Finite Group Databasebeta

← #13156600

Subgroups — #13156600

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×#88812C211yesmaximal, characteristic, complemented
2C2×#88792C212yesmaximal, complemented
3C3×#63860C212yesmaximal, complemented
4C2×C3×#3713C211yesmaximal, characteristic
5Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#10591470—32nomaximal
7#10590353—31nomaximal
8C2×C3×#634C2212yes—
9C2×C32×D12C2212yes—
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4×Dic3C2211yescharacteristic
13#88812—22no—
14#88792—24no—
15C3×(C2×D3)⋊Dic3—24no—
16C3×#3713—22no—
17C2×#3754—32no—
18C3×#936—32no—
19C2×C3×#213—32no—
20C3×#875—32no—
21#63860D312yescomplemented
22C2×#3719—32no—
23C2×#3718—31no—
24C2×#3715—32no—
25C2×#3715—32no—
26C2×#3715—62no—
27C2×#3713D311yescharacteristic
28C4×C3⋊(C3⋊C8)—31no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C22×C32×D3—22no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—22no—
34C3×#634—22no—
35C3×#634—22no—
36C2×C32×Dic3D412yes—
37C32×D12—22no—
38C32×D12—22no—
39C3⋊(C3⋊(C3⋊C8))D412yes—
40C3×C3⋊(C3⋊C8)D412yes—
41#5155—91no—
42#5059—92no—
43C2×C32×D4—32no—
44C2×#634C2×D312yes—
45C2×C3×D12—22no—
46C2×C3×D12C2×D312yes—
47C2×C3×D12C2×D312yes—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—32no—
50C2×C3×Dic6—62no—
51C32×C42—31no—
52#3754—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×#217—64no—
60C3×#213—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(C2×D3)⋊Dic3—24no—
67#3719—64no—
68#3718—62no—
69#3715—64no—
70#3715—64no—
71#3715—68no—
72#3713—22no—
73C22×C33C2×D411yescharacteristic
74C2×C32×D3—42no—
75C2×C32×D3—44no—
76C33×C4C2×D411yescharacteristic
77C33×C4C2×D411yescharacteristic
78C3×C3⋊Dic3—42no—
79C32×Dic3—22no—
80C2×C3×D8—91no—
81C3×#124—92no—
82C2×#216—92no—
83C2×#215—92no—
84#936—32no—
85C2×#214—92no—
86C2×#214—92no—
87C2×#214—182no—
88C2×#213—32no—
89C2×Dic12—91no—
90#875—32no—
91C8×Dic3—91no—
92C4×C3⋊C8—92no—
93C4×C3⋊C8—181no—
94C23×C32—62no—
95C22×C3×D3—22no—
96C22×C3×D3—22no—
97C22×C3×D3—42no—
98C32×D4—62no—
99C32×D4—62no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—32no—
103C2×C32×C4—41no—
104C2×C32×C4C22×D312yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—22no—
107#634—22no—
108#634—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—122no—
113C2×C3×Dic3#6714yes—
114C3×D12—24no—
115C3×D12—22no—
116C3×D12—22no—
117C3×D12—22no—
118C3×D12—22no—
119C3×D12—42no—
120C3×Dic6—64no—
121C3×Dic6—62no—
122C3×Dic6—62no—
123C3×Dic6—62no—
124C3×Dic6—62no—
125C3×Dic6—122no—
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—
137C32×D3—82no—
138C2×C3×Q8—92no—
139C2×C3×D4—32no—
140C2×C3×D4—32no—
141C2×C3×D4—62no—
142C2×D12D3212yes—
143C2×Dic6—32no—
144C2×Dic6—32no—
145C2×Dic6—62no—
146C3×D8—182no—
147C2×C3×C8—91no—
148C3×C22⋊C4—184no—
149C3×C42—32no—
150C3×C42—61no—
151#217—64no—
152#216—184no—
153#215—184no—
154#214—184no—
155#214—184no—
156#214—188no—
157#213—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—
167Dic12—182no—
168C22×C32—21no—
169C22×C32—22no—
170C22×C32—41no—
171C22×C32—122no—
172C22×C32—124no—
173C22×C32D3×D411yescharacteristic
