Finite Group Databasebeta

← #13161302

Subgroups — #13161302

Order 1728.

1124 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 162 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#11723985C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3C2×C32×#215C211yesmaximal, characteristic, complemented
4#11723529C211yesmaximal, characteristic
5C2×#88796C211yesmaximal, characteristic, complemented
6C2×#88794C211yesmaximal, characteristic, complemented
7#11722018C211yesmaximal, characteristic, complemented
8#10592874—31nomaximal
9#10591620—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—22no—
15#88857—22no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3755—22no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#215—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88796—22no—
22#88794—22no—
23C2×C32×SD16—31no—
24#64153—31no—
25#64153—34no—
26#64153—64no—
27C2×#3755D311yescharacteristic, complemented
28#64149—31no—
29C2×#3754—31no—
30C2×#3744—31no—
31C2×C3×#215—34no—
32C2×C3×#215D314yescomplemented
33#63914—34no—
34C2×#3716—34no—
35C2×#3716—34no—
36C8×C3⋊Dic3—31no—
37#63553—34no—
38C22×C3×C3⋊D3—21no—
39C22×C32×D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)D412yes—
46#6308—21no—
47#6308—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C32×D12—21no—
51C32×D12—21no—
52#6290—44no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54C32×C3⋊C8D412yescomplemented
55#5175—94no—
56#5175—91no—
57C2×C32×Q8—24no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×D4—31no—
61C2×#638—62no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×#634—31no—
64C2×#634—34no—
65C2×#634—64no—
66C2×C3×D12—34no—
67C2×C3×D12C2×D314yes—
68C2×#625—68no—
69C32×SD16—62no—
70C2×C32×C8—31no—
71#3755—22no—
72#3754—62no—
73#3750—62no—
74#3750—68no—
75#3750—616no—
76C4×C3⋊Dic3—31no—
77C4×C3⋊Dic3—34no—
78C4×C3⋊Dic3—64no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80#3744—62no—
81C3×#215—28no—
82C3×#215—68no—
83C2×C3×C3⋊C8—34no—
84C2×C3×C3⋊C8C2×D314yes—
85#3716—68no—
86#3716—68no—
87C22×C33C2×D411yescharacteristic
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3—42no—
90C2×C32×D3—41no—
91C2×C32×D3—42no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4C2×D411yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C3⋊(C3⋊Dic3)SD1614yescomplemented
97C2×C3×SD16—34no—
98C2×C3×SD16—91no—
99#944—94no—
100#944—91no—
101#944—184no—
102C2×#215—34no—
103C2×#215C3⋊D311yescharacteristic, complemented
104#940—94no—
105#940—91no—
106C2×#214—94no—
107C2×#214—91no—
108C2×#204—94no—
109C2×#204—91no—
110C8×Dic3—94no—
111C8×Dic3—91no—
112C23×C32—61no—
113C22×C3⋊D3—21no—
114C22×C3×D3—24no—
115C22×C3×D3—64no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—24no—
119C32×Q8—24no—
120C32×Q8—44no—
121C32×Q8—44no—
122C32×D4—61no—
123C32×D4—61no—
124C2×C32×C4—28no—
125C2×C32×C4—24no—
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130#638—124no—
131C2×C3⋊Dic3—32no—
132C2×C3⋊Dic3—38no—
133C2×C3⋊Dic3—68no—
134C2×C3⋊Dic3—68no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—61no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139#634—61no—
140#634—61no—
141#634—64no—
142#634—64no—
143#634—124no—
144#634—124no—
145C3×D12—24no—
146C3×D12—24no—
147C3×D12—64no—
148C3×D12—64no—
149#625—1216no—
150C32×C8—32no—
151C3⋊(C3⋊C8)#6712yescomplemented
152C3×C3⋊C8—38no—
153C3×C3⋊C8#6718yescomplemented
154#492—271noSylow
155C2×C33#12711yescharacteristic
156C2×C33C2×SD1612yes—
157C3×C3⋊D3—81no—
158C32×D3—81no—
159C2×C3×Q8—24no—
160C2×C3×Q8D3214yes—
161C2×C3×Q8C2×C3⋊D311yescharacteristic
162C2×C3×D4—34no—
163C2×C3×D4—91no—
164C2×#67—182no—
165C2×#67—188no—
166C2×D12—34no—
167C2×D12C2×C3⋊D311yescharacteristic
168C2×Dic6—94no—
169C2×Dic6—91no—
170C2×Dic6—184no—
171C3×SD16—68no—
172C3×SD16—182no—
173C2×C3×C8—34no—
174C2×C3×C8—91no—
175#215—22no—
176#215—68no—
177#214—188no—
178#214—182no—
179Dic3⋊C4—188no—
180Dic3⋊C4—182no—
181Dic3⋊C4—1816no—
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×C32—122no—
191C22×C32—121no—
192C22×C32D3×D414yes—
