Finite Group Databasebeta

← #13161247

Subgroups — #13161247

Order 1728.

888 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 172 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
2C3×Q8×C3⋊Dic3C211yesmaximal, characteristic
3C2×C32×#216C211yesmaximal, characteristic
4#11723455C211yesmaximal, characteristic
5#11722051C211yesmaximal, characteristic
6#11722042C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10592881—31nomaximal
9#10591625—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13#88863—22no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×#216—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88807—21no—
22#88807—21no—
23C2×C32×Q16—31no—
24Q8×C3⋊Dic3D311yescharacteristic
25#64153—31no—
26C2×#3756—31no—
27C2×#3756—34no—
28C2×#3756—64no—
29C3×Q8×Dic3—34no—
30C2×C3×#216D314yes—
31#63866—34no—
32#63662—31no—
33#63643—31no—
34#63641—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—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C2×C3×C3⋊Dic3D411yescharacteristic
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C32×C3⋊C8—21no—
53#5182—94no—
54#5031—91no—
55C2×C32×Q8—24no—
56C2×C32×Q8—31no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C2×D314yes—
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×Dic6C2×D314yes—
63C32×Q16—62no—
64C2×C32×C8—31no—
65#3756—616no—
66#3756—62no—
67#3756—68no—
68C4⋊(C3⋊Dic3)—21no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70#3750—61no—
71#3750—61no—
72C4×C3⋊Dic3—21no—
73C4×C3⋊Dic3C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77C3×#216—28no—
78C3×C4⋊Dic3—34no—
79C3×C4⋊Dic3—64no—
80C3×C4×Dic3—34no—
81C3×C4×Dic3—64no—
82C2×C3×C3⋊C8C2×D314yes—
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—21no—
91C3×C3⋊Dic3—41no—
92C3×C3⋊Dic3Q1612yescomplemented
93C32×Dic3—41no—
94C2×C3×Q16—34no—
95C3×C4×Q8—91no—
96Q8×Dic3—34no—
97#944—94no—
98C2×#216—94no—
99C2×#216—91no—
100C2×#216—184no—
101C2×#216C3⋊D311yescharacteristic
102#869—91no—
103#835—94no—
104#816—94no—
105Dic3⋊C8—94no—
106#810—91no—
107#810—91no—
108C4⋊(C3⋊C8)—91no—
109C32×Q8—21no—
110C32×Q8—28no—
111C32×Q8—24no—
112C32×Q8—44no—
113C32×Q8—61no—
114C32×Q8—61no—
115C32×Q8D1212yes—
116C32×Q8#6718yes—
117C2×C32×C4—21no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
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—
128C2×C3⋊Dic3#6711yescharacteristic
129C2×C3⋊Dic3#6711yescharacteristic
130#634—68no—
131#634—61no—
132#634—61no—
133#634—64no—
134#634—64no—
135#634—124no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—64no—
140C3×Dic6—24no—
141C3×Dic6—24no—
142C32×C8—61no—
143C3⋊(C3⋊C8)—68no—
144C3⋊(C3⋊C8)—61no—
145C3⋊(C3⋊C8)—64no—
146C3×C3⋊C8—24no—
147#462—271noSylow
148C2×C33#12111yescharacteristic
149C2×C33C2×Q1611yescharacteristic
150C2×C33#13711yescharacteristic
151C2×C3×Q8—24no—
152C2×C3×Q8—34no—
153C2×C3×Q8D3214yes—
154C2×C3×Q8C2×C3⋊D311yescharacteristic
155C2×Dic6—94no—
156C2×Dic6—91no—
157C2×Dic6—184no—
158C2×Dic6C2×C3⋊D311yescharacteristic
159C3×Q16—68no—
160C2×C3×C8—34no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—181no—
163C3×C42—91no—
164C3×C42—181no—
165#216—22no—
166#216—1816no—
167#216—188no—
168#216—182no—
169C4⋊Dic3—34no—
170C4⋊Dic3—64no—
171C4⋊Dic3—181no—
172C4⋊Dic3—181no—
173Dic3⋊C4—184no—
174Dic3⋊C4—184no—
175C4×Dic3—34no—
176C4×Dic3—64no—
177C2×C3⋊C8—94no—