174C22×C32C2×#6712yes—
175C2×C3×D3—44no—
176C2×C3×D3—44no—
177C2×C3×D3—42no—
178C2×C3×D3—42no—
179C2×C3×D3—44no—
180C2×C3×D3—84no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—62no—
188C32×C4D3×D412yes—
189C32×C4D3×D412yes—
190C32×C4D3×D411yescharacteristic
191C32×C4C2×#6711yescharacteristic
192C3⋊Dic3—42no—
193C3×Dic3—24no—
194C3×Dic3—24no—
195C3×Dic3—124no—
196C3×Dic3—122no—
197C3×Dic3—122no—
198C2×Q16—271no—
199C2×SD16—272no—
200C2×D8—91no—
201#124—92no—
202C4×C8—271no—
203C33#49511yescharacteristic, complemented, Sylow
204C23×C3—62no—
205C23×C3—62no—
206C23×C3—122no—
207C22×D3—22no—
208C3×Q8—182no—
209C3×Q8—182no—
210C3×D4—64no—
211C3×D4—62no—
212C3×D4—62no—
213C3×D4—62no—
214C3×D4—62no—
215C3×D4—122no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—34no—
219C2×C3×C4—41no—
220C2×C3×C4—62no—
221C2×C3×C4—182no—
222C2×C3×C4C2×D3211yescharacteristic
223C2×C3×C4C2×D3212yes—
224C2×Dic3—62no—
225C2×Dic3—62no—
226C2×Dic3—122no—
227C2×Dic3#62512yes—
228D12—22no—
229D12—22no—
230Dic6—64no—
231Dic6—62no—
232Dic6—62no—
233Dic6—62no—
234Dic6—62no—
235Dic6—122no—
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—242no—
251C2×C32#91212yes—
252C2×C32#94012yes—
253C2×C32#94011yescharacteristic
254C2×C32#95014yes—
255C3×D3—84no—
256C3×D3—82no—
257C3×D3—82no—
258C2×Q8—92no—
259C2×D4—32no—
260Q16—542no—
261SD16—544no—
262D8—182no—
263C2×C8—91no—
264C2×C8—271no—
265C22⋊C4—184no—
266C42—31no—
267C22×C3—22no—
268C22×C3—21no—
269C22×C3—41no—
270C22×C3—124no—
271C22×C3—124no—
272C22×C3—122no—
273C22×C3—124no—
274C22×C3—122no—
275C22×C3—244no—
276C22×C3C2×#62511yescharacteristic
277C22×C3D3×#6712yes—
278C2×D3—44no—
279C2×D3—42no—
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—362no—
289C3×C4#380211yescharacteristic
290C3×C4#380211yescharacteristic
291C3×C4D3×#6712yes—
292C3×C4#381112yes—
293Dic3—22no—
294Dic3—124no—
295Dic3—122no—
296Dic3—122no—
297C32—21no—
298C32—22no—
299C32—41no—
300C32#505912yescomplemented
301C32#515511yescharacteristic, complemented
302C23—62no—
303Q8—182no—
304Q8—182no—
305D4—62no—
306D4—62no—
307C2×C4—32no—
308C2×C4—182no—
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—244no—
319C2×C3—242no—
320C2×C3—242no—
321C2×C3#6382212yes—
322C2×C3#6391411yescharacteristic
323C2×C3#6393314yes—
324C2×C3#6397912yes—
325D3—82no—
326C22—122no—
327C22—124no—
328C22D3×#62511yescharacteristic
329C4—62no—
330C4—362no—
331C4#8895311yescharacteristic
332C4#8895411yescharacteristic
333C3—22no—
334C3—21no—
335C3—41no—
336C3#1059035311yescharacteristic, complemented
337C3#1059147012yescomplemented
338C2—242no—
339C2#1172285012yes—
340C2#1172302911yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×#3713C2
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
1. C2×#88812C2semidirect
2. C2×#88792C2semidirect
3. C3×#63860C2semidirect
5. Dic3×C3⋊(C3⋊C8)C2semidirect
21. #63860D3semidirect
203. C33#495semidirect
300. C32#5059semidirect
301. C32#5155semidirect
336. C3#10590353semidirect
337. C3#10591470semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
C2×#887922
C3×#638602
C2×C3×#37132
Dic3×C3⋊(C3⋊C8)2
#105914703
#105903533