Finite Group Databasebeta

← C4×#88793 #13158979

Subgroups — C4×#88793 #13158979

Order 1728.

908 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic, complemented
3C2×#88793C211yesmaximal, characteristic
4C4×C3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
5#11722053C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic
8C4×#3755—31nomaximal
9C4×#3715—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3—21no—
12C2×C32×Dic6C2211yescharacteristic
13C33×C42C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3—21no—
21#88793C414yesdirect product factor
22C32×C4×Q8—31no—
23C2×#3755—31no—
24C4×C4⋊(C3⋊D3)D311yescharacteristic
25C3×C4×D12—34no—
26C3×C4×Dic6D314yescomplemented
27C2×#3715—34no—
28#63662—31no—
29#63636—31no—
30#63635—31no—
31C4×C3⋊(C3⋊C8)—31no—
32C4×C3⋊(C3⋊C8)—34no—
33C4×C3⋊(C3⋊C8)—64no—
34#63574—34no—
35#63565—34no—
36#63562—34no—
37C22×C3×C3⋊D3—21no—
38C2×C33×C4D411yescharacteristic
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C3×C4⋊(C3⋊D3)—21no—
43C3×C4⋊(C3⋊D3)C2×C412yes—
44C3×C4×C3⋊D3—42no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—21no—
47C32×Dic6—21no—
48C32×Dic6C2×C412yescomplemented
49C3⋊(C3⋊(C3⋊C8))—21no—
50C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
51C4×#215—94no—
52C4×#214—91no—
53C2×C32×Q8—31no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×C3×D12—34no—
57C2×C3×C4×D3—64no—
58C2×C3×Dic6C2×D314yes—
59C32×C4⋊C4—31no—
60C32×C4⋊C4—61no—
61C32×C42—24no—
62C32×C42—61no—
63C32×C42C2×D314yes—
64C32×C42C2×D311yescharacteristic
65#3755—34no—
66#3752—21no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—34no—
72C2×C3⋊(C3⋊C8)—64no—
73C2×C3⋊(C3⋊C8)—64no—
74C3×(C2×D3)⋊C4—64no—
75C3×C4⋊Dic3—34no—
76C3×C4⋊Dic3C2×D314yes—
77C3×Dic3⋊C4—24no—
78C3×C4×Dic3—24no—
79#3715—316no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4—21no—
85C33×C4SD1612yescomplemented
86C33×C4C22×C411yescharacteristic
87C33×C4#4111yescharacteristic
88C3×C3⋊Dic3—41no—
89C32×Dic3—22no—
90C32×Dic3—41no—
91C3×C4×Q8—34no—
92C3×C4×D4—91no—
93C2×#215—94no—
94C2×#214—91no—
95C4×D12—34no—
96C4×Dic6C3⋊D311yescharacteristic, complemented
97#835—94no—
98#832—91no—
99#810—91no—
100#809—94no—
101#808—94no—
102#808—91no—
103C4×C3⋊C8—94no—
104C4×C3⋊C8—91no—
105C4×C3⋊C8—184no—
106C22×C3⋊D3—21no—
107C22×C3×D3—64no—
108C32×Q8—32no—
109C32×Q8—61no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—61no—
114C2×C32×C4—61no—
115C2×C32×C4#6714yes—
116C2×C32×C4#6714yes—
117C2×C32×C4#6711yescharacteristic
118C2×C32×C4#6711yescharacteristic
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)C4×D312yes—
124C4×C3⋊D3—42no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—64no—
128C3×D12—38no—
129C3×D12—64no—
130C3×C4×D3—128no—
131C3×Dic6—24no—
132C3×Dic6C4×D318yescomplemented
133C3⋊(C3⋊C8)—32no—
134C3⋊(C3⋊C8)—38no—
135C3⋊(C3⋊C8)—68no—
136C3⋊(C3⋊C8)—68no—
137C3⋊(C3⋊C8)—64no—
138C3⋊(C3⋊C8)—61no—
139C4×SD16—271noSylow
140C2×C33C4×D411yescharacteristic
141C2×C33C2×SD1611yescharacteristic
142C2×C33#13511yescharacteristic
143C3×C3⋊D3—42no—
144C2×C3×Q8—34no—
145C2×C3×D4—91no—
146C22×C3×C4—181no—
147C2×D12—34no—
148C2×C4×D3—64no—
149C2×Dic6C2×C3⋊D311yescharacteristic
150C3×C4⋊C4—34no—
151C3×C4⋊C4—64no—
152C3×C4⋊C4—91no—
153C3×C22⋊C4—181no—
154C3×C42—24no—
155C3×C42—64no—
156C3×C42D3214yes—
157C3×C42C2×C3⋊D311yescharacteristic
158#215—916no—
159#214—94no—
160(C2×D3)⋊C4—64no—
161C4⋊Dic3—34no—
162C4⋊Dic3C2×C3⋊D311yescharacteristic
163Dic3⋊C4—21no—
164C4×Dic3—21no—
165C2×C3⋊C8—94no—
166C2×C3⋊C8—91no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170C2×C3⋊C8—184no—
171C22×C32—24no—
172C22×C32C2×#6714yes—
173C22×C32C2×#6711yescharacteristic
174C2×C3⋊D3—22no—
175C2×C3⋊D3—41no—
176C2×C3⋊D3—41no—
177C2×C3×D3—68no—
178C2×C3×D3—124no—
179C2×C3×D3—124no—
180C32×C4—28no—
181C32×C4—24no—
182C32×C4—21no—
183C32×C4—28no—
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—62no—
187C32×C4—121no—
188C32×C4#21412yescomplemented
189C32×C4#21518yescomplemented
