Finite Group Databasebeta

← #13162014

Subgroups — #13162014

Order 1728.

810 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C3×#64153C211yesmaximal, characteristic, complemented
2C2×C3×#3755C212yesmaximal, complemented
3#11723697C211yesmaximal, characteristic
4C2×#88811C212yesmaximal, complemented
5#11722070C211yesmaximal, characteristic, complemented
6#10591620—31nomaximal
7#10591618—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×C3×#634C2211yescharacteristic
11C2×#6295—22no—
12C3×#3755—24no—
13C3×#3750—22no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#88811—24no—
17#64153D311yescharacteristic, complemented
18C2×#3755D312yescomplemented
19C3×#944—32no—
20C3×#944—61no—
21C2×C3×#215—32no—
22C2×C3×#215—32no—
23C2×C3×#215—62no—
24#63914—31no—
25#63911—32no—
26C2×#3717—32no—
27C2×#3716—32no—
28C2×#3715—32no—
29#63553—31no—
30Dic3×C3⋊C8—32no—
31C22×C3×C3⋊D3—22no—
32C33×Q8—22no—
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D412yes—
37C3×C4⋊(C3⋊D3)—22no—
38C3×C4⋊(C3⋊D3)—22no—
39C3×#634—22no—
40#6295—44no—
41C3×C3⋊(C3⋊C8)D414yescomplemented
42#5175—92no—
43#5085—91no—
44C2×C32×Q8—21no—
45C2×C32×Q8—22no—
46C2×C32×Q8—41no—
47C2×C32×Q8C2×D311yescharacteristic
48C2×C32×Q8C2×D312yes—
49C2×C4⋊(C3⋊D3)C2×D312yes—
50C2×#634C2×D311yescharacteristic
51C2×C3×D12—32no—
52C2×C3×D12—32no—
53C2×C3×D12—62no—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—61no—
56C2×#626—64no—
57C2×#625—62no—
58#3755—24no—
59#3750—22no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D312yes—
62C3×#215—64no—
63C3×#215—64no—
64C3×#215—68no—
65C3×Dic3⋊C4—64no—
66C3×Dic3⋊C4—64no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—62no—
72#3717—64no—
73#3716—64no—
74#3715—64no—
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—42no—
77C2×C3×C3⋊D3—44no—
78C33×C4—41no—
79C33×C4C2×D412yes—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3SD1614yescomplemented
82C2×C3×SD16—92no—
83C3×C4⋊Q8—91no—
84#944—32no—
85#944—61no—
86C2×#215—32no—
87C2×#215—32no—
88C2×#215—62no—
89C2×#215—92no—
90#940—92no—
91C2×#214—92no—
92C2×#204—92no—
93#874—91no—
94C8×Dic3—92no—
95C4×C3⋊C8—91no—
96C22×C3⋊D3—22no—
97C22×C3×D3—62no—
98C22×C3×D3—62no—
99C22×C3×D3—122no—
100C32×Q8—22no—
101C32×Q8—24no—
102C32×Q8—22no—
103C32×Q8—22no—
104C32×Q8—42no—
105C32×Q8—42no—
106C32×Q8—44no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4—41no—
113C2×C32×C4—42no—
114C2×C32×C4—42no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C32×C4C22×D312yes—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3#6712yes—
119C4⋊(C3⋊D3)—22no—
120C4⋊(C3⋊D3)—22no—
121#634—22no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C3×D12—64no—
127C3×D12—62no—
128C3×D12—62no—
129C3×D12—62no—
130C3×D12—62no—
131C3×D12—122no—
132C3×Dic6—62no—
133C3×Dic6—62no—
134C3×Dic6—122no—
135#626—128no—
136#625—124no—
137C3⋊(C3⋊C8)#6714yescomplemented
138C3×C3⋊C8—34no—
139C3×C3⋊C8—34no—
140C3×C3⋊C8—64no—
141#492—271noSylow
142C2×C33#12711yescharacteristic
143C2×C33C2×SD1612yes—
144C3×C3⋊D3—82no—
145C2×C3×Q8—21no—
146C2×C3×Q8—22no—
147C2×C3×Q8—41no—
148C2×C3×Q8—91no—
149C2×C3×Q8D3211yescharacteristic
150C2×C3×Q8D3212yes—
151C2×C3×D4—92no—
152C2×#67—184no—
153C2×D12—32no—
154C2×D12—32no—
155C2×D12—62no—
156C2×D12—182no—
157C2×Dic6—32no—
158C2×Dic6—61no—
159C3×SD16—184no—
160C2×C3×C8—92no—
161C3×C4⋊C4—182no—
162C3×C42—91no—
163#215—64no—
164#215—64no—
165#215—68no—
166#215—184no—
167#214—184no—
168Dic3⋊C4—64no—
169Dic3⋊C4—64no—
170C4×Dic3—32no—
171C4×Dic3—61no—
172C2×C3⋊C8—32no—
173C2×C3⋊C8—32no—
174C2×C3⋊C8—62no—
175#204—184no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32D3×D412yes—
180C22×C32C2×#6711yescharacteristic
181C2×C3⋊D3—42no—
