Finite Group Databasebeta

← #13161241

Subgroups — #13161241

Order 1728.

1040 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1040 conjugacy classes of subgroups — too many to draw. Only its 188 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×#88862C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3C32×Q8×Dic3C211yesmaximal, characteristic, complemented
4#11723517C211yesmaximal, characteristic
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592726—31nomaximal
9#10591617—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#6292—21no—
14C2×#6292—21no—
15#88862—22no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3755—22no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4⋊Dic3—21no—
20C32×C4⋊Dic3C2211yescharacteristic
21C32×C4×Dic3—21no—
22C32×C4×Dic3C2211yescharacteristic
23C32×C4×Q8—31no—
24C2×#3755—31no—
25C2×#3755—34no—
26C2×#3755—64no—
27C2×#3755D311yescharacteristic, complemented
28#64083—31no—
29C3×Q8×Dic3D314yescomplemented
30C2×C3×#215—34no—
31#63911—34no—
32#63636—31no—
33#63636—31no—
34C4⋊(C3⋊(C3⋊C8))—31no—
35Dic3⋊(C3⋊C8)—34no—
36#63564—34no—
37#63562—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C33×Q8—21no—
41C33×Q8D412yescomplemented
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C2×C32×Dic3—21no—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51#6292—42no—
52#6292—42no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54C3×C3⋊(C3⋊C8)—21no—
55#5163—91no—
56#5027—94no—
57C2×C32×Q8—24no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4⋊(C3⋊D3)—34no—
62C2×C4⋊(C3⋊D3)—64no—
63C2×C4⋊(C3⋊D3)—61no—
64C2×C4⋊(C3⋊D3)—61no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C3×D12—34no—
67C2×#626—64no—
68C2×#626—64no—
69C32×C4⋊C4—31no—
70C32×C4⋊C4—61no—
71C32×C42—31no—
72C32×C42—61no—
73#3755—22no—
74#3755—616no—
75#3755—68no—
76#3755—62no—
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—34no—
79C2×C3⋊(C3⋊C8)—64no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×#215—68no—
82C3×C4⋊Dic3—24no—
83C3×C4⋊Dic3C2×D314yes—
84C3×C4×Dic3—24no—
85C3×C4×Dic3C2×D314yes—
86C2×C3×C3⋊C8—34no—
87C22×C33C2×D411yescharacteristic
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3⋊(C3⋊D3)—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C33×C4—22no—
95C33×C4C2×D411yescharacteristic
96C33×C4#4111yescharacteristic
97C32×Dic3—21no—
98C32×Dic3—41no—
99C32×Dic3SD1612yescomplemented
100C2×C3×SD16—91no—
101C3×C4×Q8—34no—
102Q8×Dic3C3⋊D311yescharacteristic, complemented
103C2×#215—34no—
104C2×#215—94no—
105C2×#215—91no—
106C2×#215—184no—
107#940—91no—
108#874—94no—
109#832—91no—
110#821—91no—
111Dic3⋊C8—91no—
112#809—94no—
113#809—94no—
114C4⋊(C3⋊C8)—94no—
115C22×C3⋊D3—21no—
116C22×C3⋊D3—61no—
117C22×C3⋊D3—64no—
118C22×C3⋊D3—124no—
119C22×C3×D3—64no—
120C32×Q8—21no—
121C32×Q8—28no—
122C32×Q8—24no—
123C32×Q8—44no—
124C32×Q8D1218yescomplemented
125C32×Q8#6712yescomplemented
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4—31no—
130C2×C32×C4—31no—
