Finite Group Databasebeta

← #13161254

Subgroups — #13161254

Order 1728.

888 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 188 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×#88863C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic
3C32×Q8×Dic3C211yesmaximal, characteristic
4#11723443C211yesmaximal, characteristic
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
6#11722041C211yesmaximal, characteristic
7#11722039C211yesmaximal, characteristic
8#10592730—31nomaximal
9#10591625—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13#88863—22no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3756—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3—21no—
20C32×C4×Dic3C2211yescharacteristic
21Dic3⋊(C3⋊Dic3)—21no—
22Dic3⋊(C3⋊Dic3)—21no—
23C32×C4×Q8—31no—
24C2×#3756—31no—
25C2×#3756—34no—
26C2×#3756—64no—
27C2×#3756D311yescharacteristic
28#64078—31no—
29C3×Q8×Dic3D314yes—
30C2×C3×#216—34no—
31#63866—34no—
32#63637—31no—
33#63637—31no—
34C4⋊(C3⋊(C3⋊C8))—31no—
35Dic3⋊(C3⋊C8)—34no—
36#63567—34no—
37#63565—34no—
38C33×Q8—21no—
39C33×Q8D412yes—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43#6308—21no—
44#6308—21no—
45C2×C3×C3⋊Dic3—21no—
46C3×#634—21no—
47C3×#634—21no—
48C2×C32×Dic3—21no—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)—21no—
53#5182—91no—
54#5031—94no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C2×D311yescharacteristic
58C2×#634—31no—
59C2×#634—34no—
60C2×#634—64no—
61C2×#634C2×D311yescharacteristic
62C2×C3×Dic6—34no—
63C32×C4⋊C4—31no—
64C32×C4⋊C4—61no—
65C32×C42—31no—
66C32×C42—61no—
67#3756—22no—
68#3756—616no—
69#3756—68no—
70#3756—62no—
71C4⋊(C3⋊Dic3)—61no—
72C4⋊(C3⋊Dic3)—61no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C3×#216—68no—
78C3×C4⋊Dic3—24no—
79C3×C4⋊Dic3C2×D314yes—
80C3×C4×Dic3—24no—
81C3×C4×Dic3C2×D314yes—
82C2×C3×C3⋊C8—34no—
83Dic3⋊Dic3—64no—
84Dic3⋊Dic3—64no—
85C22×C33C2×D411yescharacteristic
86C33×C4—22no—
87C33×C4C2×D411yescharacteristic
88C33×C4#4111yescharacteristic
89C3⋊(C3⋊Dic3)—41no—
90C3×C3⋊Dic3—41no—
91C32×Dic3—21no—
92C32×Dic3—41no—
93C32×Dic3Q1612yescomplemented
94C2×C3×Q16—91no—
95C3×C4×Q8—34no—
96Q8×Dic3C3⋊D311yescharacteristic
97#944—91no—
98C2×#216—34no—
99C2×#216—94no—
100C2×#216—91no—
101C2×#216—184no—
102#869—94no—
103#835—91no—
104#816—91no—
105Dic3⋊C8—91no—
106#810—94no—
107#810—94no—
108C4⋊(C3⋊C8)—94no—
109C32×Q8—21no—
110C32×Q8—28no—
111C32×Q8—24no—
112C32×Q8—44no—
113C32×Q8D1218yes—
114C32×Q8#6712yes—
115C2×C32×C4—21no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—31no—
119C2×C32×C4—31no—
120C2×C32×C4—44no—
121C2×C32×C4—61no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—124no—
128#634—21no—
129#634—21no—
130#634—68no—
131#634—64no—
132#634—61no—
133#634—61no—
134#634—64no—
135#634—124no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—64no—
138C2×C3×Dic3#6714yes—
139C2×C3×Dic3#6714yes—
140C3×Dic6—64no—
141C3×Dic6—64no—
142C3⋊(C3⋊C8)—21no—
143C3⋊(C3⋊C8)—68no—
144C3⋊(C3⋊C8)—64no—
145C3⋊(C3⋊C8)—61no—
146C3×C3⋊C8—64no—
147#462—271noSylow
148C2×C33#12111yescharacteristic
149C2×C33C2×Q1611yescharacteristic
150C2×C33#13711yescharacteristic
151C2×C3×Q8—24no—
152C2×C3×Q8—91no—
153C2×C3×Q8D3214yes—
154C2×C3×Q8C2×C3⋊D311yescharacteristic
155C2×Dic6—34no—
156C2×Dic6—94no—
157C2×Dic6—91no—
158C2×Dic6—184no—
159C3×Q16—182no—
160C2×C3×C8—91no—
161C3×C4⋊C4—34no—
162C3×C4⋊C4—64no—
163C3×C42—34no—
164C3×C42—64no—
165#216—68no—
166#216—1816no—
167#216—188no—
168#216—182no—
169C4⋊Dic3—21no—
170C4⋊Dic3—184no—
171C4⋊Dic3—184no—
172C4⋊Dic3C2×C3⋊D311yescharacteristic
173Dic3⋊C4—181no—
174Dic3⋊C4—181no—
175C4×Dic3—21no—
