Finite Group Databasebeta

← #13161032

Subgroups — #13161032

Order 1728.

1096 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 168 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
1D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C3×#3754C211yesmaximal, characteristic
3C2×C32×#214C211yesmaximal, characteristic
4#11723451C211yesmaximal, characteristic, complemented
5#11722050C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8#10592861—31nomaximal
9#10591497—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)—21no—
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14#88864—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3754—22no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#214—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88806—21no—
22#88806—21no—
23C2×C32×SD16—31no—
24#64153—31no—
25D4×C3⋊Dic3—31no—
26D4×C3⋊Dic3—34no—
27D4×C3⋊Dic3—64no—
28C2×#3754D311yescharacteristic
29C2×C3×#214—34no—
30C2×C3×#214D314yes—
31#63873—34no—
32#63662—31no—
33#63643—31no—
34#63641—31no—
35#63572—34no—
36#63566—34no—
37#63566—34no—
38C23×C33—21no—
39C33×D4—21no—
40C33×D4D412yes—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)—42no—
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C32×C3⋊C8—21no—
54#5162—94no—
55#5162—91no—
56C2×C32×Q8—31no—
57C2×C32×D4—24no—
58C2×C32×D4C2×D314yes—
59C2×C32×D4C2×D311yescharacteristic
60C22×C3⋊Dic3—61no—
61C22×C3⋊Dic3—64no—
62C22×C3⋊Dic3—68no—
63C2×#634C2×D311yescharacteristic
64C2×C3×Dic6—34no—
65C2×C3×Dic6C2×D314yes—
66C32×SD16—62no—
67C2×C32×C8—31no—
68#3757—61no—
69#3757—64no—
70#3757—68no—
71#3754—22no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—34no—
74C4⋊(C3⋊Dic3)—64no—
75#3750—61no—
76#3750—61no—
77C4×C3⋊Dic3—31no—
78C4×C3⋊Dic3—34no—
79C4×C3⋊Dic3—64no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×#214—28no—
82C3×#214—68no—
83C2×C3×C3⋊C8—34no—
84C2×C3×C3⋊C8C2×D314yes—
85#3724—64no—
86#3724—64no—
87C22×C33—22no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33C2×D411yescharacteristic
91C33×C4C2×D411yescharacteristic
92C33×C4#4111yescharacteristic
93C3⋊(C3⋊Dic3)—21no—
94C3⋊(C3⋊Dic3)—41no—
95C3⋊(C3⋊Dic3)SD1612yescomplemented
96C3×C3⋊Dic3—41no—
97C32×Dic3—41no—
98C2×C3×SD16—34no—
99C2×C3×SD16—91no—
100#944—94no—
101#944—91no—
102D4×Dic3—94no—
103D4×Dic3—91no—
104D4×Dic3—184no—
105C2×#214—34no—
106C2×#214C3⋊D311yescharacteristic
107#835—94no—
108#835—91no—
109#816—94no—
110#816—91no—
111Dic3⋊C8—94no—
112Dic3⋊C8—91no—
113C23×C32—28no—
114C23×C32—24no—
115C23×C32—21no—
116C32×Q8—61no—
117C32×Q8—61no—
118C32×D4—28no—
119C32×D4—24no—
120C32×D4—21no—
121C32×D4—28no—
122C32×D4#6718yes—
123C32×D4#6712yes—
124C2×C32×C4—24no—
125C2×C32×C4—61no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—31no—
130C2×C3⋊Dic3—31no—
131C2×C3⋊Dic3—34no—
132C2×C3⋊Dic3—34no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—68no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—61no—
138C2×C3⋊Dic3—122no—
139C2×C3⋊Dic3—128no—
140C2×C3⋊Dic3—1216no—
141#634—21no—
142#634—21no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—64no—
145C3×Dic6—24no—
146C3×Dic6—24no—
147C3×Dic6—64no—
148C3×Dic6—64no—
149C32×C8—61no—
150C3⋊(C3⋊C8)—21no—
151C3×C3⋊C8—24no—
152C3×C3⋊C8—64no—
153#461—271noSylow
154C2×C33—42no—
155C2×C33#12111yescharacteristic
156C2×C33C2×SD1611yescharacteristic
157C2×C33#13711yescharacteristic
158C2×C3×Q8—34no—
159C2×C3×Q8—91no—
160C2×C3×D4—24no—
161C2×C3×D4D3214yes—
162C2×C3×D4C2×C3⋊D311yescharacteristic
163C22×Dic3—184no—
164C22×Dic3—181no—
165C22×Dic3—188no—
166C2×Dic6—34no—
167C2×Dic6C2×C3⋊D311yescharacteristic
168C3×SD16—68no—
169C3×SD16—182no—
170C2×C3×C8—34no—
171C2×C3×C8—91no—
172#217—184no—
173#217—181no—
174#217—188no—
175#214—22no—
176#214—68no—
177C4⋊Dic3—94no—
178C4⋊Dic3—91no—
179C4⋊Dic3—184no—
180Dic3⋊C4—181no—
181Dic3⋊C4—181no—
182Dic3⋊C4—184no—
183Dic3⋊C4—184no—
184C4×Dic3—94no—
185C4×Dic3—91no—
186C4×Dic3—184no—
187C2×C3⋊C8—34no—
188C2×C3⋊C8C2×C3⋊D311yescharacteristic
189C22×C32—216no—
