Finite Group Databasebeta

← #13159278

Subgroups — #13159278

Order 1728.

884 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 142 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#11723931C211yesmaximal, characteristic, complemented
2#11723529C211yesmaximal, characteristic
3C3×#63636C211yesmaximal, characteristic, complemented
4C32×#809C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7#11722021C211yesmaximal, characteristic, complemented
8#10592463—31nomaximal
9#10590554—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×#6290—21no—
13C2×#6290—21no—
14C33×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17#88857—21no—
18C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#64149—31no—
22#64099—31no—
23#64099—34no—
24#64099—64no—
25#63914—34no—
26C32×#102—31no—
27#63659—31no—
28#63648—31no—
29#63642—31no—
30#63636D311yescharacteristic, complemented
31C3×#809—34no—
32C3×#809D314yescomplemented
33#63563—34no—
34#63563—34no—
35#63555—34no—
36C22×C3×C3⋊D3—21no—
37C22×C32×D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C32×D12—21no—
46C32×D12—21no—
47#6290—42no—
48#6290—42no—
49C3×C3⋊(C3⋊C8)—21no—
50C32×C3⋊C8—21no—
51#4764—94no—
52#4764—91no—
53C2×C32×D4—31no—
54C2×#638—61no—
55C2×#638—61no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C3×D12—34no—
58C2×C3×D12C2×D314yes—
59C2×#625—64no—
60C2×#625—64no—
61C2×C32×C8—31no—
62C32×C4⋊C4—24no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—61no—
68C4⋊(C3⋊Dic3)—64no—
69C4⋊(C3⋊Dic3)—64no—
70C4⋊(C3⋊Dic3)—68no—
71#3750—61no—
72#3750—64no—
73#3750—68no—
74C4×C3⋊Dic3—31no—
75C4×C3⋊Dic3—34no—
76C4×C3⋊Dic3—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C2×C3×C3⋊C8—34no—
79C2×C3×C3⋊C8C2×D314yes—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C2×C32×D3—41no—
85C2×C32×D3—41no—
86C2×C32×D3—41no—
87C33×C4—41no—
88C33×C4#4111yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—21no—
91C3⋊(C3⋊Dic3)—21no—
92C3⋊(C3⋊Dic3)—41no—
93#940—94no—
94#940—91no—
95#890—94no—
96#890—91no—
97#890—184no—
98C3×#102—34no—
99C3×#102—91no—
100#832—94no—
101#832—91no—
102#821—94no—
103#821—91no—
104#815—94no—
105#815—91no—
106#809—34no—
107#809C3⋊D311yescharacteristic, complemented
108C23×C32—61no—
109C22×C3⋊D3—21no—
110C22×C3×D3—24no—
111C22×C3×D3—64no—
112C32×D4—61no—
113C32×D4—61no—
114C2×C32×C4—28no—
115C2×C32×C4—24no—
116C2×C32×C4—21no—
117C2×C32×C4—24no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120#638—122no—
121#638—122no—
122C2×C3⋊Dic3—31no—
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—34no—
125C2×C3⋊Dic3—34no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—68no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3—61no—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)—21no—
133C3×D12—24no—
134C3×D12—24no—
135C3×D12—64no—
136C3×D12—64no—
137#625—128no—
138#625—128no—
139C32×C8—61no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—24no—
142C3×C3⋊C8—64no—
143#490—271noSylow
144C2×C33#12411yescharacteristic
145C2×C33AGL(1,ℤ/8)11yescharacteristic
