Finite Group Databasebeta

← #13159657

Subgroups — #13159657

Order 1728.

1040 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1040 conjugacy classes of subgroups — too many to draw. Only its 160 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×C32×D24C211yesmaximal, characteristic, complemented
2#11723524C211yesmaximal, characteristic, complemented
3D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C3×#63641C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592849—31nomaximal
9#10590753—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6291—21no—
14C2×#6291—21no—
15C2×D3×C3⋊Dic3—21no—
16C2×C33×C8C2211yescharacteristic
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×D24—22no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88790—22no—
23C2×C32×D8—31no—
24#64149—31no—
25D4×C3⋊Dic3—31no—
26C2×#3753—31no—
27C2×C3×D24D314yescomplemented
28#63911—34no—
29Dic3×D12—34no—
30C2×#3714—34no—
31#63659—31no—
32#63648—31no—
33#63648—34no—
34#63648—64no—
35#63641D311yescharacteristic, complemented
36C3×Dic3⋊C8—34no—
37#63563—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C32×D3—21no—
40C22×C32×D3—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C32×D12—21no—
48C32×D12—21no—
49C32×D12—21no—
50C32×D12D412yescomplemented
51#6291—42no—
52#6291—42no—
53D3×C3⋊Dic3—42no—
54C33×C8—21no—
55C3×C3⋊(C3⋊C8)—21no—
56#5150—94no—
57#4843—91no—
58C2×C32×D4—31no—
59C2×C32×D4—31no—
60C2×#638—61no—
61C2×#638—61no—
62C22×C3⋊Dic3—61no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—34no—
65C2×C4⋊(C3⋊D3)—64no—
66C2×C3×D12C2×D314yes—
67C2×C3×D12C2×D314yes—
68C2×#626—64no—
69C2×#626—64no—
70C2×D3×Dic3—64no—
71C32×D8—62no—
72C2×C32×C8—24no—
73C2×C32×C8C2×D314yes—
74C2×C32×C8C2×D311yescharacteristic
75#3757—61no—
76#3753—62no—
77C4⋊(C3⋊Dic3)—31no—
78C4⋊(C3⋊Dic3)—34no—
79C4⋊(C3⋊Dic3)—64no—
80C4×C3⋊Dic3C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×C4×Dic3—34no—
83C2×C3×C3⋊C8—34no—
84C3×D24—28no—
85(C2×D3)⋊Dic3—64no—
86#3714—68no—
87C22×C33C2×D411yescharacteristic
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3⋊(C3⋊D3)—41no—
91C2×C32×D3—22no—
92C2×C32×D3—41no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3⋊(C3⋊Dic3)—41no—
100C3×C3⋊Dic3—21no—
101C3×C3⋊Dic3D812yescomplemented
102C2×C3×D8—34no—
103#940—94no—
104D4×Dic3—94no—
105C2×#213—94no—
106C2×D24—91no—
107C2×D24C3⋊D311yescharacteristic, complemented
108#874—91no—
109C4×D12—91no—
110C3×C4⋊C8—91no—
111#832—94no—
112#821—91no—
113#821—94no—
114#821—91no—
115#821—184no—
116Dic3⋊C8—34no—
117C23×C32—61no—
118C23×C32—61no—
119C22×C3⋊D3—61no—
120C22×C3⋊D3—64no—
121C22×C3⋊D3—124no—
122C22×C3×D3—24no—
123C22×C3×D3—24no—
124C32×D4—32no—
125C32×D4—61no—
126C32×D4—61no—
127C32×D4—61no—
128C2×C32×C4—24no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131#638—122no—
132#638—122no—
133C2×C3⋊Dic3—61no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—124no—
136C2×C3⋊Dic3—122no—
