Finite Group Databasebeta

← #13161223

Subgroups — #13161223

Order 1728.

944 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 176 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×#88862C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C32×Q8×Dic3C211yesmaximal, characteristic, complemented
4#11723441C211yesmaximal, characteristic
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic
8#10592727—31nomaximal
9#10591607—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13#88862—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
21#88804—21no—
22#88804—21no—
23C32×C4×Q8—31no—
24C2×#3756D311yescharacteristic, complemented
25C2×#3755—31no—
26C2×#3755—34no—
27C2×#3755—64no—
28#64084—31no—
29C3×Q8×Dic3D314yescomplemented
30C2×C3×#216—34no—
31#63864—34no—
32#63637—31no—
33#63636—31no—
34C4⋊(C3⋊(C3⋊C8))—31no—
35Dic3⋊(C3⋊C8)—34no—
36#63565—34no—
37#63564—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C33×Q8—21no—
40C33×Q8D412yes—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—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#5185—91no—
54#5028—94no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—34no—
60C2×C4⋊(C3⋊D3)—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#3755—616no—
69#3755—68no—
70#3755—62no—
71#3752—61no—
72#3752—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—
83#3722—64no—
84#3722—64no—
85C22×C33C2×D411yescharacteristic
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C33×C4—22no—
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3×C3⋊Dic3—41no—
93C32×Dic3—21no—
94C32×Dic3—21no—
95C32×Dic3—41no—
96C2×C3×Q16—91no—
97C3×C4×Q8—34no—
98#947—91no—
99Q8×Dic3C3⋊D311yescharacteristic, complemented
100C2×#216—34no—
101C2×#215—94no—
102C2×#215—91no—
103C2×#215—184no—
104#875—94no—
105#835—91no—
106#821—91no—
107Dic3⋊C8—91no—
108#810—94no—
109#809—94no—
110C4⋊(C3⋊C8)—94no—
111C22×C3⋊D3—61no—
112C22×C3⋊D3—64no—
113C22×C3⋊D3—124no—
114C32×Q8—21no—
115C32×Q8—28no—
116C32×Q8—24no—
117C32×Q8—44no—
118C32×Q8D1218yes—
119C32×Q8#6712yes—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—31no—
124C2×C32×C4—31no—
125C2×C32×C4—44no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130C4⋊(C3⋊D3)—68no—
131C4⋊(C3⋊D3)—64no—
132C4⋊(C3⋊D3)—61no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—124no—
136#634—21no—
137#634—21no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3#6714yes—
141C2×C3×Dic3#6714yes—
142C3×Dic6—64no—
143C3×Dic6—64no—
144C3⋊(C3⋊C8)—21no—
145C3⋊(C3⋊C8)—68no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—61no—
148C3×C3⋊C8—64no—
149#464—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33#13511yescharacteristic
152C2×C33#13711yescharacteristic
153C3⋊(C3⋊D3)—81no—
154C2×C3×Q8—24no—
155C2×C3×Q8—91no—
156C2×C3×Q8D3214yes—
157C2×C3×Q8C2×C3⋊D311yescharacteristic
158C2×D12—94no—
159C2×D12—91no—
160C2×D12—184no—
161C2×Dic6—34no—
162C3×Q16—182no—
163C2×C3×C8—91no—
164C3×C4⋊C4—34no—
165C3×C4⋊C4—64no—
166C3×C42—34no—
167C3×C42—64no—
168#216—68no—
169#215—1816no—
170#215—188no—
171#215—182no—
172(C2×D3)⋊C4—184no—
173(C2×D3)⋊C4—184no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—181no—
176C4⋊Dic3—21no—
177C4⋊Dic3C2×C3⋊D311yescharacteristic
178C4×Dic3—21no—
179C4×Dic3C2×C3⋊D311yescharacteristic
180C2×C3⋊C8—34no—
181C2×C3⋊C8—94no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—184no—
184C22×C32—24no—
185C22×C32D3×D411yescharacteristic
186C22×C32C2×#6714yes—
187C2×C3⋊D3—128no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—121no—
190C2×C3⋊D3—124no—
191C2×C3⋊D3—121no—
192C2×C3⋊D3—124no—
193C2×C3⋊D3—124no—
194C2×C3⋊D3—244no—
195C2×C3⋊D3—244no—
196C32×C4—22no—
197C32×C4—24no—
198C32×C4—24no—
199C32×C4—28no—
200C32×C4—48no—
201C32×C4—61no—
202C32×C4—61no—
