165C3×C4×D3—64no—
166C3×C4×D3—64no—
167C3×Dic6—24no—
168C3×Dic6—24no—
169#626—616no—
170C32×C8—21no—
171C32×C8—24no—
172C32×C8—44no—
173C3⋊(C3⋊C8)—21no—
174C3×C3⋊C8—64no—
175#450—271noSylow
176C2×C33#12011yescharacteristic
177C2×C33C2×SD1611yescharacteristic
178C2×C33AGL(1,ℤ/8)11yescharacteristic
179C3⋊(C3⋊D3)—42no—
180C3×C3⋊D3—22no—
181C32×D3—42no—
182C23×D3—181no—
183C23×D3—184no—
184C2×C3×Q8—34no—
185C2×C3×D4—34no—
186C22×C3×C4—91no—
187C2×#67—184no—
188D3×D4—188no—
189D3×D4—188no—
190C2×D12—94no—
191C2×D12—91no—
192C2×D12—184no—
193C2×D12—182no—
194C2×D12—181no—
195C2×D12—182no—
196C2×D12C2×C3⋊D311yescharacteristic
197C2×C4×D3—34no—
198C2×Dic6C2×C3⋊D311yescharacteristic, complemented
199C3×SD16—68no—
200C2×C3×C8—24no—
201C2×C3×C8—91no—
202C2×C3×C8D3214yescomplemented
203C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
204#217—181no—
205#215—188no—
206(C2×D3)⋊C4—184no—
207C4⋊Dic3—94no—
208C4⋊Dic3—91no—
209C4⋊Dic3—184no—
210Dic3⋊C4—181no—
211Dic3⋊C4—184no—
212C2×C3⋊C8—34no—
213#204—22no—
214#204—182no—
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—62no—
224C2×C3⋊D3—68no—
225C2×C3⋊D3—128no—
226C2×C3⋊D3—128no—
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#20412yescomplemented
234C2×C3×D3—28no—
235C2×C3×D3—38no—
236C2×C3×D3—44no—
237C2×C3×D3—44no—
238C2×C3×D3—64no—
239C2×C3×D3—64no—
240D32—1232no—
241C32×C4—24no—
242C32×C4—24no—
243C32×C4—121no—
244C32×C4D3×D414yes—
245C32×C4D3×D411yescharacteristic
246C32×C4D3×D411yescharacteristic
247C32×C4C2×#6714yes—
248C3⋊Dic3—21no—
249C3⋊Dic3—121no—
250C3⋊Dic3—124no—
251C3⋊Dic3—244no—
252C3×Dic3—44no—
253C3×Dic3—64no—
254C22×D4—271no—
255C2×SD16—31no—
256C2×SD16—271no—
257C22⋊Q8—271no—
258#102—271no—
259#102—271no—
260C22⋊C8—91no—
261C33#45011yescharacteristic, complemented, Sylow
262C23×C3—64no—
263C23×C3—91no—
264C22×D3—21no—
265C22×D3—34no—
266C22×D3—184no—
267C22×D3—1816no—
268C22×D3—184no—
269C22×D3—181no—
270C22×D3—364no—
271C22×D3—368no—
272C22×D3—362no—
273C22×D3—362no—
274C22×D3—368no—
275C3×Q8—64no—
276C3×Q8—64no—
277C3×D4—38no—
278C3×D4—64no—
279C2×C3×C4—24no—
280C2×C3×C4—64no—
281C2×C3×C4—91no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4C2×D3214yes—
285C2×C3×C4C22×C3⋊D311yescharacteristic
286#67—1816no—
287C2×Dic3—21no—
288C2×Dic3—34no—
289C2×Dic3—184no—
290C2×Dic3—181no—
291C2×Dic3—364no—
292D12—21no—
293D12—98no—
294D12—92no—
295D12—188no—
296D12—184no—
297D12—184no—
298D12—181no—
299D12—364no—
300D12#63812yes—
301C4×D3—64no—
302C4×D3—64no—
303Dic6—21no—
304Dic6—21no—
305C3×C8—24no—
306C3×C8—21no—
307C3×C8—44no—
308C3×C8—181no—
309C3⋊C8—64no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—122no—
314C2×C32#88211yescharacteristic
315C2×C32C2×#20411yescharacteristic
316C2×C32#90811yescharacteristic
317C2×C32D3×SD1614yes—
318C2×C32#91414yes—
319C2×C32#93714yes—
320C3⋊D3—22no—
321C3⋊D3—122no—
322C3⋊D3—128no—
323C3⋊D3—248no—
324C3×D3—48no—
325C3×D3—68no—
326C24—541no—
327C2×Q8—31no—
328C2×D4—31no—
329C2×D4—271no—
330C2×D4—541no—
331C2×D4—542no—
332C2×D4—542no—
333C22×C4—91no—
334SD16—62no—
335SD16—542no—
336C2×C8—91no—
337C2×C8D3×C3⋊D311yescharacteristic, complemented
