Finite Group Databasebeta

← #13159311

Subgroups — #13159311

Order 1728.

1096 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1096 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
1#11723934C211yesmaximal, characteristic
2#11723451C211yesmaximal, characteristic, complemented
3C2×#88797C211yesmaximal, characteristic
4C2×#88795C211yesmaximal, characteristic
5C3×#63637C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7#11722050C211yesmaximal, characteristic, complemented
8#10592517—31nomaximal
9#10590565—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15#88859—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20#88806—21no—
21#88797—22no—
22#88795—22no—
23#64153—31no—
24C2×#3755—31no—
25C2×#3744—31no—
26#64102—31no—
27#64102—34no—
28#64102—64no—
29#63873—34no—
30C2×#3717—34no—
31C2×#3717—34no—
32C32×#103—31no—
33#63641—31no—
34#63637D311yescharacteristic, complemented
35C3×#810—34no—
36C3×#810D314yescomplemented
37#63572—34no—
38C22×C3⋊(C3⋊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))D412yes—
45C4×C3⋊(C3⋊D3)—42no—
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#4818—94no—
55#4818—91no—
56C2×C32×Q8—31no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—34no—
59C2×C4⋊(C3⋊D3)—64no—
60C2×C4×C3⋊D3—61no—
61C2×C4×C3⋊D3—64no—
62C2×C4×C3⋊D3—68no—
63C2×#634C2×D311yescharacteristic
64C2×C3×Dic6—34no—
65C2×C3×Dic6C2×D314yes—
66C2×C32×C8—31no—
67C32×C4⋊C4—24no—
68C32×C4⋊C4C2×D314yes—
69C32×C4⋊C4C2×D311yescharacteristic
70#3755—62no—
71#3752—61no—
72#3752—64no—
73#3752—68no—
74#3750—61no—
75#3750—61no—
76C4×C3⋊Dic3—31no—
77C4×C3⋊Dic3—34no—
78C4×C3⋊Dic3—64no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80#3744—62no—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8C2×D314yes—
83#3724—64no—
84#3724—64no—
85#3717—68no—
86#3717—68no—
87C22×C33C2×D411yescharacteristic
88C2×C3⋊(C3⋊D3)—22no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3⋊(C3⋊D3)—41no—
91C33×C4—41no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—21no—
95C3⋊(C3⋊Dic3)SD1612yescomplemented
96C3×C3⋊Dic3—41no—
97C32×Dic3—41no—
98#944—94no—
99#944—91no—
100C2×#215—94no—
101C2×#215—91no—
102C2×#204—94no—
103C2×#204—91no—
104#893—94no—
105#893—91no—
106#893—184no—
107C3×#103—34no—
108C3×#103—91no—
109Dic3⋊C8—94no—
110Dic3⋊C8—91no—
111#810—34no—
112#810C3⋊D311yescharacteristic, complemented
113C22×C3⋊D3—68no—
114C22×C3⋊D3—64no—
115C22×C3⋊D3—61no—
116C32×Q8—61no—
117C32×Q8—61no—
118C2×C32×C4—28no—
119C2×C32×C4—24no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—31no—
127C2×C3⋊Dic3—31no—
128C2×C3⋊Dic3—34no—
129C2×C3⋊Dic3—34no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—64no—
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—122no—
139C4×C3⋊D3—128no—
140C4×C3⋊D3—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#12111yescharacteristic
155C2×C33C2×SD1611yescharacteristic
156C2×C33#13711yescharacteristic
157C3⋊(C3⋊D3)—42no—
158C2×C3×Q8—34no—
159C2×C3×Q8—91no—
160C2×D12—94no—
161C2×D12—91no—
162C2×D12—184no—
163C2×C4×D3—184no—
164C2×C4×D3—181no—
165C2×C4×D3—188no—
166C2×Dic6—34no—
167C2×Dic6C2×C3⋊D311yescharacteristic
168C2×C3×C8—34no—
169C2×C3×C8—91no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4D3214yes—
172C3×C4⋊C4C2×C3⋊D311yescharacteristic
173#215—182no—
174#215—188no—
175(C2×D3)⋊C4—184no—
176(C2×D3)⋊C4—181no—
177(C2×D3)⋊C4—188no—
178Dic3⋊C4—181no—
179Dic3⋊C4—181no—
180Dic3⋊C4—184no—
181Dic3⋊C4—184no—
182C4×Dic3—94no—
183C4×Dic3—91no—
184C4×Dic3—184no—
185C2×C3⋊C8—34no—
186C2×C3⋊C8C2×C3⋊D311yescharacteristic
187#204—188no—
188#204—182no—
189C22×C32—24no—
190C22×C32D3×D414yes—
191C22×C32D3×D411yescharacteristic
192C2×C3⋊D3—616no—
193C2×C3⋊D3—68no—
194C2×C3⋊D3—62no—
195C2×C3⋊D3—128no—
196C2×C3⋊D3—128no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—121no—
199C2×C3⋊D3—124no—
