Finite Group Databasebeta

← #13161300

Subgroups — #13161300

Order 1728.

1048 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1048 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
1#11723988C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C2×C32×#216C211yesmaximal, characteristic, complemented
4#11723451C211yesmaximal, characteristic, complemented
5C2×#88797C211yesmaximal, characteristic
6C2×#88795C211yesmaximal, characteristic
7#11722021C211yesmaximal, characteristic, complemented
8#10592889—31nomaximal
9#10591629—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14#88859—22no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3756—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#216—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—22no—
21#88797—22no—
22#88795—22no—
23C2×C32×Q16—31no—
24#64156—31no—
25#64156—34no—
26#64156—64no—
27#64153—31no—
28C2×#3756D311yescharacteristic, complemented
29C2×#3755—31no—
30C2×#3744—31no—
31C2×C3×#216—34no—
32C2×C3×#216D314yescomplemented
33#63873—34no—
34C2×#3717—34no—
35C2×#3717—34no—
36#63642—31no—
37#63555—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)D412yes—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C2×C3×C3⋊Dic3—21no—
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#5190—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—22no—
69#3755—62no—
70#3752—62no—
71#3752—68no—
72#3752—616no—
73#3750—62no—
74C4×C3⋊Dic3—31no—
75C4×C3⋊Dic3—34no—
76C4×C3⋊Dic3—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78#3744—62no—
79C3×#216—28no—
80C3×#216—68no—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8C2×D314yes—
83#3724—68no—
84#3717—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—
96C2×C3×Q16—91no—
97#947—94no—
98#947—91no—
99#947—184no—
100#944—94no—
101#944—91no—
102C2×#216—34no—
103C2×#216C3⋊D311yescharacteristic, complemented
104C2×#215—94no—
105C2×#215—91no—
106C2×#204—94no—
107C2×#204—91no—
108#815—94no—
109#815—91no—
110C22×C3⋊D3—68no—
111C22×C3⋊D3—64no—
112C22×C3⋊D3—61no—
113C32×Q8—21no—
114C32×Q8—21no—
115C32×Q8—24no—
116C32×Q8—24no—
117C32×Q8—44no—
118C32×Q8—44no—
119C32×Q8—61no—
120C32×Q8—61no—
121C2×C32×C4—28no—
122C2×C32×C4—24no—
123C2×C32×C4—21no—
124C2×C32×C4—24no—
125C2×C32×C4—61no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—32no—
130C2×C3⋊Dic3—38no—
131C2×C3⋊Dic3—68no—
132C4⋊(C3⋊D3)—61no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—64no—
136C4⋊(C3⋊D3)—124no—
137C4⋊(C3⋊D3)—124no—
138#634—21no—
139#634—21no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—64no—
142C3×Dic6—24no—
143C3×Dic6—24no—
144C3×Dic6—64no—
145C3×Dic6—64no—
146C32×C8—32no—
147C3⋊(C3⋊C8)#6712yes—
148C3×C3⋊C8—38no—
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—
165C3×Q16—182no—
166C2×C3×C8—34no—
167C2×C3×C8—91no—
168#216—22no—
169#216—68no—
170#215—188no—
171#215—182no—
172(C2×D3)⋊C4—188no—
173(C2×D3)⋊C4—182no—
174(C2×D3)⋊C4—1816no—
175Dic3⋊C4—182no—
176Dic3⋊C4—188no—
177C4×Dic3—94no—
178C4×Dic3—91no—
179C4×Dic3—184no—
180C2×C3⋊C8—34no—
181C2×C3⋊C8C2×C3⋊D311yescharacteristic
182#204—188no—
183#204—182no—
184C22×C32—24no—
185C22×C32D3×D414yes—
186C22×C32D3×D411yescharacteristic
187C2×C3⋊D3—128no—
188C2×C3⋊D3—1216no—
189C2×C3⋊D3—128no—
190C2×C3⋊D3—122no—
191C2×C3⋊D3—124no—
192C2×C3⋊D3—121no—
