Finite Group Databasebeta

← #13161256

Subgroups — #13161256

Order 1728.

860 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 146 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
2#11723536C211yesmaximal, characteristic
3C3×#63662C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5#11722042C211yesmaximal, characteristic
6#11722041C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic
8#10592491—31nomaximal
9#10591538—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×#6293—21no—
13#88857—21no—
14#88857—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#64153—31no—
22#64153—34no—
23#64153—64no—
24#64097—31no—
25#63919—34no—
26C32×#103—31no—
27#63662D311yescharacteristic
28#63643—31no—
29C8×C3⋊Dic3—31no—
30#63637—31no—
31C3×#835—34no—
32C3×#835D314yes—
33#63567—34no—
34#63567—34no—
35#63553—34no—
36C33×Q8—21no—
37C33×Q8—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43#6308—21no—
44#6308—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47#6293—42no—
48C3×C3⋊(C3⋊C8)—21no—
49C32×C3⋊C8—21no—
50#4792—94no—
51#4792—91no—
52C2×C32×Q8—24no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C2×D311yescharacteristic
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—61no—
58C2×#634—64no—
59C2×#627—64no—
60C2×C32×C8—31no—
61C32×C4⋊C4—31no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63#3750—61no—
64#3750—61no—
65#3750—61no—
66#3750—64no—
67#3750—64no—
68#3750—68no—
69#3750—68no—
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—64no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3C2×D314yes—
76C2×C3×C3⋊C8—34no—
77C2×C3×C3⋊C8C2×D314yes—
78#3724—64no—
79C22×C33C2×D411yescharacteristic
80C33×C4—41no—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3⋊(C3⋊Dic3)SD1612yes—
85C3⋊(C3⋊Dic3)Q1612yes—
86C3×C3⋊Dic3—41no—
87C32×Dic3—41no—
88#944—94no—
89#944—91no—
90#944—184no—
91#888—94no—
92#888—91no—
93C3×#103—34no—
94C3×#103—91no—
95#835—34no—
96#835C3⋊D311yescharacteristic
97#816—94no—
98#816—91no—
99C8×Dic3—94no—
100C8×Dic3—91no—
101#810—94no—
102#810—91no—
103C32×Q8—21no—
104C32×Q8—21no—
105C32×Q8—24no—
106C32×Q8—24no—
107C32×Q8—44no—
108C32×Q8—44no—
109C2×C32×C4—28no—
110C2×C32×C4—24no—
111C2×C32×C4—21no—
112C2×C32×C4—24no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—34no—
120C2×C3⋊Dic3—34no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—68no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—61no—
126#634—61no—
127#634—61no—
128#634—64no—
129#634—64no—
130#634—124no—
131#634—124no—
132#634—122no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—64no—
135#627—128no—
136C32×C8—61no—
137C3⋊(C3⋊C8)—21no—
138C3×C3⋊C8—24no—
139C3×C3⋊C8—64no—
140#487—271noSylow
141C2×C33#12411yescharacteristic
142C2×C33C2×SD1611yescharacteristic
143C2×C33C2×Q1611yescharacteristic
144C2×C3×Q8—24no—
145C2×C3×Q8D3214yes—
146C2×C3×Q8C2×C3⋊D311yescharacteristic
147C2×Dic6—94no—
148C2×Dic6—91no—
149C2×Dic6—184no—
150C2×Dic6—181no—
151C2×Dic6—184no—
152C2×C3×C8—34no—
153C2×C3×C8—91no—
154C3×C4⋊C4—34no—
155C3×C4⋊C4—91no—
156C4⋊Dic3—34no—
157C4⋊Dic3C2×C3⋊D311yescharacteristic
158Dic3⋊C4—181no—
159Dic3⋊C4—184no—
160Dic3⋊C4—184no—
161Dic3⋊C4—184no—
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164Dic3⋊C4—188no—
165Dic3⋊C4—188no—
166C4×Dic3—94no—
167C4×Dic3—91no—
168C4×Dic3—184no—
169C2×C3⋊C8—34no—
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171C22×C32—24no—
172C22×C32D3×D414yes—