200C2×C3⋊D3—121no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—48no—
204C32×C4—44no—
205C32×C4—41no—
206C32×C4—121no—
207C32×C4#23514yes—
208C32×C4#23511yescharacteristic
209C32×C4D3×D414yes—
210C32×C4D3×D411yescharacteristic
211C3⋊Dic3—32no—
212C3⋊Dic3—38no—
213C3⋊Dic3—41no—
214C3⋊Dic3—68no—
215C3⋊Dic3—61no—
216C3⋊Dic3—64no—
217C3⋊Dic3—124no—
218C3×Dic3—44no—
219C3×Dic3—124no—
220C2×SD16—271no—
221C2×SD16—271no—
222C4⋊Q8—271no—
223C4×D4—271no—
224C4⋊C8—271no—
225#103—31no—
226#103—91no—
227C33#46111yescharacteristic, complemented, Sylow
228C22×D3—188no—
229C22×D3—184no—
230C22×D3—181no—
231C3×Q8—64no—
232C3×Q8—64no—
233C3×Q8—181no—
234C3×Q8—181no—
235C2×C3×C4—28no—
236C2×C3×C4—24no—
237C2×C3×C4—21no—
238C2×C3×C4—24no—
239C2×C3×C4—64no—
240C2×C3×C4—181no—
241C2×C3×C4C2×D3214yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243C2×Dic3—21no—
244C2×Dic3—64no—
245C2×Dic3—94no—
246C2×Dic3—91no—
247C2×Dic3—94no—
248C2×Dic3—91no—
249C2×Dic3—184no—
250C2×Dic3—184no—
251D12—98no—
252D12—92no—
253D12—188no—
254D12—188no—
255D12—184no—
256D12—181no—
257C4×D3—362no—
258C4×D3—368no—
259C4×D3—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#88414yes—
272C2×C32#88411yescharacteristic
273C2×C32D3×SD1614yes—
274C2×C32D3×SD1611yescharacteristic
275C2×C32#91814yes—
276C2×C32#91811yescharacteristic
277C3⋊D3—1216no—
278C3⋊D3—128no—
279C3⋊D3—122no—
280C2×Q8—31no—
281C2×Q8—91no—
282C2×D4—271no—
283C22×C4—541no—
284SD16—542no—
285SD16—542no—
286C2×C8—31no—
287C2×C8—91no—
288C4⋊C4—541no—
289C4⋊C4—541no—
290C4⋊C4D3×C3⋊D311yescharacteristic
291C22⋊C4—541no—
292C42—271no—
293C22×C3—24no—
294C22×C3#381114yes—
295C22×C3D4×C3⋊D311yescharacteristic
296C2×D3—1816no—
297C2×D3—182no—
298C2×D3—188no—
299C2×D3—368no—
300C2×D3—368no—
301C2×D3—361no—
302C2×D3—361no—
303C2×D3—364no—
304C2×D3—364no—
305C3×C4—24no—
306C3×C4—24no—
307C3×C4—48no—
308C3×C4—41no—
309C3×C4—44no—
310C3×C4—124no—
311C3×C4—361no—
312C3×C4#379814yes—
313C3×C4#380214yes—
314C3×C4#382811yescharacteristic
315C3×C4D4×C3⋊D311yescharacteristic
316Dic3—41no—
317Dic3—92no—
318Dic3—98no—
319Dic3—124no—
320Dic3—188no—
321Dic3—181no—
322Dic3—184no—
323Dic3—364no—
324C32—24no—
325C32#481814yescomplemented
326C32#481811yescharacteristic, complemented
327C23—541no—
328Q8—61no—
329Q8—61no—
330Q8—181no—
331Q8—181no—
332D4—272no—
333D4—541no—
334C2×C4—21no—
335C2×C4—61no—
336C2×C4—181no—
337C2×C4—271no—
338C2×C4—271no—
339C2×C4—1082no—
340C2×C4C2×D3×C3⋊D311yescharacteristic
341C8—61no—
342C8—181no—
343C2×C3—24no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3#6391214yes—
347C2×C3#6393014yes—
348C2×C3#6394514yes—
349C2×C3#6409311yescharacteristic
350C2×C3SD16×C3⋊D311yescharacteristic
351C2×C3#6412711yescharacteristic
352D3—3616no—
353D3—362no—
354D3—368no—
355C22—542no—
356C22—1081no—
357C22—1081no—
358C22#8903911yescharacteristic
359C4—41no—
360C4—121no—
361C4—272no—
362C4—361no—
363C4—541no—
364C4#8902011yescharacteristic
365C4#8902611yescharacteristic
366C3—24no—
367C3#1059056514yescomplemented
368C3#1059251711yescharacteristic, complemented
369C2—1082no—
370C2#1172352711yescharacteristic
371C2#1172355311yescharacteristic
372C2#1172357811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#461
LCSG ▷ C2×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
2. #11723451C2semidirect
5. C3×#63637C2semidirect
6. C32×#810C2semidirect
7. #11722050C2semidirect
34. #63637D3semidirect
36. C3×#810D3semidirect
95. C3⋊(C3⋊Dic3)SD16semidirect
112. #810C3⋊D3semidirect
227. C33#461semidirect
325. C32#4818semidirect
326. C32#4818semidirect
367. C3#10590565semidirect
368. C3#10592517semidirect

Maximal subgroups

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

subgroupindex
#117239342
#117234512
C2×#887972
C2×#887952
C3×#636372
C32×#8102
#117220502
#105925173
#105905653