Finite Group Databasebeta

← #13159227

Subgroups — #13159227

Order 1728.

840 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C32×#887C211yesmaximal, characteristic, complemented
2#11723439C211yesmaximal, characteristic
3C2×#88799C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic
5#11722180C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic
7Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592727—31nomaximal
9#10590532—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×Dic3⋊C4—21no—
17C32×C4×Dic3—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88803—21no—
20#88803—21no—
21#88799—22no—
22#88793—22no—
23C32×C4×Q8—31no—
24C2×#3756—31no—
25C2×#3755—31no—
26#64084—31no—
27C3×#887D314yescomplemented
28#63860—34no—
29C2×#3719—34no—
30C2×#3715—34no—
31#63637—31no—
32#63637—34no—
33#63637—64no—
34#63636D311yescharacteristic
35C4⋊(C3⋊(C3⋊C8))—31no—
36C3×#809—34no—
37Dic3⋊(C3⋊C8)—34no—
38C22×C3×C3⋊D3—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C2×C32×Dic3—21no—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49C32×Dic6—21no—
50C32×Dic6D412yes—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)—21no—
53#5028—94no—
54#4767—91no—
55C2×C32×Q8—31no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×#634—31no—
58C2×#634—34no—
59C2×#634—64no—
60C2×C3×D12—34no—
61C2×C3×Dic6C2×D314yes—
62C32×C4⋊C4—24no—
63C32×C4⋊C4—61no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C42—31no—
67C32×C42—61no—
68#3756—62no—
69#3755—62no—
70#3752—61no—
71#3752—61no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×Dic3⋊C4—24no—
77C3×C4×Dic3—24no—
78C3×C4×Dic3C2×D314yes—
79C2×C3×C3⋊C8—34no—
80(C2×D3)⋊Dic3—64no—
81(C2×D3)⋊Dic3—64no—
82#3719—68no—
83#3715—68no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—41no—
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C33×C4—41no—
89C33×C4C2×D411yescharacteristic
90C33×C4#4111yescharacteristic
91C3⋊(C3⋊Dic3)—41no—
92C32×Dic3—22no—
93C32×Dic3—21no—
94C32×Dic3—21no—
95C3×C4×Q8—34no—
96C2×#216—94no—
97C2×#215—94no—
98#936—91no—
99C2×#214—91no—
100C2×Dic12—91no—
101#887C3⋊D311yescharacteristic, complemented
102#875—94no—
103C3×#102—91no—
104Dic3⋊C8—91no—
105#810—94no—
106#810—91no—
107#810—184no—
108#809—34no—
109C4⋊(C3⋊C8)—94no—
110C22×C3⋊D3—21no—
111C22×C3×D3—64no—
112C32×Q8—32no—
113C32×Q8—61no—
114C2×C32×C4—21no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—31no—
118C2×C32×C4—31no—
119C2×C32×C4—44no—
120C2×C32×C4—61no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—124no—
126C4⋊(C3⋊D3)—21no—
127C4⋊(C3⋊D3)—21no—
128#634—68no—
129#634—64no—
130#634—61no—
131#634—61no—
132#634—64no—
133#634—124no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3#6714yes—
136C2×C3×Dic3#6714yes—
137C3×D12—64no—
138C3×D12—64no—
139C3×Dic6—24no—
140C3×Dic6D1218yes—
141C3⋊(C3⋊C8)—21no—
142C3⋊(C3⋊C8)—61no—
143C3⋊(C3⋊C8)—64no—
144C3⋊(C3⋊C8)—124no—
145C3×C3⋊C8—64no—
146#464—271noSylow
147C2×C33#12111yescharacteristic
148C2×C33#13511yescharacteristic
149C2×C33#13711yescharacteristic
150C3×C3⋊D3—81no—
151C2×C3×Q8—34no—
152C2×C3×D4—91no—
153C2×D12—34no—
154C2×Dic6—94no—
155C2×Dic6—91no—
156C2×Dic6—184no—
157C2×Dic6C2×C3⋊D311yescharacteristic
158C2×C3×C8—91no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—64no—
161C3×C4⋊C4D3214yes—
162C3×C4⋊C4C2×C3⋊D311yescharacteristic
163C3×C42—34no—
164C3×C42—64no—
165#217—181no—
166#217—181no—
167#216—188no—
168#215—188no—
169#214—182no—
170(C2×D3)⋊C4—184no—
171(C2×D3)⋊C4—184no—
172Dic3⋊C4—21no—
173C4×Dic3—21no—
174C4×Dic3C2×C3⋊D311yescharacteristic
175C2×C3⋊C8—34no—
176C2×C3⋊C8—94no—
177C2×C3⋊C8—91no—
178C2×C3⋊C8—184no—
179Dic12—182no—
180C22×C32—24no—
181C22×C32D3×D411yescharacteristic
182C22×C32C2×#6714yes—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C2×C3×D3—124no—
