Finite Group Databasebeta

← #13159366

Subgroups — #13159366

Order 1728.

1168 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 164 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
1#11723934C211yesmaximal, characteristic, complemented
2#11723499C211yesmaximal, characteristic, complemented
3C2×#88790C211yesmaximal, characteristic
4C3×#63635C211yesmaximal, characteristic, complemented
5C32×#809C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722021C211yesmaximal, characteristic, complemented
8#10592484—31nomaximal
9#10590462—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×C32×D12C2211yescharacteristic
13C2×D3×C3⋊Dic3—21no—
14C33×C4⋊C4C2211yescharacteristic
15#88859—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88790C414yescomplemented
22D4×C3⋊Dic3—31no—
23C2×#3753—31no—
24#64102—31no—
25#64102—34no—
26#64102—64no—
27#63898—34no—
28C2×#3714—34no—
29C32×#102—31no—
30#63648—31no—
31#63642—31no—
32#63635D311yescharacteristic, complemented
33C3×#809D314yescomplemented
34C3×#808—34no—
35#63564—34no—
36#63555—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C22×C32×D3—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
45C4×C3⋊(C3⋊D3)—42no—
46C2×C3×C3⋊Dic3—21no—
47C32×D12—21no—
48C32×D12C2×C412yescomplemented
49D3×C3⋊Dic3—42no—
50C3×C3⋊(C3⋊C8)C2×C412yescomplemented
51C32×C3⋊C8—21no—
52#4869—91no—
53#4785—94no—
54C2×C32×D4—31no—
55C22×C3⋊Dic3—61no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×C4×C3⋊D3—61no—
60C2×C4×C3⋊D3—64no—
61C2×C4×C3⋊D3—68no—
62C2×C3×D12C2×D314yes—
63C2×D3×Dic3—64no—
64C2×C32×C8—31no—
65C32×C4⋊C4—24no—
66C32×C4⋊C4C2×D314yes—
67C32×C4⋊C4C2×D311yescharacteristic
68#3757—61no—
69#3753—34no—
70#3752—61no—
71#3752—64no—
72#3752—68no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C4×C3⋊Dic3—31no—
75C4×C3⋊Dic3—34no—
76C4×C3⋊Dic3—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×C4⋊Dic3—34no—
79C2×C3×C3⋊C8—34no—
80C2×C3×C3⋊C8C2×D314yes—
81(C2×D3)⋊Dic3—64no—
82#3714—316no—
83C22×C33C2×D411yescharacteristic
84C2×C3⋊(C3⋊D3)—22no—
85C2×C3⋊(C3⋊D3)—41no—
86C2×C3⋊(C3⋊D3)—41no—
87C2×C32×D3—22no—
88C2×C32×D3—41no—
89C2×C32×D3—41no—
90C33×C4—41no—
91C33×C4C22×C411yescharacteristic
92C33×C4#4111yescharacteristic
93C3⋊(C3⋊Dic3)—21no—
94C3⋊(C3⋊Dic3)—21no—
95C3×C3⋊Dic3—41no—
96D4×Dic3—94no—
97C2×#213—94no—
98C2×D24—91no—
99#893—91no—
100#893—94no—
101#893—91no—
102#893—184no—
103C3×#106—91no—
104C3×#102—34no—
105#821—94no—
106#815—94no—
107#815—91no—
108#809—91no—
109#809C3⋊D311yescharacteristic, complemented
110#808—34no—
111C23×C32—61no—
112C22×C3⋊D3—68no—
113C22×C3⋊D3—64no—
114C22×C3⋊D3—61no—
115C22×C3×D3—24no—
116C32×D4—32no—
117C32×D4—61no—
118C2×C32×C4—28no—
119C2×C32×C4—24no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—31no—
127C2×C3⋊Dic3—34no—
128C2×C3⋊Dic3—34no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—122no—
