Finite Group Databasebeta

← #13159703

Subgroups — #13159703

Order 1728.

880 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 214 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×C32×#204C211yesmaximal, characteristic
2#11723499C211yesmaximal, characteristic
3#11723397C211yesmaximal, characteristic, complemented
4#11722185C211yesmaximal, characteristic, complemented
5C3×#63645C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic
8#10592863—31nomaximal
9#10590779—34nomaximal
10C2×C32×D12C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×D3×C3⋊Dic3—21no—
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C32×#204C414yescomplemented
19Dic3⋊(C3⋊Dic3)—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22C2×C32×SD16—31no—
23Q8×C3⋊Dic3—31no—
24D4×C3⋊Dic3—31no—
25C2×C3×#204D314yes—
26#63898—34no—
27#63843—34no—
28#63662—31no—
29#63659—31no—
30#63645D311yescharacteristic, complemented
31#63642—31no—
32#63642—34no—
33#63642—64no—
34C3×#818—34no—
35#63565—34no—
36#63562—34no—
37C22×C32×D3—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3—21no—
44C32×D12—21no—
45C32×D12C2×C412yes—
46C32×Dic6—21no—
47C32×Dic6C2×C412yescomplemented
48D3×C3⋊Dic3—42no—
49C33×C8C2×C412yescomplemented
50C3⋊(C3⋊(C3⋊C8))—21no—
51#5164—94no—
52#4889—91no—
53C2×C32×Q8—31no—
54C2×C32×D4—31no—
55C22×C3⋊Dic3—61no—
56C2×C3×D12C2×D314yes—
57C2×C3×Dic6C2×D314yes—
58C2×D3×Dic3—64no—
59C32×SD16—34no—
60C2×C32×C8—24no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C2×D314yes—
63#3757—61no—
64C4⋊(C3⋊Dic3)—61no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4×C3⋊Dic3—31no—
68C4×C3⋊Dic3—34no—
69C4×C3⋊Dic3—61no—
70C4×C3⋊Dic3—64no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3—34no—
76C3×#204Dic3116yescomplemented
77Dic3⋊Dic3—64no—
78(C2×D3)⋊Dic3—64no—
79Dic32—64no—
80C22×C33C2×D411yescharacteristic
81C2×C32×D3—22no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C33×C4C22×C411yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—21no—
87C3⋊(C3⋊Dic3)—21no—
88C3×C3⋊Dic3—41no—
89C3×C3⋊Dic3—41no—
90C32×Dic3—22no—
91C2×C3×SD16—34no—
92Q8×Dic3—94no—
93D4×Dic3—94no—
94C2×#204C3⋊D311yescharacteristic
95#893—91no—
96#887—91no—
97C3×#107—91no—
98#835—94no—
99#832—94no—
100#818—34no—
101#815—94no—
102#815—91no—
103#815—184no—
104#810—91no—
105#809—91no—
106C23×C32—61no—
107C22×C3×D3—24no—
108C32×Q8—32no—
109C32×Q8—61no—
110C32×D4—32no—
111C32×D4—61no—
112C2×C32×C4—24no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—34no—
121C2×C3⋊Dic3—34no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—122no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—64no—
128C3×D12—24no—
129C3×D12C2×Dic318yes—
130C3×Dic6—24no—
131C3×Dic6C2×Dic318yescomplemented
132D3×Dic3—128no—
133C32×C8—28no—
134C32×C8C4×D312yescomplemented
135C32×C8C2×Dic318yescomplemented
136C3⋊(C3⋊C8)—68no—
137C3⋊(C3⋊C8)—64no—
138C3⋊(C3⋊C8)—61no—
139#440—271noSylow
140C2×C33C4×D411yescharacteristic
141C2×C33AGL(1,ℤ/8)11yescharacteristic
142C2×C33#13711yescharacteristic
143C32×D3—42no—
144C2×C3×Q8—34no—
145C2×C3×D4—34no—
146C22×Dic3—184no—
147C2×D12C2×C3⋊D311yescharacteristic
148C2×C4×D3—181no—
149C2×Dic6C2×C3⋊D311yescharacteristic
150C3×SD16—316no—
151C2×C3×C8—24no—
152C2×C3×C8D3214yes—
153C2×C3×C8C2×C3⋊D311yescharacteristic
154C3×C4⋊C4—91no—
155C3×C4⋊C4—91no—
156#217—184no—
157(C2×D3)⋊C4—181no—
158C4⋊Dic3—34no—
159C4⋊Dic3—34no—
160C4⋊Dic3—184no—
161Dic3⋊C4—181no—
162C4×Dic3—94no—
163C4×Dic3—91no—
164C4×Dic3—184no—
165C4×Dic3—181no—
166C4×Dic3—184no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170#204C3⋊Dic314yescomplemented
171C22×C32—24no—
172C22×C32—62no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32D3×D414yes—
176C22×C32D3×D411yescharacteristic
177C2×C3×D3—28no—
178C2×C3×D3—44no—
179C2×C3×D3—44no—
180C32×C4—24no—
181C32×C4—24no—
182C32×C4—62no—
183C32×C4C2×C4×D311yescharacteristic
