Finite Group Databasebeta

← #13159427

Subgroups — #13159427

Order 1728.

1340 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 178 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×#89019C211yesmaximal, characteristic
2C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
3C2×#88785C211yesmaximal, characteristic
4C3×#63637C211yesmaximal, characteristic, complemented
5C32×#810C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8C3⋊D3×#103—31nomaximal
9#10590601—34nomaximal
10C2×C4×C3⋊(C3⋊D3)—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6293—21no—
15C2×#6289—21no—
16#89019—22no—
17#89019C414yescomplemented
18C33×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20#88857—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88785—22no—
24C2×Q8×C3⋊D3—31no—
25C2×#3797—34no—
26C2×C8×C3⋊D3—31no—
27C4⋊C4×C3⋊D3—31no—
28C4⋊C4×C3⋊D3—34no—
29C4⋊C4×C3⋊D3—64no—
30C2×#3710—34no—
31C32×#103—31no—
32#63662—31no—
33#63643—31no—
34#63637D311yescharacteristic, complemented
35C3×#810—34no—
36C3×#810D314yescomplemented
37#63566—34no—
38#63566—34no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C4×C3⋊(C3⋊D3)—42no—
45C4×C3⋊(C3⋊D3)C2×C412yes—
46C4×C3⋊(C3⋊D3)D412yes—
47C2×C3×C3⋊Dic3—21no—
48C3×#634—21no—
49C3×#634C2×C412yescomplemented
50C2×C32×Dic3—21no—
51C32×Dic6—21no—
52C32×Dic6C2×C412yescomplemented
53#6293—24no—
54#6289—24no—
55C3×C3⋊(C3⋊C8)—21no—
56C32×C3⋊C8—21no—
57D3×#103—94no—
58D3×#103—91no—
59C2×C32×Q8—31no—
60Q8×C3⋊D3—34no—
61Q8×C3⋊D3—62no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—34no—
64C2×C4×C3⋊D3—61no—
65C2×C4×C3⋊D3—64no—
66C2×C4×C3⋊D3—61no—
67C2×C4×C3⋊D3—64no—
68C2×C4×C3⋊D3—68no—
69C2×#634—61no—
70C2×#634C2×D311yescharacteristic
71C2×C3×Dic6—34no—
72C2×C3×Dic6C2×D314yes—
73C2×#627—64no—
74C2×#624—64no—
75#3797—316no—
76#3797—68no—
77C2×C32×C8—31no—
78C32×C4⋊C4—24no—
79C32×C4⋊C4C2×D314yes—
80C32×C4⋊C4C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—34no—
83C4⋊(C3⋊Dic3)—64no—
84#3750—61no—
85#3750—64no—
86#3750—68no—
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C8×C3⋊D3—62no—
89C2×C3×C3⋊C8—34no—
90C2×C3×C3⋊C8C2×D314yes—
91#3710—68no—
92C22×C33C2×D411yescharacteristic
93C2×C3⋊(C3⋊D3)C22⋊C412yes—
94C2×C3⋊(C3⋊D3)SD1612yes—
95C2×C3⋊(C3⋊D3)Q1612yes—
96C33×C4—41no—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3⋊(C3⋊Dic3)—41no—
100C3⋊(C3⋊Dic3)C22⋊C412yes—
101C3×C3⋊Dic3—22no—
102C32×Dic3—22no—
103C2×D3×Q8—94no—
104C2×D3×Q8—91no—
105C2×D3×C8—94no—
106C2×D3×C8—91no—
107D3×C4⋊C4—94no—
108D3×C4⋊C4—91no—
109D3×C4⋊C4—184no—
110C3×#103—34no—
111C3×#103—91no—
112#835—94no—
113#835—91no—
114#816—94no—
115#816—91no—
116#810—34no—
117#810C3⋊D311yescharacteristic, complemented
118C22×C3⋊D3—31no—
119C22×C3⋊D3—34no—
120C22×C3⋊D3—64no—
121C32×Q8—32no—
122C32×Q8—61no—
123C2×C32×C4—28no—
124C2×C32×C4—24no—
125C2×C32×C4—21no—
126C2×C32×C4—24no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—31no—