190C22×C32—28no—
191C22×C32—22no—
192C22×C32—24no—
193C22×C32—48no—
194C22×C32—48no—
195C22×C32—44no—
196C22×C32—41no—
197C22×C32—44no—
198C22×C32—41no—
199C22×C32D3×D414yes—
200C22×C32D3×D411yescharacteristic
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—121no—
204C32×C4#23714yes—
205C32×C4#23711yescharacteristic
206C32×C4C2×#6714yes—
207C32×C4C2×#6711yescharacteristic
208C3⋊Dic3—32no—
209C3⋊Dic3—38no—
210C3⋊Dic3—41no—
211C3⋊Dic3—68no—
212C3⋊Dic3—61no—
213C3⋊Dic3—64no—
214C3⋊Dic3—124no—
215C3⋊Dic3—128no—
216C3⋊Dic3—124no—
217C3⋊Dic3—121no—
218C3×Dic3—44no—
219C3×Dic3—124no—
220C2×SD16—31no—
221C2×SD16—91no—
222C4⋊Q8—271no—
223C4×D4—271no—
224C4⋊C8—271no—
225#103—271no—
226#103—271no—
227C33#46111yescharacteristic, complemented, Sylow
228C23×C3—28no—
229C23×C3—24no—
230C23×C3—21no—
231C3×Q8—64no—
232C3×Q8—64no—
233C3×Q8—181no—
234C3×Q8—181no—
235C3×D4—28no—
236C3×D4—24no—
237C3×D4—21no—
238C3×D4—28no—
239C3×D4#62518yes—
240C3×D4#63812yes—
241C2×C3×C4—24no—
242C2×C3×C4—64no—
243C2×C3×C4—181no—
244C2×C3×C4C2×D3214yes—
245C2×C3×C4C22×C3⋊D311yescharacteristic
246C2×Dic3—21no—
247C2×Dic3—64no—
248C2×Dic3—94no—
249C2×Dic3—91no—
250C2×Dic3—94no—
251C2×Dic3—91no—
252C2×Dic3—184no—
253C2×Dic3—184no—
254C2×Dic3—188no—
255C2×Dic3—184no—
256C2×Dic3—181no—
257C2×Dic3—362no—
258C2×Dic3—368no—
259C2×Dic3—3616no—
260Dic6—21no—
261Dic6—21no—
262Dic6—64no—
263Dic6—64no—
264C3×C8—64no—
265C3×C8—181no—
266C3⋊C8—21no—
267C3⋊C8—64no—
268C2×C32—24no—
269C2×C32—24no—
270C2×C32—24no—
271C2×C32—416no—
272C2×C32—48no—
273C2×C32—42no—
274C2×C32D3×SD1614yes—
275C2×C32D3×SD1611yescharacteristic
276C2×C32#91514yes—
277C2×C32#91511yescharacteristic
278C2×C32#93914yes—
279C2×C32#93911yescharacteristic
280C2×Q8—31no—
281C2×Q8—91no—
282C2×D4D3×C3⋊D311yescharacteristic
283C22×C4—541no—
284SD16—62no—
285SD16—182no—
286C2×C8—31no—
287C2×C8—91no—
288C4⋊C4—271no—
289C4⋊C4—541no—
290C4⋊C4—541no—
291C22⋊C4—541no—
292C42—271no—
293C22×C3—216no—
294C22×C3—22no—
295C22×C3—28no—
296C22×C3—24no—
297C22×C3—48no—
298C22×C3—48no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3#381114yes—
304C22×C3D4×C3⋊D311yescharacteristic
305C3×C4—24no—
306C3×C4—24no—
307C3×C4—124no—
308C3×C4—361no—
309C3×C4#380514yes—
310C3×C4C2×#62514yes—
311C3×C4#383011yescharacteristic
312C3×C4C2×#63811yescharacteristic
313Dic3—41no—
314Dic3—92no—
315Dic3—98no—
316Dic3—124no—
317Dic3—188no—
318Dic3—181no—
319Dic3—184no—
320Dic3—364no—
321Dic3—368no—
322Dic3—361no—
323Dic3—364no—
324C32—24no—
325C32#516214yescomplemented
326C32#516211yescharacteristic, complemented
327C23—21no—
328Q8—61no—
329Q8—61no—
330Q8—181no—
331Q8—181no—
332D4—21no—
333D4#629012yes—
334C2×C4—61no—
335C2×C4—181no—
336C2×C4—271no—
337C2×C4—271no—
338C2×C4—541no—
339C2×C4—1082no—
340C2×C4C2×D3×C3⋊D311yescharacteristic
341C8—61no—
342C8—181no—
343C2×C3—24no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—416no—
347C2×C3—48no—
348C2×C3—42no—
349C2×C3#6393014yes—
350C2×C3#6393114yes—
351C2×C3#6398014yes—
352C2×C3SD16×C3⋊D311yescharacteristic
353C2×C3#6412411yescharacteristic
354C2×C3#6414811yescharacteristic
355C22—22no—
356C22—41no—
357C22—41no—
358C22#8903911yescharacteristic
359C4—121no—
360C4—272no—
361C4—361no—
362C4—541no—
363C4—1081no—
364C4#8902911yescharacteristic
365C4C2×#629011yescharacteristic
366C3—24no—
367C3#1059149714yescomplemented
368C3#1059286111yescharacteristic, complemented
369C2—42no—
370C2#1172355311yescharacteristic
371C2#1172355411yescharacteristic
372C2#1172363711yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#461
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #11723451C2semidirect
5. #11722050C2semidirect
6. #11722040C2semidirect
7. #11722038C2semidirect
95. C3⋊(C3⋊Dic3)SD16semidirect
227. C33#461semidirect
325. C32#5162semidirect
326. C32#5162semidirect
367. C3#10591497semidirect
368. C3#10592861semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊(C3⋊Dic3)2
C2×C3×#37542
C2×C32×#2142
#117234512
#117220502
#117220402
#117220382
#105928613
#105914973