184C32×C4#23714yes—
185C32×C4#23911yescharacteristic
186C32×C4C22×Dic314yes—
187C3⋊Dic3—41no—
188C3⋊Dic3—41no—
189C3⋊Dic3—61no—
190C3⋊Dic3—61no—
191C3⋊Dic3—64no—
192C3⋊Dic3—64no—
193C3⋊Dic3—124no—
194C3⋊Dic3—124no—
195C3×Dic3—28no—
196C3×Dic3—124no—
197C3×Dic3—124no—
198C2×SD16—31no—
199C4×Q8—271no—
200C4×D4—271no—
201#107—91no—
202#103—271no—
203#102—271no—
204#97—271no—
205C33#44011yescharacteristic, complemented, Sylow
206C23×C3—64no—
207C22×D3—21no—
208C3×Q8—38no—
209C3×Q8—64no—
210C3×D4—38no—
211C3×D4—64no—
212C2×C3×C4—24no—
213C2×C3×C4—64no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216C2×C3×C4C2×D3214yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—64no—
220C2×Dic3—64no—
221C2×Dic3—94no—
222C2×Dic3—94no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—184no—
226C2×Dic3—184no—
227C2×Dic3—368no—
228D12—21no—
229D12C2×C3⋊Dic312yes—
230C4×D3—362no—
231Dic6—21no—
232Dic6C2×C3⋊Dic312yescomplemented
233C3×C8—28no—
234C3×C8D3×Dic318yescomplemented
235C3×C8C2×C3⋊Dic312yescomplemented
236C3⋊C8—188no—
237C3⋊C8—184no—
238C3⋊C8—181no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32—122no—
243C2×C32#89311yescharacteristic
244C2×C32#91111yescharacteristic
245C2×C32#91414yes—
246C2×C32#91514yes—
247C2×C32#91811yescharacteristic
248C2×C32D4×Dic314yes—
249C3×D3—48no—
250C2×Q8—31no—
251C2×D4—31no—
252C22×C4—541no—
253SD16—34no—
254C2×C8—271no—
255C2×C8D3×C3⋊D311yescharacteristic
256C4⋊C4—91no—
257C4⋊C4—91no—
258C4⋊C4—541no—
259C22⋊C4—541no—
260C42—271no—
261C42—541no—
262C22×C3—24no—
263C22×C3—68no—
264C22×C3—124no—
265C22×C3—124no—
266C22×C3#380214yes—
267C22×C3D4×C3⋊D311yescharacteristic
268C2×D3—22no—
269C2×D3—41no—
270C2×D3—41no—
271C3×C4—24no—
272C3×C4—24no—
273C3×C4—68no—
274C3×C4—361no—
275C3×C4—361no—
276C3×C4#379614yes—
277C3×C4C2×D3×Dic314yes—
278C3×C4#383011yescharacteristic
279C3×C4C22×C3⋊Dic311yescharacteristic
280Dic3—22no—
281Dic3—124no—
282Dic3—124no—
283Dic3—184no—
284Dic3—184no—
285Dic3—181no—
286Dic3—181no—
287Dic3—364no—
288Dic3—364no—
289C32—24no—
290C32#488911yescharacteristic, complemented
291C32#516414yescomplemented
292C23—61no—
293Q8—32no—
294Q8—61no—
295D4—32no—
296D4—61no—
297C2×C4—61no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4—271no—
301C2×C4—271no—
302C2×C4—1082no—
303C2×C4C2×D3×C3⋊D311yescharacteristic
304C8—541no—
305C8D3×C3⋊Dic312yescomplemented
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—128no—
310C2×C3#6379814yes—
311C2×C3#6381114yes—
312C2×C3#6389814yes—
313C2×C3#6412311yescharacteristic
314C2×C3#6412411yescharacteristic
315C2×C3D4×C3⋊Dic311yescharacteristic
316D3—42no—
317C22—62no—
318C22—121no—
319C22—121no—
320C22#8902611yescharacteristic
321C4—62no—
322C4—361no—
323C4—361no—
324C4—541no—
325C4—541no—
326C4#8901511yescharacteristic
327C4C2×D3×C3⋊Dic311yescharacteristic
328C3—24no—
329C3#1059077914yescomplemented
330C3#1059286311yescharacteristic, complemented
331C2—122no—
332C2#1172332311yescharacteristic
333C2#1172334411yescharacteristic
334C2#1172349911yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#440
LCSG ▷ 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
3. #11723397C2semidirect
4. #11722185C2semidirect
5. C3×#63645C2semidirect
6. #11722037C2semidirect
18. C32×#204C4semidirect
30. #63645D3semidirect
47. C32×Dic6C2×C4semidirect
49. C33×C8C2×C4semidirect
76. C3×#204Dic3semidirect
131. C3×Dic6C2×Dic3semidirect
134. C32×C8C4×D3semidirect
135. C32×C8C2×Dic3semidirect
170. #204C3⋊Dic3semidirect
205. C33#440semidirect
232. Dic6C2×C3⋊Dic3semidirect
234. C3×C8D3×Dic3semidirect
235. C3×C8C2×C3⋊Dic3semidirect
290. C32#4889semidirect
291. C32#5164semidirect
305. C8D3×C3⋊Dic3semidirect
329. C3#10590779semidirect
330. C3#10592863semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2042
#117234992
#117233972
#117221852
C3×#636452
#117220372
#117220312
#105928633
#105907793