132C2×C3⋊Dic3—34no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—68no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—61no—
137C4×C3⋊D3—32no—
138C4×C3⋊D3—32no—
139C4×C3⋊D3—38no—
140C4×C3⋊D3—38no—
141C4×C3⋊D3—68no—
142C4×C3⋊D3—68no—
143C4×C3⋊D3—64no—
144C4×C3⋊D3—122no—
145C4×C3⋊D3—128no—
146C4×C3⋊D3—1216no—
147#634—21no—
148#634—64no—
149#634C4×D312yescomplemented
150C2×C3×Dic3—24no—
151C2×C3×Dic3—64no—
152C3×Dic6—24no—
153C3×Dic6—38no—
154C3×Dic6—64no—
155C3×Dic6C4×D318yescomplemented
156#627—616no—
157#624—616no—
158C32×C8—61no—
159C3⋊(C3⋊C8)—21no—
160C3×C3⋊C8—24no—
161C3×C3⋊C8—64no—
162C2×#103—271noSylow
163C2×C33C2×C22⋊C411yescharacteristic
164C2×C33C2×SD1611yescharacteristic
165C2×C33C2×Q1611yescharacteristic
166C3⋊(C3⋊D3)#10314yescomplemented
167C2×C3×Q8—34no—
168C2×C3×Q8—91no—
169D3×Q8—916no—
170D3×Q8—94no—
171D3×Q8—182no—
172D3×Q8—188no—
173C2×C4×D3—94no—
174C2×C4×D3—91no—
175C2×C4×D3—184no—
176C2×C4×D3—181no—
177C2×C4×D3—184no—
178C2×C4×D3—184no—
179C2×C4×D3—181no—
180C2×C4×D3—188no—
181C2×Dic6—34no—
182C2×Dic6—181no—
183C2×Dic6—184no—
184C2×Dic6C2×C3⋊D311yescharacteristic
185C2×C3×C8—34no—
186C2×C3×C8—91no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4D3214yes—
189C3×C4⋊C4C2×C3⋊D311yescharacteristic
190C4⋊Dic3—94no—
191C4⋊Dic3—91no—
192C4⋊Dic3—184no—
193Dic3⋊C4—184no—
194Dic3⋊C4—181no—
195Dic3⋊C4—188no—
196C2×C3⋊C8—34no—
197C2×C3⋊C8C2×C3⋊D311yescharacteristic
198D3×C8—182no—
199D3×C8—188no—
200C22×C32—24no—
201C22×C32D3×D414yes—
202C22×C32D3×D411yescharacteristic
203C2×C3⋊D3—32no—
204C2×C3⋊D3—32no—
205C2×C3⋊D3—32no—
206C2×C3⋊D3—38no—
207C2×C3⋊D3—38no—
208C2×C3⋊D3—38no—
209C2×C3⋊D3—68no—
210C2×C3⋊D3—68no—
211C2×C3⋊D3—68no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—48no—
215C32×C4—44no—
216C32×C4—41no—
217C32×C4—62no—
218C32×C4C2×C4×D314yes—
219C32×C4C2×C4×D311yescharacteristic
220C32×C4D3×D414yes—
221C32×C4D3×D411yescharacteristic
222C3⋊Dic3—22no—
223C3⋊Dic3—32no—
224C3⋊Dic3—38no—
225C3⋊Dic3—68no—
226C3⋊Dic3—128no—
227C3⋊Dic3—124no—
228C3⋊Dic3—121no—
229C3×Dic3—28no—
230C3×Dic3—68no—
231C22×Q8—271no—
232C22×C8—271no—
233C2×C4⋊C4—271no—
234#103—31no—
235#103—91no—
236#103—271no—
237#103—271no—
238C33C2×#10311yescharacteristic, complemented, Sylow
239C22×D3—94no—
240C22×D3—91no—
241C22×D3—184no—
242C3×Q8—38no—
243C3×Q8—64no—
244C3×Q8—92no—
245C3×Q8—181no—
246C2×C3×C4—28no—
247C2×C3×C4—24no—
248C2×C3×C4—21no—
249C2×C3×C4—24no—
250C2×C3×C4—64no—
251C2×C3×C4—181no—
252C2×C3×C4C2×D3214yes—
253C2×C3×C4C22×C3⋊D311yescharacteristic
254C2×Dic3—21no—
255C2×Dic3—64no—
256C2×Dic3—94no—
257C2×Dic3—91no—
258C2×Dic3—184no—
259C2×Dic3—188no—
260C2×Dic3—184no—
261C2×Dic3—181no—
262C4×D3—98no—
263C4×D3—98no—
264C4×D3—92no—
265C4×D3—92no—
266C4×D3—188no—
267C4×D3—188no—
268C4×D3—184no—
269C4×D3—1816no—
270C4×D3—362no—
271C4×D3—368no—
272C4×D3—3616no—
273Dic6—21no—
274Dic6—38no—
275Dic6—64no—
276Dic6—184no—
277Dic6—1816no—
278Dic6C4×C3⋊D312yescomplemented
279C3×C8—64no—
280C3×C8—181no—
281C3⋊C8—21no—