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×C4C2×#6714yes—
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—41no—
204C3⋊Dic3—62no—
205C3⋊Dic3—68no—
206C3⋊Dic3—128no—
207C3×Dic3—44no—
208C3×Dic3—124no—
209C2×Q16—31no—
210C2×Q16—91no—
211C2×SD16—271no—
212C2×SD16—271no—
213C4⋊Q8—271no—
214#124—271no—
215#97—271no—
216C33#49711yescharacteristic, complemented, Sylow
217C22×D3—188no—
218C22×D3—184no—
219C22×D3—181no—
220C3×Q8—24no—
221C3×Q8—24no—
222C3×Q8—21no—
223C3×Q8—21no—
224C3×Q8—44no—
225C3×Q8—44no—
226C3×Q8—64no—
227C3×Q8—64no—
228C3×Q8—181no—
229C3×Q8—181no—
230C2×C3×C4—28no—
231C2×C3×C4—24no—
232C2×C3×C4—21no—
233C2×C3×C4—24no—
234C2×C3×C4—64no—
235C2×C3×C4—181no—
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238C2×Dic3—21no—
239C2×Dic3—64no—
240C2×Dic3—98no—
241C2×Dic3—92no—
242C2×Dic3—188no—
243D12—184no—
244D12—184no—
245D12—181no—
246D12—181no—
247D12—364no—
248D12—364no—
249Dic6—21no—
250Dic6—21no—
251Dic6—64no—
252Dic6—64no—
253C3×C8—38no—
254C3×C8—92no—
255C3⋊C8—38no—
256C3⋊C8#63812yes—
257C2×C32—28no—
258C2×C32—24no—
259C2×C32#91818yes—
260C2×C32#91812yes—
261C2×C32#94014yes—
262C2×C32#94011yescharacteristic
263C3⋊D3—248no—
264C3⋊D3—244no—
265C3⋊D3—241no—
266C2×Q8—31no—
267C2×Q8—91no—
268C2×Q8D3×C3⋊D311yescharacteristic
269C2×D4—271no—
270Q16—62no—
271Q16—182no—
272SD16—542no—
273SD16—542no—
274C2×C8—31no—
275C2×C8—91no—
276C4⋊C4—542no—
277C22⋊C4—542no—
278C42—271no—
279C22×C3—24no—
280C22×C3#381114yes—
281C22×C3D4×C3⋊D311yescharacteristic
282C2×D3—368no—
283C2×D3—3616no—
284C2×D3—362no—
285C2×D3—361no—
286C2×D3—368no—
287C2×D3—364no—
288C3×C4—24no—
289C3×C4—24no—
290C3×C4—48no—
291C3×C4—41no—
292C3×C4—44no—
293C3×C4—124no—
294C3×C4—361no—
295C3×C4D3×#6714yes—
296C3×C4D3×#6714yes—
297C3×C4D4×C3⋊D311yescharacteristic
298C3×C4C2×#63811yescharacteristic
299Dic3—41no—
300Dic3—124no—
301Dic3—182no—
302Dic3—188no—
303Dic3—368no—
304C32—24no—
305C32#519014yescomplemented
306C32#519011yescharacteristic, complemented
307C23—541no—
308Q8—21no—
309Q8—21no—
310Q8—61no—
311Q8—61no—
312Q8—181no—
313Q8—181no—
314D4—541no—
315D4—541no—
316C2×C4—21no—
317C2×C4—61no—
318C2×C4—181no—
319C2×C4—272no—
320C2×C4C2×D3×C3⋊D311yescharacteristic
321C8—32no—
322C8—92no—
323C2×C3—28no—
324C2×C3—24no—
325C2×C3#6394518yes—
326C2×C3#6397914yes—
327C2×C3#6412712yes—
328C2×C3#6414911yescharacteristic
329D3—728no—
330D3—721no—
331D3—724no—
332C22—1082no—
333C22—1081no—
334C22#8903911yescharacteristic
335C4—41no—
336C4—121no—
337C4—361no—
338C4—542no—
339C4D3×#63811yescharacteristic
340C4C3⋊D3×#6711yescharacteristic
341C3—24no—
342C3#1059162914yescomplemented
343C3#1059288911yescharacteristic, complemented
344C2—2161no—
345C2#1172357812yes—
346C2#1172363311yescharacteristic
subgroupG/H
centerC22#89039
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. C2×C3×#3756C2semidirect
3. C2×C32×#216C2semidirect
4. #11723451C2semidirect
7. #11722021C2semidirect
28. C2×#3756D3semidirect
32. C2×C3×#216D3semidirect
103. C2×#216C3⋊D3semidirect
216. C33#497semidirect
305. C32#5190semidirect
306. C32#5190semidirect
342. C3#10591629semidirect
343. C3#10592889semidirect

Maximal subgroups

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

subgroupindex
#117239882
C2×C3×#37562
C2×C32×#2162
#117234512
C2×#887972
C2×#887952
#117220212
#105928893
#105916293