187C2×C3×D3—124no—
188C2×C3×D3—124no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—41no—
192C32×C4—44no—
193C32×C4—62no—
194C32×C4—61no—
195C32×C4—61no—
196C32×C4—84no—
197C32×C4C2×D1214yes—
198C32×C4#23514yes—
199C32×C4#23511yescharacteristic
200C32×C4D3×D411yescharacteristic
201C3⋊Dic3—128no—
202C3⋊Dic3—121no—
203C3⋊Dic3—124no—
204C3×Dic3—28no—
205C3×Dic3—24no—
206C3×Dic3—24no—
207C2×Q16—271no—
208C2×SD16—271no—
209#124—271no—
210C4×Q8—31no—
211C4⋊C8—271no—
212#103—271no—
213#102—91no—
214C33#46411yescharacteristic, complemented, Sylow
215C23×C3—181no—
216C22×D3—64no—
217C3×Q8—38no—
218C3×Q8—64no—
219C3×D4—181no—
220C3×D4—181no—
221C2×C3×C4—24no—
222C2×C3×C4—21no—
223C2×C3×C4—24no—
224C2×C3×C4—34no—
225C2×C3×C4—34no—
226C2×C3×C4—44no—
227C2×C3×C4—64no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—184no—
232C2×Dic3—181no—
233C2×Dic3—364no—
234C2×Dic3#63811yescharacteristic
235C2×Dic3#63811yescharacteristic
236D12—64no—
237D12—64no—
238Dic6—21no—
239Dic6—188no—
240Dic6—184no—
241Dic6—184no—
242Dic6—181no—
243Dic6—181no—
244Dic6—364no—
245Dic6C4⋊(C3⋊D3)12yes—
246C3×C8—181no—
247C3⋊C8—64no—
248C3⋊C8—184no—
249C3⋊C8—181no—
250C3⋊C8—364no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32#88411yescharacteristic
255C2×C32#91211yescharacteristic
256C2×C32#91511yescharacteristic
257C2×C32#93014yes—
258C2×C32#94214yes—
259C2×C32#95014yes—
260C3⋊D3—81no—
261C3×D3—244no—
262C2×Q8—31no—
263C2×Q8—271no—
264C2×D4—91no—
265Q16—542no—
266SD16—542no—
267C2×C8—91no—
268C2×C8—271no—
269C4⋊C4—61no—
270C4⋊C4D3×C3⋊D311yescharacteristic
271C22⋊C4—541no—
272C22⋊C4—541no—
273C42—31no—
274C42—61no—
275C22×C3—24no—
276C22×C3—361no—
277C22×C3—361no—
278C22×C3—361no—
279C22×C3D3×#6714yes—
280C22×C3C2×#63811yescharacteristic
281C2×D3—124no—
282C2×D3—124no—
283C2×D3—124no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—41no—
287C3×C4—44no—
288C3×C4—68no—
289C3×C4—64no—
290C3×C4—64no—
291C3×C4—84no—
292C3×C4#379814yes—
293C3×C4D3×D1214yes—
294C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
295C3×C4#382811yescharacteristic
296Dic3—22no—
297Dic3—21no—
298Dic3—21no—
299Dic3—368no—
300Dic3—361no—
301Dic3—364no—
302C32—24no—
303C32#476711yescharacteristic, complemented
304C32#502814yescomplemented
305C23—181no—
306Q8—32no—
307Q8—61no—
308Q8—541no—
309Q8—541no—
310D4—181no—
311D4—181no—
312C2×C4—21no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—61no—
316C2×C4—541no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—181no—
319C8—541no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—721no—
324C2×C3#6390614yes—
325C2×C3#6393314yes—
326C2×C3#6394614yes—
327C2×C3#6413911yescharacteristic
328C2×C3#6415111yescharacteristic
329C2×C3#6415911yescharacteristic
330D3—244no—
331C22—361no—
332C22—361no—
333C22—361no—
334C22D3×#63811yescharacteristic
335C4—41no—
336C4—61no—
337C4—61no—
338C4—62no—
339C4—1081no—
340C4#8902011yescharacteristic
341C4D3×C4⋊(C3⋊D3)11yescharacteristic
342C3—24no—
343C3#1059053214yescomplemented
344C3#1059272711yescharacteristic, complemented
345C2—721no—
346C2#1172351411yescharacteristic
347C2#1172355611yescharacteristic
348C2#1172357911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
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. C32×#887C2semidirect
3. C2×#88799C2semidirect
5. #11722180C2semidirect
7. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
27. C3×#887D3semidirect
101. #887C3⋊D3semidirect
214. C33#464semidirect
303. C32#4767semidirect
304. C32#5028semidirect
343. C3#10590532semidirect
344. C3#10592727semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
#117234392
C2×#887992
C2×#887932
#117221802
C3×#636362
Dic3⋊(C3⋊(C3⋊C8))2
#105927273
#105905323