Finite Group Databasebeta

← #13161165

Subgroups — #13161165

Order 1728.

816 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 160 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
1C3×#64153C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic
3Dic6×C3⋊Dic3C211yesmaximal, characteristic
4C2×#88798C211yesmaximal, characteristic
5#11722205C211yesmaximal, characteristic
6#11722051C211yesmaximal, characteristic
7#11722040C211yesmaximal, characteristic
8#10592881—31nomaximal
9#10591574—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3750—21no—
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×#216—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22#88798—22no—
23C2×C32×Q16—31no—
24Q8×C3⋊Dic3—31no—
25#64153D311yescharacteristic
26C2×#3756—31no—
27C3×#944—34no—
28C2×C3×#216D314yes—
29Dic3×Dic6—34no—
30C2×#3718—34no—
31#63662—31no—
32#63662—34no—
33#63662—64no—
34#63643—31no—
35#63641—31no—
36Dic3⋊(C3⋊C8)—34no—
37#63566—34no—
38C33×Q8—21no—
39C33×Q8—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×#634—21no—
47C3×#634—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6D412yes—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C32×C3⋊C8—21no—
53#5182—94no—
54#5090—91no—
55C2×C32×Q8—24no—
56C2×C32×Q8—31no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—34no—
61C2×C3×Dic6C2×D314yes—
62C32×Q16—62no—
63C2×C32×C8—31no—
64#3756—62no—
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—61no—
68C4⋊(C3⋊Dic3)—64no—
69#3750—21no—
70#3750—21no—
71C4×C3⋊Dic3—61no—
72C4×C3⋊Dic3C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76C3×#216—28no—
77C3×Dic3⋊C4—64no—
78C3×Dic3⋊C4—64no—
79C3×C4×Dic3—34no—
80C2×C3×C3⋊C8C2×D314yes—
81Dic3⋊Dic3—64no—
82Dic32—64no—
83#3718—68no—
84C22×C33C2×D411yescharacteristic
85C33×C4—41no—
86C33×C4C2×D411yescharacteristic
87C33×C4#4111yescharacteristic
88C3⋊(C3⋊Dic3)—41no—
89C3×C3⋊Dic3—21no—
90C3×C3⋊Dic3—41no—
91C3×C3⋊Dic3Q1612yescomplemented
92C32×Dic3—22no—
93C2×C3×Q16—34no—
94C3×C4⋊Q8—91no—
95Q8×Dic3—94no—
96#944—34no—
97C2×#216—94no—
98C2×#216—91no—
99C2×#216C3⋊D311yescharacteristic
100C4×Dic6—91no—
101#835—94no—
102#835—91no—
103#835—91no—
104#835—184no—
105#816—94no—
106Dic3⋊C8—94no—
107C4⋊(C3⋊C8)—91no—
108C32×Q8—28no—
109C32×Q8—21no—
110C32×Q8—24no—
111C32×Q8—21no—
112C32×Q8—24no—
113C32×Q8—32no—
114C32×Q8—44no—
115C32×Q8—61no—
116C2×C32×C4—21no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—44no—
120C2×C32×C4—61no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—61no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3—124no—
127C2×C3⋊Dic3#6711yescharacteristic
128C2×C3⋊Dic3#6711yescharacteristic
129#634—21no—
130#634—21no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—64no—
135C3×Dic6—24no—
136C3×Dic6—64no—
137C3×Dic6—64no—
138C3×Dic6#6718yes—
139C32×C8—61no—
140C3⋊(C3⋊C8)—61no—
141C3⋊(C3⋊C8)—64no—
142C3⋊(C3⋊C8)—124no—
143C3×C3⋊C8—24no—
144#462—271noSylow
145C2×C33#12111yescharacteristic
146C2×C33C2×Q1611yescharacteristic
147C2×C33#13711yescharacteristic
148C2×C3×Q8—24no—
149C2×C3×Q8—34no—
150C2×C3×Q8—91no—
151C2×C3×Q8D3214yes—
152C2×C3×Q8C2×C3⋊D311yescharacteristic
153C2×Dic6—34no—
154C2×Dic6C2×C3⋊D311yescharacteristic
155C3×Q16—68no—
156C2×C3×C8—34no—
157C3×C4⋊C4—181no—
158C3×C4⋊C4—181no—
159C3×C42—91no—
160#216—22no—
161#216—188no—
162#216—182no—
163C4⋊Dic3—94no—
164C4⋊Dic3—91no—
165C4⋊Dic3—184no—
166C4⋊Dic3—184no—
167Dic3⋊C4—64no—
168Dic3⋊C4—64no—
169Dic3⋊C4—181no—
170C4×Dic3—34no—
171C4×Dic3—181no—
172C4×Dic3—184no—
173C2×C3⋊C8—94no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—184no—
