Finite Group Databasebeta

← #13156602

Subgroups — #13156602

Order 1728.

722 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 108 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×#88810C211yesmaximal, characteristic
2C2×#88793C212yesmaximal, complemented
3C3×#63864C212yesmaximal, complemented
4C2×C3×#3718C211yesmaximal, characteristic, complemented
5Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#10591609—32nomaximal
7#10590353—31nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×C32×Dic6C2212yes—
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4×Dic3C2211yescharacteristic
13#88810—22no—
14#88793—24no—
15C3×#3722—24no—
16C3×#3718—22no—
17C2×#3755—32no—
18C3×#947—32no—
19C2×C3×#216—32no—
20C3×#875—32no—
21#63864D312yescomplemented
22C2×#3718D311yescharacteristic, complemented
23C2×#3715—32no—
24C2×#3715—32no—
25C2×#3715—62no—
26C2×#3714—32no—
27C2×#3713—31no—
28C4×C3⋊(C3⋊C8)—31no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C22×C3×C3⋊D3—22no—
32C2×C33×C4C2311yescharacteristic
33C3×C4⋊(C3⋊D3)—22no—
34C3×C4⋊(C3⋊D3)—22no—
35C2×C32×Dic3—22no—
36C2×C32×Dic3D412yes—
37C32×Dic6—22no—
38C32×Dic6—22no—
39C3⋊(C3⋊(C3⋊C8))D412yes—
40C3×C3⋊(C3⋊C8)D412yes—
41#5191—91no—
42#5059—92no—
43C2×C32×Q8—32no—
44C2×C4⋊(C3⋊D3)C2×D312yes—
45C2×C3×D12—32no—
46C2×C3×D12—32no—
47C2×C3×D12—62no—
48C2×C3×Dic6—22no—
49C2×C3×Dic6C2×D312yes—
50C2×C3×Dic6C2×D312yes—
51C32×C42—31no—
52#3755—64no—
53C2×C3⋊(C3⋊C8)—32no—
54C2×C3⋊(C3⋊C8)—31no—
55C2×C3⋊(C3⋊C8)—61no—
56C2×C3⋊(C3⋊C8)—62no—
57C2×C3⋊(C3⋊C8)—121no—
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C3×#216—64no—
60C3×(C2×D3)⋊C4—64no—
61C3×(C2×D3)⋊C4—64no—
62C3×C4×Dic3—21no—
63C3×C4×Dic3C2×D312yes—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—61no—
66#3722—24no—
67#3718—22no—
68#3715—64no—
69#3715—64no—
70#3715—68no—
71#3714—64no—
72#3713—62no—
73C22×C33C2×D411yescharacteristic
74C2×C3×C3⋊D3—42no—
75C2×C3×C3⋊D3—44no—
76C33×C4C2×D411yescharacteristic
77C33×C4C2×D411yescharacteristic
78C32×Dic3—22no—
79C32×Dic3—42no—
80C2×C3×Q16—91no—
81C3×#124—92no—
82#947—32no—
83C2×#216—32no—
84C2×#215—92no—
85C2×#215—92no—
86C2×#215—182no—
87C2×#214—92no—
88C2×#213—92no—
89C2×D24—91no—
90#875—32no—
91C8×Dic3—91no—
92C4×C3⋊C8—92no—
93C4×C3⋊C8—181no—
94C22×C3⋊D3—22no—
95C22×C3×D3—62no—
96C22×C3×D3—62no—
97C22×C3×D3—122no—
98C32×Q8—62no—
99C32×Q8—62no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—32no—
103C2×C32×C4—41no—
104C2×C32×C4—62no—
105C2×C32×C4C22×D312yes—
106C2×C32×C4C22×D311yescharacteristic
107C4⋊(C3⋊D3)—22no—
108C4⋊(C3⋊D3)—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—42no—
113C2×C3×Dic3#6714yes—
114C3×D12—64no—
115C3×D12—62no—
116C3×D12—62no—
117C3×D12—62no—
118C3×D12—62no—
119C3×D12—122no—
120C3×Dic6—24no—
121C3×Dic6—22no—
122C3×Dic6—22no—
123C3×Dic6—22no—
124C3×Dic6—22no—
125C3×Dic6—42no—
126C3⋊(C3⋊C8)—32no—
127C3⋊(C3⋊C8)—34no—
128C3⋊(C3⋊C8)—64no—
129C3⋊(C3⋊C8)—62no—
130C3⋊(C3⋊C8)—122no—
131C3⋊(C3⋊C8)#6712yes—
132C3×C3⋊C8—34no—
133C3×C3⋊C8—62no—
134#495—271noSylow
135C2×C33#12711yescharacteristic
136C2×C33#13512yes—
137C3×C3⋊D3—82no—
138C2×C3×Q8—32no—
139C2×C3×Q8—32no—
140C2×C3×Q8—62no—
141C2×C3×D4—92no—
142C2×D12—32no—
143C2×D12—32no—
144C2×D12—62no—
145C2×Dic6D3212yes—
146C3×Q16—182no—
147C2×C3×C8—91no—
148C3×C22⋊C4—184no—
149C3×C42—32no—
150C3×C42—61no—
151#216—64no—
152#215—184no—
153#215—184no—
154#215—188no—
155#214—184no—
156#213—184no—
157(C2×D3)⋊C4—64no—
158(C2×D3)⋊C4—64no—
159C4×Dic3D3211yescharacteristic
160C2×C3⋊C8—32no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8—92no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—181no—
165C2×C3⋊C8—182no—
