Finite Group Databasebeta

← #13161309

Subgroups — #13161309

Order 1728.

816 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 168 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×C3×#3756C211yesmaximal, characteristic
2C32×#944C211yesmaximal, characteristic
3Dic3×#634C211yesmaximal, characteristic
4C2×#88798C211yesmaximal, characteristic
5#11722205C211yesmaximal, characteristic
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
7#11722038C211yesmaximal, characteristic
8#10592789—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×#3756—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×Dic3⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20#88807—21no—
21Dic3×C3⋊Dic3—21no—
22#88798—22no—
23C32×C4⋊Q8—31no—
24C2×#3756—31no—
25C2×#3756D311yescharacteristic
26C4×#634—31no—
27C3×#944D314yes—
28C2×C3×#216—34no—
29Dic3×Dic6—34no—
30C2×#3718—34no—
31#63662—31no—
32#63662—31no—
33#63662—34no—
34#63662—64no—
35C4⋊(C3⋊(C3⋊C8))—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—
44C3×#634—21no—
45C3×#634D412yes—
46C2×C32×Dic3—21no—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49C32×Dic6—21no—
50C32×Dic6—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)—21no—
53#5182—91no—
54#5090—94no—
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×C4⋊C4—61no—
63C32×C4⋊C4—61no—
64C32×C42—31no—
65#3756—22no—
66#3756—62no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—34no—
69C4⋊(C3⋊Dic3)—64no—
70#3750—61no—
71C4×C3⋊Dic3—61no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×#216—68no—
77C3×Dic3⋊C4—24no—
78C3×Dic3⋊C4—24no—
79C3×C4×Dic3C2×D314yes—
80C2×C3×C3⋊C8—34no—
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—22no—
90C32×Dic3—21no—
91C32×Dic3—41no—
92C32×Dic3Q1612yescomplemented
93C2×C3×Q16—91no—
94C3×C4⋊Q8—34no—
95Q8×Dic3—91no—
96#944C3⋊D311yescharacteristic
97C2×#216—34no—
98C2×#216—94no—
99C2×#216—91no—
100C4×Dic6—94no—
101#835—94no—
102#835—94no—
103#835—91no—
104#835—184no—
105#816—91no—
106Dic3⋊C8—91no—
107C4⋊(C3⋊C8)—94no—
108C32×Q8—28no—
109C32×Q8—24no—
110C32×Q8—21no—
111C32×Q8—21no—
112C32×Q8—24no—
113C32×Q8—44no—
114C32×Q8—61no—
115C32×Q8—61no—
116C2×C32×C4—21no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—31no—
120C2×C32×C4—31no—
121C2×C32×C4—44no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—124no—
129#634—21no—
130#634#6712yes—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—64no—
133C2×C3×Dic3#6714yes—
134C2×C3×Dic3#6714yes—
135C3×Dic6—24no—
136C3×Dic6—24no—
137C3×Dic6—38no—
138C3×Dic6—64no—
139C3⋊(C3⋊C8)—21no—
140C3⋊(C3⋊C8)—61no—
141C3⋊(C3⋊C8)—64no—
142C3⋊(C3⋊C8)—124no—
143C3×C3⋊C8—64no—
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—182no—
156C2×C3×C8—91no—
157C3×C4⋊C4—64no—
158C3×C4⋊C4—64no—
159C3×C42—34no—
160#216—68no—
161#216—182no—
162#216—188no—
163C4⋊Dic3—94no—
164C4⋊Dic3—91no—
165C4⋊Dic3—184no—
166C4⋊Dic3—181no—
167Dic3⋊C4—21no—
168Dic3⋊C4—21no—
169Dic3⋊C4—184no—
170C4×Dic3—184no—
171C4×Dic3—181no—
172C4×Dic3C2×C3⋊D311yescharacteristic
173C2×C3⋊C8—34no—
174C2×C3⋊C8—94no—
175C2×C3⋊C8—91no—
176C2×C3⋊C8—184no—
177C22×C32—24no—
178C22×C32D3×D411yescharacteristic
179C22×C32C2×#6714yes—
180C32×C4—24no—
181C32×C4—24no—
182C32×C4—32no—
183C32×C4—48no—
184C32×C4—44no—
