Finite Group Databasebeta

← #13161242

Subgroups — #13161242

Order 1728.

1048 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 168 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
1Q8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3C2×C32×#215C211yesmaximal, characteristic, complemented
4#11723529C211yesmaximal, characteristic
5#11722050C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592862—31nomaximal
9#10591622—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6290—21no—
14C2×#6290—21no—
15C4⋊(C3⋊(C3⋊Dic3))—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C4×C3⋊(C3⋊Dic3)—21no—
18C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3755—22no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×#215—22no—
22C2×C32×C3⋊C8C2211yescharacteristic
23C2×C32×SD16—31no—
24Q8×C3⋊Dic3—31no—
25Q8×C3⋊Dic3—34no—
26Q8×C3⋊Dic3—64no—
27C2×#3755D311yescharacteristic, complemented
28#64149—31no—
29C2×C3×#215—34no—
30C2×C3×#215D314yescomplemented
31#63914—34no—
32#63659—31no—
33#63648—31no—
34#63641—31no—
35#63572—34no—
36#63563—34no—
37#63563—34no—
38C22×C3×C3⋊D3—21no—
39C22×C32×D3—21no—
40C33×Q8—21no—
41C33×Q8D412yescomplemented
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C32×D12—21no—
50C32×D12—21no—
51#6290—42no—
52#6290—42no—
53C3×C3⋊(C3⋊C8)—21no—
54C32×C3⋊C8—21no—
55#5163—94no—
56#5163—91no—
57C2×C32×Q8—24no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×D4—31no—
61C2×#638—61no—
62C2×#638—61no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C3×D12—34no—
65C2×C3×D12C2×D314yes—
66C2×#625—64no—
67C2×#625—64no—
68C32×SD16—62no—
69C2×C32×C8—31no—
70#3755—22no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—34no—
73C4⋊(C3⋊Dic3)—61no—
74C4⋊(C3⋊Dic3)—64no—
75C4⋊(C3⋊Dic3)—64no—
76C4⋊(C3⋊Dic3)—68no—
77C4×C3⋊Dic3—31no—
78C4×C3⋊Dic3—34no—
79C4×C3⋊Dic3—61no—
80C4×C3⋊Dic3—64no—
81C4×C3⋊Dic3—64no—
82C4×C3⋊Dic3—68no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×#215—28no—
85C3×#215—68no—
86C2×C3×C3⋊C8—34no—
87C2×C3×C3⋊C8C2×D314yes—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C32×D3—41no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C33×C4—22no—
96C33×C4C2×D411yescharacteristic
97C33×C4#4111yescharacteristic
98C3⋊(C3⋊Dic3)—21no—
99C3⋊(C3⋊Dic3)—41no—
100C3⋊(C3⋊Dic3)SD1612yescomplemented
101C2×C3×SD16—34no—
102C2×C3×SD16—91no—
103Q8×Dic3—94no—
104Q8×Dic3—91no—
105Q8×Dic3—184no—
106C2×#215—34no—
107C2×#215C3⋊D311yescharacteristic, complemented
108#940—94no—
109#940—91no—
110#832—94no—
111#832—91no—
112#821—94no—
113#821—91no—
114Dic3⋊C8—94no—
115Dic3⋊C8—91no—
116C23×C32—61no—
117C22×C3⋊D3—21no—
118C22×C3×D3—24no—
119C22×C3×D3—64no—
120C32×Q8—28no—
121C32×Q8—24no—
122C32×Q8—21no—
123C32×Q8—28no—
124C32×Q8#6718yescomplemented
125C32×Q8#6712yescomplemented
126C32×D4—61no—
127C32×D4—61no—
128C2×C32×C4—28no—
129C2×C32×C4—24no—
130C2×C32×C4—21no—
