Finite Group Databasebeta

← #13156564

Subgroups — #13156564

Order 1728.

602 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723697C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic, complemented
3#11722033C212yesmaximal, complemented
4#11722020C211yesmaximal, characteristic, complemented
5C3×#63564C212yesmaximal, complemented
6#10590888—31nomaximal
7#10590550—32nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×#6295—22no—
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14C3×Dic3⋊Dic3—22no—
15C3×#890—32no—
16#63914—31no—
17#63911—32no—
18#63872D311yescharacteristic, complemented
19#63642—31no—
20#63636—32no—
21C3×#821—32no—
22C3×#809—32no—
23#63564D312yescomplemented
24#63562—32no—
25#63562—32no—
26#63562—62no—
27#63554—32no—
28#63554—61no—
29C22×C3×C3⋊D3—22no—
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3D412yes—
32C3×C4⋊(C3⋊D3)—22no—
33C3×C4⋊(C3⋊D3)—22no—
34C2×C32×Dic3—22no—
35#6295—44no—
36C3⋊(C3⋊(C3⋊C8))—21no—
37C32×C3⋊C8—21no—
38#5067—91no—
39#4764—92no—
40C2×C4⋊(C3⋊D3)C2×D312yes—
41C2×C3×D12—32no—
42C2×C3×D12—32no—
43C2×C3×D12—62no—
44C2×#626—62no—
45C2×#626—62no—
46C2×#625—62no—
47C2×C32×C8—31no—
48C32×C4⋊C4—32no—
49C4×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊(C3⋊C8)—32no—
51C2×C3⋊(C3⋊C8)—31no—
52C2×C3⋊(C3⋊C8)—61no—
53C2×C3⋊(C3⋊C8)—62no—
54C2×C3⋊(C3⋊C8)—121no—
55C3×C4⋊Dic3—22no—
56C3×C4⋊Dic3—62no—
57C3×C4⋊Dic3C2×D312yes—
58C3×C4⋊Dic3C2×D312yes—
59C3×Dic3⋊C4—62no—
60C3×C4×Dic3—32no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8—21no—
63C2×C3×C3⋊C8C2×D312yes—
64Dic3⋊Dic3—22no—
65C22×C33C2×D411yescharacteristic
66C2×C3×C3⋊D3—42no—
67C2×C3×C3⋊D3—42no—
68C2×C3×C3⋊D3—42no—
69C33×C4#4111yescharacteristic
70C33×C4#4111yescharacteristic
71C3×C3⋊Dic3—22no—
72C32×Dic3—42no—
73C3×#125—91no—
74#940—92no—
75#890—32no—
76#874—91no—
77C3×#102—92no—
78#832—92no—
79#821—32no—
80#815—92no—
81#815—181no—
82#809—32no—
83#809—92no—
84#809—92no—
85#809—182no—
86#803—91no—
87C22×C3⋊D3—22no—
88C22×C3×D3—62no—
89C22×C3×D3—62no—
90C22×C3×D3—122no—
91C2×C32×C4—21no—
92C2×C32×C4—22no—
93C2×C32×C4—41no—
94C2×C32×C4—62no—
95C2×C32×C4C22×D312yes—
96C2×C32×C4C22×D311yescharacteristic
97C2×C3⋊Dic3#6712yes—
98C4⋊(C3⋊D3)—22no—
99C4⋊(C3⋊D3)—22no—
100C2×C3×Dic3—22no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—32no—
103C2×C3×Dic3—32no—
104C2×C3×Dic3—42no—
105C2×C3×Dic3—62no—
106C3×D12—64no—
107C3×D12—62no—
108C3×D12—62no—
109C3×D12—62no—
110C3×D12—62no—
111C3×D12—122no—
112#626—124no—
113#626—124no—
114#625—124no—
115C32×C8—61no—
116C3⋊(C3⋊C8)—61no—
117C3⋊(C3⋊C8)—62no—
118C3⋊(C3⋊C8)—62no—
119C3⋊(C3⋊C8)—122no—
120C3⋊(C3⋊C8)—122no—
121C3×C3⋊C8—22no—
122C3×C3⋊C8—22no—
123#490—271noSylow
124C2×C33#12411yescharacteristic
125C2×C33AGL(1,ℤ/8)12yes—
126C3×C3⋊D3—82no—
127C2×C3×D4—92no—
128C2×#67—182no—
129C2×#67—182no—
130C2×D12—32no—
131C2×D12—32no—
132C2×D12—62no—
133C2×D12—182no—
134C2×C3×C8—32no—
135C2×C3×C8—61no—
136C3×C4⋊C4—32no—
137C3×C4⋊C4—32no—
138C3×C4⋊C4—62no—
139C3×C4⋊C4—182no—
140C3×C42—91no—
141C4⋊Dic3—62no—
142C4⋊Dic3D3212yes—
143Dic3⋊C4—62no—
144C4×Dic3—32no—
145C4×Dic3—61no—
146C2×C3⋊C8—91no—
147C2×C3⋊C8—92no—
148C2×C3⋊C8—181no—
149C2×C3⋊C8—182no—
150C2×C3⋊C8—361no—
151C2×C3⋊C8D3211yescharacteristic
152C22×C32—21no—
153C22×C32—22no—
154C22×C32—41no—
155C22×C32D3×D412yes—
156C22×C32C2×#6711yescharacteristic
157C2×C3⋊D3—42no—
158C2×C3⋊D3—42no—
159C2×C3⋊D3—42no—
160C2×C3×D3—124no—
161C2×C3×D3—122no—
162C2×C3×D3—122no—
163C2×C3×D3—122no—
164C2×C3×D3—122no—
165C2×C3×D3—122no—
166C2×C3×D3—122no—
167C2×C3×D3—242no—
168C2×C3×D3—242no—
