Finite Group Databasebeta

← C2×#11720769 #13147307

Subgroups — C2×#11720769 #13147307

Order 1728.

785 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 785 conjugacy classes of subgroups — too many to draw. Only its 31 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⋊(C22⋊A4)C211yesmaximal, characteristic, complemented
2#11720769C212yesmaximal, direct product factor
3C2×#64162C311yesmaximal, characteristic, complemented
4C2×C3⋊(#120⋊C3)—31nomaximal
5C32⋊(C22⋊A4)C2211yescharacteristic, complemented
6C2×(C3⋊D3)⋊A4—41nomaximal
7C25×C32C2×C311yescharacteristic, complemented
8C2×C3×C22⋊A4—21no—
9C2×C3×C22⋊A4D311yescharacteristic, complemented
10C22×#638—31no—
11#64162—32no—
12#64162C2×C312yescomplemented
13C2×#3757—31no—
14C3⋊(#120⋊C3)—32no—
15C2×C22⋊H(3,𝔽3)—41no—
16C2×C22⋊H(3,𝔽3)—82no—
17(C3⋊D3)⋊A4—42no—
18C2×#953—31no—
19C2×#953—91no—
20C2×#120⋊C3—91no—
21C24×C32—31no—
22C24×C32—31no—
23C24×C32—62no—
24C24×C32—62no—
25C24×C32C22×C311yescharacteristic, complemented
26C23×C3⋊D3A411yescharacteristic, complemented
27C3×C22⋊A4—21no—
28C3×C22⋊A4C2×D311yescharacteristic, complemented
29C2×D3×A4—121no—
30C2×#638—32no—
31C2×#638—32no—
32C2×#638—62no—
33C2×#638—62no—
34C22×C3⋊Dic3—31no—
35#3757—32no—
36#3757—32no—
37C2×(C3⋊D3)⋊C3—161no—
38C22⋊H(3,𝔽3)—41no—
39C22⋊H(3,𝔽3)—82no—
40C25×C3—31no—
41C25×C3C3×D311yescharacteristic
42C2×C22⋊A4—31no—
43C2×C22⋊A4—61no—
44C22×#67—91no—
45C22×#67—93no—
46#953—32no—
47#953—92no—
48#953—96no—
49#953—92no—
50C2×#217—93no—
51C2×#217—91no—
52#120⋊C3—92no—
53C23×C32—22no—
54C23×C32—32no—
55C23×C32—32no—
56C23×C32—31no—
57C23×C32—31no—
58C23×C32—62no—
59C23×C32—62no—
60C23×C32—62no—
61C23×C32—62no—
62C23×C32—62no—
63C23×C32—64no—
64C23×C32—62no—
65C23×C32—62no—
66C23×C32—64no—
67C23×C32C2×A411yescharacteristic
68C22×C3⋊D3—61no—
69C22×C3⋊D3—61no—
70C22×C3⋊D3C2×A412yescomplemented
71C2×C3×A4—41no—
72C2×C3×A4—82no—
73C2×C3×A4—84no—
74C2×C3×A4—81no—
75D3×A4—122no—
76#638—64no—
77#638—64no—
78C2×C3⋊Dic3—32no—
79C2×C3⋊Dic3—61no—
80C2×C3⋊Dic3—61no—
81C2×#120—91noSylow
82C2×H(3,𝔽3)—161no—
83(C3⋊D3)⋊C3—162no—
84C24×C3—33no—
85C24×C3—31no—
86C24×C3—33no—
87C24×C3—31no—
88C24×C3—31no—
89C24×C3—62no—
90C24×C3—62no—
91C24×C3—66no—
92C24×C3—66no—
93C24×C3C2×C3×D311yescharacteristic
94C23×D3—31no—
95C23×D3—91no—
96C22⋊A4—31no—
97C22⋊A4—61no—
98C22×A4—361no—
99C2×#67—92no—
100C2×#67—92no—
101C2×#67—96no—
102C2×#67—96no—
103C2×#67—182no—
104C2×#67—186no—
105C2×#67—186no—
106C2×#67—182no—
107C22×Dic3—93no—
108C22×Dic3—91no—
109#217—96no—
110#217—92no—
111#217—96no—
112#217—92no—
113C22×C32—22no—
114C22×C32—31no—
115C22×C32—31no—
116C22×C32—32no—
117C22×C32—32no—
118C22×C32—62no—
119C22×C32—62no—
120C22×C32—64no—
121C22×C32—62no—
122C22×C32—62no—
123C22×C32—62no—
124C22×C32—62no—
125C22×C32—64no—
126C22×C32—62no—
127C22×C32C22×A411yescharacteristic
128C2×C3⋊D3—41no—
129C2×C3⋊D3—62no—
130C2×C3⋊D3—121no—
131C2×C3×D3—481no—
132C3×A4—41no—
133C3×A4—82no—
134C3×A4—84no—
135C3×A4—81no—
136C3⋊Dic3—62no—
137C25(C3⋊D3)⋊C311yescharacteristic, complemented
138C22×D4—271no—
139#120—92no—
140#120—272no—
141C2×C22⋊C4—271no—
142H(3,𝔽3)—161noSylow
143C23×C3—22no—
144C23×C3—33no—
145C23×C3—33no—
146C23×C3—32no—
147C23×C3—36no—
148C23×C3—36no—
149C23×C3—32no—
150C23×C3—31no—
151C23×C3—31no—
152C23×C3—31no—
153C23×C3—62no—
154C23×C3—62no—
155C23×C3—62no—
156C23×C3—66no—
157C23×C3—62no—
158C23×C3—66no—
159C23×C3—66no—
160C23×C3—62no—
161C23×C3—62no—
162C23×C3—64no—
163C23×C3—66no—
164C23×C3—612no—
165C23×C3—62no—
