Finite Group Databasebeta

← #13159958

Subgroups — #13159958

Order 1728.

896 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 146 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×#3745C211yesmaximal, characteristic
2#11723501C211yesmaximal, characteristic, complemented
3#11723436C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722184C211yesmaximal, characteristic, complemented
6C32×#816C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592518—31nomaximal
9#10590751—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C2×C33×C8C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3745—22no—
18C32×C4⋊Dic3C2211yescharacteristic
19#88803—21no—
20#88803—21no—
21#88803—21no—
22#88793—22no—
23#64156—31no—
24C2×#3755—31no—
25C2×#3745D311yescharacteristic
26#64102—31no—
27C2×C3×D24—34no—
28#63898—34no—
29#63863—34no—
30C2×#3715—34no—
31C32×#103—31no—
32#63641—31no—
33#63641—34no—
34#63641—64no—
35#63636—31no—
36C3×#816D314yescomplemented
37#63562—34no—
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)D412yes—
47C2×C32×Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51Dic3×C3⋊D3—42no—
52C33×C8—21no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54#5153—91no—
55#4819—94no—
56C2×C32×Q8—31no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×C4×C3⋊D3—61no—
60C2×C3×D12—34no—
61C2×C3×D12—34no—
62C2×C3×Dic6C2×D314yes—
63C2×D3×Dic3—64no—
64C2×C32×C8—24no—
65C2×C32×C8C2×D314yes—
66C2×C32×C8C2×D311yescharacteristic
67C32×C4⋊C4—31no—
68#3755—62no—
69#3752—61no—
70#3752—61no—
71#3752—61no—
72C4×C3⋊Dic3—31no—
73C4×C3⋊Dic3—34no—
74C4×C3⋊Dic3—64no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78#3745—22no—
79C3×C4⋊Dic3C2×D314yes—
80C3×D24—68no—
81(C2×D3)⋊Dic3—64no—
82(C2×D3)⋊Dic3—64no—
83(C2×D3)⋊Dic3—64no—
84#3715—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—21no—
95C3⋊(C3⋊Dic3)—21no—
96C32×Dic3—41no—
97C32×Dic3—41no—
98C2×C3×D8—91no—
99#947—94no—
100C2×#215—94no—
101#936—91no—
102D4×Dic3—91no—
103C2×#214—91no—
104C2×D24—34no—
105#893—94no—
106C3×#103—34no—
107#832—91no—
108#816C3⋊D311yescharacteristic, complemented
109Dic3⋊C8—94no—
110Dic3⋊C8—91no—
111Dic3⋊C8—184no—
112#809—94no—
113C22×C3⋊D3—21no—
114C22×C3⋊D3—21no—
115C22×C3×D3—64no—
116C22×C3×D3—64no—
117C32×Q8—61no—
118C32×Q8—61no—
119C2×C32×C4—24no—
120C2×C32×C4—61no—
121C2×C32×C4—61no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—34no—
127C2×C3⋊Dic3—34no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—64no—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)#6712yes—
134C4×C3⋊D3—122no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—24no—
137C3×D12—38no—
138C3×D12—64no—
139C3×D12—64no—
140C3×D12—64no—
141C3×Dic6—24no—
142C3×Dic6—24no—
143D3×Dic3—128no—
144C32×C8—21no—
145C32×C8—24no—
146C32×C8—44no—
147C3⋊(C3⋊C8)—61no—
148C3⋊(C3⋊C8)—64no—
149C3⋊(C3⋊C8)—124no—
150#463—271noSylow
151C2×C33#12111yescharacteristic
152C2×C33#13511yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C3×C3⋊D3—42no—
155C3×C3⋊D3—81no—
156C2×C3×Q8—34no—
157C2×C3×D4—91no—
158C2×C3×D4—91no—
159C22×Dic3—181no—
160C2×D12—34no—
161C2×D12—34no—
162C2×C4×D3—184no—
163C2×Dic6C2×C3⋊D311yescharacteristic
164C3×D8—182no—
165C2×C3×C8—24no—
166C2×C3×C8D3214yes—
167C2×C3×C8C2×C3⋊D311yescharacteristic
168C3×C4⋊C4—34no—
169#217—181no—
170#217—181no—
171#217—181no—
172#215—188no—
173#214—182no—
174(C2×D3)⋊C4—184no—
175(C2×D3)⋊C4—184no—
176(C2×D3)⋊C4—184no—
177C4⋊Dic3C2×C3⋊D311yescharacteristic
178C4×Dic3—94no—
179C4×Dic3—91no—
180C4×Dic3—184no—
181C2×C3⋊C8—94no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—184no—
184D24—68no—
185C22×C32—24no—
186C22×C32D3×D414yes—
187C22×C32D3×D411yescharacteristic
188C2×C3⋊D3—22no—
189C2×C3⋊D3—41no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3×D3—68no—
195C2×C3×D3—124no—
196C2×C3×D3—124no—
197C2×C3×D3—124no—