185C32×C4—41no—
186C32×C4—61no—
187C32×C4—121no—
188C32×C4D3×D414yes—
189C32×C4#23714yes—
190C32×C4#23711yescharacteristic
191C32×C4C2×#6711yescharacteristic
192C3⋊Dic3—22no—
193C3⋊Dic3—121no—
194C3⋊Dic3—124no—
195C3⋊Dic3—244no—
196C3×Dic3—24no—
197C3×Dic3—44no—
198C3×Dic3—68no—
199C3×Dic3#21618yescomplemented
200C2×Q16—91no—
201C2×Q16—271no—
202C4⋊Q8—31no—
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—44no—
214C3×Q8—64no—
215C3×Q8—64no—
216C3×Q8—92no—
217C3×Q8—181no—
218C2×C3×C4—24no—
219C2×C3×C4—21no—
220C2×C3×C4—24no—
221C2×C3×C4—34no—
222C2×C3×C4—34no—
223C2×C3×C4—44no—
224C2×C3×C4—64no—
225C2×C3×C4—181no—
226C2×C3×C4C2×D3214yes—
227C2×C3×C4C22×C3⋊D311yescharacteristic
228C2×Dic3—21no—
229C2×Dic3—64no—
230C2×Dic3—184no—
231C2×Dic3—181no—
232C2×Dic3—364no—
233C2×Dic3#63811yescharacteristic
234C2×Dic3#63811yescharacteristic
235Dic6—21no—
236Dic6—21no—
237Dic6—38no—
238Dic6—64no—
239C3×C8—181no—
240C3⋊C8—64no—
241C3⋊C8—184no—
242C3⋊C8—181no—
243C3⋊C8—364no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32D3×Q1611yescharacteristic
248C2×C32#91811yescharacteristic
249C2×C32#93814yes—
250C2×C32#93911yescharacteristic
251C2×C32#94214yes—
252C2×C32C2×#21614yes—
253C2×Q8—31no—
254C2×Q8—91no—
255C2×Q8D3×C3⋊D311yescharacteristic
256Q16—182no—
257Q16—542no—
258C2×C8—91no—
259C2×C8—271no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—271no—
263C4⋊C4—541no—
264C42—31no—
265C42—541no—
266C22×C3—24no—
267C22×C3D3×#6714yes—
268C22×C3C2×#63811yescharacteristic
269C3×C4—24no—
270C3×C4—24no—
271C3×C4—38no—
272C3×C4—48no—
273C3×C4—41no—
274C3×C4—44no—
275C3×C4—64no—
276C3×C4—124no—
277C3×C4—182no—
278C3×C4#380514yes—
279C3×C4D3×#6714yes—
280C3×C4D4×C3⋊D311yescharacteristic
281C3×C4#383011yescharacteristic
282Dic3—21no—
283Dic3—41no—
284Dic3—68no—
285Dic3—361no—
286Dic3—364no—
287Dic3—724no—
288Dic3#375612yescomplemented
289C32—24no—
290C32#509014yescomplemented
291C32#518211yescharacteristic, complemented
292Q8—21no—
293Q8—21no—
294Q8—61no—
295Q8—61no—
296Q8—92no—
297Q8—181no—
298C2×C4—21no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—61no—
302C2×C4—181no—
303C2×C4—541no—
304C2×C4C2×D3×C3⋊D311yescharacteristic
305C8—181no—
306C8—541no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3D3×#21614yes—
311C2×C3#6394814yes—
312C2×C3#6397014yes—
313C2×C3#6414711yescharacteristic
314C2×C3#6415111yescharacteristic
315C2×C3C2×#375611yescharacteristic
316C22D3×#63811yescharacteristic
317C4—32no—
318C4—41no—
319C4—61no—
320C4—121no—
321C4—182no—
322C4—1081no—
323C4#8902911yescharacteristic
324C4C3⋊D3×#6711yescharacteristic
325C3—24no—
326C3#1059157414yescomplemented
327C3#1059278911yescharacteristic, complemented
328C2D3×#375611yescharacteristic
329C2#1172358111yescharacteristic
330C2#1172363111yescharacteristic
subgroupG/H
centerC22D3×#638
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
92. C32×Dic3Q16semidirect
199. C3×Dic3#216semidirect
207. C33#462semidirect
288. Dic3#3756semidirect
290. C32#5090semidirect
291. C32#5182semidirect
326. C3#10591574semidirect
327. C3#10592789semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37562
C32×#9442
Dic3×#6342
C2×#887982
#117222052
Dic3⋊(C3⋊(C3⋊C8))2
#117220382
#105927893
#105915743