190C32×C4C2×C4×D314yes—
191C32×C4C2×C4×D311yescharacteristic
192C32×C4#23514yes—
193C32×C4#23511yescharacteristic
194C3⋊Dic3—41no—
195C3×Dic3—28no—
196C3×Dic3—44no—
197C3×Dic3—124no—
198C2×SD16—271no—
199C4×Q8—31no—
200C4×D4—91no—
201#106—271no—
202#103—271no—
203#102—271no—
204C4×C8—271no—
205C33C4×SD1611yescharacteristic, complemented, Sylow
206C23×C3—181no—
207C22×D3—64no—
208C3×Q8—38no—
209C3×Q8—64no—
210C3×D4—92no—
211C3×D4—181no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—24no—
215C2×C3×C4—64no—
216C2×C3×C4—64no—
217C2×C3×C4—181no—
218C2×C3×C4—362no—
219C2×C3×C4#62514yes—
220C2×C3×C4#62514yes—
221C2×C3×C4#63811yescharacteristic
222C2×C3×C4#63811yescharacteristic
223C2×C3×C4C2×D3214yes—
224C2×C3×C4C22×C3⋊D311yescharacteristic
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—64no—
228D12—38no—
229D12—64no—
230C4×D3—128no—
231Dic6—21no—
232Dic6C4×C3⋊D312yescomplemented
233C3⋊C8—98no—
234C3⋊C8—92no—
235C3⋊C8—188no—
236C3⋊C8—188no—
237C3⋊C8—184no—
238C3⋊C8—181no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32C4×#6714yes—
243C2×C32C4×#6711yescharacteristic
244C2×C32C2×#21411yescharacteristic
245C2×C32C2×#21514yes—
246C2×C32#95014yes—
247C2×C32#95011yescharacteristic
248C3⋊D3—42no—
249C3×D3—128no—
250C2×Q8—31no—
251C2×D4—91no—
252C22×C4—181no—
253SD16—274no—
254C2×C8—271no—
255C2×C8—271no—
256C4⋊C4—31no—
257C4⋊C4—61no—
258C4⋊C4—91no—
259C22⋊C4—181no—
260C42—61no—
261C42D3×C3⋊D311yescharacteristic
262C22×C3—24no—
263C22×C3—182no—
264C22×C3—361no—
265C22×C3—361no—
266C22×C3C2×#62514yes—
267C22×C3C2×#63811yescharacteristic
268C2×D3—68no—
269C2×D3—124no—
270C2×D3—124no—
271C3×C4—28no—
272C3×C4—24no—
273C3×C4—21no—
274C3×C4—24no—
275C3×C4—24no—
276C3×C4—28no—
277C3×C4—68no—
278C3×C4—124no—
279C3×C4—361no—
280C3×C4#371518yescomplemented
281C3×C4#375512yescomplemented
282C3×C4#379814yes—
283C3×C4C4×D3214yes—
284C3×C4C2×C4×C3⋊D311yescharacteristic
285C3×C4#382811yescharacteristic
286Dic3—22no—
287Dic3—41no—
288Dic3—124no—
289C32—24no—
290C32C4×#21411yescharacteristic, complemented
291C32C4×#21514yescomplemented
292C23—181no—
293Q8—32no—
294Q8—61no—
295D4—92no—
296D4—181no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—181no—
300C2×C4—362no—
301C2×C4#629011yescharacteristic
302C2×C4#629011yescharacteristic
303C2×C4C2×D3×C3⋊D311yescharacteristic
304C8—272no—
305C8—541no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—362no—
310C2×C3#6382214yes—
311C2×C3C2×#371514yes—
312C2×C3C4×#62514yes—
313C2×C3C4×#63811yescharacteristic
314C2×C3C2×#375511yescharacteristic
315C2×C3#6415911yescharacteristic
316D3—128no—
317C22—182no—
318C22—361no—
319C22—361no—
320C22C2×#629011yescharacteristic
321C4—21no—
322C4—62no—
323C4—121no—
324C4—361no—
325C4#8879312yesdirect product factor
326C4#8902011yescharacteristic
327C4C4×D3×C3⋊D311yescharacteristic
328C3—24no—
329C3C4×#371514yescomplemented
330C3C4×#375511yescharacteristic, complemented
331C2—362no—
332C2#1172336311yescharacteristic
333C2C2×#8879311yescharacteristic
334C2C4×#629011yescharacteristic
subgroupG/H
centerC2×C4#6290
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33C4×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C32×C4×Dic6C2semidirect
4. C4×C3⋊(C3⋊(C3⋊C8))C2semidirect
5. #11722053C2semidirect
6. #11722037C2semidirect
21. #88793C4direct (N × H)
26. C3×C4×Dic6D3semidirect
48. C32×Dic6C2×C4semidirect
50. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
85. C33×C4SD16semidirect
96. C4×Dic6C3⋊D3semidirect
132. C3×Dic6C4×D3semidirect
188. C32×C4#214semidirect
189. C32×C4#215semidirect
205. C33C4×SD16semidirect
232. Dic6C4×C3⋊D3semidirect
280. C3×C4#3715semidirect
281. C3×C4#3755semidirect
290. C32C4×#214semidirect
291. C32C4×#215semidirect
325. C4#88793direct (N × H)
329. C3C4×#3715semidirect
330. C3C4×#3755semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊(C3⋊D3)2
C32×C4×Dic62
C2×#887932
C4×C3⋊(C3⋊(C3⋊C8))2
#117220532
#117220372
#117220332
C4×#37553
C4×#37153