Finite Group Databasebeta

← #13161222

Subgroups — #13161222

Order 1728.

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

normal subgroups only

This group has 912 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×#88863C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic
3C32×Q8×Dic3C211yesmaximal, characteristic, complemented
4#11723439C211yesmaximal, characteristic
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#11722041C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic
8#10592727—31nomaximal
9#10591604—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13#88863—22no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3755—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3—21no—
20C32×C4×Dic3C2211yescharacteristic
21#88803—21no—
22#88803—21no—
23C32×C4×Q8—31no—
24C2×#3756—31no—
25C2×#3756—34no—
26C2×#3756—64no—
27C2×#3755D311yescharacteristic
28#64084—31no—
29C3×Q8×Dic3D314yescomplemented
30C2×C3×#215—34no—
31#63860—34no—
32#63637—31no—
33#63636—31no—
34C4⋊(C3⋊(C3⋊C8))—31no—
35Dic3⋊(C3⋊C8)—34no—
36#63567—34no—
37#63562—34no—
38C22×C3×C3⋊D3—21no—
39C33×Q8—21no—
40C33×Q8D412yes—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C2×C32×Dic3—21no—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)—21no—
53#5166—91no—
54#5028—94no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×D12—34no—
63C32×C4⋊C4—31no—
64C32×C4⋊C4—61no—
65C32×C42—31no—
66C32×C42—61no—
67#3756—616no—
68#3756—68no—
69#3756—62no—
70#3755—22no—
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×#215—68no—
78C3×C4⋊Dic3—24no—
79C3×C4⋊Dic3C2×D314yes—
80C3×C4×Dic3—24no—
81C3×C4×Dic3C2×D314yes—
82C2×C3×C3⋊C8—34no—
83(C2×D3)⋊Dic3—64no—
84(C2×D3)⋊Dic3—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×SD16—91no—
97C3×C4×Q8—34no—
98Q8×Dic3C3⋊D311yescharacteristic, complemented
99C2×#216—94no—
100C2×#216—91no—
101C2×#216—184no—
102C2×#215—34no—
103#936—91no—
104#875—94no—
105#832—91no—
106#816—91no—
107Dic3⋊C8—91no—
108#810—94no—
109#809—94no—
110C4⋊(C3⋊C8)—94no—
111C22×C3⋊D3—21no—
112C22×C3×D3—64no—
113C32×Q8—21no—
114C32×Q8—28no—
115C32×Q8—24no—
116C32×Q8—44no—
117C32×Q8D1218yes—
118C32×Q8#6712yes—
119C2×C32×C4—21no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—31no—
123C2×C32×C4—31no—
124C2×C32×C4—44no—
125C2×C32×C4—61no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—61no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3—124no—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)—21no—
133#634—68no—
134#634—64no—
135#634—61no—
136#634—61no—
137#634—64no—
138#634—124no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3#6714yes—
141C2×C3×Dic3#6714yes—
142C3×D12—64no—
143C3×D12—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×Q8D3214yes—
156C2×C3×Q8C2×C3⋊D311yescharacteristic
157C2×C3×D4—91no—
158C2×D12—34no—
159C2×Dic6—94no—
160C2×Dic6—91no—
161C2×Dic6—184no—
162C3×SD16—182no—
163C2×C3×C8—91no—
164C3×C4⋊C4—34no—
165C3×C4⋊C4—64no—
166C3×C42—34no—
167C3×C42—64no—
168#217—181no—
169#217—181no—
170#216—1816no—
171#216—188no—
172#216—182no—
173#215—68no—
174(C2×D3)⋊C4—184no—
175(C2×D3)⋊C4—184no—
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—41no—
188C2×C3⋊D3—41no—
189C2×C3⋊D3—41no—
190C2×C3×D3—124no—
191C2×C3×D3—124no—
192C2×C3×D3—124no—
193C32×C4—22no—
194C32×C4—24no—
195C32×C4—24no—
196C32×C4—28no—
197C32×C4—48no—
198C32×C4—61no—
199C32×C4—61no—
200C32×C4—121no—
201C32×C4C2×D1214yes—
202C32×C4#23514yes—
203C32×C4#23711yescharacteristic
204C32×C4C2×#6711yescharacteristic
