Finite Group Databasebeta

← C2×#11720851 #13149835

Subgroups — C2×#11720851 #13149835

Order 1728.

963 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 963 conjugacy classes of subgroups — too many to draw. Only its 171 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×((C3⋊D3)⋊C3)⋊Q8C212yesmaximal
2C2×#88530C211yesmaximal, characteristic, complemented
3C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
4C2×#3746⋊C3C211yesmaximal, characteristic, complemented
5C2×#88439C212yesmaximal, complemented
6#11720851C218yesmaximal, direct product factor
7C2×#63802—31nomaximal
8C2×D3×Dic12—31nomaximal
9C2×#6229C2212yes—
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×#634⋊C3C2212yes—
12C2×((C3⋊D3)⋊C3)⋊C4—22no—
13C2×#6196—22no—
14((C3⋊D3)⋊C3)⋊Q8—24no—
15((C3⋊D3)⋊C3)⋊Q8C2218yes—
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17#88530C2214yescomplemented
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C8×(C3⋊D3)⋊C3C2214yes—
20#3746⋊C3C2214yescomplemented
21#88439C2218yescomplemented
22C2×#3797—32no—
23C2×D3×Dic6—32no—
24C2×#3746D311yescharacteristic, complemented
25C2×C8×C3⋊D3D311yescharacteristic
26C2×C3×Dic12—31no—
27C2×C3×Dic12—31no—
28C2×C3×Dic12—31no—
29C2×C3×Dic12—61no—
30C2×C3×D3×C8—31no—
31C2×#3719—32no—
32C2×#3718—32no—
33#63802—38no—
34D3×Dic12—38no—
35C22×(C3⋊D3)⋊C3D411yescharacteristic
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084—22no—
38#6229—22no—
39#6229C2314yes—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C4×(C3⋊D3)⋊C3D412yes—
43C4×(C3⋊D3)⋊C3C2312yes—
44#634⋊C3—22no—
45#634⋊C3C2314yes—
46((C3⋊D3)⋊C3)⋊C4—28no—
47#6196—28no—
48C8×H(3,𝔽3)C2312yes—
49(C3⋊(C3⋊C8))⋊C3C2312yes—
50C2×D3×Q16—91no—
51C2×D3×Q16—91no—
52C22×Dic12—91no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×#634C2×D312yes—
55C2×C3×C4×D3—31no—
56C2×C3×Dic6—32no—
57C2×C3×Dic6—32no—
58C2×C3×Dic6—32no—
59C2×C3×Dic6—62no—
60C2×#627—62no—
61C2×#627—62no—
62C2×#624—62no—
63C2×D3×Dic3—62no—
64#3797—38no—
65#3797—64no—
66D3×Dic6—38no—
67D3×Dic6—64no—
68C2×C32×C8—21no—
69C2×C32×C8C2×D311yescharacteristic
70C2×C32×C8C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72#3746C2×D314yescomplemented
73C8×C3⋊D3C2×D314yes—
74C2×C3×C3⋊C8—31no—
75C3×Dic12—34no—
76C3×Dic12—34no—
77C3×Dic12—34no—
78C3×Dic12—64no—
79C3×D3×C8—34no—
80#3719—38no—
81#3718—38no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C2×(C3⋊D3)⋊C3Q1614yescomplemented
84C2×(C3⋊D3)⋊C3C2×D412yes—
85C4×H(3,𝔽3)C2×D411yescharacteristic
86C4×H(3,𝔽3)C2411yescharacteristic
87#1084—24no—
88(C3⋊Dic3)⋊C3—24no—
89(C3⋊Dic3)⋊C3C2×D412yes—
90C2×D3×Q8—92no—
91C2×D3×Q8—92no—
92C22×Dic6—92no—
93C2×C3×Q16—91no—
94C2×C3×Q16—91no—
95C22×C3×C8—91no—
96C2×#216—92no—
97C2×#216—92no—
98D3×Q16—98no—
99D3×Q16—98no—
100C2×Dic12—31no—
101C2×Dic12—31no—
102C2×Dic12—31no—
103C2×Dic12—61no—
104C2×Dic12—91no—
