Finite Group Databasebeta

← #13159300

Subgroups — #13159300

Order 1728.

816 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 176 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
2#11723443C211yesmaximal, characteristic
3C2×#88799C211yesmaximal, characteristic
4C2×#88798C211yesmaximal, characteristic
5#11722180C211yesmaximal, characteristic
6C3×#63637C211yesmaximal, characteristic
7Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
8#10592730—31nomaximal
9#10590561—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
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
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#88799—22no—
22#88798—22no—
23C32×C4×Q8—31no—
24C2×#3756—31no—
25C2×#3756—31no—
26#64078—31no—
27C3×#887D314yes—
28#63866—34no—
29C2×#3719—34no—
30C2×#3718—34no—
31#63637—31no—
32#63637—34no—
33#63637—64no—
34#63637D311yescharacteristic
35C4⋊(C3⋊(C3⋊C8))—31no—
36C3×#810—34no—
37Dic3⋊(C3⋊C8)—34no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41#6308—21no—
42#6308—21no—
43C2×C3×C3⋊Dic3—21no—
44C3×#634—21no—
45C3×#634—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#5031—94no—
54#4795—91no—
55C2×C32×Q8—31no—
56C2×#634—31no—
57C2×#634—34no—
58C2×#634—64no—
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—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#3756—62no—
70C4⋊(C3⋊Dic3)—61no—
71C4⋊(C3⋊Dic3)—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—
80Dic3⋊Dic3—64no—
81Dic3⋊Dic3—64no—
82#3719—68no—
83#3718—68no—
84C22×C33C2×D411yescharacteristic
85C33×C4—41no—
86C33×C4C2×D411yescharacteristic
87C33×C4#4111yescharacteristic
88C3⋊(C3⋊Dic3)—41no—
89C3×C3⋊Dic3—41no—
90C32×Dic3—22no—
91C32×Dic3—21no—
92C32×Dic3Q1612yescomplemented
93C3×C4×Q8—34no—
94#944—91no—
95C2×#216—94no—
96C2×#216—94no—
97C2×#216—91no—
98C2×Dic12—91no—
99#887C3⋊D311yescharacteristic
100#869—94no—
101C3×#103—91no—
102Dic3⋊C8—91no—
103#810—34no—
104#810—94no—
105#810—91no—
106#810—184no—
107C4⋊(C3⋊C8)—94no—
108C32×Q8—32no—
109C32×Q8—61no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—31no—
114C2×C32×C4—31no—
115C2×C32×C4—44no—
116C2×C32×C4—61no—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—124no—
123#634—21no—
124#634—21no—
125#634—68no—
126#634—64no—
127#634—61no—
128#634—61no—
129#634—64no—
130#634—124no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—64no—
133C2×C3×Dic3#6714yes—
134C2×C3×Dic3#6714yes—
135C3×Dic6—24no—
136C3×Dic6—64no—
137C3×Dic6—64no—
138C3×Dic6D1218yes—
139C3⋊(C3⋊C8)—21no—
140C3⋊(C3⋊C8)—61no—
141C3⋊(C3⋊C8)—64no—
142C3⋊(C3⋊C8)—124no—
143C3×C3⋊C8—64no—
144#462—271noSylow
145C2×C33#12111yescharacteristic
146C2×C33C2×Q1611yescharacteristic
147C2×C33#13711yescharacteristic
148C2×C3×Q8—34no—
149C2×C3×Q8—91no—
150C2×Dic6—34no—
151C2×Dic6—94no—
152C2×Dic6—91no—
153C2×Dic6—184no—
154C2×Dic6C2×C3⋊D311yescharacteristic
155C2×C3×C8—91no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—64no—
158C3×C4⋊C4D3214yes—
159C3×C4⋊C4C2×C3⋊D311yescharacteristic
160C3×C42—34no—
161C3×C42—64no—
162#216—188no—
163#216—188no—
164#216—182no—
165C4⋊Dic3—184no—
166C4⋊Dic3—184no—
167Dic3⋊C4—21no—
168Dic3⋊C4—181no—
169Dic3⋊C4—181no—
170C4×Dic3—21no—
171C4×Dic3C2×C3⋊D311yescharacteristic
172C2×C3⋊C8—34no—
173C2×C3⋊C8—94no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—184no—
176Dic12—182no—
177C22×C32—24no—
178C22×C32D3×D411yescharacteristic
179C22×C32C2×#6714yes—
180C32×C4—24no—
181C32×C4—24no—
182C32×C4—32no—