132C4⋊(C3⋊D3)—32no—
133C4⋊(C3⋊D3)—38no—
134C4⋊(C3⋊D3)—68no—
135C4⋊(C3⋊D3)—68no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)—61no—
138C4×C3⋊D3—1216no—
139C4×C3⋊D3—128no—
140C4×C3⋊D3—122no—
141C2×C3×Dic3—64no—
142C3×D12—24no—
143C3×D12C4×D318yescomplemented
144D3×Dic3—128no—
145C32×C8—61no—
146C3⋊(C3⋊C8)C4×D312yescomplemented
147C3×C3⋊C8—24no—
148C3×C3⋊C8—38no—
149#442—271noSylow
150C2×C33C4×D411yescharacteristic
151C2×C33AGL(1,ℤ/8)11yescharacteristic
152C2×C33AGL(1,ℤ/8)11yescharacteristic
153C3⋊(C3⋊D3)—42no—
154C32×D3—42no—
155C2×C3×D4—34no—
156C22×Dic3—184no—
157C2×D12—94no—
158C2×D12—91no—
159C2×D12—184no—
160C2×D12C2×C3⋊D311yescharacteristic
161C2×C4×D3—184no—
162C2×C4×D3—181no—
163C2×C4×D3—181no—
164C2×C4×D3—188no—
165C2×C3×C8—34no—
166C2×C3×C8—91no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4D3214yes—
170C3×C4⋊C4C2×C3⋊D311yescharacteristic
171#217—184no—
172#213—916no—
173(C2×D3)⋊C4—184no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—188no—
177C4⋊Dic3—34no—
178C4×Dic3—94no—
179C4×Dic3—91no—
180C4×Dic3—184no—
181C2×C3⋊C8—34no—
182C2×C3⋊C8C2×C3⋊D311yescharacteristic
183D24—94no—
184C22×C32—24no—
185C22×C32—62no—
186C22×C32—121no—
187C22×C32—121no—
188C22×C32D3×D414yes—
189C22×C32D3×D411yescharacteristic
190C2×C3⋊D3—616no—
191C2×C3⋊D3—68no—
192C2×C3⋊D3—62no—
193C2×C3⋊D3—128no—
194C2×C3⋊D3—128no—
195C2×C3⋊D3—124no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—121no—
199C2×C3×D3—28no—
200C2×C3×D3—44no—
201C2×C3×D3—44no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—48no—
205C32×C4—44no—
206C32×C4—41no—
207C32×C4C2×C4×D314yes—
208C32×C4C2×C4×D311yescharacteristic
209C32×C4#23714yes—
210C32×C4#23911yescharacteristic
211C3⋊Dic3—41no—
212C3⋊Dic3—68no—
213C3⋊Dic3—64no—
214C3⋊Dic3—61no—
215C3⋊Dic3—61no—
216C3⋊Dic3—64no—
217C3⋊Dic3—124no—
218C3×Dic3—124no—
219C2×D8—271no—
220C4×D4—271no—
221C4×D4—271no—
222#106—91no—
223#102—31no—
224#102—271no—
225#97—271no—
226C33#44211yescharacteristic, complemented, Sylow
227C23×C3—64no—
228C22×D3—21no—
229C22×D3—188no—
230C22×D3—184no—
231C22×D3—181no—
232C3×D4—38no—
233C3×D4—64no—
234C2×C3×C4—28no—
235C2×C3×C4—24no—
236C2×C3×C4—21no—
237C2×C3×C4—24no—
238C2×C3×C4—181no—
239C2×C3×C4C2×D3214yes—
240C2×C3×C4C22×C3⋊D311yescharacteristic
241C2×Dic3—64no—
242C2×Dic3—94no—
243C2×Dic3—94no—
244C2×Dic3—91no—
245C2×Dic3—91no—
246C2×Dic3—184no—
247C2×Dic3—184no—
248C2×Dic3—368no—
249D12—21no—
250D12—98no—
251D12—92no—
252D12—188no—
253D12—188no—
254D12—184no—
255D12—181no—
256D12C4×C3⋊D312yescomplemented
257C4×D3—3616no—
258C4×D3—362no—
259C4×D3—368no—
260C4×D3—362no—
261C3×C8—64no—
262C3×C8—92no—
263C3⋊C8—21no—
264C3⋊C8—38no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—122no—
269C2×C32#67⋊C414yes—
270C2×C32#89311yescharacteristic
271C2×C32#91114yes—
272C2×C32#91414yes—
273C2×C32#91411yescharacteristic
274C2×C32#91411yescharacteristic