105C2×Dic12—98no—
106C2×Dic12—92no—
107C2×D3×C8—31no—
108C2×D3×C8—91no—
109C22×C3⋊D3D1211yescharacteristic
110C22×C3×D3—31no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3D1211yescharacteristic
116C4×C3⋊D3D1212yes—
117C4×C3⋊D3C22×D312yes—
118#634—22no—
119#634C22×D314yes—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—122no—
125C3×C4×D3—32no—
126C3×C4×D3—32no—
127C3×Dic6—34no—
128C3×Dic6—34no—
129C3×Dic6—34no—
130C3×Dic6—64no—
131C3×Dic6—62no—
132C3×Dic6—62no—
133C3×Dic6—62no—
134C3×Dic6—122no—
135#627—68no—
136#627—68no—
137#624—68no—
138D3×Dic3—68no—
139C32×C8—22no—
140C32×C8C22×D312yes—
141C32×C8C22×D312yes—
142C3⋊(C3⋊C8)C22×D312yes—
143C3×C3⋊C8—32no—
144C22×Q16—271noSylow
145C2×H(3,𝔽3)C2×Q1612yescomplemented
146C2×H(3,𝔽3)C22×D411yescharacteristic
147(C3⋊D3)⋊C3C2×Q1614yescomplemented
148C2×C3×Q8—92no—
149C2×C3×Q8—92no—
150C22×C3×C4—91no—
151C22×Dic3—182no—
152D3×Q8—98no—
153D3×Q8—98no—
154D3×Q8—184no—
155D3×Q8—184no—
156C2×C4×D3—31no—
157C2×C4×D3—91no—
158C2×C4×D3—182no—
159C2×C4×D3—182no—
160C2×Dic6—32no—
161C2×Dic6—32no—
162C2×Dic6—32no—
163C2×Dic6—62no—
164C2×Dic6—92no—
165C2×Dic6—98no—
166C2×Dic6—184no—
167C2×Dic6—182no—
168C2×Dic6—182no—
169C2×Dic6—182no—
170C3×Q16—94no—
171C3×Q16—94no—
172C2×C3×C8—31no—
173C2×C3×C8—31no—
174C2×C3×C8—61no—
175C2×C3×C8—94no—
176C2×C3×C8—91no—
177C2×C3×C8D3211yescharacteristic
178#216—98no—
179#216—98no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—91no—
182Dic12—34no—
183Dic12—34no—
184Dic12—34no—
185Dic12—64no—
186Dic12—94no—
187Dic12—98no—
188D3×C8—34no—
189D3×C8—94no—
190C22×C32—21no—
191C22×C32C2×D1211yescharacteristic
192C22×C32D3×D411yescharacteristic
193C2×C3⋊D3Dic1214yescomplemented
194C2×C3⋊D3C2×D1212yes—
195C2×C3×D3—34no—
196C2×C3×D3—32no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4C2×D1211yescharacteristic
200C32×C4D3×D411yescharacteristic
201C32×C4C23×D311yescharacteristic
202C32×C4C23×D311yescharacteristic
203C3⋊Dic3—24no—
204C3⋊Dic3C2×D1212yes—
205C3×Dic3—32no—
206C3×Dic3—64no—
207C3×Dic3—64no—
208C3×Dic3—64no—
209C3×Dic3—124no—
210C22×Q8—272no—
211C2×Q16—91no—
212C2×Q16—91no—
213C2×Q16—278no—
214C2×Q16—272no—
215C22×C8—91no—
216H(3,𝔽3)C22×Q1611yescharacteristic, complemented, Sylow
217C23×C3—91no—
218C22×D3—31no—
219C22×D3—91no—
220C3×Q8—94no—
221C3×Q8—94no—
222C3×Q8—182no—
223C3×Q8—182no—
224C2×C3×C4—31no—
225C2×C3×C4—31no—
226C2×C3×C4—61no—
227C2×C3×C4—92no—
228C2×C3×C4—92no—
229C2×C3×C4—91no—
230C2×C3×C4—182no—
231C2×C3×C4—182no—
232C2×C3×C4C2×D3211yescharacteristic
233C2×Dic3—31no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—91no—
238C2×Dic3—122no—
239C2×Dic3—182no—
240C2×Dic3—188no—
241C4×D3—32no—
242C4×D3—32no—
243C4×D3—92no—
244C4×D3—92no—
245C4×D3—188no—
246C4×D3—188no—
247Dic6—34no—
248Dic6—34no—
249Dic6—34no—
250Dic6—64no—
251Dic6—62no—
252Dic6—62no—