137C2×C3⋊Dic3D1211yescharacteristic
138C2×C3⋊Dic3D1211yescharacteristic
139C4⋊(C3⋊D3)—68no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—61no—
143C4⋊(C3⋊D3)—64no—
144C4⋊(C3⋊D3)—124no—
145C2×C3×Dic3—34no—
146C2×C3×Dic3—34no—
147C3×D12—24no—
148C3×D12—24no—
149C3×D12—24no—
150C3×D12#6718yescomplemented
151#626—128no—
152#626—128no—
153D3×Dic3—128no—
154C32×C8—21no—
155C32×C8—24no—
156C32×C8—44no—
157C3⋊(C3⋊C8)—21no—
158C3×C3⋊C8—64no—
159#459—271noSylow
160C2×C33#12111yescharacteristic
161C2×C33C2×D811yescharacteristic
162C2×C33AGL(1,ℤ/8)11yescharacteristic
163C3⋊(C3⋊D3)—81no—
164C32×D3—42no—
165C32×D3—81no—
166C2×C3×D4—34no—
167C2×C3×D4—34no—
168C2×#67—184no—
169C2×#67—184no—
170C22×Dic3—184no—
171C2×D12—94no—
172C2×D12—91no—
173C2×D12—184no—
174C2×D12—181no—
175C2×D12—181no—
176C2×D12C2×C3⋊D311yescharacteristic
177C2×D12C2×C3⋊D311yescharacteristic
178C2×C4×D3—181no—
179C3×D8—68no—
180C2×C3×C8—24no—
181C2×C3×C8—91no—
182C2×C3×C8D3214yes—
183C2×C3×C8C2×C3⋊D311yescharacteristic
184C3×C42—91no—
185#217—184no—
186#213—188no—
187(C2×D3)⋊C4—181no—
188C4⋊Dic3—94no—
189C4⋊Dic3—91no—
190C4⋊Dic3—184no—
191C4×Dic3—34no—
192C2×C3⋊C8—34no—
193D24—22no—
194D24—182no—
195C22×C32—24no—
196C22×C32—62no—
197C22×C32—121no—
198C22×C32—121no—
199C22×C32—121no—
200C22×C32—121no—
201C22×C32—121no—
202C22×C32C2×D1211yescharacteristic
203C22×C32D3×D414yes—
204C2×C3⋊D3—128no—
205C2×C3⋊D3—121no—
206C2×C3⋊D3—121no—
207C2×C3⋊D3—121no—
208C2×C3⋊D3—124no—
209C2×C3⋊D3—124no—
210C2×C3⋊D3—124no—
211C2×C3⋊D3—244no—
212C2×C3⋊D3—244no—
213C2×C3×D3—28no—
214C2×C3×D3—44no—
215C2×C3×D3—44no—
216C2×C3×D3—44no—
217C2×C3×D3—44no—
218C2×C3×D3—44no—
219C32×C4—24no—
220C32×C4—24no—
221C32×C4D3×D411yescharacteristic
222C32×C4#23714yes—
223C32×C4#23911yescharacteristic
224C32×C4C2×#6714yes—
225C3⋊Dic3—21no—
226C3⋊Dic3—121no—
227C3⋊Dic3—124no—
228C3⋊Dic3—244no—
229C3⋊Dic3D2412yescomplemented
230C3×Dic3—38no—
231C3×Dic3—64no—
232C2×D8—31no—
233C2×D8—271no—
234#127—271no—
235C4×D4—271no—
236C4⋊C8—91no—
237#102—271no—
238#102—271no—
239C33#45911yescharacteristic, complemented, Sylow
240C23×C3—64no—
241C23×C3—64no—
242C22×D3—21no—
243C22×D3—21no—
244C22×D3—184no—
245C22×D3—181no—
246C22×D3—364no—
247C3×D4—38no—
248C3×D4—64no—
249C3×D4—64no—
250C3×D4—64no—
251C2×C3×C4—24no—
252C2×C3×C4—91no—
253C2×C3×C4—91no—
254C2×C3×C4C2×D3214yes—
255C2×C3×C4C22×C3⋊D311yescharacteristic
256#67—368no—
257#67—368no—
258C2×Dic3—34no—
259C2×Dic3—34no—
260C2×Dic3—184no—
261C2×Dic3—181no—
262C2×Dic3—364no—
263C2×Dic3—368no—
264D12—21no—
265D12—21no—
266D12—21no—
267D12—188no—
268D12—184no—
269D12—184no—
270D12—181no—
271D12—181no—
272D12—364no—
273D12—362no—
274D12—362no—
275D12#63812yescomplemented
276C4×D3—362no—
277C3×C8—24no—
278C3×C8—21no—
279C3×C8—44no—
280C3×C8—181no—
281C3⋊C8—64no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—122no—
286C2×C32—241no—
287C2×C32#89511yescharacteristic
288C2×C32C2×D2411yescharacteristic