193C22×C32D3×D411yescharacteristic
194C2×C3⋊D3—42no—
195C2×C3⋊D3—41no—
196C2×C3×D3—48no—
197C2×C3×D3—44no—
198C2×C3×D3—128no—
199C2×C3×D3—124no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—48no—
203C32×C4—44no—
204C32×C4—41no—
205C32×C4D3×D414yes—
206C32×C4D3×D411yescharacteristic
207C32×C4C2×#6714yes—
208C32×C4C2×#6711yescharacteristic
209C3⋊Dic3—34no—
210C3⋊Dic3—316no—
211C3⋊Dic3—616no—
212C3⋊Dic3—128no—
213C3⋊Dic3—124no—
214C3⋊Dic3—121no—
215C2×SD16—31no—
216C2×SD16—91no—
217C2×SD16—271no—
218C2×SD16—271no—
219C4⋊Q8—271no—
220#127—271no—
221C4×C8—271no—
222C33#49211yescharacteristic, complemented, Sylow
223C23×C3—64no—
224C23×C3—181no—
225C22×D3—21no—
226C22×D3—64no—
227C3×Q8—24no—
228C3×Q8—24no—
229C3×Q8—21no—
230C3×Q8—21no—
231C3×Q8—44no—
232C3×Q8—44no—
233C3×D4—64no—
234C3×D4—64no—
235C3×D4—181no—
236C3×D4—181no—
237C2×C3×C4—28no—
238C2×C3×C4—24no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4C2×D3214yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243#67—364no—
244#67—3616no—
245C2×Dic3—98no—
246C2×Dic3—92no—
247C2×Dic3—188no—
248C2×Dic3—188no—
249C2×Dic3—184no—
250C2×Dic3—181no—
251D12—21no—
252D12—21no—
253D12—64no—
254D12—64no—
255Dic6—184no—
256Dic6—184no—
257Dic6—181no—
258Dic6—181no—
259Dic6—364no—
260Dic6—364no—
261C3×C8—38no—
262C3×C8—92no—
263C3⋊C8—38no—
264C3⋊C8#63812yescomplemented
265C2×C32—28no—
266C2×C32—24no—
267C2×C32—241no—
268C2×C32D3×SD1618yes—
269C2×C32D3×SD1612yes—
270C2×C32#94014yes—
271C2×C32#94011yescharacteristic
272C3⋊D3—81no—
273C3×D3—84no—
274C3×D3—244no—
275C2×Q8—271no—
276C2×Q8D3×C3⋊D311yescharacteristic
277C2×D4—31no—
278C2×D4—91no—
279C2×D4—542no—
280SD16—62no—
281SD16—182no—
282SD16—542no—
283SD16—542no—
284C2×C8—31no—
285C2×C8—91no—
286C4⋊C4—542no—
287C42—271no—
288C22×C3—24no—
289C22×C3—128no—
290C22×C3—124no—
291C22×C3—362no—
292C22×C3—361no—
293C22×C3#381114yes—
294C22×C3D4×C3⋊D311yescharacteristic
295C2×D3—42no—
296C2×D3—41no—
297C2×D3—128no—
298C2×D3—124no—
299C3×C4—24no—
300C3×C4—24no—
301C3×C4—48no—
302C3×C4—41no—
303C3×C4—44no—
304C3×C4D3×#6714yes—
305C3×C4D3×#6714yes—
306C3×C4D4×C3⋊D311yescharacteristic
307C3×C4C2×#63811yescharacteristic
308Dic3—94no—
309Dic3—916no—
310Dic3—1816no—
311Dic3—368no—
312Dic3—361no—
313Dic3—364no—
314C32—24no—
315C32#517514yescomplemented
316C32#517511yescharacteristic, complemented
317C23—61no—
318C23—181no—
319Q8—21no—
320Q8—21no—
321Q8—541no—
322Q8—541no—
323D4—61no—
324D4—61no—
325D4—181no—
326D4—181no—
327D4—1084no—
328C2×C4—21no—
329C2×C4—272no—
330C2×C4—541no—
331C2×C4C2×D3×C3⋊D311yescharacteristic
332C8—32no—
333C8—92no—
334C2×C3—28no—
335C2×C3—24no—
336C2×C3—244no—
337C2×C3—721no—
338C2×C3#6394018yes—
339C2×C3#6397914yes—
340C2×C3SD16×C3⋊D312yes—
341C2×C3#6414911yescharacteristic
342D3—81no—
343D3—244no—
344C22—122no—
345C22—121no—
346C22—362no—
347C22—361no—
348C22#8903911yescharacteristic
349C4—41no—
350C4—274no—
351C4—1081no—
352C4D3×#63811yescharacteristic
353C4C3⋊D3×#6711yescharacteristic
354C3—24no—
355C3#1059162014yescomplemented
356C3#1059287411yescharacteristic, complemented
357C2—241no—
358C2—721no—
359C2#1172357112yes—
360C2#1172363311yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723529C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#492
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11723985C2semidirect
2. C2×C3×#3755C2semidirect
3. C2×C32×#215C2semidirect
5. C2×#88796C2semidirect
6. C2×#88794C2semidirect
7. #11722018C2semidirect
27. C2×#3755D3semidirect
32. C2×C3×#215D3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
54. C32×C3⋊C8D4semidirect
96. C3⋊(C3⋊Dic3)SD16semidirect
103. C2×#215C3⋊D3semidirect
151. C3⋊(C3⋊C8)#67semidirect
153. C3×C3⋊C8#67semidirect
222. C33#492semidirect
264. C3⋊C8#638semidirect
315. C32#5175semidirect
316. C32#5175semidirect
355. C3#10591620semidirect
356. C3#10592874semidirect

Maximal subgroups

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

subgroupindex
#117239852
C2×C3×#37552
C2×C32×#2152
#117235292
C2×#887962
C2×#887942
#117220182
#105928743
#105916203