Finite Group Databasebeta

← C2×#11724125 #13180564

Subgroups — C2×#11724125 #13180564

Order 1728.

968 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 156 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×#88957C212yesmaximal
2C2×#88940C211yesmaximal, characteristic
3C2×#88934C212yesmaximal
4C2×C3×#3773C211yesmaximal, characteristic
5C2×Dic3×#157C211yesmaximal, characteristic
6#11724125C218yesmaximal, direct product factor
7C2×#64222—31nomaximal
8C22×C3⋊#157C2211yescharacteristic
9C22×C3×#157C2211yescharacteristic
10C2×#6294C2212yes—
11C2×#6293—22no—
12C2×Dic3×C3⋊D3C2211yescharacteristic
13C2×C3×#627—22no—
14C2×C3×#624C2212yes—
15#88957—24no—
16#88957C2218yes—
17#88940C2214yes—
18#88934C2218yes—
19C3×#3773C2214yes—
20Dic3×#157C2214yes—
21C2×#3797—32no—
22C2×D3×Dic6—62no—
23C2×C4×#157—31no—
24C2×#3773—32no—
25C2×#3773—31no—
26C2×#3773D311yescharacteristic
27#64222—38no—
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3⋊#157Q814yescomplemented
30C2×C3⋊#157C2312yes—
31C2×C3×#157Q814yescomplemented
32C2×C3×#157C2312yes—
33C2×C3⋊(C3⋊Dic3)D411yescharacteristic
34C2×C3×C3⋊Dic3—22no—
35C2×C32×Dic3—22no—
36C2×C32×Dic3D411yescharacteristic
37#6294—22no—
38#6294C2314yes—
39#6293—28no—
40Dic3×C3⋊D3D412yes—
41Dic3×C3⋊D3C2312yes—
42C3×#627—28no—
43C3×#624—22no—
44C3×#624C2314yes—
45C2×#888—91nomaximal
46C22×#157—31no—
47C22×#157C2×D311yescharacteristic
48C2×C4×C3⋊D3—31no—
49C2×#634—62no—
50C2×C3×C4×D3—62no—
51C2×C3×Dic6—62no—
52C2×C3×Dic6—62no—
53C2×#627—22no—
54C2×#627—62no—
55C2×#627—62no—
56C2×#627—122no—
57C2×#624—32no—
58C2×#624C2×D312yes—
59C2×D3×Dic3—62no—
60C2×D3×Dic3—62no—
61#3797—38no—
62#3797—64no—
63D3×Dic6—68no—
64D3×Dic6—68no—
65C4×#157—34no—
66#3773—34no—
67#3773—38no—
68#3773C2×D314yes—
69C22×C33C2×D411yescharacteristic
70C2×C3×C3⋊D3C2×D411yescharacteristic
71C2×C3×C3⋊D3C2×Q812yes—
72C2×C3×C3⋊D3C2×Q812yes—
73C2×C3×C3⋊D3C2411yescharacteristic
74C3⋊#157C2×Q814yescomplemented
75C3×#157C2×Q814yescomplemented
76C3⋊(C3⋊Dic3)C2×D412yes—
77C3×C3⋊Dic3—24no—
78C32×Dic3—24no—
79C32×Dic3C2×D412yes—
80C2×D3×Q8—182no—
81C22×Dic6—92no—
82C2×C3×C4⋊C4—91no—
83C2×C4⋊Dic3—91no—
84C2×Dic3⋊C4—92no—
85C2×C4×Dic3—91no—
86#888—98no—
87C22×C3⋊D3C22×D311yescharacteristic
88C22×C3×D3—62no—
89C2×#157—34no—
90C2×#157—32no—
91C2×#157Dic614yescomplemented
92C2×#157C22×D312yes—
93C2×C32×C4—31no—
94C2×C32×C4—62no—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—31no—
97C2×C3⋊Dic3—62no—
98C2×C3⋊Dic3—122no—
99C4×C3⋊D3—32no—
