Finite Group Databasebeta

← #13161220

Subgroups — #13161220

Order 1728.

1016 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 158 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×#88862C211yesmaximal, characteristic
2C3×#64153C211yesmaximal, characteristic, complemented
3C2×C32×#216C211yesmaximal, characteristic, complemented
4#11723440C211yesmaximal, characteristic
5C2×#88798C211yesmaximal, characteristic, complemented
6C2×#88797C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic, complemented
8#10592889—31nomaximal
9#10591608—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14#88862—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3750—22no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×#216—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20#88804—22no—
21#88798—22no—
22#88797—22no—
23C2×C32×Q16—31no—
24#64156—31no—
25#64153D311yescharacteristic, complemented
26C2×#3756—31no—
27C2×#3755—31no—
28C2×#3755—34no—
29C2×#3755—64no—
30C2×#3744—31no—
31C3×#944—34no—
32C2×C3×#216D314yescomplemented
33#63864—34no—
34C2×#3718—34no—
35C2×#3717—34no—
36#63642—31no—
37#63554—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3D412yes—
47C3×#634—21no—
48C3×#634—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C3⋊(C3⋊(C3⋊C8))D412yes—
53C32×C3⋊C8D412yes—
54#5190—94no—
55#5086—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8—31no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4⋊(C3⋊D3)—34no—
62C2×C4⋊(C3⋊D3)—64no—
63C2×#634C2×D311yescharacteristic
64C2×C3×Dic6—34no—
65C2×C3×Dic6C2×D314yes—
66C32×Q16—62no—
67C2×C32×C8—31no—
68#3756—62no—
69#3755—62no—
70#3755—68no—
71#3755—616no—
72#3752—62no—
73#3750—22no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78#3744—62no—
79C3×#216—28no—
80C3×Dic3⋊C4—68no—
81C3×C4×Dic3—34no—
82C2×C3×C3⋊C8C2×D314yes—
83#3722—68no—
84#3718—68no—
85#3717—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—42no—
89C33×C4—41no—
90C33×C4C2×D411yescharacteristic
91C33×C4C2×D411yescharacteristic
92C3×C3⋊Dic3—22no—
93C3×C3⋊Dic3—41no—
94C32×Dic3—41no—
95C2×C3×Q16—34no—
96C3×C4⋊Q8—91no—
97#947—94no—
98#944—34no—
99C2×#216—94no—
100C2×#216—91no—
101C2×#216C3⋊D311yescharacteristic, complemented
102C2×#215—94no—
103C2×#215—91no—
104C2×#215—91no—
105C2×#215—184no—
106C2×#204—94no—
107#875—91no—
108#815—94no—
109#803—91no—
110C22×C3⋊D3—61no—
111C22×C3⋊D3—64no—
112C22×C3⋊D3—124no—
113C32×Q8—28no—
114C32×Q8—24no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—24no—
118C32×Q8—44no—
119C32×Q8—61no—
120C32×Q8—61no—
121C2×C32×C4—21no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—44no—
125C2×C32×C4—61no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3#6712yes—
130C4⋊(C3⋊D3)—68no—
131C4⋊(C3⋊D3)—61no—
132C4⋊(C3⋊D3)—64no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—124no—
136#634—21no—
137#634—21no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—38no—
140C2×C3×Dic3—64no—
141C3×Dic6—24no—
142C3×Dic6—24no—
143C3×Dic6—64no—
144C3×Dic6—64no—
145C32×C8—32no—
146C3⋊(C3⋊C8)—32no—
147C3⋊(C3⋊C8)—38no—
148C3⋊(C3⋊C8)—68no—
149C3×C3⋊C8#6718yes—
150#497—271noSylow
151C2×C33#12711yescharacteristic
152C2×C33#13712yes—
153C3⋊(C3⋊D3)—81no—
154C2×C3×Q8—24no—
155C2×C3×Q8—34no—
156C2×C3×Q8—91no—
157C2×C3×Q8D3214yes—
158C2×C3×Q8C2×C3⋊D311yescharacteristic
159C2×D12—94no—
160C2×D12—91no—
161C2×D12—184no—
162C2×Dic6—34no—
163C2×Dic6C2×C3⋊D311yescharacteristic
164C3×Q16—68no—
165C2×C3×C8—34no—
166C3×C4⋊C4—182no—
167C3×C42—91no—
168#216—22no—
169#216—182no—
170#216—188no—
171#215—188no—
172#215—182no—
173#215—1816no—
174#215—182no—
175(C2×D3)⋊C4—182no—
176(C2×D3)⋊C4—188no—
177Dic3⋊C4—68no—
178C4×Dic3—34no—
