Finite Group Databasebeta

← #13161162

Subgroups — #13161162

Order 1728.

1000 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 182 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
3#11723501C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic, complemented
5#11722057C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic
7#11722021C211yesmaximal, characteristic, complemented
8#10592516—31nomaximal
9#10591572—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C4⋊(C3⋊(C3⋊Dic3))—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3755C414yescomplemented
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88803—21no—
22Q8×C3⋊Dic3—31no—
23Q8×C3⋊Dic3—34no—
24Q8×C3⋊Dic3—64no—
25C2×#3755D311yescharacteristic
26#64102—31no—
27C2×C3×#215—34no—
28#63898—34no—
29C32×#103—31no—
30#63648—31no—
31#63642—31no—
32#63634—31no—
33C3×#835D314yescomplemented
34#63575—34no—
35#63563—34no—
36#63555—34no—
37C22×C3×C3⋊D3—21no—
38C33×Q8—21no—
39C33×Q8C2×C412yescomplemented
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)C2×C412yes—
47C2×C32×Dic3—21no—
48Dic3×C3⋊D3—42no—
49C3×C3⋊(C3⋊C8)C2×C412yescomplemented
50C32×C3⋊C8—21no—
51#5164—91no—
52#4817—94no—
53C2×C32×Q8—24no—
54C2×C32×Q8C2×D314yes—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C4×C3⋊D3—61no—
58C2×C3×D12—34no—
59C2×D3×Dic3—64no—
60C2×C32×C8—31no—
61C32×C4⋊C4—31no—
62#3755Dic314yescomplemented
63#3752—61no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—61no—
67C4⋊(C3⋊Dic3)—64no—
68C4⋊(C3⋊Dic3)—64no—
69C4⋊(C3⋊Dic3)—68no—
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—61no—
73C4×C3⋊Dic3—64no—
74C4×C3⋊Dic3—64no—
75C4×C3⋊Dic3—68no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C3×#215—316no—
78C3×C4⋊Dic3C2×D314yes—
79C2×C3×C3⋊C8—34no—
80C2×C3×C3⋊C8C2×D314yes—
81(C2×D3)⋊Dic3—64no—
82C22×C33C2×D411yescharacteristic
83C2×C3×C3⋊D3—22no—
84C2×C3×C3⋊D3—41no—
85C2×C3×C3⋊D3—41no—
86C33×C4—22no—
87C33×C4C22×C411yescharacteristic
88C33×C4#4111yescharacteristic
89C3⋊(C3⋊Dic3)—21no—
90C3⋊(C3⋊Dic3)—21no—
91C3⋊(C3⋊Dic3)—41no—
92C32×Dic3—41no—
93C2×C3×SD16—91no—
94Q8×Dic3—94no—
95Q8×Dic3—91no—
96Q8×Dic3—184no—
97C2×#215—34no—
98D4×Dic3—91no—
99#893—94no—
100C3×#103—34no—
101#835C3⋊D311yescharacteristic, complemented
102#832—91no—
103#821—94no—
104#818—91no—
105#815—94no—
106#815—91no—
107#807—94no—
108C22×C3⋊D3—21no—
109C22×C3×D3—64no—
110C32×Q8—28no—
111C32×Q8—24no—
112C32×Q8—21no—
113C32×Q8—28no—
114C32×Q8C4×D318yescomplemented
115C32×Q8C2×Dic312yescomplemented
116C2×C32×C4—28no—
117C2×C32×C4—24no—
118C2×C32×C4—21no—
119C2×C32×C4—24no—
120C2×C32×C4—61no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—34no—
126C2×C3⋊Dic3—34no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—68no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—61no—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)C2×Dic312yes—
134C4×C3⋊D3—122no—
135C2×C3×Dic3—24no—
136C3×D12—38no—
137C3×D12—64no—
138D3×Dic3—128no—
139C32×C8—61no—
140C3⋊(C3⋊C8)C2×Dic312yescomplemented
141C3×C3⋊C8—24no—
142C3×C3⋊C8—38no—
143#440—271noSylow
144C2×C33C4×D411yescharacteristic
145C2×C33AGL(1,ℤ/8)11yescharacteristic
146C2×C33#13711yescharacteristic
147C3×C3⋊D3—42no—
148C2×C3×Q8—24no—
149C2×C3×Q8D3214yes—
150C2×C3×Q8C2×C3⋊D311yescharacteristic
151C2×C3×D4—91no—
152C22×Dic3—181no—
153C2×D12—34no—
154C2×C4×D3—184no—
155C3×SD16—94no—
156C2×C3×C8—34no—
157C2×C3×C8—91no—
158C3×C4⋊C4—34no—
159#217—181no—
160#215—316no—
161(C2×D3)⋊C4—184no—
162C4⋊Dic3—94no—
163C4⋊Dic3—91no—
164C4⋊Dic3—184no—
165C4⋊Dic3—184no—
166C4⋊Dic3—181no—
167C4⋊Dic3—188no—
168C4⋊Dic3C2×C3⋊D311yescharacteristic
169C4×Dic3—94no—
170C4×Dic3—91no—
171C4×Dic3—184no—
172C4×Dic3—184no—
173C4×Dic3—181no—
174C4×Dic3—188no—
175C2×C3⋊C8—34no—
176C2×C3⋊C8C2×C3⋊D311yescharacteristic
177C22×C32—24no—
178C22×C32D3×D414yes—
179C22×C32D3×D411yescharacteristic
180C2×C3⋊D3—22no—
181C2×C3⋊D3—41no—
182C2×C3⋊D3—41no—
183C2×C3×D3—68no—
184C2×C3×D3—124no—
185C2×C3×D3—124no—