166C2×C3⋊C8—361no—
167D24—182no—
168C22×C32—21no—
169C22×C32—22no—
170C22×C32—41no—
171C22×C32D3×D411yescharacteristic
172C22×C32C2×#6712yes—
173C2×C3⋊D3—42no—
174C2×C3⋊D3—44no—
175C2×C3×D3—124no—
176C2×C3×D3—124no—
177C2×C3×D3—122no—
178C2×C3×D3—122no—
179C2×C3×D3—124no—
180C2×C3×D3—244no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—62no—
188C32×C4—122no—
189C32×C4D3×D412yes—
190C32×C4D3×D412yes—
191C32×C4D3×D411yescharacteristic
192C32×C4C2×#6711yescharacteristic
193C3×Dic3—24no—
194C3×Dic3—24no—
195C3×Dic3—44no—
196C3×Dic3—42no—
197C3×Dic3—42no—
198C2×Q16—91no—
199C2×SD16—272no—
200C2×D8—271no—
201#124—92no—
202C4×C8—271no—
203C33#49511yescharacteristic, complemented, Sylow
204C23×C3—182no—
205C22×D3—62no—
206C22×D3—62no—
207C22×D3—122no—
208C3×Q8—64no—
209C3×Q8—62no—
210C3×Q8—62no—
211C3×Q8—62no—
212C3×Q8—62no—
213C3×Q8—122no—
214C3×D4—182no—
215C3×D4—182no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—34no—
219C2×C3×C4—41no—
220C2×C3×C4—62no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—122no—
224C2×C3×C4C2×D3211yescharacteristic
225C2×C3×C4C2×D3212yes—
226C2×Dic3—22no—
227C2×Dic3#62512yes—
228D12—64no—
229D12—62no—
230D12—62no—
231D12—62no—
232D12—62no—
233D12—122no—
234Dic6—22no—
235Dic6—22no—
236C3×C8—92no—
237C3⋊C8—34no—
238C3⋊C8—62no—
239C3⋊C8—92no—
240C3⋊C8—94no—
241C3⋊C8—182no—
242C3⋊C8—184no—
243C3⋊C8—362no—
244C2×C32—22no—
245C2×C32—21no—
246C2×C32—24no—
247C2×C32—22no—
248C2×C32—41no—
249C2×C32—42no—
250C2×C32#91912yes—
251C2×C32#94012yes—
252C2×C32#94011yescharacteristic
253C2×C32#95014yes—
254C3⋊D3—82no—
255C3×D3—244no—
256C3×D3—242no—
257C3×D3—242no—
258C2×Q8—32no—
259C2×D4—92no—
260Q16—182no—
261SD16—544no—
262D8—542no—
263C2×C8—91no—
264C2×C8—271no—
265C22⋊C4—184no—
266C42—31no—
267C22×C3—21no—
268C22×C3—22no—
269C22×C3—41no—
270C22×C3—364no—
271C22×C3—362no—
272C22×C3C2×#62511yescharacteristic
273C22×C3D3×#6712yes—
274C2×D3—124no—
275C2×D3—124no—
276C2×D3—122no—
277C2×D3—124no—
278C2×D3—122no—
279C2×D3—244no—
280C3×C4—21no—
281C3×C4—22no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—64no—
287C3×C4—64no—
288C3×C4—124no—
289C3×C4—122no—
290C3×C4—122no—
291C3×C4#380211yescharacteristic
292C3×C4#380211yescharacteristic
293C3×C4D3×#6712yes—
294C3×C4#381112yes—
295Dic3—22no—
296Dic3—42no—
297C32—22no—
298C32—21no—
299C32—41no—
300C32#505912yescomplemented
301C32#519111yescharacteristic, complemented
302C23—182no—
303Q8—62no—
304Q8—62no—
305D4—182no—
306D4—182no—
307C2×C4—32no—
308C2×C4—62no—
309C2×C4D3311yescharacteristic
310C8—92no—
311C8—272no—
312C2×C3—22no—
313C2×C3—21no—
314C2×C3—22no—
315C2×C3—24no—
316C2×C3—41no—
317C2×C3—42no—
318C2×C3—722no—
319C2×C3#6382212yes—
320C2×C3#6391411yescharacteristic
321C2×C3#6394914yes—
322C2×C3#6397912yes—
323D3—244no—
324D3—242no—
325D3—242no—
326C22—362no—
327C22—364no—
328C22D3×#62511yescharacteristic
329C4—62no—
330C4—122no—
331C4#8895311yescharacteristic
332C4#8895411yescharacteristic
333C3—22no—
334C3—21no—
335C3—41no—
336C3#1059035311yescharacteristic, complemented
337C3#1059160912yescomplemented
338C2—722no—
339C2#1172285212yes—
340C2#1172302911yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×#88810C2
hypocenterC33#495
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. C2×#88793C2semidirect
3. C3×#63864C2semidirect
4. C2×C3×#3718C2semidirect
5. Dic3×C3⋊(C3⋊C8)C2semidirect
21. #63864D3semidirect
22. C2×#3718D3semidirect
203. C33#495semidirect
300. C32#5059semidirect
301. C32#5191semidirect
336. C3#10590353semidirect
337. C3#10591609semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
C2×#887932
C3×#638642
C2×C3×#37182
Dic3×C3⋊(C3⋊C8)2
#105916093
#105903533