253Dic6—62no—
254Dic6—94no—
255Dic6—122no—
256Dic6—182no—
257Dic6—188no—
258Dic6—188no—
259Dic6—188no—
260C3×C8—32no—
261C3×C8—32no—
262C3×C8—62no—
263C3×C8—92no—
264C3×C8C2×D3212yes—
265C3⋊C8—32no—
266C3⋊C8—92no—
267C2×C32—21no—
268C2×C32—22no—
269C2×C32C2×Dic1212yescomplemented
270C2×C32D3×Q1612yescomplemented
271C2×C32C22×D1211yescharacteristic
272C2×C32C2×D3×D411yescharacteristic
273C3⋊D3C2×Dic1214yescomplemented
274C3×D3—34no—
275C2×Q8—92no—
276C2×Q8—92no—
277C2×Q8—278no—
278C2×Q8—542no—
279C2×Q8—544no—
280C22×C4—91no—
281C22×C4—542no—
282Q16—94no—
283Q16—94no—
284Q16—278no—
285C2×C8—91no—
286C2×C8—94no—
287C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—61no—
291C22×C3—92no—
292C22×C3—94no—
293C22×C3D3×D1211yescharacteristic
294C2×D3—32no—
295C2×D3—34no—
296C2×D3—92no—
297C2×D3—94no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—61no—
303C3×C4—61no—
304C3×C4—92no—
305C3×C4—184no—
306C3×C4—184no—
307C3×C4D3×D1211yescharacteristic
308C3×C4C22×D3211yescharacteristic
309Dic3—32no—
310Dic3—64no—
311Dic3—64no—
312Dic3—64no—
313Dic3—92no—
314Dic3—124no—
315Dic3—184no—
316C32—21no—
317C32C22×Dic1211yescharacteristic, complemented
318C32C2×D3×Q1611yescharacteristic, complemented
319C23—91no—
320Q8—94no—
321Q8—94no—
322Q8—182no—
323Q8—182no—
324Q8—548no—
325C2×C4—91no—
326C2×C4—92no—
327C2×C4—92no—
328C2×C4—182no—
329C2×C4—182no—
330C2×C4—548no—
331C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
332C8—92no—
333C8C2×((C3⋊D3)⋊C3)⋊C212yes—
334C2×C3—32no—
335C2×C3—32no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—62no—
339C2×C3—61no—
340C2×C3—94no—
341C2×C3D3×Dic1212yes—
342C2×C3C2×D3×D1211yescharacteristic
343D3—34no—
344D3—94no—
345C22—94no—
346C22—92no—
347C22#8865411yescharacteristic
348C4—92no—
349C4—184no—
350C4—184no—
351C4#8865411yescharacteristic
352C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
353C3—31no—
354C3—31no—
355C3—61no—
356C3C2×D3×Dic1211yescharacteristic
357C2—94no—
358C2#1172085112yesdirect product factor
359C2C2×#8865411yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×Q16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88530C2semidirect
4. C2×#3746⋊C3C2semidirect
5. C2×#88439C2semidirect
6. #11720851C2direct (N × H)
17. #88530C22semidirect
20. #3746⋊C3C22semidirect
21. #88439C22semidirect
24. C2×#3746D3semidirect
72. #3746C2×D3semidirect
83. C2×(C3⋊D3)⋊C3Q16semidirect
145. C2×H(3,𝔽3)C2×Q16semidirect
147. (C3⋊D3)⋊C3C2×Q16semidirect
193. C2×C3⋊D3Dic12semidirect
216. H(3,𝔽3)C22×Q16semidirect
269. C2×C32C2×Dic12semidirect
270. C2×C32D3×Q16semidirect
273. C3⋊D3C2×Dic12semidirect
317. C32C22×Dic12semidirect
318. C32C2×D3×Q16semidirect
358. C2#11720851direct (N × H)

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊Q82
C2×#885302
C2×C8×(C3⋊D3)⋊C32
C2×#3746⋊C32
C2×#884392
#117208512
C2×#638023
C2×D3×Dic123