Finite Group Databasebeta

← #13161161

Subgroups — #13161161

Order 1728.

1012 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 216 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×C32×#215C211yesmaximal, characteristic
3#11723499C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic, complemented
5#11722052C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic, complemented
8SD16×C3⋊Dic3—31nomaximal
9#10591571—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×D3×C3⋊Dic3—21no—
13C4⋊(C3⋊(C3⋊Dic3))—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#215C414yescomplemented
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22C2×C32×SD16—31no—
23Q8×C3⋊Dic3—31no—
24Q8×C3⋊Dic3—34no—
25Q8×C3⋊Dic3—64no—
26D4×C3⋊Dic3—31no—
27C2×C3×#215D314yes—
28#63898—34no—
29#63662D311yescharacteristic, complemented
30#63659—31no—
31#63644—31no—
32C8×C3⋊Dic3—31no—
33C3×#835—34no—
34#63573—34no—
35#63563—34no—
36#63553—34no—
37C22×C32×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
45C2×C3×C3⋊Dic3—21no—
46C32×D12—21no—
47C32×D12C2×C412yes—
48D3×C3⋊Dic3—42no—
49C3×C3⋊(C3⋊C8)—21no—
50C32×C3⋊C8C2×C412yescomplemented
51Dic3×SD16—94no—
52#4788—91no—
53C2×C32×Q8—24no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C2×D314yes—
56C2×C32×D4—31no—
57C22×C3⋊Dic3—61no—
58C2×C3×D12C2×D314yes—
59C2×D3×Dic3—64no—
60C32×SD16—34no—
61C2×C32×C8—31no—
62#3757—61no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—61no—
66C4⋊(C3⋊Dic3)—64no—
67C4⋊(C3⋊Dic3)—64no—
68C4⋊(C3⋊Dic3)—68no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
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×#215Dic3116yescomplemented
78C3×C4⋊Dic3—34no—
79C2×C3×C3⋊C8—34no—
80C2×C3×C3⋊C8C2×D314yes—
81(C2×D3)⋊Dic3—64no—
82C22×C33C2×D411yescharacteristic
83C2×C32×D3—22no—
84C2×C32×D3—41no—
85C2×C32×D3—41no—
86C33×C4—22no—
87C33×C4C22×C411yescharacteristic
88C33×C4#4111yescharacteristic
89C3⋊(C3⋊Dic3)—21no—
90C3⋊(C3⋊Dic3)—41no—
91C3⋊(C3⋊Dic3)SD1612yescomplemented
92C3×C3⋊Dic3—41no—
93C2×C3×SD16—34no—
94Q8×Dic3—94no—
95Q8×Dic3—91no—
96Q8×Dic3—184no—
97C2×#215C3⋊D311yescharacteristic
98D4×Dic3—94no—
99#893—91no—
100C3×#103—91no—
101#835—34no—
102#832—94no—
103#821—91no—
104#817—94no—
105C8×Dic3—94no—
106C8×Dic3—91no—
107#808—91no—
108C23×C32—61no—
109C22×C3×D3—24no—
110C32×Q8—28no—
111C32×Q8—24no—
112C32×Q8—21no—
113C32×Q8—28no—
114C32×Q8C4×D312yescomplemented
115C32×Q8C2×Dic318yescomplemented
116C32×D4—32no—
117C32×D4—61no—
118C2×C32×C4—28no—
119C2×C32×C4—24no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—31no—
127C2×C3⋊Dic3—34no—
128C2×C3⋊Dic3—34no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—68no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—61no—
134C2×C3⋊Dic3—122no—
135C2×C3×Dic3—64no—
136C3×D12—24no—
137C3×D12C2×Dic318yes—
138D3×Dic3—128no—
139C32×C8—32no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—64no—
142C3×C3⋊C8C2×Dic318yescomplemented
143C4×SD16—271noSylow
144C2×C33C4×D411yescharacteristic
145C2×C33C2×SD1611yescharacteristic
146C2×C33#13511yescharacteristic
147C32×D3—42no—
148C2×C3×Q8—24no—
149C2×C3×Q8D3214yes—
150C2×C3×Q8C2×C3⋊D311yescharacteristic
151C2×C3×D4—34no—
152C22×Dic3—184no—
153C2×D12C2×C3⋊D311yescharacteristic
154C2×C4×D3—181no—
155C3×SD16—316no—
156C2×C3×C8—34no—
157C2×C3×C8—91no—
158C3×C4⋊C4—91no—
159#217—184no—
160#215C3⋊Dic314yescomplemented
161(C2×D3)⋊C4—181no—
162C4⋊Dic3—34no—
163C4⋊Dic3—94no—
164C4⋊Dic3—91no—
165C4⋊Dic3—184no—
166C4⋊Dic3—184no—
167C4⋊Dic3—181no—
168C4⋊Dic3—188no—
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×C32—62no—
179C22×C32—121no—
180C22×C32—121no—
181C22×C32D3×D414yes—
182C22×C32D3×D411yescharacteristic
183C2×C3×D3—28no—
184C2×C3×D3—44no—
185C2×C3×D3—44no—
186C32×C4—216no—
187C32×C4—22no—
188C32×C4—28no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4C2×C4×D311yescharacteristic
192C32×C4#23714yes—
193C32×C4#23711yescharacteristic
194C32×C4C22×Dic314yes—
