Finite Group Databasebeta

← #13151322

Subgroups — #13151322

Order 1728.

762 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#11721846C211yesmaximal, characteristic
2#64144⋊C3C212yesmaximal
3#11721299C214yesmaximal, complemented
4#11721297C214yesmaximal, complemented
5#11721098C212yesmaximal, complemented
6#11721080C212yesmaximal, complemented
7#10595276—32nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#6229C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C2×#6196C2212yes—
12#88552C2214yes—
13C4×#1084C2211yescharacteristic
14#3757⋊C3C2212yes—
15#3757⋊C3C2214yes—
16#3750⋊C3C2214yes—
17#88449C2212yes—
18#88449C2212yes—
19#88448C2214yes—
20#88448C2214yes—
21#88447C2212yes—
22#64144D312yes—
23C3×#936—32no—
24C3×#936—61no—
25C3×#935—32no—
26#63982—34no—
27#63964—34no—
28#63956—34no—
29#63956—34no—
30#63867—34no—
31#63839—34no—
32C23×H(3,𝔽3)C2312yes—
33D4×H(3,𝔽3)—41no—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×#1084C2312yes—
36C2×#1084C2312yes—
37#6229—41no—
38C2×(C3⋊Dic3)⋊C3—42no—
39C2×(C3⋊Dic3)⋊C3C2314yes—
40C2×(C3⋊Dic3)⋊C3C2314yes—
41#6196—42no—
42#5672—91no—
43#5651—92no—
44C2×C32×D4—21no—
45C2×C32×D4C2×D312yes—
46C22×C3⋊Dic3C2×D312yes—
47C22×C3×Dic3—32no—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—61no—
50C2×#627—34no—
51#3757C2×D312yes—
52#3757C2×D314yes—
53#3750C2×D314yes—
54C3×#217—32no—
55C3×#217—34no—
56C3×#217—34no—
57C3×#217—34no—
58C3×#217—64no—
59C3×Dic3⋊C4—34no—
60C3×C4×Dic3—32no—
61C3×C4×Dic3—61no—
62#3724—34no—
63#3724—34no—
64Dic3⋊Dic3—34no—
65Dic3⋊Dic3—34no—
66Dic3⋊Dic3—34no—
67Dic3⋊Dic3—34no—
68Dic32—34no—
69C22×H(3,𝔽3)—22no—
70C22×H(3,𝔽3)—24no—
71C22×H(3,𝔽3)C2411yescharacteristic
72C4×H(3,𝔽3)—21no—
73#1084—22no—
74#1084—22no—
75(C3⋊Dic3)⋊C3—24no—
76(C3⋊Dic3)⋊C3—24no—
77C3×#124—91no—
78C3×#123—92no—
79#944—92no—
80#936—32no—
81#936—61no—
82#935—32no—
83#935—92no—
84#924—94no—
85#889—94no—
86#888—94no—
87#879—94no—
88#879—94no—
89#879—94no—
90#878—94no—
91#878—94no—
92#870—92no—
93C23×C32—22no—
94C23×C32C22×D312yes—
95C23×C32C22×D312yes—
96C32×D4—42no—
97C32×D4—42no—
98C2×C32×C4—21no—
99C2×C32×C4C22×D312yes—
100C2×C3⋊Dic3—42no—
101C2×C3⋊Dic3C22×D314yes—
102C2×C3⋊Dic3C22×D314yes—
103C2×C3×Dic3—34no—
104C2×C3×Dic3—34no—
105C2×C3×Dic3—34no—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—122no—
110C3×Dic6—122no—
111C3×Dic6—122no—
112#627—124no—
113#563—271noSylow
114C2×H(3,𝔽3)—42no—
115C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
116C2×H(3,𝔽3)2−1+412yes—
117C2×C3×Q8—91no—
118C2×C3×D4—32no—
119C2×C3×D4—61no—
120C2×C3×D4D3211yescharacteristic
121C22×C3×C4—92no—
122C22×Dic3—32no—
123C22×Dic3—92no—
124C2×Dic6—32no—
125C2×Dic6—61no—
126C2×Dic6—92no—
127C2×Dic6—94no—
128C3×C4⋊C4—94no—
129C3×C22⋊C4—94no—
130C3×C22⋊C4—92no—
131C3×C22⋊C4—94no—
132C3×C42—91no—
133#217—34no—
134#217—34no—
135#217—34no—
136#217—32no—
137#217—64no—
138#217—92no—
139#217—94no—
140C4⋊Dic3—92no—
141C4⋊Dic3—92no—
142C4⋊Dic3—94no—
143C4⋊Dic3—94no—
144Dic3⋊C4—34no—
145Dic3⋊C4—94no—
146Dic3⋊C4—94no—
147Dic3⋊C4—94no—