176C4×Dic3C2×C3⋊D311yescharacteristic
177C2×C3⋊C8—34no—
178C2×C3⋊C8—94no—
179C2×C3⋊C8—91no—
180C2×C3⋊C8—184no—
181C22×C32—24no—
182C22×C32D3×D411yescharacteristic
183C22×C32C2×#6714yes—
184C32×C4—22no—
185C32×C4—24no—
186C32×C4—24no—
187C32×C4—28no—
188C32×C4—32no—
189C32×C4—48no—
190C32×C4—61no—
191C32×C4—121no—
192C32×C4C2×D1214yes—
193C32×C4#23514yes—
194C32×C4#23711yescharacteristic
195C32×C4C2×#6711yescharacteristic
196C3⋊Dic3—41no—
197C3⋊Dic3—128no—
198C3⋊Dic3—124no—
199C3⋊Dic3—121no—
200C3×Dic3—24no—
201C3×Dic3—44no—
202C3×Dic3—124no—
203C3×Dic3#21618yescomplemented
204C2×Q16—91no—
205C2×Q16—271no—
206C4⋊Q8—271no—
207C4×Q8—31no—
208C4⋊C8—271no—
209#103—271no—
210#103—271no—
211C33#46211yescharacteristic, complemented, Sylow
212C3×Q8—24no—
213C3×Q8—21no—
214C3×Q8—28no—
215C3×Q8—44no—
216C3×Q8—181no—
217C3×Q8—181no—
218C3×Q8#62618yes—
219C3×Q8C4⋊(C3⋊D3)12yes—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—34no—
224C2×C3×C4—34no—
225C2×C3×C4—44no—
226C2×C3×C4—64no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—64no—
232C2×Dic3—184no—
233C2×Dic3—181no—
234C2×Dic3—364no—
235C2×Dic3#63811yescharacteristic
236C2×Dic3#63811yescharacteristic
237Dic6—64no—
238Dic6—64no—
239Dic6—188no—
240Dic6—184no—
241Dic6—184no—
242Dic6—181no—
243Dic6—181no—
244Dic6—364no—
245C3×C8—181no—
246C3⋊C8—64no—
247C3⋊C8—188no—
248C3⋊C8—184no—
249C3⋊C8—181no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32D3×Q1611yescharacteristic
254C2×C32#91811yescharacteristic
255C2×C32#93014yes—
256C2×C32#93911yescharacteristic
257C2×C32C2×#21614yes—
258C2×C32#95114yes—
259C2×Q8—91no—
260C2×Q8—271no—
261C2×Q8D3×C3⋊D311yescharacteristic
262Q16—182no—
263Q16—542no—
264C2×C8—91no—
265C2×C8—271no—
266C4⋊C4—31no—
267C4⋊C4—61no—
268C4⋊C4—541no—
269C4⋊C4—541no—
270C42—31no—
271C42—61no—
272C22×C3—24no—
273C22×C3D3×#6714yes—
274C22×C3C2×#63811yescharacteristic
275C3×C4—22no—
276C3×C4—28no—
277C3×C4—24no—
278C3×C4—24no—
279C3×C4—38no—
280C3×C4—48no—
281C3×C4—64no—
282C3×C4—124no—
283C3×C4—361no—
284C3×C4#380414yes—
285C3×C4C2×#62614yes—
286C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
287C3×C4#382811yescharacteristic
288Dic3—21no—
289Dic3—41no—
290Dic3—124no—
291Dic3—368no—
292Dic3—361no—
293Dic3—364no—
294Dic3#375612yescomplemented
295C32—24no—
296C32#503114yescomplemented
297C32#518211yescharacteristic, complemented
298Q8—21no—
299Q8—181no—
300Q8—181no—
301Q8—541no—
302Q8—541no—
303Q8#629212yes—
304C2×C4—21no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—61no—
308C2×C4—181no—
309C2×C4—541no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—181no—
312C8—541no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3D3×#21614yes—
317C2×C3#6394314yes—
318C2×C3#6397814yes—
319C2×C3#6413911yescharacteristic
320C2×C3C2×#375611yescharacteristic
321C2×C3#6416011yescharacteristic
322C22D3×#63811yescharacteristic
323C4—22no—
324C4—32no—
325C4—61no—
326C4—121no—
327C4—361no—
328C4—1081no—
329C4#8903111yescharacteristic
330C4C2×#629211yescharacteristic
331C3—24no—
332C3#1059162514yescomplemented
333C3#1059273011yescharacteristic, complemented
334C2D3×#375611yescharacteristic
335C2#1172357411yescharacteristic
336C2#1172363211yescharacteristic
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
93. C32×Dic3Q16semidirect
203. C3×Dic3#216semidirect
211. C33#462semidirect
294. Dic3#3756semidirect
296. C32#5031semidirect
297. C32#5182semidirect
332. C3#10591625semidirect
333. C3#10592730semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C2×C3×#37562
C32×Q8×Dic32
#117234432
Dic3⋊(C3⋊(C3⋊C8))2
#117220412
#117220392
#105927303
#105916253