275C3⋊D3—1216no—
276C3⋊D3—122no—
277C3⋊D3—128no—
278C3×D3—48no—
279C2×D4—31no—
280C2×D4—271no—
281C22×C4—541no—
282C22×C4—541no—
283D8—274no—
284C2×C8—31no—
285C2×C8—91no—
286C4⋊C4—91no—
287C4⋊C4D3×C3⋊D311yescharacteristic
288C22⋊C4—541no—
289C22⋊C4—541no—
290C42—271no—
291C22×C3—24no—
292C22×C3—68no—
293C22×C3—124no—
294C22×C3—124no—
295C22×C3#381114yes—
296C22×C3D4×C3⋊D311yescharacteristic
297C2×D3—22no—
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—1816no—
301C2×D3—182no—
302C2×D3—188no—
303C2×D3—368no—
304C2×D3—368no—
305C2×D3—361no—
306C2×D3—361no—
307C2×D3—364no—
308C2×D3—364no—
309C3×C4—24no—
310C3×C4—24no—
311C3×C4—48no—
312C3×C4—41no—
313C3×C4—44no—
314C3×C4—361no—
315C3×C4#379614yes—
316C3×C4C4×D3214yes—
317C3×C4C2×C4×C3⋊D311yescharacteristic
318C3×C4#383011yescharacteristic
319Dic3—124no—
320Dic3—188no—
321Dic3—181no—
322Dic3—181no—
323Dic3—184no—
324Dic3—184no—
325Dic3—364no—
326C32—24no—
327C32#478514yescomplemented
328C32#486911yescharacteristic, complemented
329C23—61no—
330C23—541no—
331D4—32no—
332D4—61no—
333D4—272no—
334D4—541no—
335C2×C4—21no—
336C2×C4—181no—
337C2×C4—271no—
338C2×C4—271no—
339C2×C4—1082no—
340C2×C4—1082no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—61no—
343C8—92no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—128no—
348C2×C3#6388114yes—
349C2×C3#6393714yes—
350C2×C3#6394114yes—
351C2×C3#6409011yescharacteristic
352C2×C3#6412011yescharacteristic
353C2×C3#6412311yescharacteristic
354D3—42no—
355D3—3616no—
356D3—362no—
357D3—368no—
358C22—62no—
359C22—121no—
360C22—121no—
361C22—542no—
362C22—1081no—
363C22—1081no—
364C22#8903911yescharacteristic
365C4—41no—
366C4—361no—
367C4—541no—
368C4—541no—
369C4#8901511yescharacteristic
370C4C4×D3×C3⋊D311yescharacteristic
371C3—24no—
372C3#1059046214yescomplemented
373C3#1059248411yescharacteristic, complemented
374C2—122no—
375C2—1082no—
376C2#1172346211yescharacteristic
377C2#1172356411yescharacteristic
378C2#1172357211yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#88790C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723934C2semidirect
2. #11723499C2semidirect
4. C3×#63635C2semidirect
5. C32×#809C2semidirect
6. #11722036C2semidirect
7. #11722021C2semidirect
21. #88790C4semidirect
32. #63635D3semidirect
33. C3×#809D3semidirect
44. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
48. C32×D12C2×C4semidirect
50. C3×C3⋊(C3⋊C8)C2×C4semidirect
109. #809C3⋊D3semidirect
143. C3×D12C4×D3semidirect
146. C3⋊(C3⋊C8)C4×D3semidirect
226. C33#442semidirect
256. D12C4×C3⋊D3semidirect
327. C32#4785semidirect
328. C32#4869semidirect
372. C3#10590462semidirect
373. C3#10592484semidirect

Maximal subgroups

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

subgroupindex
#117239342
#117234992
C2×#887902
C3×#636352
C32×#8092
#117220362
#117220212
#105924843
#105904623