100C4×C3⋊D3—32no—
101#634—68no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—22no—
105C2×C3×Dic3—42no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—122no—
109C3×C4×D3—64no—
110C3×C4×D3—64no—
111C3×Dic6—68no—
112C3×Dic6—68no—
113#627—28no—
114#627—68no—
115#627—68no—
116#627—128no—
117#624—22no—
118#624—34no—
119#624—62no—
120#624C22×D314yes—
121D3×Dic3—64no—
122D3×Dic3—64no—
123D3×Dic3—64no—
124D3×Dic3—64no—
125C2×C4⋊Q8—271noSylow
126C2×C33C4⋊Q812yescomplemented
127C2×C33C22×D411yescharacteristic
128C3×C3⋊D3C4⋊Q812yescomplemented
129C3×C3⋊D3C22×Q811yescharacteristic
130C3×C3⋊D3C22×Q811yescharacteristic
131C2×C3×Q8—182no—
132C22×C3×C4—92no—
133C22×C3×C4—91no—
134C22×Dic3—92no—
135C22×Dic3—91no—
136C22×Dic3—91no—
137D3×Q8—188no—
138D3×Q8—188no—
139C2×C4×D3—62no—
140C2×C4×D3—182no—
141C2×C4×D3—182no—
142C2×Dic6—62no—
143C2×Dic6—62no—
144C2×Dic6—98no—
145C2×Dic6—184no—
146C2×Dic6—182no—
147C2×Dic6—182no—
148C2×Dic6—182no—
149C2×Dic6—362no—
150C3×C4⋊C4—94no—
151C4⋊Dic3—94no—
152Dic3⋊C4—98no—
153C4×Dic3—94no—
154C22×C32—22no—
155C22×C32—42no—
156C22×C32D3×D411yescharacteristic
157C2×C3⋊D3C2×Dic612yes—
158C2×C3⋊D3D3×D411yescharacteristic
159C2×C3⋊D3D3×Q812yes—
160C2×C3⋊D3C23×D311yescharacteristic
161C2×C3×D3—62no—
162C2×C3×D3—64no—
163C2×C3×D3—64no—
164C2×C3×D3—62no—
165#157—34no—
166#157C2×Dic614yescomplemented
167C32×C4—32no—
168C32×C4—64no—
169C3⋊Dic3—24no—
170C3⋊Dic3—32no—
171C3⋊Dic3—64no—
172C3⋊Dic3—124no—
173C3×Dic3—24no—
174C3×Dic3—24no—
175C3×Dic3—24no—
176C3×Dic3—44no—
177C3×Dic3—64no—
178C3×Dic3—64no—
179C3×Dic3—124no—
180C22×Q8—272no—
181C4⋊Q8—278no—
182C2×C4⋊C4—91no—
183C2×C4⋊C4—272no—
184C2×C4⋊C4—271no—
185C2×C42—271no—
186C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
187C23×C3—91no—
188C22×D3—62no—
189C3×Q8—188no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—94no—
194C2×C3×C4—94no—
195C2×C3×C4—92no—
196C2×C3×C4—122no—
197C2×C3×C4—182no—
198C2×C3×C4—182no—
199C2×Dic3—22no—
200C2×Dic3—62no—
201C2×Dic3—62no—
202C2×Dic3—94no—
203C2×Dic3—94no—
204C2×Dic3—92no—
205C2×Dic3—91no—
206C2×Dic3—92no—
207C2×Dic3—92no—
208C2×Dic3—122no—
209C2×Dic3—182no—
210C2×Dic3—182no—
211C2×Dic3—362no—
212C2×Dic3GO+(4,𝔽2)11yescharacteristic
213C4×D3—64no—
214C4×D3—64no—
215C4×D3—184no—
216C4×D3—184no—
217C4×D3—184no—
218C4×D3—184no—
219Dic6—68no—
220Dic6—68no—
221Dic6—188no—
222Dic6—188no—
223Dic6—188no—
224Dic6—368no—
225C2×C32—22no—
226C2×C32—24no—
227C2×C32—44no—
228C2×C32—42no—
229C2×C32#88812yescomplemented