182C2×C3⋊D3—44no—
183C2×C3×D3—124no—
184C2×C3×D3—124no—
185C2×C3×D3—122no—
186C2×C3×D3—122no—
187C2×C3×D3—124no—
188C2×C3×D3—244no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—42no—
193C32×C4—42no—
194C32×C4—44no—
195C32×C4—44no—
196C32×C4—41no—
197C32×C4—42no—
198C32×C4D3×D412yes—
199C32×C4D3×D412yes—
200C32×C4C2×#6712yes—
201C3⋊Dic3—41no—
202C3⋊Dic3#21514yescomplemented
203C3×Dic3—38no—
204C3×Dic3—64no—
205C3×Dic3—122no—
206C3×Dic3—122no—
207C2×SD16—92no—
208C2×SD16—272no—
209C4⋊Q8—91no—
210#127—271no—
211C4×C8—271no—
212C33#49211yescharacteristic, complemented, Sylow
213C23×C3—182no—
214C22×D3—62no—
215C22×D3—62no—
216C22×D3—122no—
217C3×Q8—24no—
218C3×Q8—22no—
219C3×Q8—22no—
220C3×Q8—22no—
221C3×Q8—42no—
222C3×Q8—42no—
223C3×Q8—44no—
224C3×Q8—182no—
225C3×D4—182no—
226C3×D4—182no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—41no—
235C2×C3×C4—92no—
236C2×C3×C4—181no—
237C2×C3×C4C2×D3211yescharacteristic
238C2×C3×C4C2×D3212yes—
239#67—368no—
240C2×Dic3—34no—
241C2×Dic3—62no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244D12—64no—
245D12—62no—
246D12—62no—
247D12—62no—
248D12—62no—
249D12—122no—
250D12—364no—
251Dic6—62no—
252Dic6—62no—
253Dic6—122no—
254C3×C8—94no—
255C3⋊C8—34no—
256C3⋊C8—34no—
257C3⋊C8—64no—
258C2×C32—24no—
259C2×C32—22no—
260C2×C32—21no—
261C2×C32—22no—
262C2×C32—41no—
263C2×C32—42no—
264C2×C32D3×SD1614yes—
265C2×C32#94012yes—
266C2×C32#94011yescharacteristic
267C2×C32C2×#21512yes—
268C3⋊D3—82no—
269C3×D3—244no—
270C3×D3—242no—
271C3×D3—242no—
272C2×Q8—91no—
273C2×Q8#111311yescharacteristic
274C2×D4—92no—
275C2×D4—542no—
276SD16—184no—
277SD16—544no—
278C2×C8—92no—
279C4⋊C4—182no—
280C42—91no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—41no—
284C22×C3—364no—
285C22×C3—362no—
286C22×C3D3×#6712yes—
287C22×C3#381111yescharacteristic
288C2×D3—124no—
289C2×D3—124no—
290C2×D3—124no—
291C2×D3—122no—
292C2×D3—122no—
293C2×D3—244no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—22no—
297C3×C4—42no—
298C3×C4—44no—
299C3×C4—44no—
300C3×C4—42no—
301C3×C4—41no—
302C3×C4—42no—
303C3×C4—94no—
304C3×C4—361no—
305C3×C4D3×#6712yes—
306C3×C4D3×#6712yes—
307C3×C4#381112yes—
308Dic3—38no—
309Dic3—64no—
310Dic3—122no—
311Dic3—122no—
312C32—22no—
313C32—21no—
314C32—41no—
315C32#508511yescharacteristic, complemented
316C32#517512yescomplemented
317C23—182no—
318Q8—22no—
319Q8—182no—
320D4—182no—
321D4—182no—
322D4—1084no—
323C2×C4—21no—
324C2×C4—92no—
325C2×C4—181no—
326C2×C4C2×#111311yescharacteristic
327C8—94no—
328C2×C3—22no—
329C2×C3—24no—
330C2×C3—21no—
331C2×C3—22no—
332C2×C3—42no—
333C2×C3—41no—
334C2×C3—722no—
335C2×C3D3×#21514yes—
336C2×C3#6394012yes—
337C2×C3#6397912yes—
338C2×C3#6397911yescharacteristic
339D3—244no—
340D3—242no—
341D3—242no—
342C22—364no—
343C22—362no—
344C22#8904911yescharacteristic
345C4—41no—
346C4—94no—
347C4—361no—
348C4#8904912yes—
349C3—22no—
350C3—21no—
351C3—41no—
352C3#1059161812yescomplemented
353C3#1059162011yescharacteristic, complemented
354C2—722no—
355C2#1172370812yes—
356C2#1172372811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723697C2
hypercenterC2×Q8#1113
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. C3×#64153C2semidirect
2. C2×C3×#3755C2semidirect
4. C2×#88811C2semidirect
5. #11722070C2semidirect
17. #64153D3semidirect
18. C2×#3755D3semidirect
41. C3×C3⋊(C3⋊C8)D4semidirect
81. C3×C3⋊Dic3SD16semidirect
137. C3⋊(C3⋊C8)#67semidirect
202. C3⋊Dic3#215semidirect
212. C33#492semidirect
315. C32#5085semidirect
316. C32#5175semidirect
352. C3#10591618semidirect
353. C3#10591620semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C2×C3×#37552
#117236972
C2×#888112
#117220702
#105916203
#105916183