195C3⋊Dic3—32no—
196C3⋊Dic3—38no—
197C3⋊Dic3—41no—
198C3⋊Dic3—68no—
199C3⋊Dic3—61no—
200C3⋊Dic3—64no—
201C3⋊Dic3—124no—
202C3⋊Dic3—128no—
203C3⋊Dic3—124no—
204C3⋊Dic3—121no—
205C3×Dic3—124no—
206C2×SD16—31no—
207C4×Q8—271no—
208C4×D4—271no—
209#106—271no—
210#103—91no—
211#102—271no—
212C4×C8—271no—
213C33C4×SD1611yescharacteristic, complemented, Sylow
214C23×C3—64no—
215C22×D3—21no—
216C3×Q8—28no—
217C3×Q8—24no—
218C3×Q8—21no—
219C3×Q8—28no—
220C3×Q8D3×Dic318yescomplemented
221C3×Q8C2×C3⋊Dic312yescomplemented
222C3×D4—38no—
223C3×D4—64no—
224C2×C3×C4—28no—
225C2×C3×C4—24no—
226C2×C3×C4—21no—
227C2×C3×C4—24no—
228C2×C3×C4—181no—
229C2×C3×C4C2×D3214yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231C2×Dic3—64no—
232C2×Dic3—94no—
233C2×Dic3—91no—
234C2×Dic3—94no—
235C2×Dic3—91no—
236C2×Dic3—184no—
237C2×Dic3—184no—
238C2×Dic3—188no—
239C2×Dic3—184no—
240C2×Dic3—181no—
241C2×Dic3—368no—
242D12—21no—
243D12C2×C3⋊Dic312yes—
244C4×D3—362no—
245C3×C8—38no—
246C3×C8—181no—
247C3⋊C8—64no—
248C3⋊C8C2×C3⋊Dic312yescomplemented
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—122no—
253C2×C32#67⋊C411yescharacteristic
254C2×C32D3×SD1614yes—
255C2×C32D3×SD1611yescharacteristic
256C2×C32#91614yes—
257C2×C32#91911yescharacteristic
258C2×C32D4×Dic314yes—
259C3×D3—48no—
260C2×Q8D3×C3⋊D311yescharacteristic
261C2×D4—31no—
262C22×C4—541no—
263SD16—34no—
264C2×C8—31no—
265C2×C8—91no—
266C4⋊C4—91no—
267C4⋊C4—271no—
268C4⋊C4—541no—
269C22⋊C4—541no—
270C42—271no—
271C42—541no—
272C22×C3—24no—
273C22×C3—68no—
274C22×C3—124no—
275C22×C3—124no—
276C22×C3#381114yes—
277C22×C3D4×C3⋊D311yescharacteristic
278C2×D3—22no—
279C2×D3—41no—
280C2×D3—41no—
281C3×C4—216no—
282C3×C4—22no—
283C3×C4—28no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—361no—
287C3×C4C2×D3×Dic314yes—
288C3×C4#380514yes—
289C3×C4#383011yescharacteristic
290C3×C4C22×C3⋊Dic311yescharacteristic
291Dic3—92no—
292Dic3—98no—
293Dic3—124no—
294Dic3—188no—
295Dic3—181no—
296Dic3—184no—
297Dic3—364no—
298Dic3—368no—
299Dic3—361no—
300Dic3—364no—
301C32—24no—
302C32#478811yescharacteristic, complemented
303C32Dic3×SD1614yescomplemented
304C23—61no—
305Q8—21no—
306Q8D3×C3⋊Dic312yescomplemented
307D4—32no—
308D4—61no—
309C2×C4—21no—
310C2×C4—181no—
311C2×C4—271no—
312C2×C4—271no—
313C2×C4—541no—
314C2×C4—1082no—
315C2×C4C2×D3×C3⋊D311yescharacteristic
316C8—32no—
317C8—181no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—128no—
322C2×C3#6394014yes—
323C2×C3#6395014yes—
324C2×C3#67⋊Dic314yes—
325C2×C3SD16×C3⋊D311yescharacteristic
326C2×C3#6412511yescharacteristic
327C2×C3D4×C3⋊Dic311yescharacteristic
328D3—42no—
329C22—62no—
330C22—121no—
331C22—121no—
332C22#8903911yescharacteristic
333C4—22no—
334C4—272no—
335C4—361no—
336C4—541no—
337C4—1081no—
338C4C2×D3×C3⋊Dic311yescharacteristic
339C4#8902911yescharacteristic
340C3—24no—
341C3#1059157114yescomplemented
342C3SD16×C3⋊Dic311yescharacteristic, complemented
343C2—122no—
344C2#1172357111yescharacteristic
345C2#1172358711yescharacteristic
346C2#67⋊(C3⋊Dic3)11yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C4×SD16
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. C3×#63662C2semidirect
5. #11722052C2semidirect
7. #11722018C2semidirect
19. C32×#215C4semidirect
29. #63662D3semidirect
39. C33×Q8C2×C4semidirect
50. C32×C3⋊C8C2×C4semidirect
77. C3×#215Dic3semidirect
91. C3⋊(C3⋊Dic3)SD16semidirect
114. C32×Q8C4×D3semidirect
115. C32×Q8C2×Dic3semidirect
142. C3×C3⋊C8C2×Dic3semidirect
160. #215C3⋊Dic3semidirect
213. C33C4×SD16semidirect
220. C3×Q8D3×Dic3semidirect
221. C3×Q8C2×C3⋊Dic3semidirect
248. C3⋊C8C2×C3⋊Dic3semidirect
302. C32#4788semidirect
303. C32Dic3×SD16semidirect
306. Q8D3×C3⋊Dic3semidirect
341. C3#10591571semidirect
342. C3SD16×C3⋊Dic3semidirect

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×C32×#2152
#117234992
C3×#636622
#117220522
#117220322
#117220182
SD16×C3⋊Dic33
#105915713