183C32×C4—41no—
184C32×C4—44no—
185C32×C4—62no—
186C32×C4—61no—
187C32×C4—84no—
188C32×C4C2×D1214yes—
189C32×C4#23514yes—
190C32×C4#23511yescharacteristic
191C32×C4D3×D411yescharacteristic
192C3⋊Dic3—41no—
193C3⋊Dic3—128no—
194C3⋊Dic3—121no—
195C3⋊Dic3—124no—
196C3×Dic3—28no—
197C3×Dic3—24no—
198C3×Dic3—124no—
199C3×Dic3#21618yescomplemented
200C2×Q16—271no—
201C2×Q16—271no—
202C4⋊Q8—271no—
203C4×Q8—31no—
204C4⋊C8—271no—
205#103—91no—
206#103—271no—
207C33#46211yescharacteristic, complemented, Sylow
208C3×Q8—38no—
209C3×Q8—64no—
210C3×Q8—181no—
211C3×Q8—181no—
212C2×C3×C4—24no—
213C2×C3×C4—21no—
214C2×C3×C4—24no—
215C2×C3×C4—34no—
216C2×C3×C4—34no—
217C2×C3×C4—44no—
218C2×C3×C4—64no—
219C2×C3×C4—181no—
220C2×C3×C4C2×D3214yes—
221C2×C3×C4C22×C3⋊D311yescharacteristic
222C2×Dic3—21no—
223C2×Dic3—64no—
224C2×Dic3—184no—
225C2×Dic3—181no—
226C2×Dic3—364no—
227C2×Dic3#63811yescharacteristic
228C2×Dic3#63811yescharacteristic
229Dic6—21no—
230Dic6—64no—
231Dic6—64no—
232Dic6—188no—
233Dic6—184no—
234Dic6—184no—
235Dic6—181no—
236Dic6—181no—
237Dic6—364no—
238Dic6C4⋊(C3⋊D3)12yes—
239C3×C8—181no—
240C3⋊C8—64no—
241C3⋊C8—184no—
242C3⋊C8—181no—
243C3⋊C8—364no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32#88411yescharacteristic
248C2×C32#91511yescharacteristic
249C2×C32D3×Q1611yescharacteristic
250C2×C32#93014yes—
251C2×C32C2×#21614yes—
252C2×C32#95114yes—
253C2×Q8—31no—
254C2×Q8—91no—
255C2×Q8—271no—
256Q16—542no—
257Q16—542no—
258C2×C8—91no—
259C2×C8—271no—
260C4⋊C4—61no—
261C4⋊C4—541no—
262C4⋊C4—541no—
263C4⋊C4D3×C3⋊D311yescharacteristic
264C42—31no—
265C42—61no—
266C22×C3—24no—
267C22×C3D3×#6714yes—
268C22×C3C2×#63811yescharacteristic
269C3×C4—24no—
270C3×C4—24no—
271C3×C4—38no—
272C3×C4—41no—
273C3×C4—44no—
274C3×C4—68no—
275C3×C4—64no—
276C3×C4—84no—
277C3×C4—361no—
278C3×C4#379814yes—
279C3×C4D3×D1214yes—
280C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
281C3×C4#382811yescharacteristic
282Dic3—22no—
283Dic3—21no—
284Dic3—124no—
285Dic3—368no—
286Dic3—361no—
287Dic3—364no—
288Dic3#375612yescomplemented
289C32—24no—
290C32#479511yescharacteristic, complemented
291C32#503114yescomplemented
292Q8—32no—
293Q8—61no—
294Q8—181no—
295Q8—181no—
296Q8—541no—
297Q8—541no—
298C2×C4—21no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—61no—
302C2×C4—181no—
303C2×C4—541no—
304C2×C4C2×D3×C3⋊D311yescharacteristic
305C8—181no—
306C8—541no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3#6390614yes—
311C2×C3#6392914yes—
312C2×C3D3×#21614yes—
313C2×C3#6413911yescharacteristic
314C2×C3C2×#375611yescharacteristic
315C2×C3#6416011yescharacteristic
316C22D3×#63811yescharacteristic
317C4—32no—
318C4—41no—
319C4—61no—
320C4—62no—
321C4—361no—
322C4—1081no—
323C4#8902011yescharacteristic
324C4D3×C4⋊(C3⋊D3)11yescharacteristic
325C3—24no—
326C3#1059056114yescomplemented
327C3#1059273011yescharacteristic, complemented
328C2#1172351411yescharacteristic
329C2#1172355011yescharacteristic
330C2D3×#375611yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#462
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
92. C32×Dic3Q16semidirect
199. C3×Dic3#216semidirect
207. C33#462semidirect
288. Dic3#3756semidirect
290. C32#4795semidirect
291. C32#5031semidirect
326. C3#10590561semidirect
327. C3#10592730semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
#117234432
C2×#887992
C2×#887982
#117221802
C3×#636372
Dic3⋊(C3⋊(C3⋊C8))2
#105927303
#105905613