198C2×C3×D3—124no—
199C2×C3×D3—124no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—121no—
203C32×C4—121no—
204C32×C4#23514yes—
205C32×C4D3×D414yes—
206C32×C4#23711yescharacteristic
207C32×C4C2×#6711yescharacteristic
208C3⋊Dic3—68no—
209C3⋊Dic3—61no—
210C3⋊Dic3—61no—
211C3⋊Dic3—64no—
212C3⋊Dic3—64no—
213C3⋊Dic3—124no—
214C3×Dic3—44no—
215C3×Dic3—44no—
216C2×SD16—271no—
217C2×D8—91no—
218#124—271no—
219C4×D4—271no—
220C4⋊C8—271no—
221#103—31no—
222#102—271no—
223C33#46311yescharacteristic, complemented, Sylow
224C23×C3—181no—
225C23×C3—181no—
226C22×D3—64no—
227C22×D3—64no—
228C3×Q8—64no—
229C3×Q8—64no—
230C3×D4—92no—
231C3×D4—181no—
232C3×D4—181no—
233C3×D4—181no—
234C2×C3×C4—24no—
235C2×C3×C4—64no—
236C2×C3×C4—64no—
237C2×C3×C4C2×D3214yes—
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—94no—
242C2×Dic3—94no—
243C2×Dic3—91no—
244C2×Dic3—91no—
245C2×Dic3—184no—
246C2×Dic3—184no—
247C2×Dic3—362no—
248D12—38no—
249D12—64no—
250D12—64no—
251D12—64no—
252C4×D3—368no—
253Dic6—21no—
254Dic6—21no—
255C3×C8—24no—
256C3×C8—21no—
257C3×C8—44no—
258C3⋊C8—184no—
259C3⋊C8—181no—
260C3⋊C8—364no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32#88414yes—
265C2×C32#91111yescharacteristic
266C2×C32#91211yescharacteristic
267C2×C32#91414yes—
268C2×C32#91914yes—
269C2×C32#93911yescharacteristic
270C3⋊D3—42no—
271C3⋊D3—81no—
272C3×D3—128no—
273C3×D3—244no—
274C2×Q8—31no—
275C2×D4—91no—
276C2×D4—91no—
277C22×C4—541no—
278SD16—542no—
279D8—182no—
280C2×C8—271no—
281C2×C8D3×C3⋊D311yescharacteristic
282C4⋊C4—31no—
283C22⋊C4—541no—
284C22⋊C4—541no—
285C22⋊C4—541no—
286C42—271no—
287C22×C3—24no—
288C22×C3—182no—
289C22×C3—361no—
290C22×C3—361no—
291C22×C3—361no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3#380214yes—
295C22×C3D4×C3⋊D311yescharacteristic
296C2×D3—68no—
297C2×D3—124no—
298C2×D3—124no—
299C2×D3—124no—
300C2×D3—124no—
301C2×D3—124no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—124no—
305C3×C4—124no—
306C3×C4#379514yes—
307C3×C4D3×#6714yes—
308C3×C4#382811yescharacteristic
309C3×C4D4×C3⋊D311yescharacteristic
310Dic3—41no—
311Dic3—41no—
312Dic3—188no—
313Dic3—184no—
314Dic3—184no—
315Dic3—181no—
316Dic3—181no—
317Dic3—364no—
318C32—24no—
319C32#481914yescomplemented
320C32#515311yescharacteristic, complemented
321C23—181no—
322C23—181no—
323Q8—61no—
324Q8—61no—
325D4—92no—
326D4—181no—
327D4—181no—
328D4—181no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—271no—
332C2×C4—271no—
333C2×C4—1082no—
334C2×C4C2×D3×C3⋊D311yescharacteristic
335C8—21no—
336C8—541no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—362no—
341C2×C3—721no—
342C2×C3#6379814yes—
343C2×C3#6381014yes—
344C2×C3#6388514yes—
345C2×C3#6409311yescharacteristic
346C2×C3#6412311yescharacteristic
347C2×C3#6412811yescharacteristic
348D3—128no—
349D3—244no—
350C22—182no—
351C22—361no—
352C22—361no—
353C22—361no—
354C22—361no—
355C22—361no—
356C22#8902611yescharacteristic
357C4—121no—
358C4—121no—
359C4—541no—
360C4—541no—
361C4#8902111yescharacteristic
362C4C3⋊D3×#6711yescharacteristic
363C3—24no—
364C3#1059075114yescomplemented
365C3#1059251811yescharacteristic, complemented
366C2—362no—
367C2—721no—
368C2#1172332311yescharacteristic
369C2#1172334511yescharacteristic
370C2#1172351911yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3745C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11723501C2semidirect
3. #11723436C2semidirect
4. C2×#88793C2semidirect
5. #11722184C2semidirect
6. C32×#816C2semidirect
7. #11722033C2semidirect
36. C3×#816D3semidirect
108. #816C3⋊D3semidirect
223. C33#463semidirect
319. C32#4819semidirect
320. C32#5153semidirect
364. C3#10590751semidirect
365. C3#10592518semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37452
#117235012
#117234362
C2×#887932
#117221842
C32×#8162
#117220332
#105925183
#105907513