Finite Group Databasebeta

← C2×#11724212 #13180871

Subgroups — C2×#11724212 #13180871

Order 1728.

810 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 86 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×A4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×C3⋊SL(2,ℤ/4)C212yesmaximal
3#11724212C214yesmaximal, direct product factor
4C2×#64204—32nomaximal
5C2×#63982—31nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×#6296—41nomaximal
8A4×C3⋊Dic3C2212yes—
9C3×C3⋊SL(2,ℤ/4)C2214yes—
10C23×C3⋊Dic3D311yescharacteristic
11C22×#627—31no—
12C2×Dic3×A4—32no—
13C2×Dic3×A4—61no—
14C2×C3⋊SL(2,ℤ/4)D312yes—
15C2×C3×SL(2,ℤ/4)—32no—
16C2×C3×SL(2,ℤ/4)—62no—
17#64204—38no—
18C2×C3×#217—32no—
19#63982—34no—
20#63982—34no—
21C2×#3724—31no—
22C2×Dic3⋊Dic3—32no—
23C2×C32×A4Q812yescomplemented
24C2×C32×A4C2311yescharacteristic
25C2×C3×C3⋊Dic3—42no—
26C2×C3×C3⋊Dic3—41no—
27#6296—44no—
28C2×A4⋊Q8—91no—
29C2×#878—92no—
30C24×C32C2×D311yescharacteristic
31C22×C3×A4—22no—
32C22×C3×A4—21no—
33C22×C3×A4—41no—
34C22×C3×A4C2×D312yes—
35C22×C3⋊Dic3—31no—
36C22×C3⋊Dic3—31no—
37C22×C3⋊Dic3—32no—
38C22×C3⋊Dic3C2×D312yes—
39C22×C3×Dic3—32no—
40C22×C3×Dic3—32no—
41C2×#627—34no—
42C2×#627—61no—
43C2×#627—62no—
44C2×#627—61no—
45C2×#627—122no—
46Dic3×A4—34no—
47Dic3×A4—62no—
48C3⋊SL(2,ℤ/4)C2×D314yes—
49C3×SL(2,ℤ/4)—34no—
50C3×SL(2,ℤ/4)—64no—
51C3×#217—34no—
52C3×#217—34no—
53#3724—34no—
54Dic3⋊Dic3—38no—
55C22×C33—41no—
56C32×A4C2×Q811yescharacteristic, complemented
57C3×C3⋊Dic3—44no—
58C3×C3⋊Dic3—42no—
59C23×Dic3—32no—
60C23×Dic3—61no—
61C22×Dic6—92no—
62C2×C4×A4—91no—
63C2×SL(2,ℤ/4)—32no—
64C2×SL(2,ℤ/4)—62no—
65A4⋊Q8—94no—
66C2×C3×C22⋊C4—92no—
67C2×#217—32no—
68C2×C4⋊Dic3—92no—
69C2×Dic3⋊C4—92no—
70C2×Dic3⋊C4—92no—
71#878—98no—
72#878—98no—
73C23×C32—31no—
74C23×C32—32no—
75C23×C32—31no—
76C23×C32Dic612yescomplemented
77C23×C32C22×D311yescharacteristic
78C2×C3×A4—24no—
79C2×C3×A4—22no—
80C2×C3×A4—22no—
81C2×C3×A4—21no—
82C2×C3×A4—42no—
83C2×C3×A4—41no—
84C2×C3×A4Dic614yescomplemented
85C2×C3×A4C22×D312yes—
86C2×C3⋊Dic3—31no—
87C2×C3⋊Dic3—32no—
88C2×C3⋊Dic3—32no—
89C2×C3⋊Dic3—32no—
90C2×C3⋊Dic3—42no—
91C2×C3⋊Dic3—61no—
92C2×C3⋊Dic3S411yescharacteristic
93C2×C3×Dic3—34no—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—64no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—122no—
99C2×C3×Dic3—122no—
100C2×C3×Dic3—241no—
101C2×C3×Dic3—242no—
102#627—64no—
103#627—64no—
104#627—128no—
105C2×C22⋊Q8—271noSylow
106C2×C33—42no—
107C2×C33—41no—
108C24×C3—21no—
109C24×C3D3212yes—