338C4⋊C4—271no—
339C4⋊C4—541no—
340C22⋊C4—541no—
341C22×C3—24no—
342C22×C3—68no—
343C22×C3—92no—
344C22×C3—124no—
345C22×C3—124no—
346C22×C3—181no—
347C22×C3—181no—
348C22×C3D3×D1214yes—
349C22×C3D4×C3⋊D311yescharacteristic
350C2×D3—22no—
351C2×D3—38no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—64no—
355C2×D3—64no—
356C2×D3—182no—
357C2×D3—188no—
358C2×D3—368no—
359C2×D3—368no—
360C2×D3—3632no—
361C2×D3—364no—
362C2×D3—364no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—724no—
366C2×D3—724no—
367C3×C4—24no—
368C3×C4—24no—
369C3×C4—124no—
370C3×C4—181no—
371C3×C4#380214yes—
372C3×C4D3×#6714yes—
373C3×C4D4×C3⋊D311yescharacteristic
374C3×C4C2×#63811yescharacteristic
375Dic3—41no—
376Dic3—64no—
377Dic3—364no—
378Dic3—361no—
379Dic3—724no—
380C32—24no—
381C32#473111yescharacteristic, complemented
382C32#516914yescomplemented
383C23—61no—
384C23—91no—
385C23—544no—
386C23—541no—
387C23—1082no—
388C23—1082no—
389Q8—61no—
390Q8—61no—
391D4—32no—
392D4—61no—
393D4—272no—
394D4—544no—
395D4—541no—
396C2×C4—61no—
397C2×C4—91no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—541no—
401C2×C4C2×D3×C3⋊D311yescharacteristic
402C8—21no—
403C8—181no—
404C2×C3—24no—
405C2×C3—24no—
406C2×C3—24no—
407C2×C3—128no—
408C2×C3—182no—
409C2×C3D3×#20414yes—
410C2×C3#6379414yes—
411C2×C3#6392214yes—
412C2×C3SD16×C3⋊D311yescharacteristic
413C2×C3#6412311yescharacteristic
414C2×C3#6414611yescharacteristic
415D3—42no—
416D3—68no—
417D3—362no—
418D3—368no—
419D3—728no—
420C22—62no—
421C22—92no—
422C22—121no—
423C22—121no—
424C22—181no—
425C22—181no—
426C22—542no—
427C22—1088no—
428C22—1081no—
429C22—1081no—
430C22C3⋊D3×D1211yescharacteristic
431C4—121no—
432C4—181no—
433C4—1081no—
434C4#8902611yescharacteristic
435C4D3×#63811yescharacteristic
436C3—24no—
437C3#1059094214yescomplemented
438C3#1059286811yescharacteristic, complemented
439C2—122no—
440C2—182no—
441C2—1082no—
442C2C3⋊D3×#20411yescharacteristic
443C2#1172331911yescharacteristic
444C2#1172354011yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3⋊D3×D12C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#450
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
2. C2×C32×#204C2semidirect
3. #11723435C2semidirect
4. C2×#88795C2semidirect
5. #11722191C2semidirect
6. C3×#63647C2semidirect
7. #11722032C2semidirect
14. C2×C32×Dic6C22semidirect
18. C2×C33×C8C22semidirect
19. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
20. C2×C3×C3⋊(C3⋊C8)C22semidirect
30. C2×C3×#204D3semidirect
37. #63647D3semidirect
76. C2×C3×Dic6C2×D3semidirect
82. C2×C32×C8C2×D3semidirect
83. C2×C32×C8C2×D3semidirect
90. C2×C3⋊(C3⋊C8)C2×D3semidirect
102. C2×C3×C3⋊D3SD16semidirect
119. C2×#204C3⋊D3semidirect
198. C2×Dic6C2×C3⋊D3semidirect
202. C2×C3×C8D32semidirect
203. C2×C3×C8C2×C3⋊D3semidirect
233. C2×C3⋊D3#204semidirect
261. C33#450semidirect
337. C2×C8D3×C3⋊D3semidirect
381. C32#4731semidirect
382. C32#5169semidirect
437. C3#10590942semidirect
438. C3#10592868semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
C2×C32×#2042
#117234352
C2×#887952
#117221912
C3×#636472
#117220322
#105928683
#105909423