173C22×C32D3×D411yescharacteristic
174C32×C4—24no—
175C32×C4—24no—
176C32×C4—48no—
177C32×C4—44no—
178C32×C4—41no—
179C32×C4—121no—
180C32×C4#23514yes—
181C32×C4#23511yescharacteristic
182C32×C4#23714yes—
183C32×C4#23711yescharacteristic
184C3⋊Dic3—32no—
185C3⋊Dic3—32no—
186C3⋊Dic3—38no—
187C3⋊Dic3—38no—
188C3⋊Dic3—41no—
189C3⋊Dic3—68no—
190C3⋊Dic3—68no—
191C3⋊Dic3—128no—
192C3⋊Dic3—124no—
193C3⋊Dic3—121no—
194C3×Dic3—44no—
195C3×Dic3—124no—
196C4⋊Q8—271no—
197C4⋊Q8—271no—
198#103—31no—
199#103—91no—
200#103—271no—
201#103—271no—
202C4×C8—271no—
203C33#48711yescharacteristic, complemented, Sylow
204C3×Q8—24no—
205C3×Q8—24no—
206C3×Q8—21no—
207C3×Q8—21no—
208C3×Q8—44no—
209C3×Q8—44no—
210C2×C3×C4—28no—
211C2×C3×C4—24no—
212C2×C3×C4—21no—
213C2×C3×C4—24no—
214C2×C3×C4—64no—
215C2×C3×C4—181no—
216C2×C3×C4C2×D3214yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—64no—
220C2×Dic3—94no—
221C2×Dic3—94no—
222C2×Dic3—91no—
223C2×Dic3—91no—
224C2×Dic3—184no—
225C2×Dic3—184no—
226C2×Dic3—188no—
227C2×Dic3—184no—
228C2×Dic3—181no—
229Dic6—184no—
230Dic6—184no—
231Dic6—181no—
232Dic6—181no—
233Dic6—364no—
234Dic6—364no—
235Dic6—362no—
236Dic6—368no—
237C3×C8—64no—
238C3×C8—181no—
239C3⋊C8—21no—
240C3⋊C8—64no—
241C2×C32—24no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32#88514yes—
245C2×C32#88511yescharacteristic
246C2×C32D3×SD1614yes—
247C2×C32D3×SD1611yescharacteristic
248C2×C32D3×Q1614yes—
249C2×C32D3×Q1611yescharacteristic
250C2×Q8—271no—
251C2×Q8—541no—
252C2×Q8D3×C3⋊D311yescharacteristic
253C2×C8—31no—
254C2×C8—91no—
255C4⋊C4—31no—
256C4⋊C4—91no—
257C4⋊C4—541no—
258C4⋊C4—541no—
259C4⋊C4—541no—
260C42—271no—
261C22×C3—24no—
262C22×C3#381114yes—
263C22×C3D4×C3⋊D311yescharacteristic
264C3×C4—24no—
265C3×C4—24no—
266C3×C4—48no—
267C3×C4—41no—
268C3×C4—44no—
269C3×C4—124no—
270C3×C4—361no—
271C3×C4#380414yes—
272C3×C4#380414yes—
273C3×C4#382811yescharacteristic
274C3×C4#383011yescharacteristic
275Dic3—41no—
276Dic3—92no—
277Dic3—92no—
278Dic3—98no—
279Dic3—98no—
280Dic3—124no—
281Dic3—188no—
282Dic3—188no—
283Dic3—368no—
284Dic3—361no—
285Dic3—364no—
286C32—24no—
287C32#479214yescomplemented
288C32#479211yescharacteristic, complemented
289Q8—21no—
290Q8—21no—
291Q8—541no—
292Q8—541no—
293Q8—1082no—
294C2×C4—21no—
295C2×C4—61no—
296C2×C4—181no—
297C2×C4—271no—
298C2×C4—271no—
299C2×C4—541no—
300C2×C4C2×D3×C3⋊D311yescharacteristic
301C8—61no—
302C8—181no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—24no—
306C2×C3#6394014yes—
307C2×C3#6394414yes—
308C2×C3#6395314yes—
309C2×C3#6409411yescharacteristic
310C2×C3SD16×C3⋊D311yescharacteristic
311C2×C3Q16×C3⋊D311yescharacteristic
312C22#8903911yescharacteristic
313C4—41no—
314C4—121no—
315C4—272no—
316C4—272no—
317C4—361no—
318C4—1081no—
319C4#8902811yescharacteristic
320C4#8903111yescharacteristic
321C3—24no—
322C3#1059153814yescomplemented
323C3#1059249111yescharacteristic, complemented
324C2#1172357111yescharacteristic
325C2#1172357711yescharacteristic
326C2#1172359211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#487
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
203. C33#487semidirect
287. C32#4792semidirect
288. C32#4792semidirect
322. C3#10591538semidirect
323. C3#10592491semidirect

Maximal subgroups

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

subgroupindex
#117239852
#117235362
C3×#636622
C32×#8352
#117220422
#117220412
#117220182
#105924913
#105915383