131C2×C32×C4—44no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—68no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—124no—
143C4⋊(C3⋊D3)—122no—
144C4⋊(C3⋊D3)—122no—
145C2×C3×Dic3—24no—
146C2×C3×Dic3#6714yes—
147C2×C3×Dic3#6714yes—
148C3×D12—64no—
149C3×D12—64no—
150#626—128no—
151#626—128no—
152C3⋊(C3⋊C8)—21no—
153C3⋊(C3⋊C8)—68no—
154C3⋊(C3⋊C8)—64no—
155C3⋊(C3⋊C8)—61no—
156C3×C3⋊C8—64no—
157#460—271noSylow
158C2×C33#12111yescharacteristic
159C2×C33C2×SD1611yescharacteristic
160C2×C33AGL(1,ℤ/8)11yescharacteristic
161C3⋊(C3⋊D3)—81no—
162C3×C3⋊D3—81no—
163C2×C3×Q8—24no—
164C2×C3×Q8D3214yes—
165C2×C3×Q8C2×C3⋊D311yescharacteristic
166C2×C3×D4—91no—
167C2×#67—181no—
168C2×#67—181no—
169C2×D12—34no—
170C2×D12—94no—
171C2×D12—91no—
172C2×D12—184no—
173C2×D12—184no—
174C2×D12—184no—
175C3×SD16—182no—
176C2×C3×C8—91no—
177C3×C4⋊C4—34no—
178C3×C4⋊C4—64no—
179C3×C42—34no—
180C3×C42—64no—
181#215—68no—
182#215—1816no—
183#215—188no—
184#215—182no—
185C4⋊Dic3—21no—
186C4⋊Dic3C2×C3⋊D311yescharacteristic
187C4×Dic3—21no—
188C4×Dic3C2×C3⋊D311yescharacteristic
189C2×C3⋊C8—34no—
190C2×C3⋊C8—94no—
191C2×C3⋊C8—91no—
192C2×C3⋊C8—184no—
193C22×C32—24no—
194C22×C32D3×D411yescharacteristic
195C22×C32C2×#6714yes—
196C2×C3⋊D3—41no—
197C2×C3⋊D3—41no—
198C2×C3⋊D3—41no—
199C2×C3⋊D3—128no—
200C2×C3⋊D3—121no—
201C2×C3⋊D3—121no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—124no—
205C2×C3⋊D3—124no—
206C2×C3⋊D3—244no—
207C2×C3⋊D3—244no—
208C2×C3×D3—124no—
209C2×C3×D3—124no—
210C2×C3×D3—124no—
211C32×C4—22no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—28no—
215C32×C4—32no—
216C32×C4—48no—
217C32×C4—61no—
218C32×C4—121no—
219C32×C4C2×D1214yes—
220C32×C4#23514yes—
221C32×C4#23711yescharacteristic
222C32×C4C2×#6711yescharacteristic
223C3×Dic3—24no—
224C3×Dic3—44no—
225C3×Dic3#21518yescomplemented
226C2×SD16—91no—
227C2×SD16—271no—
228#127—271no—
229C4×Q8—31no—
230C4⋊C8—271no—
231#102—271no—
232#102—271no—
233C33#46011yescharacteristic, complemented, Sylow
234C23×C3—181no—
235C22×D3—64no—
236C22×D3—184no—
237C22×D3—181no—
238C22×D3—364no—
239C3×Q8—24no—
240C3×Q8—21no—
241C3×Q8—28no—
242C3×Q8—44no—
243C3×Q8#62618yescomplemented
244C3×Q8C4⋊(C3⋊D3)12yescomplemented
245C3×D4—181no—
246C3×D4—181no—
247C2×C3×C4—24no—
248C2×C3×C4—21no—
249C2×C3×C4—24no—
250C2×C3×C4—34no—
251C2×C3×C4—34no—
252C2×C3×C4—44no—
253C2×C3×C4—64no—
254C2×C3×C4C2×D3214yes—
255C2×C3×C4C22×C3⋊D311yescharacteristic
256#67—362no—
257#67—362no—
258C2×Dic3—21no—
259C2×Dic3#63811yescharacteristic
260C2×Dic3#63811yescharacteristic
261D12—64no—
262D12—64no—
263D12—188no—
264D12—184no—
265D12—184no—
266D12—181no—
267D12—181no—
268D12—364no—
269D12—368no—
270D12—368no—
271C3×C8—181no—
272C3⋊C8—64no—
273C3⋊C8—188no—
274C3⋊C8—184no—
275C3⋊C8—181no—
276C2×C32—24no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32D3×SD1611yescharacteristic
280C2×C32#91411yescharacteristic
281C2×C32#93014yes—
282C2×C32#93911yescharacteristic
283C2×C32C2×#21514yes—