179C2×C3⋊C8—94no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—184no—
182C2×C3⋊C8C2×C3⋊D311yescharacteristic
183#204—188no—
184C22×C32—24no—
185C22×C32D3×D414yes—
186C22×C32C2×#6711yescharacteristic
187C2×C3⋊D3—128no—
188C2×C3⋊D3—124no—
189C2×C3⋊D3—121no—
190C2×C3⋊D3—122no—
191C2×C3⋊D3—128no—
192C2×C3⋊D3—248no—
193C32×C4—24no—
194C32×C4—24no—
195C32×C4—48no—
196C32×C4—44no—
197C32×C4—41no—
198C32×C4—121no—
199C32×C4D3×D414yes—
200C32×C4D3×D411yescharacteristic
201C32×C4D3×D411yescharacteristic
202C32×C4C2×#6714yes—
203C3⋊Dic3—22no—
204C3⋊Dic3—41no—
205C3×Dic3—44no—
206C3×Dic3—68no—
207C3×Dic3—124no—
208C2×Q16—31no—
209C2×Q16—271no—
210C2×SD16—271no—
211C2×SD16—271no—
212C4⋊Q8—91no—
213#124—271no—
214#97—271no—
215C33#49711yescharacteristic, complemented, Sylow
216C22×D3—184no—
217C22×D3—181no—
218C22×D3—364no—
219C3×Q8—28no—
220C3×Q8—24no—
221C3×Q8—24no—
222C3×Q8—21no—
223C3×Q8—21no—
224C3×Q8—44no—
225C3×Q8—64no—
226C3×Q8—64no—
227C3×Q8—181no—
228C3×Q8—181no—
229C2×C3×C4—24no—
230C2×C3×C4—21no—
231C2×C3×C4—24no—
232C2×C3×C4—44no—
233C2×C3×C4—64no—
234C2×C3×C4—92no—
235C2×C3×C4—181no—
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238C2×Dic3—21no—
239C2×Dic3—38no—
240C2×Dic3—64no—
241D12—188no—
242D12—184no—
243D12—184no—
244D12—181no—
245D12—181no—
246D12—364no—
247Dic6—21no—
248Dic6—21no—
249Dic6—64no—
250Dic6—64no—
251C3×C8—38no—
252C3⋊C8—98no—
253C3⋊C8—92no—
254C3⋊C8—188no—
255C3⋊C8#63812yes—
256C2×C32—28no—
257C2×C32—24no—
258C2×C32#91818yes—
259C2×C32#94014yes—
260C2×C32#94011yescharacteristic
261C2×C32#94212yes—
262C3⋊D3—248no—
263C3⋊D3—244no—
264C3⋊D3—241no—
265C2×Q8—31no—
266C2×Q8—91no—
267C2×Q8D3×C3⋊D311yescharacteristic
268C2×D4—271no—
269Q16—62no—
270Q16—542no—
271SD16—542no—
272SD16—542no—
273C2×C8—31no—
274C2×C8—271no—
275C4⋊C4—182no—
276C22⋊C4—542no—
277C42—91no—
278C22×C3—24no—
279C22×C3D3×#6714yes—
280C22×C3D4×C3⋊D311yescharacteristic
281C2×D3—368no—
282C2×D3—362no—
283C2×D3—361no—
284C2×D3—368no—
285C2×D3—364no—
286C2×D3—728no—
287C3×C4—24no—
288C3×C4—24no—
289C3×C4—48no—
290C3×C4—41no—
291C3×C4—44no—
292C3×C4—124no—
293C3×C4—182no—
294C3×C4—361no—
295C3×C4D3×#6714yes—
296C3×C4#381114yes—
297C3×C4D4×C3⋊D311yescharacteristic
298C3×C4C2×#63811yescharacteristic
299Dic3—41no—
300Dic3—68no—
301Dic3—124no—
302C32—24no—
303C32#508611yescharacteristic, complemented
304C32#519014yescomplemented
305C23—541no—
306Q8—21no—
307Q8—21no—
308Q8—61no—
309Q8—61no—
310Q8—181no—
311Q8—181no—
312D4—541no—
313D4—541no—
314C2×C4—21no—
315C2×C4—61no—
316C2×C4—92no—
317C2×C4—181no—
318C2×C4C2×D3×C3⋊D311yescharacteristic
319C8—32no—
320C8—272no—
321C2×C3—28no—
322C2×C3—24no—
323C2×C3#6394818yes—
324C2×C3#6397914yes—
325C2×C3#6412712yes—
326C2×C3#6414911yescharacteristic
327D3—728no—
328D3—724no—
329D3—721no—
330C22—1082no—
331C22—1081no—
332C22C3⋊D3×#6711yescharacteristic
333C4—41no—
334C4—121no—
335C4—182no—
336C4—361no—
337C4D3×#63811yescharacteristic
338C4#8903911yescharacteristic
339C3—24no—
340C3#1059160814yescomplemented
341C3#1059288911yescharacteristic, complemented
342C2—2161no—
343C2#1172358612yes—
344C2#1172363311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#497
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#64153C2semidirect
3. C2×C32×#216C2semidirect
5. C2×#88798C2semidirect
7. #11722020C2semidirect
25. #64153D3semidirect
32. C2×C3×#216D3semidirect
101. C2×#216C3⋊D3semidirect
215. C33#497semidirect
303. C32#5086semidirect
304. C32#5190semidirect
340. C3#10591608semidirect
341. C3#10592889semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C3×#641532
C2×C32×#2162
#117234402
C2×#887982
C2×#887972
#117220202
#105928893
#105916083