205C3⋊Dic3—128no—
206C3⋊Dic3—124no—
207C3⋊Dic3—121no—
208C3×Dic3—24no—
209C3×Dic3—24no—
210C3×Dic3—44no—
211C2×Q16—271no—
212C2×SD16—91no—
213#124—271no—
214C4×Q8—31no—
215C4⋊C8—271no—
216#103—271no—
217#102—271no—
218C33#46411yescharacteristic, complemented, Sylow
219C23×C3—181no—
220C22×D3—64no—
221C3×Q8—24no—
222C3×Q8—21no—
223C3×Q8—28no—
224C3×Q8—44no—
225C3×Q8#62618yes—
226C3×Q8C4⋊(C3⋊D3)12yes—
227C3×D4—181no—
228C3×D4—181no—
229C2×C3×C4—24no—
230C2×C3×C4—21no—
231C2×C3×C4—24no—
232C2×C3×C4—34no—
233C2×C3×C4—34no—
234C2×C3×C4—44no—
235C2×C3×C4—64no—
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238C2×Dic3—21no—
239C2×Dic3—184no—
240C2×Dic3—181no—
241C2×Dic3—364no—
242C2×Dic3#63811yescharacteristic
243C2×Dic3#63811yescharacteristic
244D12—64no—
245D12—64no—
246Dic6—188no—
247Dic6—184no—
248Dic6—184no—
249Dic6—181no—
250Dic6—181no—
251Dic6—364no—
252C3×C8—181no—
253C3⋊C8—64no—
254C3⋊C8—188no—
255C3⋊C8—184no—
256C3⋊C8—181no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32—24no—
260C2×C32#91511yescharacteristic
261C2×C32#91611yescharacteristic
262C2×C32#93014yes—
263C2×C32#93911yescharacteristic
264C2×C32#94214yes—
265C2×C32#95014yes—
266C3⋊D3—81no—
267C3×D3—244no—
268C2×Q8—271no—
269C2×Q8D3×C3⋊D311yescharacteristic
270C2×D4—91no—
271Q16—542no—
272SD16—182no—
273C2×C8—91no—
274C2×C8—271no—
275C4⋊C4—31no—
276C4⋊C4—61no—
277C22⋊C4—541no—
278C22⋊C4—541no—
279C42—31no—
280C42—61no—
281C22×C3—24no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3—361no—
285C22×C3D3×#6714yes—
286C22×C3C2×#63811yescharacteristic
287C2×D3—124no—
288C2×D3—124no—
289C2×D3—124no—
290C3×C4—22no—
291C3×C4—28no—
292C3×C4—24no—
293C3×C4—24no—
294C3×C4—48no—
295C3×C4—64no—
296C3×C4—64no—
297C3×C4—124no—
298C3×C4#380414yes—
299C3×C4C2×#62614yes—
300C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
301C3×C4#382811yescharacteristic
302Dic3—21no—
303Dic3—21no—
304Dic3—41no—
305Dic3—368no—
306Dic3—361no—
307Dic3—364no—
308C32—24no—
309C32#502814yescomplemented
310C32#516611yescharacteristic, complemented
311C23—181no—
312Q8—21no—
313Q8—541no—
314Q8—541no—
315Q8#629212yes—
316D4—181no—
317D4—181no—
318C2×C4—21no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—61no—
322C2×C4—541no—
323C2×C4C2×D3×C3⋊D311yescharacteristic
324C8—181no—
325C8—541no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—721no—
330C2×C3#6394614yes—
331C2×C3#6394714yes—
332C2×C3#6397814yes—
333C2×C3#6413911yescharacteristic
334C2×C3#6415111yescharacteristic
335C2×C3#6415911yescharacteristic
336D3—244no—
337C22—361no—
338C22—361no—
339C22—361no—
340C22D3×#63811yescharacteristic
341C4—22no—
342C4—61no—
343C4—61no—
344C4—121no—
345C4—1081no—
346C4#8903111yescharacteristic
347C4C2×#629211yescharacteristic
348C3—24no—
349C3#1059160414yescomplemented
350C3#1059272711yescharacteristic, complemented
351C2—721no—
352C2#1172357911yescharacteristic
353C2#1172358011yescharacteristic
354C2#1172363211yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(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
1. C2×#88863C2semidirect
3. C32×Q8×Dic3C2semidirect
5. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
6. #11722041C2semidirect
29. C3×Q8×Dic3D3semidirect
98. Q8×Dic3C3⋊D3semidirect
218. C33#464semidirect
309. C32#5028semidirect
310. C32#5166semidirect
349. C3#10591604semidirect
350. C3#10592727semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C2×C3×#37552
C32×Q8×Dic32
#117234392
Dic3⋊(C3⋊(C3⋊C8))2
#117220412
#117220332
#105927273
#105916043