Finite Group Databasebeta

← #13159221

Subgroups — #13159221

Order 1728.

840 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 148 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×#64096C211yesmaximal, characteristic, complemented
2#11723429C211yesmaximal, characteristic
3C2×#88800C211yesmaximal, characteristic, complemented
4C2×#88792C211yesmaximal, characteristic
5#11722180C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic
7#11722051C211yesmaximal, characteristic, complemented
8#10592466—31nomaximal
9#10590532—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3750—21no—
16C3×C4×C3⋊Dic3—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19(C2×D3)⋊(C3⋊Dic3)—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88800—22no—
22#88792—22no—
23#64145—31no—
24C2×#3754—31no—
25C2×#3746—31no—
26#64096D311yescharacteristic, complemented
27C3×#887—34no—
28#63860—34no—
29C2×#3719—34no—
30C2×#3715—34no—
31C32×#102—31no—
32#63641—31no—
33#63637—31no—
34#63637—34no—
35#63637—64no—
36C3×#809D314yes—
37Dic3⋊(C3⋊C8)—34no—
38C22×C32×D3—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×#634—21no—
48C3×#634D412yes—
49C32×D12—21no—
50C32×D12—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C32×C3⋊C8—21no—
53#5028—91no—
54#4767—94no—
55C2×C32×D4—31no—
56C2×#634—31no—
57C2×#634—34no—
58C2×#634—64no—
59C2×#634C2×D311yescharacteristic
60C2×C3×D12C2×D314yes—
61C2×C3×Dic6—34no—
62C2×C32×C8—31no—
63C32×C4⋊C4—24no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C2×D311yescharacteristic
66#3757—61no—
67#3757—61no—
68#3754—62no—
69#3750—21no—
70C4×C3⋊Dic3—21no—
71C4×C3⋊Dic3C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75#3746—62no—
76C3×Dic3⋊C4—64no—
77C3×C4×Dic3—34no—
78C3×C4×Dic3—64no—
79C2×C3×C3⋊C8C2×D314yes—
80(C2×D3)⋊Dic3—64no—
81(C2×D3)⋊Dic3—64no—
82#3719—68no—
83#3715—68no—
84C22×C33C2×D411yescharacteristic
85C2×C32×D3—41no—
86C2×C32×D3—41no—
87C2×C32×D3—41no—
88C33×C4—41no—
89C33×C4C2×D411yescharacteristic
90C33×C4#4111yescharacteristic
91C3⋊(C3⋊Dic3)—41no—
92C3×C3⋊Dic3—22no—
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3—21no—
95C3×C4×Q8—91no—
96C2×#216—91no—
97C2×#215—91no—
98#936—94no—
99C2×#214—94no—
100C2×Dic12—94no—
101#887—34no—
102#875—91no—
103C3×#102—34no—
104Dic3⋊C8—94no—
105#810—94no—
106#810—91no—
107#810—184no—
108#809C3⋊D311yescharacteristic
109C4⋊(C3⋊C8)—91no—
110C23×C32—61no—
111C22×C3×D3—24no—
112C32×D4—61no—
113C32×D4—61no—
114C2×C32×C4—21no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—44no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—124no—
124C2×C3⋊Dic3#6711yescharacteristic
125C2×C3⋊Dic3#6711yescharacteristic
126#634—21no—
127#634—68no—
128#634—61no—
129#634—64no—
130#634—61no—
131#634—64no—
132#634—124no—
133#634D1212yes—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—64no—
137C3×D12—24no—
138C3×D12—24no—
139C3×Dic6—38no—
140C3×Dic6—64no—
141C32×C8—61no—
142C3⋊(C3⋊C8)—61no—
143C3⋊(C3⋊C8)—64no—
144C3⋊(C3⋊C8)—124no—
145C3×C3⋊C8—24no—
146#464—271noSylow
147C2×C33#12111yescharacteristic
148C2×C33#13511yescharacteristic
149C2×C33#13711yescharacteristic
150C32×D3—81no—
151C2×C3×Q8—91no—
152C2×C3×D4—34no—
153C2×D12C2×C3⋊D311yescharacteristic
154C2×Dic6—34no—
155C2×Dic6—94no—
156C2×Dic6—91no—
157C2×Dic6—184no—
158C2×C3×C8—34no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—181no—
161C3×C4⋊C4D3214yes—
162C3×C4⋊C4C2×C3⋊D311yescharacteristic
163C3×C42—91no—
164C3×C42—181no—
165#217—184no—
166#217—184no—
167#216—182no—
168#215—182no—
169#214—188no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172Dic3⋊C4—64no—
173C4×Dic3—34no—
174C4×Dic3—64no—
175C2×C3⋊C8—94no—
176C2×C3⋊C8—91no—
177C2×C3⋊C8—184no—