169C32×C4—22no—
170C32×C4—21no—
171C32×C4—22no—
172C32×C4—21no—
173C32×C4—41no—
174C32×C4—41no—
175C32×C4—122no—
176C32×C4#23512yes—
177C32×C4#23712yes—
178C32×C4#23911yescharacteristic
179C32×C4#23911yescharacteristic
180C3⋊Dic3—22no—
181C3×Dic3—42no—
182C3×Dic3—42no—
183C3×Dic3—62no—
184C3×Dic3—62no—
185C3×Dic3—82no—
186C3×Dic3—122no—
187#127—271no—
188#125—91no—
189#102—92no—
190#102—272no—
191#97—271no—
192C33#49011yescharacteristic, complemented, Sylow
193C23×C3—182no—
194C22×D3—62no—
195C22×D3—62no—
196C22×D3—122no—
197C3×D4—182no—
198C3×D4—182no—
199C2×C3×C4—21no—
200C2×C3×C4—22no—
201C2×C3×C4—41no—
202C2×C3×C4—62no—
203C2×C3×C4—62no—
204C2×C3×C4—92no—
205C2×C3×C4—122no—
206C2×C3×C4C2×D3212yes—
207C2×C3×C4C2×D3211yescharacteristic
208#67—364no—
209#67—364no—
210C2×Dic3—22no—
211C2×Dic3—32no—
212C2×Dic3—32no—
213C2×Dic3—62no—
214D12—64no—
215D12—62no—
216D12—62no—
217D12—62no—
218D12—62no—
219D12—122no—
220D12—364no—
221C3×C8—62no—
222C3×C8—62no—
223C3⋊C8—21no—
224C3⋊C8—182no—
225C3⋊C8—181no—
226C3⋊C8—182no—
227C3⋊C8—362no—
228C3⋊C8—362no—
229C2×C32—22no—
230C2×C32—22no—
231C2×C32—21no—
232C2×C32—22no—
233C2×C32—22no—
234C2×C32—41no—
235C2×C32—42no—
236C2×C32#88512yes—
237C2×C32#91112yes—
238C2×C32#91412yes—
239C2×C32#94711yescharacteristic
240C2×C32#94912yes—
241C3⋊D3—82no—
242C3×D3—244no—
243C3×D3—242no—
244C3×D3—242no—
245C2×D4—92no—
246C2×D4—542no—
247C2×C8—31no—
248C2×C8—271no—
249C4⋊C4—32no—
250C4⋊C4—182no—
251C42—91no—
252C22×C3—21no—
253C22×C3—22no—
254C22×C3—41no—
255C22×C3—362no—
256C22×C3—362no—
257C22×C3—362no—
258C22×C3#380211yescharacteristic
259C22×C3D3×#6712yes—
260C2×D3—124no—
261C2×D3—122no—
262C2×D3—122no—
263C2×D3—122no—
264C2×D3—122no—
265C2×D3—122no—
266C2×D3—122no—
267C2×D3—242no—
268C2×D3—242no—
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—21no—
272C3×C4—22no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—122no—
276C3×C4—122no—
277C3×C4—182no—
278C3×C4—242no—
279C3×C4#379612yes—
280C3×C4#379811yescharacteristic
281C3×C4#379912yes—
282C3×C4#380511yescharacteristic
283Dic3—42no—
284Dic3—62no—
285Dic3—62no—
286Dic3—122no—
287C32—22no—
288C32—21no—
289C32—41no—
290C32#476412yescomplemented
291C32#506711yescharacteristic, complemented
292C23—182no—
293D4—182no—
294D4—182no—
295D4—1084no—
296C2×C4—62no—
297C2×C4—92no—
298C2×C4D3311yescharacteristic
299C8—61no—
300C8—541no—
301C2×C3—22no—
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—21no—
306C2×C3—42no—
307C2×C3—41no—
308C2×C3—722no—
309C2×C3#6379812yes—
310C2×C3#6390712yes—
311C2×C3#6391511yescharacteristic
312C2×C3#6392512yes—
313C2×C3#6393912yes—
314D3—244no—
315D3—242no—
316D3—242no—
317C22—362no—
318C22—362no—
319C22—362no—
320C22#8895311yescharacteristic
321C4—122no—
322C4—182no—
323C4#8896411yescharacteristic
324C4#8896511yescharacteristic
325C3—22no—
326C3—21no—
327C3—41no—
328C3#1059055012yescomplemented
329C3#1059088811yescharacteristic, complemented
330C2—722no—
331C2#1172287612yes—
332C2#1172300711yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723697C2
hypocenterC33#490
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63872C2semidirect
3. #11722033C2semidirect
4. #11722020C2semidirect
5. C3×#63564C2semidirect
18. #63872D3semidirect
23. #63564D3semidirect
192. C33#490semidirect
290. C32#4764semidirect
291. C32#5067semidirect
328. C3#10590550semidirect
329. C3#10590888semidirect

Maximal subgroups

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

subgroupindex
#117236972
C3×#638722
#117220332
#117220202
C3×#635642
#105908883
#105905503