289C2×C32#90811yescharacteristic
290C2×C32D3×D814yes—
291C2×C32#91114yes—
292C2×C32#93914yes—
293C3⋊D3—248no—
294C3⋊D3—244no—
295C3⋊D3—241no—
296C3×D3—48no—
297C3×D3—84no—
298C2×D4—31no—
299C2×D4—31no—
300C2×D4—271no—
301C2×D4—541no—
302C2×D4—541no—
303C22×C4—541no—
304D8—62no—
305D8—542no—
306C2×C8—91no—
307C2×C8D3×C3⋊D311yescharacteristic
308C4⋊C4—271no—
309C22⋊C4—541no—
310C42—91no—
311C22×C3—24no—
312C22×C3—68no—
313C22×C3—124no—
314C22×C3—124no—
315C22×C3—124no—
316C22×C3—124no—
317C22×C3—124no—
318C22×C3D3×D1214yes—
319C22×C3D4×C3⋊D311yescharacteristic
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—368no—
327C2×D3—364no—
328C2×D3—364no—
329C2×D3—361no—
330C2×D3—361no—
331C2×D3—364no—
332C2×D3—361no—
333C2×D3—724no—
334C2×D3—724no—
335C3×C4—24no—
336C3×C4—24no—
337C3×C4—92no—
338C3×C4—181no—
339C3×C4#379614yes—
340C3×C4D3×#6714yes—
341C3×C4#383011yescharacteristic
342C3×C4C2×#63811yescharacteristic
343Dic3—38no—
344Dic3—64no—
345Dic3—364no—
346Dic3—361no—
347Dic3—724no—
348C32—24no—
349C32#484311yescharacteristic, complemented
350C32#515014yescomplemented
351C23—61no—
352C23—61no—
353C23—541no—
354D4—32no—
355D4—61no—
356D4—61no—
357D4—61no—
358D4—541no—
359D4—541no—
360D4—1082no—
361D4—1082no—
362C2×C4—91no—
363C2×C4—91no—
364C2×C4—541no—
365C2×C4—1082no—
366C2×C4C2×D3×C3⋊D311yescharacteristic
367C8—21no—
368C8—181no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3—128no—
373C2×C3—244no—
374C2×C3D3×D2414yes—
375C2×C3#6379514yes—
376C2×C3#6388614yes—
377C2×C3D8×C3⋊D311yescharacteristic
378C2×C3#6412011yescharacteristic
379C2×C3#6414811yescharacteristic
380D3—42no—
381D3—81no—
382D3—728no—
383D3—724no—
384D3—721no—
385C22—62no—
386C22—121no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—1081no—
392C22—1081no—
393C22—1081no—
394C22C3⋊D3×D1211yescharacteristic
395C4—92no—
396C4—181no—
397C4—1081no—
398C4#8901511yescharacteristic
399C4D3×#63811yescharacteristic
400C3—24no—
401C3#1059075314yescomplemented
402C3#1059284911yescharacteristic, complemented
403C2—122no—
404C2—241no—
405C2—2161no—
406C2C3⋊D3×D2411yescharacteristic
407C2#1172332011yescharacteristic
408C2#1172347511yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#459
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
1. C2×C32×D24C2semidirect
2. #11723524C2semidirect
3. D12×C3⋊Dic3C2semidirect
4. C2×#88790C2semidirect
5. #11722191C2semidirect
6. C3×#63641C2semidirect
7. #11722032C2semidirect
27. C2×C3×D24D3semidirect
35. #63641D3semidirect
50. C32×D12D4semidirect
101. C3×C3⋊Dic3D8semidirect
107. C2×D24C3⋊D3semidirect
150. C3×D12#67semidirect
229. C3⋊Dic3D24semidirect
239. C33#459semidirect
275. D12#638semidirect
349. C32#4843semidirect
350. C32#5150semidirect
401. C3#10590753semidirect
402. C3#10592849semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D242
#117235242
D12×C3⋊Dic32
C2×#887902
#117221912
C3×#636412
#117220322
#105928493
#105907533