186C32×C4—216no—
187C32×C4—28no—
188C32×C4—22no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—121no—
192C32×C4C2×C4×D314yes—
193C32×C4#23714yes—
194C32×C4#23711yescharacteristic
195C32×C4C22×Dic311yescharacteristic
196C3⋊Dic3—68no—
197C3⋊Dic3—61no—
198C3⋊Dic3—61no—
199C3⋊Dic3—64no—
200C3⋊Dic3—64no—
201C3⋊Dic3—124no—
202C3⋊Dic3—128no—
203C3⋊Dic3—124no—
204C3⋊Dic3—121no—
205C3×Dic3—44no—
206C2×SD16—91no—
207C4×Q8—271no—
208C4×D4—271no—
209#107—271no—
210#103—31no—
211#102—271no—
212#97—271no—
213C33#44011yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—64no—
216C3×Q8—28no—
217C3×Q8—24no—
218C3×Q8—21no—
219C3×Q8—28no—
220C3×Q8D3×Dic318yescomplemented
221C3×Q8C4×C3⋊D312yescomplemented
222C3×D4—92no—
223C3×D4—181no—
224C2×C3×C4—28no—
225C2×C3×C4—24no—
226C2×C3×C4—21no—
227C2×C3×C4—24no—
228C2×C3×C4—64no—
229C2×C3×C4C2×D3214yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231C2×Dic3—21no—
232C2×Dic3—94no—
233C2×Dic3—94no—
234C2×Dic3—91no—
235C2×Dic3—91no—
236C2×Dic3—184no—
237C2×Dic3—184no—
238C2×Dic3—188no—
239C2×Dic3—184no—
240C2×Dic3—181no—
241C2×Dic3—362no—
242D12—38no—
243D12—64no—
244C4×D3—368no—
245C3×C8—64no—
246C3×C8—92no—
247C3⋊C8—21no—
248C3⋊C8—38no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32#67⋊C414yes—
253C2×C32#91414yes—
254C2×C32#91411yescharacteristic
255C2×C32#91511yescharacteristic
256C2×C32#91814yes—
257C2×C32D4×Dic311yescharacteristic
258C3⋊D3—42no—
259C3×D3—128no—
260C2×Q8D3×C3⋊D311yescharacteristic
261C2×D4—91no—
262C22×C4—541no—
263SD16—94no—
264C2×C8—31no—
265C2×C8—91no—
266C4⋊C4—31no—
267C4⋊C4—271no—
268C4⋊C4—541no—
269C22⋊C4—541no—
270C42—271no—
271C42—541no—
272C22×C3—24no—
273C22×C3—182no—
274C22×C3—361no—
275C22×C3—361no—
276C22×C3#381114yes—
277C22×C3D4×C3⋊D311yescharacteristic
278C2×D3—68no—
279C2×D3—124no—
280C2×D3—124no—
281C3×C4—216no—
282C3×C4—22no—
283C3×C4—28no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—124no—
287C3×C4C2×D3×Dic314yes—
288C3×C4#380514yes—
289C3×C4C2×C4×C3⋊D311yescharacteristic
290C3×C4#383011yescharacteristic
291Dic3—41no—
292Dic3—188no—
293Dic3—181no—
294Dic3—181no—
295Dic3—184no—
296Dic3—184no—
297Dic3—364no—
298Dic3—368no—
299Dic3—361no—
300Dic3—364no—
301C32—24no—
302C32#481714yescomplemented
303C32#516411yescharacteristic, complemented
304C23—181no—
305Q8—21no—
306Q8Dic3×C3⋊D312yescomplemented
307D4—92no—
308D4—181no—
309C2×C4—21no—
310C2×C4—61no—
311C2×C4—271no—
312C2×C4—271no—
313C2×C4—541no—
314C2×C4—1082no—
315C2×C4C2×D3×C3⋊D311yescharacteristic
316C8—61no—
317C8—92no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—362no—
322C2×C3#6394114yes—
323C2×C3#6395114yes—
324C2×C3#67⋊Dic314yes—
325C2×C3#6409011yescharacteristic
326C2×C3#6412311yescharacteristic
327C2×C3#6412711yescharacteristic
328D3—128no—
329C22—182no—
330C22—361no—
331C22—361no—
332C22#8903911yescharacteristic
333C4—22no—
334C4—121no—
335C4—541no—
336C4—541no—
337C4—1081no—
338C4#8902911yescharacteristic
339C4C2×Dic3×C3⋊D311yescharacteristic
340C3—24no—
341C3#1059157214yescomplemented
342C3#1059251611yescharacteristic, complemented
343C2—362no—
344C2#1172357211yescharacteristic
345C2#1172359011yescharacteristic
346C2#1172362611yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#440
LCSG ▷ 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
4. C32×#835C2semidirect
5. #11722057C2semidirect
7. #11722021C2semidirect
17. C3×#3755C4semidirect
33. C3×#835D3semidirect
39. C33×Q8C2×C4semidirect
49. C3×C3⋊(C3⋊C8)C2×C4semidirect
62. #3755Dic3semidirect
101. #835C3⋊D3semidirect
114. C32×Q8C4×D3semidirect
115. C32×Q8C2×Dic3semidirect
140. C3⋊(C3⋊C8)C2×Dic3semidirect
213. C33#440semidirect
220. C3×Q8D3×Dic3semidirect
221. C3×Q8C4×C3⋊D3semidirect
302. C32#4817semidirect
303. C32#5164semidirect
306. Q8Dic3×C3⋊D3semidirect
341. C3#10591572semidirect
342. C3#10592516semidirect

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
#117235012
C32×#8352
#117220572
#117220342
#117220212
#105925163
#105915723