176C2×C3⋊C8C2×C3⋊D311yescharacteristic
177C22×C32—24no—
178C22×C32D3×D414yes—
179C22×C32C2×#6711yescharacteristic
180C32×C4—24no—
181C32×C4—24no—
182C32×C4—48no—
183C32×C4—41no—
184C32×C4—44no—
185C32×C4—62no—
186C32×C4D3×D411yescharacteristic
187C32×C4#23714yes—
188C32×C4#23711yescharacteristic
189C32×C4C2×#6714yes—
190C3⋊Dic3—21no—
191C3⋊Dic3—41no—
192C3⋊Dic3—121no—
193C3⋊Dic3—124no—
194C3⋊Dic3—244no—
195C3⋊Dic3#21612yescomplemented
196C3×Dic3—28no—
197C3×Dic3—38no—
198C3×Dic3—64no—
199C3×Dic3—124no—
200C2×Q16—31no—
201C2×Q16—271no—
202C4⋊Q8—91no—
203C4×Q8—271no—
204C4⋊C8—271no—
205#103—271no—
206#103—271no—
207C33#46211yescharacteristic, complemented, Sylow
208C3×Q8—28no—
209C3×Q8—24no—
210C3×Q8—24no—
211C3×Q8—21no—
212C3×Q8—21no—
213C3×Q8—38no—
214C3×Q8—44no—
215C3×Q8—64no—
216C3×Q8—181no—
217C3×Q8—181no—
218C2×C3×C4—24no—
219C2×C3×C4—21no—
220C2×C3×C4—24no—
221C2×C3×C4—44no—
222C2×C3×C4—64no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—181no—
226C2×C3×C4C2×D3214yes—
227C2×C3×C4C22×C3⋊D311yescharacteristic
228C2×Dic3—21no—
229C2×Dic3—34no—
230C2×Dic3—34no—
231C2×Dic3—64no—
232C2×Dic3—184no—
233C2×Dic3—181no—
234C2×Dic3—364no—
235Dic6—21no—
236Dic6—64no—
237Dic6—64no—
238Dic6#63812yes—
239C3×C8—64no—
240C3⋊C8—21no—
241C3⋊C8—184no—
242C3⋊C8—181no—
243C3⋊C8—364no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32D3×Q1614yes—
248C2×C32#91814yes—
249C2×C32#93811yescharacteristic
250C2×C32#93914yes—
251C2×C32#94211yescharacteristic
252C2×C32C2×#21611yescharacteristic
253C2×Q8—31no—
254C2×Q8—91no—
255C2×Q8D3×C3⋊D311yescharacteristic
256Q16—62no—
257Q16—542no—
258C2×C8—31no—
259C2×C8—271no—
260C4⋊C4—181no—
261C4⋊C4—181no—
262C4⋊C4—271no—
263C4⋊C4—541no—
264C42—91no—
265C42—541no—
266C22×C3—24no—
267C22×C3D3×#6714yes—
268C22×C3D4×C3⋊D311yescharacteristic
269C3×C4—24no—
270C3×C4—24no—
271C3×C4—48no—
272C3×C4—41no—
273C3×C4—44no—
274C3×C4—68no—
275C3×C4—92no—
276C3×C4—181no—
277C3×C4—361no—
278C3×C4#380514yes—
279C3×C4D3×#6714yes—
280C3×C4#383011yescharacteristic
281C3×C4C2×#63811yescharacteristic
282Dic3—22no—
283Dic3—38no—
284Dic3—64no—
285Dic3—124no—
286Dic3—361no—
287Dic3—364no—
288Dic3—724no—
289C32—24no—
290C32#509011yescharacteristic, complemented
291C32#518214yescomplemented
292Q8—21no—
293Q8—21no—
294Q8—32no—
295Q8—61no—
296Q8—181no—
297Q8—181no—
298C2×C4—21no—
299C2×C4—61no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—181no—
303C2×C4—541no—
304C2×C4C2×D3×C3⋊D311yescharacteristic
305C8—61no—
306C8—541no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3D3×#21614yes—
311C2×C3#6394814yes—
312C2×C3#6397014yes—
313C2×C3Q16×C3⋊D311yescharacteristic
314C2×C3#6412711yescharacteristic
315C2×C3#6414811yescharacteristic
316C22C3⋊D3×#6711yescharacteristic
317C4—41no—
318C4—62no—
319C4—92no—
320C4—181no—
321C4—361no—
322C4—1081no—
323C4#8902911yescharacteristic
324C4D3×#63811yescharacteristic
325C3—24no—
326C3#1059157414yescomplemented
327C3#1059288111yescharacteristic, complemented
328C2C3⋊D3×#21611yescharacteristic
329C2#1172358611yescharacteristic
330C2#1172361511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#462
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
91. C3×C3⋊Dic3Q16semidirect
195. C3⋊Dic3#216semidirect
207. C33#462semidirect
290. C32#5090semidirect
291. C32#5182semidirect
326. C3#10591574semidirect
327. C3#10592881semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C2×C32×#2162
Dic6×C3⋊Dic32
C2×#887982
#117222052
#117220512
#117220402
#105928813
#105915743