166C23×C3—62no—
167C23×C3—66no—
168C23×C3—66no—
169C23×C3—66no—
170C23×C3—64no—
171C23×C3—612no—
172C23×C3D3×A411yescharacteristic
173C22×D3—32no—
174C22×D3—92no—
175C22×D3—181no—
176C22×D3—183no—
177C22×D3—183no—
178C22×D3—181no—
179C2×A4—121no—
180C2×A4—242no—
181C2×A4—244no—
182C2×A4—241no—
183C2×A4—362no—
184#67—184no—
185#67—184no—
186#67—1812no—
187#67—1812no—
188C2×Dic3—96no—
189C2×Dic3—92no—
190C2×Dic3—181no—
191C2×Dic3—183no—
192C2×Dic3—183no—
193C2×Dic3—181no—
194C2×C32—31no—
195C2×C32—31no—
196C2×C32—62no—
197C2×C32—62no—
198C2×C32—161no—
199C2×C32—321no—
200C2×C32#120⋊C311yescharacteristic, complemented
201C3⋊D3—42no—
202C3×D3—482no—
203C24—31no—
204C24—31no—
205C24—62no—
206C24—62no—
207C24—91no—
208C24C2×(C3⋊D3)⋊C311yescharacteristic, complemented
209C2×D4—272no—
210C2×D4—272no—
211C2×D4—542no—
212C2×D4—542no—
213C22×C4—271no—
214C22⋊C4—272no—
215C22⋊C4—272no—
216C22×C3—22no—
217C22×C3—31no—
218C22×C3—33no—
219C22×C3—33no—
220C22×C3—31no—
221C22×C3—36no—
222C22×C3—36no—
223C22×C3—32no—
224C22×C3—32no—
225C22×C3—31no—
226C22×C3—62no—
227C22×C3—66no—
228C22×C3—66no—
229C22×C3—612no—
230C22×C3—66no—
231C22×C3—62no—
232C22×C3—62no—
233C22×C3—64no—
234C22×C3—62no—
235C22×C3—66no—
236C22×C3—62no—
237C22×C3—62no—
238C22×C3—62no—
239C22×C3—66no—
240C22×C3—66no—
241C22×C3—66no—
242C22×C3—612no—
243C22×C3—64no—
244C22×C3—62no—
245C22×C3—1441no—
246C22×C3C2×D3×A411yescharacteristic
247C2×D3—121no—
248C2×D3—182no—
249C2×D3—186no—
250C2×D3—361no—
251C2×D3—361no—
252C2×D3—363no—
253A4—121no—
254A4—242no—
255A4—244no—
256A4—241no—
257Dic3—186no—
258Dic3—182no—
259C32—161no—
260C32—321no—
261C32C2×#120⋊C311yescharacteristic, complemented
262C23—22no—
263C23—32no—
264C23—32no—
265C23—31no—
266C23—31no—
267C23—62no—
268C23—62no—
269C23—62no—
270C23—62no—
271C23—62no—
272C23—64no—
273C23—62no—
274C23—62no—
275C23—64no—
276C23—92no—
277C23—541no—
278C23—541no—
279C23(C3⋊D3)⋊A411yescharacteristic
280D4—544no—
281D4—544no—
282C2×C4—272no—
283C2×C4—541no—
284C2×C4—541no—
285C2×C3—33no—
286C2×C3—33no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—66no—
291C2×C3—66no—
292C2×C3—62no—
293C2×C3—62no—
294C2×C3—481no—
295C2×C3—961no—
296C2×C3—1442no—
297C2×C3C3⋊(#120⋊C3)11yescharacteristic
298D3—122no—
299D3—362no—
300C22—22no—
301C22—31no—
302C22—31no—
303C22—32no—
304C22—32no—
305C22—62no—
306C22—62no—
307C22—64no—
308C22—62no—
309C22—62no—
310C22—62no—
311C22—62no—
312C22—64no—
313C22—62no—
314C22—361no—
315C22—542no—
316C22—1081no—
317C22C2×(C3⋊D3)⋊A411yescharacteristic
318C4—542no—
319C3—31no—
320C3—481no—
321C3—961no—
322C3C2×C3⋊(#120⋊C3)11yescharacteristic
323C2—31no—
324C2—31no—
325C2—62no—
326C2—62no—
327C2—362no—
328C2#1172076911yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#11720769
commutator, cosocle, hypocenterC24×C32C22×C3
FrattiniC22×C3C2×D3×A4
FittingC25×C32C2×C3
socleC23×C3D3×A4
LCSG ▷ C24×C32
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C32⋊(C22⋊A4)C2semidirect
2. #11720769C2direct (N × H)
3. C2×#64162C3semidirect
5. C32⋊(C22⋊A4)C22semidirect
7. C25×C32C2×C3semidirect
9. C2×C3×C22⋊A4D3semidirect
12. #64162C2×C3semidirect
25. C24×C32C22×C3semidirect
26. C23×C3⋊D3A4semidirect
28. C3×C22⋊A4C2×D3semidirect
70. C22×C3⋊D3C2×A4semidirect
137. C25(C3⋊D3)⋊C3semidirect
200. C2×C32#120⋊C3semidirect
208. C24C2×(C3⋊D3)⋊C3semidirect
261. C32C2×#120⋊C3semidirect
328. C2#11720769direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32⋊(C22⋊A4)2
#117207692
C2×#641623
C2×C3⋊(#120⋊C3)3
C2×(C3⋊D3)⋊A44