230C2×C32C2×D3×D411yescharacteristic
231C3⋊D3#88812yescomplemented
232C3⋊D3C22×Dic611yescharacteristic
233C3⋊D3C2×D3×Q811yescharacteristic
234C3×D3—64no—
235C3×D3—62no—
236C3×D3—62no—
237C2×Q8—182no—
238C2×Q8—278no—
239C2×Q8—542no—
240C2×Q8—544no—
241C22×C4—92no—
242C22×C4—91no—
243C22×C4—271no—
244C22×C4—272no—
245C22×C4—271no—
246C4⋊C4—94no—
247C4⋊C4—278no—
248C4⋊C4—274no—
249C42—274no—
250C22×C3—22no—
251C22×C3—42no—
252C22×C3—91no—
253C22×C3—91no—
254C22×C3—92no—
255C22×C3—92no—
256C22×C3C2×GO+(4,𝔽2)11yescharacteristic
257C2×D3—62no—
258C2×D3—62no—
259C2×D3—64no—
260C2×D3—64no—
261C3×C4—64no—
262C3×C4—64no—
263C3×C4—64no—
264C3×C4—94no—
265C3×C4—124no—
266C3×C4—184no—
267Dic3—24no—
268Dic3—64no—
269Dic3—64no—
270Dic3—94no—
271Dic3—92no—
272Dic3—124no—
273Dic3—184no—
274Dic3—364no—
275Dic3C2×GO+(4,𝔽2)12yes—
276C32—22no—
277C32—42no—
278C32C2×#88811yescharacteristic, complemented
279C23—91no—
280Q8—188no—
281Q8—548no—
282C2×C4—31no—
283C2×C4—62no—
284C2×C4—94no—
285C2×C4—92no—
286C2×C4—94no—
287C2×C4—182no—
288C2×C4—182no—
289C2×C4—272no—
290C2×C4—272no—
291C2×C4—274no—
292C2×C4—272no—
293C2×C4—274no—
294C2×C4—271no—
295C2×C4—542no—
296C2×C3—24no—
297C2×C3—22no—
298C2×C3—44no—
299C2×C3—42no—
300C2×C3—91no—
301C2×C3—91no—
302C2×C3—92no—
303C2×C3#6422212yescomplemented
304C2×C3C22×GO+(4,𝔽2)11yescharacteristic
305D3—64no—
306D3—62no—
307D3—62no—
308C22—92no—
309C22—92no—
310C22—91no—
311C22—91no—
312C22D3×GO+(4,𝔽2)11yescharacteristic
313C4—32no—
314C4—64no—
315C4—94no—
316C4—184no—
317C4—274no—
318C4—272no—
319C3—22no—
320C3—42no—
321C3C2×#6422211yescharacteristic, complemented
322C2—91no—
323C2—91no—
324C2—92no—
325C2#1172412512yesdirect product factor
326C2C2×D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C2×D3×GO+(4,𝔽2)
Fitting, socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3C23
hypocenterC33C2×C4⋊Q8
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
6. #11724125C2direct (N × H)
29. C2×C3⋊#157Q8semidirect
31. C2×C3×#157Q8semidirect
74. C3⋊#157C2×Q8semidirect
75. C3×#157C2×Q8semidirect
91. C2×#157Dic6semidirect
126. C2×C33C4⋊Q8semidirect
128. C3×C3⋊D3C4⋊Q8semidirect
166. #157C2×Dic6semidirect
186. C33C2×C4⋊Q8semidirect
229. C2×C32#888semidirect
231. C3⋊D3#888semidirect
278. C32C2×#888semidirect
303. C2×C3#64222semidirect
321. C3C2×#64222semidirect
325. C2#11724125direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#889572
C2×#889402
C2×#889342
C2×C3×#37732
C2×Dic3×#1572
#117241252
C2×#642223
C2×#8889