Finite Group Databasebeta

← C2×#11722764 #13167090

Subgroups — C2×#11722764 #13167090

Order 1728.

392 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 79 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×#88988C211yesmaximal, characteristic
2C2×#88809C211yesmaximal, characteristic
3C2×C3×#3712C211yesmaximal, characteristic
4#11722764C214yesmaximal, direct product factor
5C2×#63733—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3⋊#622C412yes—
8#88988C412yes—
9#88988C2212yes—
10C2×C3×C3⋊(C3⋊C8)—21no—
11C2×C32×C3⋊C8—21no—
12#88809—21no—
13#88809C2212yes—
14C3×#3712—21no—
15C3×#3712C2212yes—
16C2×#3787—31no—
17C2×C3×ΓL(1,𝔽25)—61no—
18C2×#3712—31no—
19C2×#3712D311yescharacteristic
20C2×#3711—61no—
21#63733—34no—
22C22×C3×C3⋊D3Q811yescharacteristic
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C3×C4×C3⋊D3C2×C411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3×C4×C3⋊D3Q811yescharacteristic
28C3×C4×C3⋊D3C2311yescharacteristic
29C3⋊#622C2×C414yes—
30C3×C3⋊(C3⋊C8)—22no—
31C32×C3⋊C8—22no—
32C2×#822—91nomaximal
33C2×C4×C3⋊D3C2×D311yescharacteristic
34C2×C3×C4×D3—61no—
35C2×C3×C4×D3—61no—
36C2×#622—32no—
37#3787—32no—
38#3787—32no—
39C2×C32×C8—61no—
40C2×C3⋊(C3⋊C8)—21no—
41C2×C3×C3⋊C8—21no—
42C2×C3×C3⋊C8—21no—
43C2×C3×C3⋊C8—41no—
44C2×C3×C3⋊C8—61no—
45C2×C3×C3⋊C8—61no—
46C2×C3×C3⋊C8—121no—
47C3×ΓL(1,𝔽25)—62no—
48C3×ΓL(1,𝔽25)—62no—
49#3712—21no—
50#3712—32no—
51#3712—61no—
52#3712C2×D312yes—
53#3711—62no—
54#3711—62no—
55C22×C33C4⋊C411yescharacteristic
56C2×C3×C3⋊D3—22no—
57C2×C3×C3⋊D3C4⋊C411yescharacteristic
58C2×C3×C3⋊D3C2×Q811yescharacteristic
59C33×C4C4⋊C411yescharacteristic
60C33×C4C2×D411yescharacteristic
61C3×C3⋊Dic3C4⋊C411yescharacteristic
62C3×C3⋊Dic3C22×C411yescharacteristic
63C2×C3×M16—91no—
64C2×#208—91no—
65C22×C3⋊C8—91no—
66C2×ΓL(1,𝔽25)—61no—
67C2×ΓL(1,𝔽25)—181no—
68#822—94no—
69C22×C3⋊D3Dic611yescharacteristic
70C22×C3×D3—61no—
71C22×C3×D3—61no—
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—41no—
75C2×C32×C4—41no—
76C2×C32×C4#6711yescharacteristic
77C2×C3⋊Dic3C4×D311yescharacteristic
78C4×C3⋊D3Dic611yescharacteristic
79C4×C3⋊D3C4×D311yescharacteristic
80C4×C3⋊D3#6711yescharacteristic
81C4×C3⋊D3C22×D311yescharacteristic
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C3×C4×D3—61no—
85C3×C4×D3—61no—
86C3×C4×D3—61no—
87C3×C4×D3—61no—
88C3×C4×D3—61no—
89C3×C4×D3—61no—
90C3×C4×D3—61no—
91C3×C4×D3—61no—
92#622—34no—
93C32×C8—62no—
94C3⋊(C3⋊C8)—22no—
95C3×C3⋊C8—22no—
96C3×C3⋊C8—22no—
97C3×C3⋊C8—42no—
98C3×C3⋊C8—62no—
99C3×C3⋊C8—62no—
100C3×C3⋊C8—122no—
101C2×#108—271noSylow
102C2×C33#10812yescomplemented
103C2×C33C2×C4⋊C411yescharacteristic
104C3×C3⋊D3—21no—
105C3×C3⋊D3—21no—
106C22×C3×C4—91no—
107C2×C4×D3—61no—
108C2×C4×D3—61no—
109C3×M16—92no—
110C3×M16—181no—
111C2×C3×C8—61no—
112C2×C3×C8—61no—
113C2×C3×C8—121no—
114C2×C3×C8—181no—
115#208—92no—
116#208—181no—
117C2×C3⋊C8—21no—
118C2×C3⋊C8—61no—
119C2×C3⋊C8—61no—
120C2×C3⋊C8—92no—
121C2×C3⋊C8—92no—
122C2×C3⋊C8—92no—
123C2×C3⋊C8—121no—
124ΓL(1,𝔽25)—62no—
125ΓL(1,𝔽25)—62no—
126ΓL(1,𝔽25)—182no—
127ΓL(1,𝔽25)—182no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—41no—
131C22×C32—41no—
132C22×C32Dic3⋊C411yescharacteristic
133C2×C3⋊D3—22no—
134C2×C3⋊D3Dic3⋊C411yescharacteristic
135C2×C3⋊D3C2×Dic611yescharacteristic