284C2×C32#94914yes—
285C3⋊D3—81no—
286C3⋊D3—248no—
287C3⋊D3—241no—
288C3⋊D3—244no—
289C3×D3—244no—
290C2×Q8D3×C3⋊D311yescharacteristic
291C2×D4—91no—
292C2×D4—271no—
293C2×D4—541no—
294C2×D4—541no—
295SD16—182no—
296SD16—542no—
297C2×C8—91no—
298C2×C8—271no—
299C4⋊C4—31no—
300C4⋊C4—61no—
301C42—31no—
302C42—61no—
303C22×C3—24no—
304C22×C3—361no—
305C22×C3—361no—
306C22×C3—361no—
307C22×C3D3×#6714yes—
308C22×C3C2×#63811yescharacteristic
309C2×D3—124no—
310C2×D3—124no—
311C2×D3—124no—
312C2×D3—368no—
313C2×D3—361no—
314C2×D3—361no—
315C2×D3—361no—
316C2×D3—364no—
317C2×D3—364no—
318C2×D3—364no—
319C2×D3—724no—
320C2×D3—724no—
321C3×C4—22no—
322C3×C4—28no—
323C3×C4—24no—
324C3×C4—24no—
325C3×C4—38no—
326C3×C4—48no—
327C3×C4—64no—
328C3×C4—124no—
329C3×C4#380414yes—
330C3×C4C2×#62614yes—
331C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
332C3×C4#382811yescharacteristic
333Dic3—21no—
334Dic3—41no—
335Dic3#375512yescomplemented
336C32—24no—
337C32#502714yescomplemented
338C32#516311yescharacteristic, complemented
339C23—181no—
340C23—541no—
341Q8—21no—
342Q8#629212yescomplemented
343D4—181no—
344D4—181no—
345D4—541no—
346D4—541no—
347D4—1082no—
348D4—1082no—
349C2×C4—21no—
350C2×C4—31no—
351C2×C4—31no—
352C2×C4—61no—
353C2×C4C2×D3×C3⋊D311yescharacteristic
354C8—181no—
355C8—541no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—721no—
360C2×C3D3×#21514yes—
361C2×C3#6393914yes—
362C2×C3#6397814yes—
363C2×C3#6413911yescharacteristic
364C2×C3C2×#375511yescharacteristic
365C2×C3#6415811yescharacteristic
366D3—244no—
367D3—728no—
368D3—724no—
369D3—721no—
370C22—361no—
371C22—361no—
372C22—361no—
373C22—1081no—
374C22—1081no—
375C22—1081no—
376C22D3×#63811yescharacteristic
377C4—22no—
378C4—32no—
379C4—61no—
380C4—121no—
381C4#8903111yescharacteristic
382C4C2×#629211yescharacteristic
383C3—24no—
384C3#1059161714yescomplemented
385C3#1059272611yescharacteristic, complemented
386C2—721no—
387C2—2161no—
388C2D3×#375511yescharacteristic
389C2#1172356811yescharacteristic
390C2#1172363211yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723517C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#460
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88862C2semidirect
2. C2×C3×#3755C2semidirect
3. C32×Q8×Dic3C2semidirect
5. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
6. #11722035C2semidirect
7. #11722033C2semidirect
27. C2×#3755D3semidirect
29. C3×Q8×Dic3D3semidirect
41. C33×Q8D4semidirect
99. C32×Dic3SD16semidirect
102. Q8×Dic3C3⋊D3semidirect
124. C32×Q8D12semidirect
125. C32×Q8#67semidirect
225. C3×Dic3#215semidirect
233. C33#460semidirect
243. C3×Q8#626semidirect
244. C3×Q8C4⋊(C3⋊D3)semidirect
335. Dic3#3755semidirect
337. C32#5027semidirect
338. C32#5163semidirect
342. Q8#6292semidirect
384. C3#10591617semidirect
385. C3#10592726semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C2×C3×#37552
C32×Q8×Dic32
#117235172
Dic3⋊(C3⋊(C3⋊C8))2
#117220352
#117220332
#105927263
#105916173