Finite Group Databasebeta

← #13178755

Subgroups — #13178755

Order 1728.

1048 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 251 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
1C3×Q8×C3⋊Dic3C212yesmaximal
2C3×Q8×C3⋊Dic3C211yesmaximal, characteristic
3C4⋊#6296C213yesmaximal
4C4×#6296C213yesmaximal
5#11723680C216yesmaximal
6#10595422—32nomaximal
7#10595351—31nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×#6296C2213yes—
10C3×C4⋊(C3⋊Dic3)C2213yes—
11C3×C4⋊(C3⋊Dic3)C2216yes—
12C3×C4×C3⋊Dic3C2216yes—
13C3×C4×C3⋊Dic3C2213yes—
14#88815C2216yes—
15#88815C2216yes—
16#88815C2211yescharacteristic
17Q8×C3⋊Dic3D311yescharacteristic
18Q8×C3⋊Dic3D312yes—
19C3×Q8×Dic3—32no—
20C3×Q8×Dic3—32no—
21C3×Q8×Dic3—32no—
22C3×Q8×Dic3—62no—
23C3×Q8×Dic3—61no—
24#63919—33no—
25#63918—36no—
26C4×#627—33no—
27C4×#627—36no—
28#63872—36no—
29#63869—36no—
30#63869—36no—
31C33×Q8Q814yescomplemented
32C2×C33×C4C2313yes—
33C2×C3×C3⋊Dic3C2312yes—
34C2×C3×C3⋊Dic3C2316yes—
35C2×C3×C3⋊Dic3C2313yes—
36C2×C3×C3⋊Dic3C2311yescharacteristic
37#6296—26no—
38#5570—91no—
39#5569—92no—
40C2×C32×Q8—22no—
41C2×C32×Q8—21no—
42C2×C32×Q8—41no—
43C2×C32×Q8C2×D311yescharacteristic
44C2×C32×Q8C2×D312yes—
45C2×#627—36no—
46C2×#627—33no—
47C4⋊(C3⋊Dic3)C2×D313yes—
48C4⋊(C3⋊Dic3)C2×D316yes—
49C4×C3⋊Dic3C2×D316yes—
50C4×C3⋊Dic3C2×D313yes—
51C3×C4⋊Dic3—36no—
52C3×C4⋊Dic3—36no—
53C3×C4⋊Dic3—36no—
54C3×C4⋊Dic3—66no—
55C3×C4⋊Dic3—63no—
56C3×C4×Dic3—36no—
57C3×C4×Dic3—36no—
58C3×C4×Dic3—36no—
59C3×C4×Dic3—63no—
60C3×C4×Dic3—66no—
61#3724—36no—
62#3724—36no—
63#3724—31no—
64Dic3⋊Dic3—36no—
65Dic3⋊Dic3—312no—
66Dic3⋊Dic3—36no—
67Dic3⋊Dic3—32no—
68C22×C33C2411yescharacteristic
69C33×C4C2×Q816yes—
70C3×C3⋊Dic3—26no—
71C3×C3⋊Dic3—22no—
72C3×C3⋊Dic3—23no—
73C3×C3⋊Dic3#4112yes—
74C3×C4×Q8—92no—
75C3×C4×Q8—91no—
76Q8×Dic3—32no—
77Q8×Dic3—32no—
78Q8×Dic3—32no—
79Q8×Dic3—61no—
80Q8×Dic3—62no—
81#890—96no—
82#890—96no—
83#888—96no—
84#870—96no—
85#869—93no—
86C4×Dic6—96no—
87C4×Dic6—93no—
88C32×Q8—24no—
89C32×Q8—28no—
90C32×Q8—44no—
91C32×Q8Dic618yescomplemented
92C32×Q8Dic614yescomplemented
93C2×C32×C4—26no—
94C2×C32×C4—23no—
95C2×C32×C4—43no—
96C2×C32×C4C22×D313yes—
97C2×C32×C4C22×D316yes—
98C2×C3⋊Dic3C22×D312yes—
99C2×C3⋊Dic3C22×D313yes—
100C2×C3⋊Dic3C22×D316yes—
101C2×C3⋊Dic3C22×D311yescharacteristic
102C2×C3×Dic3—32no—
103C2×C3×Dic3—32no—
104C2×C3×Dic3—36no—
105C2×C3×Dic3—36no—
106C2×C3×Dic3—32no—
107C2×C3×Dic3—36no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—63no—
111C2×C3×Dic3—66no—
112#627—612no—
113#627—66no—
114#557—271noSylow
115C2×C33C22×Q811yescharacteristic
116C2×C33C2×#4111yescharacteristic
117C2×C332−1+411yescharacteristic
118C2×C3×Q8—21no—
119C2×C3×Q8—22no—
120C2×C3×Q8—41no—
121C2×C3×Q8D3212yes—
122C2×C3×Q8D3211yescharacteristic
123C2×Dic6—93no—
124C2×Dic6—96no—
125C3×C4⋊C4—96no—
126C3×C4⋊C4—93no—
127C3×C42—93no—
128C3×C42—96no—
129C4⋊Dic3—36no—
130C4⋊Dic3—36no—
131C4⋊Dic3—36no—
132C4⋊Dic3—63no—
133C4⋊Dic3—66no—
134C4⋊Dic3—91no—
135C4⋊Dic3—96no—
136C4⋊Dic3—96no—
137Dic3⋊C4—92no—
138Dic3⋊C4—96no—
139Dic3⋊C4—96no—
140Dic3⋊C4—912no—
141C4×Dic3—36no—
142C4×Dic3—36no—
143C4×Dic3—36no—
144C4×Dic3—66no—
145C4×Dic3—63no—
146C22×C32—22no—
147C22×C32—21no—
148C22×C32—41no—
149C22×C32C23×D311yescharacteristic