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C2×C3×D3—61no—
139C2×C3×D3—61no—
140C2×C3×D3—122no—
141C2×C3×D3—122no—
142C32×C4—21no—
143C32×C4—21no—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—41no—
147C32×C4—41no—
148C32×C4—41no—
149C32×C4—41no—
150C32×C4Dic3⋊C411yescharacteristic
151C32×C4C2×#6711yescharacteristic
152C3⋊Dic3Dic3⋊C411yescharacteristic
153C3⋊Dic3C2×C4×D311yescharacteristic
154C3×Dic3—61no—
155C3×Dic3—61no—
156C3×Dic3—61no—
157C3×Dic3—61no—
158C2×M16—91no—
159C2×M16—271no—
160C22×C8—271no—
161#108—274no—
162C33C2×#10811yescharacteristic, complemented, Sylow
163C23×C3—91no—
164C22×D3—61no—
165C22×D3—61no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—91no—
171C2×C3×C4—91no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4GO+(4,𝔽2)11yescharacteristic
176C2×Dic3—61no—
177C2×Dic3—61no—
178C4×D3—61no—
179C4×D3—61no—
180C4×D3—61no—
181C4×D3—61no—
182C4×D3—61no—
183C4×D3—61no—
184C4×D3—61no—
185C4×D3—61no—
186C3×C8—62no—
187C3×C8—62no—
188C3×C8—122no—
189C3×C8—182no—
190C3⋊C8—22no—
191C3⋊C8—62no—
192C3⋊C8—62no—
193C3⋊C8—94no—
194C3⋊C8—122no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32—22no—
198C2×C32—22no—
199C2×C32—42no—
200C2×C32—42no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32#82212yescomplemented
204C2×C32C2×Dic3⋊C411yescharacteristic
205C3⋊D3—21no—
206C3⋊D3—21no—
207C3×D3—121no—
208C3×D3—121no—
209C3×D3—121no—
210C3×D3—121no—
211C22×C4—91no—
212M16—92no—
213M16—181no—
214M16—272no—
215M16—541no—
216C2×C8—61no—
217C2×C8—181no—
218C2×C8—272no—
219C2×C8—272no—
220C2×C8—272no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—91no—
226C22×C3—91no—
227C22×C3—182no—
228C22×C3#377311yescharacteristic
229C2×D3—61no—
230C2×D3—61no—
231C2×D3—61no—
232C2×D3—61no—
233C2×D3—122no—
234C2×D3—122no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4—91no—
244C3×C4—91no—
245C3×C4#377311yescharacteristic
246C3×C4C2×GO+(4,𝔽2)11yescharacteristic
247Dic3—61no—
248Dic3—61no—
249Dic3—61no—
250Dic3—61no—
251C32—21no—
252C32—21no—
253C32—41no—
254C32—41no—
255C32C2×#82211yescharacteristic, complemented
256C23—91no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4D32⋊D311yescharacteristic
263C8—62no—
264C8—182no—
265C8—274no—
266C2×C3—22no—
267C2×C3—22no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—42no—
271C2×C3—42no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—181no—
275C2×C3—181no—
276C2×C3#6373312yescomplemented
277C2×C3C2×#377311yescharacteristic
278D3—121no—
279D3—121no—
280D3—121no—
281D3—121no—
282C22—91no—
283C22—91no—
284C22—182no—
285C22#8893411yescharacteristic
286C4—91no—
287C4—91no—
288C4#8893411yescharacteristic
289C4C2×D32⋊D311yescharacteristic
290C3—21no—
291C3—21no—
292C3—41no—
293C3—41no—
294C3C2×#6373311yescharacteristic, complemented
295C2—181no—
296C2—181no—
297C2#1172276412yesdirect product factor
298C2C2×#8893411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC3×C4×C3⋊D3C23
hypocenterC33C2×#108
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722764C2direct (N × H)
102. C2×C33#108semidirect
162. C33C2×#108semidirect
203. C2×C32#822semidirect
255. C32C2×#822semidirect
276. C2×C3#63733semidirect
294. C3C2×#63733semidirect
297. C2#11722764direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#889882
C2×#888092
C2×C3×#37122
#117227642
C2×#637333
C2×#8229