Finite Group Databasebeta

← #13156581

Subgroups — #13156581

Order 1728.

770 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 116 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
2C2×#88793C212yesmaximal, complemented
3C3×#63873C211yesmaximal, characteristic, complemented
4C2×C3×#3717C212yesmaximal, complemented
5C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#10591618—32nomaximal
7#10590884—31nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×C32×Dic6C2212yes—
10C2×#6295—22no—
11C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2211yescharacteristic
14#88793—24no—
15C3×#3724—22no—
16C3×#3717—24no—
17C2×#3755—32no—
18C3×#944—32no—
19C2×C3×#215—32no—
20C2×C3×#204—32no—
21#63914—31no—
22#63911—32no—
23#63873D311yescharacteristic, complemented
24C2×#3717D312yescomplemented
25C2×#3715—32no—
26C2×#3715—32no—
27C2×#3715—62no—
28C8×C3⋊Dic3—31no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C22×C3×C3⋊D3—22no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3D412yes—
34C3×C4⋊(C3⋊D3)—22no—
35C3×C4⋊(C3⋊D3)—22no—
36C2×C32×Dic3—22no—
37C32×Dic6—22no—
38C32×Dic6—22no—
39#6295—44no—
40C3⋊(C3⋊(C3⋊C8))D412yescomplemented
41C32×C3⋊C8D412yescomplemented
42#5175—92no—
43#5085—91no—
44C2×C32×Q8—32no—
45C2×C4⋊(C3⋊D3)C2×D312yes—
46C2×C3×D12—32no—
47C2×C3×D12—32no—
48C2×C3×D12—62no—
49C2×C3×Dic6—22no—
50C2×C3×Dic6C2×D312yes—
51C2×C3×Dic6C2×D312yes—
52C2×#626—64no—
53C2×#625—62no—
54C2×C32×C8—31no—
55#3755—64no—
56C4×C3⋊Dic3C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)—32no—
58C2×C3⋊(C3⋊C8)—31no—
59C2×C3⋊(C3⋊C8)—61no—
60C2×C3⋊(C3⋊C8)—62no—
61C2×C3⋊(C3⋊C8)—121no—
62C3×#215—64no—
63C3×Dic3⋊C4—64no—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8C2×D312yes—
68C3×#204—64no—
69#3724—22no—
70#3717—24no—
71#3715—64no—
72#3715—64no—
73#3715—68no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—42no—
76C2×C3×C3⋊D3—44no—
77C33×C4C2×D411yescharacteristic
78C33×C4C2×D411yescharacteristic
79C3×C3⋊Dic3SD1614yescomplemented
80C32×Dic3—42no—
81C2×C3×SD16—92no—
82C3×C4⋊Q8—91no—
83#944—32no—
84C2×#215—32no—
85C2×#215—92no—
86C2×#215—92no—
87C2×#215—182no—
88#940—92no—
89C2×#214—92no—
90C2×#204—32no—
91#874—91no—
92C8×Dic3—92no—
93C8×Dic3—181no—
94C4×C3⋊C8—91no—
95C22×C3⋊D3—22no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
98C22×C3×D3—122no—
99C32×Q8—62no—
100C32×Q8—62no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—41no—
104C2×C32×C4—62no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D312yes—
107C2×C3⋊Dic3#6712yes—
108C4⋊(C3⋊D3)—22no—
109C4⋊(C3⋊D3)—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—42no—
114C2×C3×Dic3—62no—
115C3×D12—64no—
116C3×D12—62no—
117C3×D12—62no—
118C3×D12—62no—
119C3×D12—62no—
120C3×D12—122no—
121C3×Dic6—24no—
122C3×Dic6—22no—
123C3×Dic6—22no—
124C3×Dic6—22no—
125C3×Dic6—22no—
126C3×Dic6—42no—
127#626—128no—
128#625—124no—
129C32×C8—32no—
130C3⋊(C3⋊C8)—32no—
131C3⋊(C3⋊C8)—34no—
132C3⋊(C3⋊C8)—64no—
133C3⋊(C3⋊C8)—62no—
134C3⋊(C3⋊C8)—122no—
135C3×C3⋊C8—22no—
136C3×C3⋊C8#6714yescomplemented
137#492—271noSylow
138C2×C33#12711yescharacteristic
139C2×C33C2×SD1612yes—
140C3×C3⋊D3—82no—
141C2×C3×Q8—32no—
142C2×C3×Q8—32no—
143C2×C3×Q8—62no—
144C2×C3×D4—92no—
145C2×#67—184no—
146C2×D12—32no—
147C2×D12—32no—
148C2×D12—62no—
149C2×D12—182no—
150C2×Dic6D3212yes—
151C3×SD16—184no—
152C2×C3×C8—32no—
153C2×C3×C8—61no—
154C3×C4⋊C4—182no—
155C3×C42—91no—
156#215—64no—
157#215—184no—
158#215—184no—
159#215—188no—
160#214—184no—
161Dic3⋊C4—64no—
162C4×Dic3—32no—
163C4×Dic3—61no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—92no—
166C2×C3⋊C8—181no—
167C2×C3⋊C8—182no—
168C2×C3⋊C8—361no—
169C2×C3⋊C8D3211yescharacteristic
170#204—64no—
171C22×C32—21no—
172C22×C32—22no—
173C22×C32—41no—
174C22×C32D3×D412yes—
175C22×C32C2×#6711yescharacteristic
176C2×C3⋊D3—42no—