131C2×C32×C4—24no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134#638—122no—
135#638—122no—
136C2×C3⋊Dic3—31no—
137C2×C3⋊Dic3—31no—
138C2×C3⋊Dic3—34no—
139C2×C3⋊Dic3—34no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—64no—
142C2×C3⋊Dic3—68no—
143C2×C3⋊Dic3—64no—
144C2×C3⋊Dic3—61no—
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)—21no—
147C3×D12—24no—
148C3×D12—24no—
149C3×D12—64no—
150C3×D12—64no—
151#625—128no—
152#625—128no—
153C32×C8—61no—
154C3⋊(C3⋊C8)—21no—
155C3×C3⋊C8—24no—
156C3×C3⋊C8—64no—
157#460—271noSylow
158C2×C33#12111yescharacteristic
159C2×C33C2×SD1611yescharacteristic
160C2×C33AGL(1,ℤ/8)11yescharacteristic
161C3×C3⋊D3—81no—
162C32×D3—81no—
163C2×C3×Q8—24no—
164C2×C3×Q8D3214yes—
165C2×C3×Q8C2×C3⋊D311yescharacteristic
166C2×C3×D4—34no—
167C2×C3×D4—91no—
168C2×#67—181no—
169C2×#67—181no—
170C2×#67—184no—
171C2×#67—184no—
172C2×D12—34no—
173C2×D12C2×C3⋊D311yescharacteristic
174C3×SD16—68no—
175C3×SD16—182no—
176C2×C3×C8—34no—
177C2×C3×C8—91no—
178#215—22no—
179#215—68no—
180C4⋊Dic3—94no—
181C4⋊Dic3—91no—
182C4⋊Dic3—184no—
183C4⋊Dic3—184no—
184C4⋊Dic3—181no—
185C4⋊Dic3—188no—
186C4×Dic3—94no—
187C4×Dic3—91no—
188C4×Dic3—184no—
189C4×Dic3—184no—
190C4×Dic3—181no—
191C4×Dic3—188no—
192C2×C3⋊C8—34no—
193C2×C3⋊C8C2×C3⋊D311yescharacteristic
194C22×C32—24no—
195C22×C32—121no—
196C22×C32—121no—
197C22×C32—121no—
198C22×C32D3×D414yes—
199C22×C32D3×D411yescharacteristic
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3×D3—44no—
204C2×C3×D3—44no—
205C2×C3×D3—44no—
206C2×C3×D3—124no—
207C2×C3×D3—124no—
208C2×C3×D3—124no—
209C32×C4—216no—
210C32×C4—28no—
211C32×C4—22no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4#23714yes—
215C32×C4#23711yescharacteristic
216C32×C4C2×#6714yes—
217C32×C4C2×#6711yescharacteristic
218C3⋊Dic3—32no—
219C3⋊Dic3—38no—
220C3⋊Dic3—68no—
221C3⋊Dic3—61no—
222C3⋊Dic3—64no—
223C3⋊Dic3—124no—
224C3⋊Dic3—128no—
225C3⋊Dic3—124no—
226C3⋊Dic3—121no—
227C2×SD16—31no—
228C2×SD16—91no—
229#127—271no—
230C4×Q8—271no—
231C4⋊C8—271no—
232#102—271no—
233#102—271no—
234C33#46011yescharacteristic, complemented, Sylow
235C23×C3—64no—
236C23×C3—181no—
237C22×D3—21no—
238C22×D3—64no—
239C3×Q8—28no—
240C3×Q8—24no—
241C3×Q8—21no—
242C3×Q8—28no—
243C3×Q8#62518yescomplemented
244C3×Q8#63812yescomplemented
245C3×D4—64no—
246C3×D4—64no—
247C3×D4—181no—
248C3×D4—181no—
249C2×C3×C4—28no—
250C2×C3×C4—24no—
251C2×C3×C4—21no—
252C2×C3×C4—24no—
253C2×C3×C4C2×D3214yes—
254C2×C3×C4C22×C3⋊D311yescharacteristic
255#67—362no—
256#67—368no—
257#67—362no—
258#67—368no—
259C2×Dic3—94no—
260C2×Dic3—91no—
261C2×Dic3—94no—
262C2×Dic3—91no—
263C2×Dic3—184no—
264C2×Dic3—184no—
265C2×Dic3—188no—
266C2×Dic3—184no—
267C2×Dic3—181no—
268D12—21no—
269D12—21no—
270D12—64no—
271D12—64no—
272C3×C8—64no—
273C3×C8—181no—
274C3⋊C8—21no—
275C3⋊C8—64no—
276C2×C32—24no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—241no—
280C2×C32D3×SD1614yes—