148Dic3⋊C4—94no—
149Dic3⋊C4—94no—
150Dic3⋊C4—94no—
151Dic3⋊C4—94no—
152C4×Dic3—32no—
153C4×Dic3—61no—
154C4×Dic3—92no—
155C4×Dic3—94no—
156C22×C32—22no—
157C22×C32—22no—
158C22×C32—24no—
159C22×C32—24no—
160C22×C32—21no—
161C22×C32—42no—
162C22×C32—44no—
163C22×C32C23×D312yes—
164C32×C4—22no—
165C32×C4—22no—
166C3⋊Dic3—24no—
167C3⋊Dic3—24no—
168C3×Dic3—64no—
169C3×Dic3—64no—
170C3×Dic3—64no—
171C3×Dic3—64no—
172C3×Dic3—122no—
173C3×Dic3—122no—
174C4⋊Q8—272no—
175#126—274no—
176#125—272no—
177#124—91no—
178#123—92no—
179C22⋊Q8—274no—
180H(3,𝔽3)#56311yescharacteristic, complemented, Sylow
181C23×C3—32no—
182C23×C3—32no—
183C23×C3—62no—
184C23×C3C2×D3212yes—
185C3×Q8—361no—
186C3×D4—41no—
187C3×D4—122no—
188C3×D4—122no—
189C2×C3×C4—32no—
190C2×C3×C4—61no—
191C2×C3×C4—94no—
192C2×C3×C4—94no—
193C2×C3×C4—92no—
194C2×C3×C4—92no—
195C2×C3×C4—362no—
196C2×C3×C4C2×D3211yescharacteristic
197C2×Dic3—34no—
198C2×Dic3—34no—
199C2×Dic3—34no—
200C2×Dic3—34no—
201C2×Dic3—62no—
202C2×Dic3—62no—
203C2×Dic3—94no—
204C2×Dic3—94no—
205C2×Dic3—122no—
206C2×Dic3—362no—
207Dic6—122no—
208Dic6—122no—
209Dic6—362no—
210Dic6—364no—
211C2×C32—22no—
212C2×C32—21no—
213C2×C32—42no—
214C2×C32—42no—
215C2×C32—82no—
216C2×C32#100412yes—
217C2×C32#101014yes—
218C2×Q8—91no—
219C2×Q8—272no—
220C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
221C22×C4—92no—
222C4⋊C4—94no—
223C4⋊C4—272no—
224C4⋊C4—274no—
225C4⋊C4—274no—
226C4⋊C4—272no—
227C22⋊C4—94no—
228C22⋊C4—94no—
229C22⋊C4—92no—
230C42—91no—
231C42—272no—
232C22×C3—24no—
233C22×C3—22no—
234C22×C3—32no—
235C22×C3—62no—
236C22×C3—62no—
237C22×C3—64no—
238C22×C3—64no—
239C22×C3—61no—
240C22×C3—122no—
241C22×C3—124no—
242C22×C3C22×D3211yescharacteristic
243C3×C4—21no—
244C3×C4—62no—
245C3×C4—62no—
246C3×C4—184no—
247C3×C4—184no—
248C3×C4—182no—
249C3×C4—182no—
250Dic3—64no—
251Dic3—64no—
252Dic3—64no—
253Dic3—64no—
254Dic3—122no—
255Dic3—122no—
256Dic3—184no—
257Dic3—184no—
258C32—21no—
259C32#565112yescomplemented
260C23C2×((C3⋊D3)⋊C3)⋊C212yes—
261Q8—361no—
262Q8—1082no—
263D4—41no—
264C2×C4—92no—
265C2×C4—94no—
266C2×C4—92no—
267C2×C4—94no—
268C2×C4—362no—
269C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C2×C3—34no—
271C2×C3—32no—
272C2×C3—42no—
273C2×C3—62no—
274C2×C3—61no—
275C2×C3—122no—
276C2×C3—122no—
277C2×C3—242no—
278C2×C3#6430912yes—
279C2×C3#6433011yescharacteristic
280C22—24no—
281C22—22no—
282C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
283C4—21no—
284C4—182no—
285C4—184no—
286C4—184no—
287C4—182no—
288C3—32no—
289C3—61no—
290C3#1059527511yescharacteristic
291C2—42no—
292C2#1172250212yes—
293C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#563
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11721299C2semidirect
4. #11721297C2semidirect
5. #11721098C2semidirect
6. #11721080C2semidirect
180. H(3,𝔽3)#563semidirect
259. C32#5651semidirect

Maximal subgroups

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

subgroupindex
#117218462
#64144⋊C32
#117212992
#117212972
#117210982
#117210802
#105952763