110C22×A4—22no—
111C22×A4—41no—
112C22×A4D3211yescharacteristic
113C22×C3×C4—92no—
114C22×C3×C4—92no—
115C22×Dic3—32no—
116C22×Dic3—32no—
117C22×Dic3—34no—
118C22×Dic3—62no—
119C22×Dic3—92no—
120C22×Dic3—94no—
121C22×Dic3—92no—
122C22×Dic3—181no—
123C22×Dic3—181no—
124C22×Dic3—182no—
125C2×Dic6—98no—
126C2×Dic6—182no—
127C2×Dic6—184no—
128C2×Dic6—182no—
129C2×Dic6—361no—
130C4×A4—92no—
131SL(2,ℤ/4)—34no—
132SL(2,ℤ/4)—64no—
133C3×C22⋊C4—94no—
134C3×C22⋊C4—94no—
135#217—34no—
136#217—34no—
137C4⋊Dic3—98no—
138Dic3⋊C4—98no—
139Dic3⋊C4—98no—
140C22×C32—32no—
141C22×C32—31no—
142C22×C32—32no—
143C22×C32—32no—
144C22×C32—31no—
145C22×C32—31no—
146C22×C32—42no—
147C22×C32—61no—
148C22×C32—81no—
149C22×C32—82no—
150C22×C32—161no—
151C22×C32C2×Dic611yescharacteristic, complemented
152C22×C32C2×S411yescharacteristic
153C3×A4—22no—
154C3×A4—21no—
155C3×A4—41no—
156C3×A4C2×Dic612yescomplemented
157C3⋊Dic3—32no—
158C3⋊Dic3—44no—
159C3⋊Dic3C2×S412yes—
160C3×Dic3—64no—
161C3×Dic3—64no—
162C3×Dic3—124no—
163C3×Dic3—124no—
164C3×Dic3—242no—
165C3×Dic3—244no—
166C22×Q8—271no—
167C23×C4—91no—
168C22⋊Q8—274no—
169C22⋊Q8—274no—
170C2×C4⋊C4—271no—
171C2×C4⋊C4—272no—
172C2×C22⋊C4—92no—
173C33—41noSylow
174C23×C3—22no—
175C23×C3—21no—
176C23×C3—32no—
177C23×C3—32no—
178C23×C3—34no—
179C23×C3—61no—
180C23×C3—61no—
181C23×C3—62no—
182C23×C3#62714yescomplemented
183C23×C3C2×D3212yes—
184C2×A4—24no—
185C2×A4—22no—
186C2×A4—42no—
187C2×A4—41no—
188C2×A4#62712yescomplemented
189C2×A4C2×D3211yescharacteristic
190C2×C3×C4—94no—
191C2×C3×C4—94no—
192C2×C3×C4—182no—
193C2×C3×C4—184no—
194C2×C3×C4—182no—
195C2×C3×C4—361no—
196C2×Dic3—32no—
197C2×Dic3—34no—
198C2×Dic3—34no—
199C2×Dic3—61no—
200C2×Dic3—62no—
201C2×Dic3—64no—
202C2×Dic3—62no—
203C2×Dic3—94no—
204C2×Dic3—94no—
205C2×Dic3—92no—
206C2×Dic3—94no—
207C2×Dic3—122no—
208C2×Dic3—182no—
209C2×Dic3—182no—
210C2×Dic3—181no—
211C2×Dic3—182no—
212C2×Dic3—182no—
213C2×Dic3—182no—
214C2×Dic3—242no—
215Dic6—188no—
216Dic6—188no—
217Dic6—364no—
218C2×C32—31no—
219C2×C32—31no—
220C2×C32—32no—
221C2×C32—42no—
222C2×C32—44no—
223C2×C32—81no—
224C2×C32—82no—
225C2×C32—82no—
226C2×C32—84no—
227C2×C32—162no—
228C2×C32—161no—
229C2×C32A4⋊Q812yescomplemented
230C2×C32C22×S411yescharacteristic
231C24#111311yescharacteristic
232C2×Q8—274no—
233C2×Q8—541no—
234C2×Q8—542no—
235C2×Q8—541no—
236C22×C4—92no—
237C22×C4—92no—
238C22×C4—92no—
239C22×C4—271no—
240C22×C4—271no—
241C22×C4—272no—
242C4⋊C4—274no—
243C4⋊C4—278no—
244C22⋊C4—94no—