203C32×C4—121no—
204C32×C4C2×D1214yes—
205C32×C4#23514yes—
206C32×C4#23711yescharacteristic
207C32×C4C2×#6711yescharacteristic
208C3⋊Dic3—41no—
209C3×Dic3—24no—
210C3×Dic3—24no—
211C3×Dic3—44no—
212C3×Dic3—124no—
213C2×Q16—91no—
214C2×SD16—271no—
215#124—271no—
216C4×Q8—31no—
217C4⋊C8—271no—
218#103—271no—
219#102—271no—
220C33#46411yescharacteristic, complemented, Sylow
221C22×D3—184no—
222C22×D3—181no—
223C22×D3—364no—
224C3×Q8—24no—
225C3×Q8—21no—
226C3×Q8—28no—
227C3×Q8—44no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×Q8#62618yes—
231C3×Q8C4⋊(C3⋊D3)12yes—
232C2×C3×C4—24no—
233C2×C3×C4—21no—
234C2×C3×C4—24no—
235C2×C3×C4—34no—
236C2×C3×C4—34no—
237C2×C3×C4—44no—
238C2×C3×C4—64no—
239C2×C3×C4—181no—
240C2×C3×C4C2×D3214yes—
241C2×C3×C4C22×C3⋊D311yescharacteristic
242C2×Dic3—21no—
243C2×Dic3—64no—
244C2×Dic3#63811yescharacteristic
245C2×Dic3#63811yescharacteristic
246D12—188no—
247D12—184no—
248D12—184no—
249D12—181no—
250D12—181no—
251D12—364no—
252Dic6—64no—
253Dic6—64no—
254C3×C8—181no—
255C3⋊C8—64no—
256C3⋊C8—188no—
257C3⋊C8—184no—
258C3⋊C8—181no—
259C2×C32—24no—
260C2×C32—24no—
261C2×C32—24no—
262C2×C32#91811yescharacteristic
263C2×C32#91911yescharacteristic
264C2×C32#93014yes—
265C2×C32#93911yescharacteristic
266C2×C32#94214yes—
267C2×C32#95014yes—
268C3⋊D3—248no—
269C3⋊D3—241no—
270C3⋊D3—244no—
271C2×Q8—91no—
272C2×Q8D3×C3⋊D311yescharacteristic
273C2×D4—271no—
274Q16—182no—
275SD16—542no—
276C2×C8—91no—
277C2×C8—271no—
278C4⋊C4—31no—
279C4⋊C4—61no—
280C22⋊C4—541no—
281C22⋊C4—541no—
282C42—31no—
283C42—61no—
284C22×C3—24no—
285C22×C3D3×#6714yes—
286C22×C3C2×#63811yescharacteristic
287C2×D3—368no—
288C2×D3—361no—
289C2×D3—361no—
290C2×D3—361no—
291C2×D3—364no—
292C2×D3—364no—
293C2×D3—364no—
294C2×D3—724no—
295C2×D3—724no—
296C3×C4—22no—
297C3×C4—28no—
298C3×C4—24no—
299C3×C4—24no—
300C3×C4—48no—
301C3×C4—64no—
302C3×C4—64no—
303C3×C4—124no—
304C3×C4—361no—
305C3×C4#380414yes—
306C3×C4C2×#62614yes—
307C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
308C3×C4#382811yescharacteristic
309Dic3—21no—
310Dic3—21no—
311Dic3—41no—
312Dic3—124no—
313C32—24no—
314C32#502814yescomplemented
315C32#518511yescharacteristic, complemented
316C23—541no—
317Q8—21no—
318Q8—181no—
319Q8—181no—
320Q8#629212yes—
321D4—541no—
322D4—541no—
323C2×C4—21no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—61no—
327C2×C4—181no—
328C2×C4C2×D3×C3⋊D311yescharacteristic
329C8—181no—
330C8—541no—
331C2×C3—24no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3#6394814yes—
335C2×C3#6394914yes—
336C2×C3#6397814yes—
337C2×C3#6413911yescharacteristic
338C2×C3#6415111yescharacteristic
339C2×C3#6415911yescharacteristic
340D3—728no—
341D3—724no—
342D3—721no—
343C22—1081no—
344C22—1081no—
345C22—1081no—
346C22D3×#63811yescharacteristic
347C4—22no—
348C4—61no—
349C4—61no—
350C4—121no—
351C4—361no—
352C4#8903111yescharacteristic
353C4C2×#629211yescharacteristic
354C3—24no—
355C3#1059160714yescomplemented
356C3#1059272711yescharacteristic, complemented
357C2—2161no—
358C2#1172358111yescharacteristic
359C2#1172358211yescharacteristic
360C2#1172363211yescharacteristic
subgroupG/H
centerC22D3×#638
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#464
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. C32×Q8×Dic3C2semidirect
5. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
6. #11722039C2semidirect
24. C2×#3756D3semidirect
29. C3×Q8×Dic3D3semidirect
99. Q8×Dic3C3⋊D3semidirect
220. C33#464semidirect
314. C32#5028semidirect
315. C32#5185semidirect
355. C3#10591607semidirect
356. C3#10592727semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C2×C3×#37562
C32×Q8×Dic32
#117234412
Dic3⋊(C3⋊(C3⋊C8))2
#117220392
#117220352
#105927273
#105916073