Finite Group Databasebeta

← #13156585

Subgroups — #13156585

Order 1728.

810 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 122 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
2#11723455C211yesmaximal, characteristic, complemented
3C2×#88796C212yesmaximal, complemented
4C2×C3×#3717C212yesmaximal, complemented
5C3×#63553C211yesmaximal, characteristic, complemented
6#10591620—31nomaximal
7#10590877—32nomaximal
8C2×#6308C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×C32×Dic6C2211yescharacteristic
11C2×#6295—22no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88807—22no—
15#88796—24no—
16C3×#3717—24no—
17#64153—31no—
18C2×#3744—32no—
19C2×C3×#215—32no—
20C2×C3×#204—32no—
21#63914—31no—
22#63911—32no—
23#63866—32no—
24#63866—61no—
25C2×#3717D312yescomplemented
26C2×#3716—32no—
27C2×#3716—32no—
28C2×#3716—62no—
29C3×C8×Dic3—32no—
30#63553D311yescharacteristic, complemented
31C22×C3×C3⋊D3—22no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34#6308—22no—
35C2×C3×C3⋊Dic3D412yes—
36C3×C4⋊(C3⋊D3)—22no—
37C3×C4⋊(C3⋊D3)—22no—
38C2×C32×Dic3—21no—
39C32×Dic6—22no—
40#6295—44no—
41C32×C3⋊C8D414yescomplemented
42#5175—92no—
43#4693—91no—
44C2×C32×Q8—31no—
45C2×C4⋊(C3⋊D3)C2×D312yes—
46C2×#634—31no—
47C2×#634—32no—
48C2×#634—62no—
49C2×#634—61no—
50C2×#634—121no—
51C2×C3×D12—32no—
52C2×C3×D12—32no—
53C2×C3×D12—62no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6C2×D312yes—
56C2×#626—64no—
57C2×#625—62no—
58C2×C32×C8—32no—
59#3750—62no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61#3744—64no—
62C3×#215—64no—
63C3×C4×Dic3—32no—
64C3×C4×Dic3—61no—
65C2×C3×C3⋊C8—22no—
66C2×C3×C3⋊C8C2×D312yes—
67C2×C3×C3⋊C8C2×D312yes—
68C3×#204—64no—
69Dic3⋊Dic3—64no—
70Dic3⋊Dic3—64no—
71#3717—24no—
72#3716—68no—
73#3716—64no—
74#3716—64no—
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—42no—
77C2×C3×C3⋊D3—44no—
78C33×C4C2×D412yes—
79C3⋊(C3⋊Dic3)—41no—
80C3×C3⋊Dic3SD1614yescomplemented
81C32×Dic3—41no—
82C2×C3×SD16—92no—
83#944—92no—
84#944—181no—
85C2×#215—32no—
86#940—92no—
87C2×#214—92no—
88C2×#204—32no—
89C2×#204—92no—
90C2×#204—92no—
91C2×#204—182no—
92#874—91no—
93#869—91no—
94C3×C4×C8—91no—
95C8×Dic3—32no—
96C22×C3⋊D3—22no—
97C22×C3×D3—62no—
98C22×C3×D3—62no—
99C22×C3×D3—122no—
100C32×Q8—62no—
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4—41no—
104C2×C32×C4—61no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D312yes—
107C2×C3⋊Dic3—61no—
108C2×C3⋊Dic3—62no—
109C2×C3⋊Dic3—62no—
110C2×C3⋊Dic3—122no—
111C2×C3⋊Dic3—122no—
112C2×C3⋊Dic3D1212yes—
113C4⋊(C3⋊D3)—22no—
114C4⋊(C3⋊D3)—22no—
115#634—62no—
116#634—64no—
117#634—62no—
118#634—62no—
119#634—122no—
120#634—122no—
121#634—124no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—62no—
126C3×D12—64no—
127C3×D12—62no—
128C3×D12—62no—
129C3×D12—62no—
130C3×D12—62no—
131C3×D12—122no—
132C3×Dic6—22no—
133C3×Dic6—22no—
134C3×Dic6—42no—
135#626—128no—
136#625—124no—
137C32×C8—34no—
138C3×C3⋊C8—24no—
139C3×C3⋊C8D1214yescomplemented
140C3×C3⋊C8#6714yescomplemented
141#492—271noSylow
142C2×C33#12711yescharacteristic
143C2×C33C2×SD1612yes—
144C3×C3⋊D3—82no—
145C2×C3×Q8—32no—
146C2×C3×Q8—61no—
147C2×C3×D4—92no—
148C2×#67—184no—
149C2×D12—32no—
150C2×D12—32no—
151C2×D12—62no—
152C2×D12—182no—
153C2×Dic6—91no—
154C2×Dic6—92no—
155C2×Dic6—182no—
156C2×Dic6—181no—
157C2×Dic6—361no—
158C2×Dic6D3211yescharacteristic
159C3×SD16—184no—
160C2×C3×C8—32no—
161C2×C3×C8—32no—
162C2×C3×C8—62no—
163C3×C42—91no—
164#215—64no—
165#214—184no—
166C4⋊Dic3—182no—
167Dic3⋊C4—184no—
168Dic3⋊C4—184no—
169C4×Dic3—32no—
170C4×Dic3—61no—
171C2×C3⋊C8D3212yes—
172#204—64no—
173#204—184no—
174#204—184no—
175#204—188no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32C2×D1211yescharacteristic
180C22×C32D3×D412yes—
181C2×C3⋊D3—42no—
182C2×C3⋊D3—44no—