245C22⋊C4—94no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—34no—
249C22×C3—32no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—34no—
253C22×C3—34no—
254C22×C3—41no—
255C22×C3—61no—
256C22×C3—61no—
257C22×C3—62no—
258C22×C3—62no—
259C22×C3—62no—
260C22×C3—61no—
261C22×C3—62no—
262C22×C3—62no—
263C22×C3—82no—
264C22×C3—161no—
265C22×C3C2×#62712yescomplemented
266C22×C3D3×S412yes—
267A4—22no—
268A4—41no—
269A4C2×#62711yescharacteristic, complemented
270C3×C4—184no—
271C3×C4—184no—
272C3×C4—362no—
273Dic3—34no—
274Dic3—62no—
275Dic3—64no—
276Dic3—64no—
277Dic3—94no—
278Dic3—124no—
279Dic3—182no—
280Dic3—244no—
281C32—42no—
282C32—81no—
283C32—82no—
284C32—161no—
285C32C2×A4⋊Q811yescharacteristic, complemented
286C23—31no—
287C23—31no—
288C23—32no—
289C23#629612yescomplemented
290C23C2×#111311yescharacteristic
291Q8—544no—
292Q8—544no—
293C2×C4—91no—
294C2×C4—94no—
295C2×C4—94no—
296C2×C4—182no—
297C2×C4—184no—
298C2×C4—182no—
299C2×C4—272no—
300C2×C4—272no—
301C2×C4—271no—
302C2×C4—272no—
303C2×C4—541no—
304C2×C3—22no—
305C2×C3—21no—
306C2×C3—32no—
307C2×C3—32no—
308C2×C3—34no—
309C2×C3—42no—
310C2×C3—41no—
311C2×C3—61no—
312C2×C3—61no—
313C2×C3—62no—
314C2×C3—84no—
315C2×C3—82no—
316C2×C3—162no—
317C2×C3—161no—
318C2×C3#6420414yescomplemented
319C2×C3C2×D3×S412yes—
320C22—32no—
321C22—31no—
322C22—32no—
323C22—32no—
324C22—31no—
325C22—31no—
326C22—61no—
327C22C2×#629611yescharacteristic, complemented
328C22#8910111yescharacteristic
329C4—92no—
330C4—184no—
331C4—184no—
332C4—272no—
333C3—21no—
334C3—41no—
335C3—82no—
336C3—161no—
337C3C2×#6420412yescomplemented
338C2—31no—
339C2—31no—
340C2—32no—
341C2#1172421212yesdirect product factor
342C2C2×#8910111yescharacteristic
subgroupG/H
center, hypercenterC22#89101
commutator, cosocleC2×C32×A4C23
FrattiniC2C2×#89101
Fitting, socleC24×C32C2×D3
prime radicalC22×C32×A4C22
hypocenterC32×A4C2×Q8
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. #11724212C2direct (N × H)
23. C2×C32×A4Q8semidirect
56. C32×A4C2×Q8semidirect
76. C23×C32Dic6semidirect
84. C2×C3×A4Dic6semidirect
151. C22×C32C2×Dic6semidirect
156. C3×A4C2×Dic6semidirect
182. C23×C3#627semidirect
188. C2×A4#627semidirect
229. C2×C32A4⋊Q8semidirect
265. C22×C3C2×#627semidirect
269. A4C2×#627semidirect
285. C32C2×A4⋊Q8semidirect
289. C23#6296semidirect
318. C2×C3#64204semidirect
327. C22C2×#6296semidirect
337. C3C2×#64204semidirect
341. C2#11724212direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×C3⋊Dic32
C2×C3×C3⋊SL(2,ℤ/4)2
#117242122
C2×#642043
C2×#639823
C2×#62964