178C2×C3⋊C8—91no—
179C2×C3⋊C8—184no—
180C2×C3⋊C8C2×C3⋊D311yescharacteristic
181C22×C32—24no—
182C22×C32D3×D414yes—
183C22×C32C2×#6711yescharacteristic
184C32×C4—22no—
185C32×C4—24no—
186C32×C4—24no—
187C32×C4—28no—
188C32×C4—48no—
189C32×C4—121no—
190C32×C4C2×D1211yescharacteristic
191C32×C4#23511yescharacteristic
192C32×C4#23714yes—
193C32×C4C2×#6714yes—
194C3⋊Dic3—21no—
195C3⋊Dic3—41no—
196C3⋊Dic3—128no—
197C3⋊Dic3—121no—
198C3⋊Dic3—124no—
199C3⋊Dic3#21612yescomplemented
200C3×Dic3—38no—
201C3×Dic3—44no—
202C3×Dic3—64no—
203C3×Dic3—124no—
204C2×Q16—31no—
205C2×Q16—271no—
206C4⋊Q8—271no—
207C4×Q8—91no—
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—64no—
217C3×Q8—64no—
218C3×Q8#62618yes—
219C3×Q8#63812yes—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—44no—
224C2×C3×C4—64no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—34no—
232C2×Dic3—34no—
233C2×Dic3—64no—
234C2×Dic3—184no—
235C2×Dic3—181no—
236C2×Dic3—364no—
237Dic6—21no—
238Dic6—21no—
239Dic6—188no—
240Dic6—184no—
241Dic6—184no—
242Dic6—181no—
243Dic6—181no—
244Dic6—364no—
245C3×C8—64no—
246C3⋊C8—21no—
247C3⋊C8—188no—
248C3⋊C8—184no—
249C3⋊C8—181no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32D3×Q1614yes—
254C2×C32#91814yes—
255C2×C32#93011yescharacteristic
256C2×C32#93914yes—
257C2×C32C2×#21611yescharacteristic
258C2×C32#95111yescharacteristic
259C2×Q8—31no—
260C2×Q8—271no—
261C2×Q8D3×C3⋊D311yescharacteristic
262Q16—62no—
263Q16—542no—
264C2×C8—31no—
265C2×C8—271no—
266C4⋊C4—91no—
267C4⋊C4—181no—
268C4⋊C4—541no—
269C4⋊C4—541no—
270C42—91no—
271C42—181no—
272C22×C3—24no—
273C22×C3D3×#6714yes—
274C22×C3D4×C3⋊D311yescharacteristic
275C3×C4—22no—
276C3×C4—28no—
277C3×C4—24no—
278C3×C4—24no—
279C3×C4—48no—
280C3×C4—92no—
281C3×C4—124no—
282C3×C4—181no—
283C3×C4—361no—
284C3×C4#380414yes—
285C3×C4C2×#62614yes—
286C3×C4#383011yescharacteristic
287C3×C4C2×#63811yescharacteristic
288Dic3—38no—
289Dic3—41no—
290Dic3—64no—
291Dic3—124no—
292Dic3—368no—
293Dic3—361no—
294Dic3—364no—
295C32—24no—
296C32#503111yescharacteristic, complemented
297C32#518214yescomplemented
298Q8—21no—
299Q8—61no—
300Q8—61no—
301Q8—541no—
302Q8—541no—
303Q8#629112yes—
304C2×C4—21no—
305C2×C4—61no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—181no—
309C2×C4—541no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—61no—
312C8—541no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3D3×#21614yes—
317C2×C3#6394314yes—
318C2×C3#6397814yes—
319C2×C3Q16×C3⋊D311yescharacteristic
320C2×C3#6412711yescharacteristic
321C2×C3#6414811yescharacteristic
322C22C3⋊D3×#6711yescharacteristic
323C4—22no—
324C4—92no—
325C4—121no—
326C4—181no—
327C4—361no—
328C4—1081no—
329C4#8902811yescharacteristic
330C4C2×#629111yescharacteristic
331C3—24no—
332C3#1059162514yescomplemented
333C3#1059288111yescharacteristic, complemented
334C2C3⋊D3×#21611yescharacteristic
335C2#1172357611yescharacteristic
336C2#1172363511yescharacteristic
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
92. C3×C3⋊Dic3Q16semidirect
199. C3⋊Dic3#216semidirect
211. C33#462semidirect
296. C32#5031semidirect
297. C32#5182semidirect
332. C3#10591625semidirect
333. C3#10592881semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C3×Q8×C3⋊Dic32
C2×C32×#2162
#117234552
#117220512
#117220422
#117220372
#105928813
#105916253