282C3⋊C8—64no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—24no—
286C2×C32D3×C22⋊C414yes—
287C2×C32D3×C22⋊C411yescharacteristic
288C2×C32D3×SD1614yes—
289C2×C32D3×SD1611yescharacteristic
290C2×C32D3×Q1614yes—
291C2×C32D3×Q1611yescharacteristic
292C3⋊D3—34no—
293C3⋊D3—316no—
294C3⋊D3—616no—
295C2×Q8—31no—
296C2×Q8—91no—
297C2×Q8—274no—
298C2×Q8—541no—
299C2×Q8—542no—
300C22×C4—271no—
301C22×C4—541no—
302C22×C4—541no—
303C2×C8—31no—
304C2×C8—91no—
305C2×C8—542no—
306C4⋊C4—271no—
307C4⋊C4—541no—
308C4⋊C4D3×C3⋊D311yescharacteristic
309C22×C3—24no—
310C22×C3#381114yes—
311C22×C3D4×C3⋊D311yescharacteristic
312C2×D3—92no—
313C2×D3—92no—
314C2×D3—92no—
315C2×D3—98no—
316C2×D3—98no—
317C2×D3—98no—
318C2×D3—188no—
319C2×D3—188no—
320C2×D3—188no—
321C3×C4—24no—
322C3×C4—24no—
323C3×C4—48no—
324C3×C4—41no—
325C3×C4—44no—
326C3×C4—68no—
327C3×C4—182no—
328C3×C4C4×D3214yes—
329C3×C4#380214yes—
330C3×C4C2×C4×C3⋊D311yescharacteristic
331C3×C4D4×C3⋊D311yescharacteristic
332Dic3—22no—
333Dic3—68no—
334Dic3—92no—
335Dic3—98no—
336Dic3—188no—
337Dic3—368no—
338Dic3—361no—
339Dic3—364no—
340C32—24no—
341C32D3×#10314yescomplemented
342C32D3×#10311yescharacteristic, complemented
343C23—271no—
344Q8—32no—
345Q8—61no—
346Q8—92no—
347Q8—181no—
348Q8—544no—
349C2×C4—21no—
350C2×C4—61no—
351C2×C4—181no—
352C2×C4—271no—
353C2×C4—272no—
354C2×C4—272no—
355C2×C4—544no—
356C2×C4—541no—
357C2×C4—1082no—
358C2×C4C2×D3×C3⋊D311yescharacteristic
359C8—61no—
360C8—181no—
361C2×C3—24no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3#6392114yes—
365C2×C3#6393014yes—
366C2×C3#6394414yes—
367C2×C3C22⋊C4×C3⋊D311yescharacteristic
368C2×C3SD16×C3⋊D311yescharacteristic
369C2×C3Q16×C3⋊D311yescharacteristic
370D3—916no—
371D3—94no—
372D3—1816no—
373C22—272no—
374C22—272no—
375C22—272no—
376C22#8903911yescharacteristic
377C4—41no—
378C4—62no—
379C4—182no—
380C4—272no—
381C4—1081no—
382C4C4×D3×C3⋊D311yescharacteristic
383C4#8902611yescharacteristic
384C3—24no—
385C3#1059060114yescomplemented
386C3C3⋊D3×#10311yescharacteristic, complemented
387C2—274no—
388C2#1172353911yescharacteristic
389C2#1172355311yescharacteristic
390C2#1172357711yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C2×#103
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
4. C3×#63637C2semidirect
5. C32×#810C2semidirect
6. #11722040C2semidirect
7. #11722038C2semidirect
17. #89019C4semidirect
34. #63637D3semidirect
36. C3×#810D3semidirect
49. C3×#634C2×C4semidirect
52. C32×Dic6C2×C4semidirect
117. #810C3⋊D3semidirect
149. #634C4×D3semidirect
155. C3×Dic6C4×D3semidirect
166. C3⋊(C3⋊D3)#103semidirect
238. C33C2×#103semidirect
278. Dic6C4×C3⋊D3semidirect
341. C32D3×#103semidirect
342. C32D3×#103semidirect
385. C3#10590601semidirect
386. C3C3⋊D3×#103semidirect

Maximal subgroups

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

subgroupindex
C2×#890192
C4⋊C4×C3⋊(C3⋊D3)2
C2×#887852
C3×#636372
C32×#8102
#117220402
#117220382
C3⋊D3×#1033
#105906013