177C2×C3⋊D3—44no—
178C2×C3×D3—124no—
179C2×C3×D3—124no—
180C2×C3×D3—122no—
181C2×C3×D3—122no—
182C2×C3×D3—124no—
183C2×C3×D3—244no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4—122no—
191C32×C4D3×D412yes—
192C32×C4D3×D411yescharacteristic
193C32×C4D3×D411yescharacteristic
194C32×C4C2×#6712yes—
195C3⋊Dic3#21514yescomplemented
196C3×Dic3—38no—
197C3×Dic3—44no—
198C3×Dic3—42no—
199C3×Dic3—42no—
200C3×Dic3—64no—
201C2×SD16—92no—
202C2×SD16—272no—
203C4⋊Q8—91no—
204#127—271no—
205C4×C8—271no—
206C33#49211yescharacteristic, complemented, Sylow
207C23×C3—182no—
208C22×D3—62no—
209C22×D3—62no—
210C22×D3—122no—
211C3×Q8—64no—
212C3×Q8—62no—
213C3×Q8—62no—
214C3×Q8—62no—
215C3×Q8—62no—
216C3×Q8—122no—
217C3×D4—182no—
218C3×D4—182no—
219C2×C3×C4—21no—
220C2×C3×C4—22no—
221C2×C3×C4—41no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—92no—
225C2×C3×C4—122no—
226C2×C3×C4C2×D3211yescharacteristic
227C2×C3×C4C2×D3212yes—
228#67—368no—
229C2×Dic3—22no—
230C2×Dic3—34no—
231C2×Dic3—62no—
232D12—64no—
233D12—62no—
234D12—62no—
235D12—62no—
236D12—62no—
237D12—122no—
238D12—364no—
239Dic6—22no—
240Dic6—22no—
241C3×C8—34no—
242C3×C8—62no—
243C3⋊C8—92no—
244C3⋊C8—94no—
245C3⋊C8—182no—
246C3⋊C8—184no—
247C3⋊C8—362no—
248C3⋊C8#62512yescomplemented
249C2×C32—22no—
250C2×C32—21no—
251C2×C32—24no—
252C2×C32—22no—
253C2×C32—41no—
254C2×C32—42no—
255C2×C32D3×SD1614yes—
256C2×C32#94012yes—
257C2×C32#94011yescharacteristic
258C2×C32C2×#21512yes—
259C3⋊D3—82no—
260C3×D3—244no—
261C3×D3—242no—
262C3×D3—242no—
263C2×Q8—32no—
264C2×D4—92no—
265C2×D4—542no—
266SD16—184no—
267SD16—544no—
268C2×C8—31no—
269C2×C8—271no—
270C4⋊C4—182no—
271C42—91no—
272C22×C3—21no—
273C22×C3—22no—
274C22×C3—41no—
275C22×C3—364no—
276C22×C3—362no—
277C22×C3#380211yescharacteristic
278C22×C3D3×#6712yes—
279C2×D3—124no—
280C2×D3—124no—
281C2×D3—122no—
282C2×D3—124no—
283C2×D3—122no—
284C2×D3—244no—
285C3×C4—21no—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—21no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—94no—
292C3×C4—124no—
293C3×C4—122no—
294C3×C4—122no—
295C3×C4#380211yescharacteristic
296C3×C4C2×#62511yescharacteristic
297C3×C4D3×#6712yes—
298C3×C4#381112yes—
299Dic3—38no—
300Dic3—42no—
301Dic3—64no—
302C32—22no—
303C32—21no—
304C32—41no—
305C32#508511yescharacteristic, complemented
306C32#517512yescomplemented
307C23—182no—
308Q8—62no—
309Q8—62no—
310D4—182no—
311D4—182no—
312D4—1084no—
313C2×C4—62no—
314C2×C4—92no—
315C2×C4D3311yescharacteristic
316C8—32no—
317C8—272no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—24no—
322C2×C3—41no—
323C2×C3—42no—
324C2×C3—722no—
325C2×C3#6379612yes—
326C2×C3#6391411yescharacteristic
327C2×C3D3×#21514yes—
328C2×C3#6397912yes—
329D3—244no—
330D3—242no—
331D3—242no—
332C22—362no—
333C22—364no—
334C22#8895311yescharacteristic
335C4—94no—
336C4—122no—
337C4D3×#62511yescharacteristic
338C4#8895411yescharacteristic
339C3—22no—
340C3—21no—
341C3—41no—
342C3#1059088411yescharacteristic, complemented
343C3#1059161812yescomplemented
344C2—722no—
345C2#1172284012yes—
346C2#1172302911yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723697C2
hypocenterC33#492
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×#63873C2semidirect
4. C2×C3×#3717C2semidirect
5. C3⋊C8×C3⋊Dic3C2semidirect
23. #63873D3semidirect
24. C2×#3717D3semidirect
40. C3⋊(C3⋊(C3⋊C8))D4semidirect
41. C32×C3⋊C8D4semidirect
79. C3×C3⋊Dic3SD16semidirect
136. C3×C3⋊C8#67semidirect
195. C3⋊Dic3#215semidirect
206. C33#492semidirect
248. C3⋊C8#625semidirect
305. C32#5085semidirect
306. C32#5175semidirect
342. C3#10590884semidirect
343. C3#10591618semidirect

Maximal subgroups

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

subgroupindex
#117236972
C2×#887932
C3×#638732
C2×C3×#37172
C3⋊C8×C3⋊Dic32
#105916183
#105908843