150C22×C32C23×D312yes—
151C32×C4—26no—
152C32×C4—212no—
153C32×C4—46no—
154C32×C4C2×Dic6112yes—
155C32×C4C2×Dic616yes—
156C3⋊Dic3—22no—
157C3⋊Dic3—26no—
158C3⋊Dic3—23no—
159C3⋊Dic3#23912yes—
160C3×Dic3—34no—
161C3×Dic3—62no—
162C3×Dic3—64no—
163C3×Dic3—66no—
164C3×Dic3—66no—
165C3×Dic3—62no—
166C3×Dic3—66no—
167C3×Dic3—62no—
168C3×Dic3—123no—
169C3×Dic3—126no—
170C4⋊Q8—273no—
171#125—276no—
172C4×Q8—92no—
173C4×Q8—91no—
174C4×Q8—273no—
175C33#55711yescharacteristic, complemented, Sylow
176C3×Q8—28no—
177C3×Q8—24no—
178C3×Q8—44no—
179C3×Q8#62714yescomplemented
180C3×Q8#62718yescomplemented
181C2×C3×C4—26no—
182C2×C3×C4—23no—
183C2×C3×C4—43no—
184C2×C3×C4—91no—
185C2×C3×C4—92no—
186C2×C3×C4—93no—
187C2×C3×C4—96no—
188C2×C3×C4C2×D3216yes—
189C2×C3×C4C2×D3213yes—
190C2×Dic3—32no—
191C2×Dic3—32no—
192C2×Dic3—32no—
193C2×Dic3—36no—
194C2×Dic3—36no—
195C2×Dic3—36no—
196C2×Dic3—61no—
197C2×Dic3—62no—
198C2×Dic3—63no—
199C2×Dic3—66no—
200Dic6—1812no—
201Dic6—186no—
202C2×C32—22no—
203C2×C32—21no—
204C2×C32—22no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—22no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32C22×Dic612yes—
212C2×C32C22×Dic611yescharacteristic
213C2×C32C2×#23911yescharacteristic
214C2×C32#100712yes—
215C2×C32D3×#4112yes—
216C2×C32#101011yescharacteristic
217C2×Q8—273no—
218C2×Q8#111311yescharacteristic
219C4⋊C4—93no—
220C4⋊C4—96no—
221C4⋊C4—271no—
222C4⋊C4—276no—
223C4⋊C4—276no—
224C42—96no—
225C42—93no—
226C22×C3—22no—
227C22×C3—21no—
228C22×C3—41no—
229C22×C3C22×D3212yes—
230C22×C3C22×D3211yescharacteristic
231C3×C4—212no—
232C3×C4—26no—
233C3×C4—46no—
234C3×C4—92no—
235C3×C4—182no—
236C3×C4—183no—
237C3×C4—186no—
238C3×C4C2×#62716yes—
239C3×C4C2×#627112yes—
240Dic3—34no—
241Dic3—62no—
242Dic3—64no—
243Dic3—62no—
244Dic3—66no—
245Dic3—62no—
246Dic3—66no—
247Dic3—66no—
248Dic3—123no—
249Dic3—126no—
250C32—21no—
251C32—22no—
252C32—41no—
253C32#556912yescomplemented
254C32#557011yescharacteristic, complemented
255Q8—546no—
256Q8#629614yescomplemented
257C2×C4—91no—
258C2×C4—92no—
259C2×C4—93no—
260C2×C4—96no—
261C2×C4C2×#111313yes—
262C2×C3—22no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—21no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3#6431512yes—
272C2×C3#6431611yescharacteristic
273C2×C3D3×#23912yes—
274C2×C3#6431911yescharacteristic
275C2×C3C22×#62711yescharacteristic
276C2×C3C22×#62712yes—
277C22C22×#111311yescharacteristic
278C4—92no—
279C4—182no—
280C4—183no—
281C4—186no—
282C4C2×#629616yes—
283C3—22no—
284C3—21no—
285C3—41no—
286C3#1059535111yescharacteristic, complemented
287C3#1059542212yescomplemented
288C2#1172462411yescharacteristic
289C2#1172462611yescharacteristic
290C2C22×#629611yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, prime radical, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8#1113
hypocenterC33#557
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
31. C33×Q8Q8semidirect
91. C32×Q8Dic6semidirect
92. C32×Q8Dic6semidirect
175. C33#557semidirect
179. C3×Q8#627semidirect
180. C3×Q8#627semidirect
253. C32#5569semidirect
254. C32#5570semidirect
256. Q8#6296semidirect
286. C3#10595351semidirect
287. C3#10595422semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
C4⋊#62962
C4×#62962
#117236802
#105954223
#105953513