281C2×C32D3×SD1611yescharacteristic
282C2×C32#91414yes—
283C2×C32#91411yescharacteristic
284C2×C32#93914yes—
285C2×C32#93911yescharacteristic
286C3⋊D3—81no—
287C3×D3—84no—
288C3×D3—244no—
289C2×Q8D3×C3⋊D311yescharacteristic
290C2×D4—31no—
291C2×D4—91no—
292C2×D4—541no—
293C2×D4—541no—
294SD16—62no—
295SD16—182no—
296C2×C8—31no—
297C2×C8—91no—
298C4⋊C4—271no—
299C4⋊C4—541no—
300C42—271no—
301C42—541no—
302C22×C3—24no—
303C22×C3—124no—
304C22×C3—124no—
305C22×C3—124no—
306C22×C3—361no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3#381114yes—
310C22×C3D4×C3⋊D311yescharacteristic
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—124no—
315C2×D3—124no—
316C2×D3—124no—
317C3×C4—216no—
318C3×C4—22no—
319C3×C4—28no—
320C3×C4—24no—
321C3×C4—24no—
322C3×C4#380514yes—
323C3×C4C2×#62514yes—
324C3×C4#383011yescharacteristic
325C3×C4C2×#63811yescharacteristic
326Dic3—92no—
327Dic3—98no—
328Dic3—188no—
329Dic3—181no—
330Dic3—184no—
331Dic3—364no—
332Dic3—368no—
333Dic3—361no—
334Dic3—364no—
335C32—24no—
336C32#516314yescomplemented
337C32#516311yescharacteristic, complemented
338C23—61no—
339C23—181no—
340Q8—21no—
341Q8#629012yescomplemented
342D4—61no—
343D4—61no—
344D4—181no—
345D4—181no—
346D4—1082no—
347D4—1082no—
348C2×C4—21no—
349C2×C4—271no—
350C2×C4—271no—
351C2×C4—541no—
352C2×C4C2×D3×C3⋊D311yescharacteristic
353C8—61no—
354C8—181no—
355C2×C3—24no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—244no—
359C2×C3—721no—
360C2×C3#6394014yes—
361C2×C3#6394114yes—
362C2×C3#6398014yes—
363C2×C3SD16×C3⋊D311yescharacteristic
364C2×C3#6412311yescharacteristic
365C2×C3#6414811yescharacteristic
366D3—81no—
367D3—244no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—361no—
372C22—361no—
373C22—361no—
374C22#8903911yescharacteristic
375C4—22no—
376C4—272no—
377C4—541no—
378C4—1081no—
379C4#8902911yescharacteristic
380C4C2×#629011yescharacteristic
381C3—24no—
382C3#1059162214yescomplemented
383C3#1059286211yescharacteristic, complemented
384C2—241no—
385C2—721no—
386C2#1172357111yescharacteristic
387C2#1172357211yescharacteristic
388C2#1172363711yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723529C2
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. Q8×C3⋊(C3⋊Dic3)C2semidirect
2. C2×C3×#3755C2semidirect
3. C2×C32×#215C2semidirect
5. #11722050C2semidirect
6. #11722034C2semidirect
7. #11722032C2semidirect
27. C2×#3755D3semidirect
30. C2×C3×#215D3semidirect
41. C33×Q8D4semidirect
100. C3⋊(C3⋊Dic3)SD16semidirect
107. C2×#215C3⋊D3semidirect
124. C32×Q8#67semidirect
125. C32×Q8#67semidirect
234. C33#460semidirect
243. C3×Q8#625semidirect
244. C3×Q8#638semidirect
336. C32#5163semidirect
337. C32#5163semidirect
341. Q8#6290semidirect
382. C3#10591622semidirect
383. C3#10592862semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊(C3⋊Dic3)2
C2×C3×#37552
C2×C32×#2152
#117235292
#117220502
#117220342
#117220322
#105928623
#105916223