183C2×C3×D3—124no—
184C2×C3×D3—122no—
185C2×C3×D3—124no—
186C2×C3×D3—124no—
187C2×C3×D3—122no—
188C2×C3×D3—244no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—42no—
193C32×C4—121no—
194C32×C4C2×D1212yes—
195C32×C4D3×D412yes—
196C32×C4C2×#6712yes—
197C3⋊Dic3—124no—
198C3⋊Dic3—122no—
199C3⋊Dic3—124no—
200C3⋊Dic3—121no—
201C3⋊Dic3—122no—
202C3⋊Dic3#20414yescomplemented
203C3×Dic3—38no—
204C3×Dic3—42no—
205C3×Dic3—42no—
206C3×Dic3—64no—
207C2×SD16—92no—
208C2×SD16—272no—
209C4⋊Q8—271no—
210#127—271no—
211C4×C8—91no—
212C33#49211yescharacteristic, complemented, Sylow
213C23×C3—182no—
214C22×D3—62no—
215C22×D3—62no—
216C22×D3—122no—
217C3×Q8—62no—
218C3×Q8—62no—
219C3×Q8—122no—
220C3×D4—182no—
221C3×D4—182no—
222C2×C3×C4—21no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—92no—
228C2×C3×C4C2×D3212yes—
229C2×C3×C4C2×D3211yescharacteristic
230#67—368no—
231C2×Dic3—21no—
232C2×Dic3—34no—
233C2×Dic3—62no—
234C2×Dic3—182no—
235C2×Dic3—182no—
236C2×Dic3—181no—
237C2×Dic3—362no—
238C2×Dic3—362no—
239D12—64no—
240D12—62no—
241D12—62no—
242D12—62no—
243D12—62no—
244D12—122no—
245D12—364no—
246Dic6—22no—
247Dic6—184no—
248Dic6—182no—
249Dic6—182no—
250Dic6—182no—
251Dic6—362no—
252Dic6—362no—
253Dic6—364no—
254C3×C8—34no—
255C3×C8—34no—
256C3×C8—64no—
257C3⋊C8#62614yescomplemented
258C2×C32—24no—
259C2×C32—22no—
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—41no—
263C2×C32—42no—
264C2×C32#87411yescharacteristic
265C2×C32C2×#20412yes—
266C2×C32D3×SD1614yes—
267C2×C32#94012yes—
268C3⋊D3—82no—
269C3×D3—244no—
270C3×D3—242no—
271C3×D3—242no—
272C2×Q8—31no—
273C2×Q8—271no—
274C2×D4—92no—
275C2×D4—542no—
276SD16—184no—
277SD16—544no—
278C2×C8—32no—
279C4⋊C4—542no—
280C42—91no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—41no—
284C22×C3—364no—
285C22×C3—362no—
286C22×C3D3×D1212yes—
287C22×C3#381111yescharacteristic
288C2×D3—124no—
289C2×D3—124no—
290C2×D3—122no—
291C2×D3—124no—
292C2×D3—122no—
293C2×D3—244no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—22no—
297C3×C4—42no—
298C3×C4—94no—
299C3×C4—122no—
300C3×C4—122no—
301C3×C4D3×D1212yes—
302C3×C4C2×#62612yes—
303C3×C4D3×#6712yes—
304Dic3—38no—
305Dic3—41no—
306Dic3—64no—
307Dic3—364no—
308Dic3—362no—
309Dic3—364no—
310Dic3—362no—
311Dic3—361no—
312C32—21no—
313C32—22no—
314C32—41no—
315C32#469311yescharacteristic, complemented
316C32#517512yescomplemented
317C23—182no—
318Q8—62no—
319Q8—542no—
320D4—182no—
321D4—182no—
322D4—1084no—
323C2×C4—61no—
324C2×C4—92no—
325C2×C4—541no—
326C2×C4D3311yescharacteristic
327C8—34no—
328C2×C3—21no—
329C2×C3—22no—
330C2×C3—22no—
331C2×C3—24no—
332C2×C3—41no—
333C2×C3—42no—
334C2×C3—722no—
335C2×C3D3×#20414yes—
336C2×C3#6391112yes—
337C2×C3#6394012yes—
338C2×C3#6397911yescharacteristic
339D3—244no—
340D3—242no—
341D3—242no—
342C22—362no—
343C22—364no—
344C22#8895511yescharacteristic
345C4—94no—
346C4—121no—
347C4—1081no—
348C4D3×#62612yes—
349C3—22no—
350C3—21no—
351C3—41no—
352C3#1059087712yescomplemented
353C3#1059162011yescharacteristic, complemented
354C2—722no—
355C2#1172284312yes—
356C2#1172303011yescharacteristic
subgroupG/H
centerC22#88955
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. #11723455C2semidirect
3. C2×#88796C2semidirect
4. C2×C3×#3717C2semidirect
5. C3×#63553C2semidirect
25. C2×#3717D3semidirect
30. #63553D3semidirect
41. C32×C3⋊C8D4semidirect
80. C3×C3⋊Dic3SD16semidirect
139. C3×C3⋊C8D12semidirect
140. C3×C3⋊C8#67semidirect
202. C3⋊Dic3#204semidirect
212. C33#492semidirect
257. C3⋊C8#626semidirect
315. C32#4693semidirect
316. C32#5175semidirect
352. C3#10590877semidirect
353. C3#10591620semidirect

Maximal subgroups

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

subgroupindex
#117236972
#117234552
C2×#887962
C2×C3×#37172
C3×#635532
#105916203
#105908773