178C2×C3⋊C8C2×C3⋊D311yescharacteristic
179Dic12—188no—
180C22×C32—24no—
181C22×C32—121no—
182C22×C32—121no—
183C22×C32—121no—
184C22×C32D3×D414yes—
185C22×C32C2×#6711yescharacteristic
186C2×C3×D3—44no—
187C2×C3×D3—44no—
188C2×C3×D3—44no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—41no—
192C32×C4—44no—
193C32×C4—84no—
194C32×C4C2×D1211yescharacteristic
195C32×C4#23514yes—
196C32×C4#23511yescharacteristic
197C32×C4D3×D414yes—
198C3⋊Dic3—22no—
199C3⋊Dic3—21no—
200C3⋊Dic3—21no—
201C3⋊Dic3—128no—
202C3⋊Dic3—124no—
203C3⋊Dic3—121no—
204C3×Dic3—68no—
205C3×Dic3—64no—
206C3×Dic3—64no—
207C2×Q16—271no—
208C2×SD16—271no—
209#124—271no—
210C4×Q8—91no—
211C4⋊C8—271no—
212#103—271no—
213#102—31no—
214C33#46411yescharacteristic, complemented, Sylow
215C23×C3—64no—
216C22×D3—21no—
217C3×Q8—92no—
218C3×Q8—181no—
219C3×D4—64no—
220C3×D4—64no—
221C2×C3×C4—24no—
222C2×C3×C4—21no—
223C2×C3×C4—24no—
224C2×C3×C4—44no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—34no—
231C2×Dic3—34no—
232C2×Dic3—64no—
233C2×Dic3—184no—
234C2×Dic3—181no—
235C2×Dic3—364no—
236D12—21no—
237D12—21no—
238Dic6—38no—
239Dic6—64no—
240Dic6—188no—
241Dic6—184no—
242Dic6—184no—
243Dic6—181no—
244Dic6—181no—
245Dic6—364no—
246C3×C8—64no—
247C3⋊C8—21no—
248C3⋊C8—184no—
249C3⋊C8—181no—
250C3⋊C8—364no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—241no—
255C2×C32#88414yes—
256C2×C32#91214yes—
257C2×C32#91514yes—
258C2×C32#93011yescharacteristic
259C2×C32#94211yescharacteristic
260C2×C32#95011yescharacteristic
261C3×D3—84no—
262C2×Q8—91no—
263C2×Q8—271no—
264C2×D4—31no—
265Q16—542no—
266SD16—542no—
267C2×C8—31no—
268C2×C8—271no—
269C4⋊C4—181no—
270C4⋊C4D3×C3⋊D311yescharacteristic
271C22⋊C4—541no—
272C22⋊C4—541no—
273C42—91no—
274C42—181no—
275C22×C3—24no—
276C22×C3—124no—
277C22×C3—124no—
278C22×C3—124no—
279C22×C3D3×#6714yes—
280C22×C3D4×C3⋊D311yescharacteristic
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—41no—
287C3×C4—44no—
288C3×C4—84no—
289C3×C4—182no—
290C3×C4—181no—
291C3×C4—181no—
292C3×C4#379814yes—
293C3×C4D3×D1214yes—
294C3×C4#382811yescharacteristic
295C3×C4D4×C3⋊D311yescharacteristic
296Dic3—68no—
297Dic3—64no—
298Dic3—64no—
299Dic3—368no—
300Dic3—361no—
301Dic3—364no—
302C32—24no—
303C32#476714yescomplemented
304C32#502811yescharacteristic, complemented
305C23—61no—
306Q8—92no—
307Q8—181no—
308Q8—541no—
309Q8—541no—
310D4—61no—
311D4—61no—
312C2×C4—21no—
313C2×C4—91no—
314C2×C4—91no—
315C2×C4—181no—
316C2×C4—541no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—61no—
319C8—541no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—244no—
324C2×C3#6390614yes—
325C2×C3#6393314yes—
326C2×C3#6394614yes—
327C2×C3#6409311yescharacteristic
328C2×C3#6412111yescharacteristic
329C2×C3#6412411yescharacteristic
330D3—81no—
331C22—121no—
332C22—121no—
333C22—121no—
334C22C3⋊D3×#6711yescharacteristic
335C4—41no—
336C4—182no—
337C4—181no—
338C4—181no—
339C4—1081no—
340C4#8902011yescharacteristic
341C4C3⋊D3×D1211yescharacteristic
342C3—24no—
343C3#1059053214yescomplemented
344C3#1059246611yescharacteristic, complemented
345C2—241no—
346C2#1172352011yescharacteristic
347C2#1172355911yescharacteristic
348C2#1172358411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#464
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
1. C3×#64096C2semidirect
3. C2×#88800C2semidirect
5. #11722180C2semidirect
7. #11722051C2semidirect
26. #64096D3semidirect
214. C33#464semidirect
303. C32#4767semidirect
304. C32#5028semidirect
343. C3#10590532semidirect
344. C3#10592466semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
#117234292
C2×#888002
C2×#887922
#117221802
C32×#8092
#117220512
#105924663
#105905323