146C2×C33AGL(1,ℤ/8)11yescharacteristic
147C3×C3⋊D3—81no—
148C32×D3—81no—
149C2×C3×D4—34no—
150C2×C3×D4—91no—
151C2×#67—181no—
152C2×#67—181no—
153C2×#67—184no—
154C2×#67—184no—
155C2×D12—34no—
156C2×D12C2×C3⋊D311yescharacteristic
157C2×C3×C8—34no—
158C2×C3×C8—91no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4D3214yes—
161C3×C4⋊C4C2×C3⋊D311yescharacteristic
162C4⋊Dic3—94no—
163C4⋊Dic3—91no—
164C4⋊Dic3—184no—
165C4⋊Dic3—184no—
166C4⋊Dic3—181no—
167C4⋊Dic3—188no—
168Dic3⋊C4—184no—
169Dic3⋊C4—181no—
170Dic3⋊C4—188no—
171C4×Dic3—94no—
172C4×Dic3—91no—
173C4×Dic3—184no—
174C2×C3⋊C8—34no—
175C2×C3⋊C8C2×C3⋊D311yescharacteristic
176C22×C32—24no—
177C22×C32—121no—
178C22×C32—121no—
179C22×C32—121no—
180C22×C32D3×D414yes—
181C22×C32D3×D411yescharacteristic
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3×D3—44no—
186C2×C3×D3—44no—
187C2×C3×D3—44no—
188C2×C3×D3—124no—
189C2×C3×D3—124no—
190C2×C3×D3—124no—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—48no—
194C32×C4—44no—
195C32×C4—41no—
196C32×C4#23514yes—
197C32×C4#23511yescharacteristic
198C32×C4#23714yes—
199C32×C4#23711yescharacteristic
200C3⋊Dic3—61no—
201C3⋊Dic3—61no—
202C3⋊Dic3—64no—
203C3⋊Dic3—64no—
204C3⋊Dic3—124no—
205C3⋊Dic3—124no—
206C3⋊Dic3—128no—
207C3⋊Dic3—124no—
208C3⋊Dic3—121no—
209#127—271no—
210#125—271no—
211#102—31no—
212#102—91no—
213#102—271no—
214#102—271no—
215#97—271no—
216C33#49011yescharacteristic, complemented, Sylow
217C23×C3—64no—
218C23×C3—181no—
219C22×D3—21no—
220C22×D3—64no—
221C3×D4—64no—
222C3×D4—64no—
223C3×D4—181no—
224C3×D4—181no—
225C2×C3×C4—28no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—24no—
229C2×C3×C4C2×D3214yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231#67—362no—
232#67—362no—
233#67—368no—
234#67—368no—
235C2×Dic3—94no—
236C2×Dic3—94no—
237C2×Dic3—91no—
238C2×Dic3—91no—
239C2×Dic3—184no—
240C2×Dic3—184no—
241C2×Dic3—188no—
242C2×Dic3—184no—
243C2×Dic3—181no—
244D12—21no—
245D12—21no—
246D12—64no—
247D12—64no—
248C3×C8—64no—
249C3×C8—181no—
250C3⋊C8—21no—
251C3⋊C8—64no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—24no—
255C2×C32—241no—
256C2×C32#88514yes—
257C2×C32#88511yescharacteristic
258C2×C32#91114yes—
259C2×C32#91111yescharacteristic
260C2×C32#91414yes—
261C2×C32#91411yescharacteristic
262C3⋊D3—81no—
263C3×D3—84no—
264C3×D3—244no—
265C2×D4—31no—
266C2×D4—91no—
267C2×D4—541no—
268C2×D4—541no—
269C2×C8—31no—
270C2×C8—91no—
271C4⋊C4—271no—
272C4⋊C4—541no—
273C4⋊C4—541no—
274C4⋊C4D3×C3⋊D311yescharacteristic
275C42—271no—
276C22×C3—24no—
277C22×C3—124no—
278C22×C3—124no—
279C22×C3—124no—
280C22×C3—361no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3#381114yes—
284C22×C3D4×C3⋊D311yescharacteristic
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—124no—
289C2×D3—124no—
290C2×D3—124no—
291C3×C4—24no—
292C3×C4—24no—
293C3×C4—48no—
294C3×C4—41no—
295C3×C4—44no—
296C3×C4#379514yes—
297C3×C4#379514yes—
298C3×C4#382811yescharacteristic
299C3×C4#383011yescharacteristic
300Dic3—181no—
301Dic3—181no—
302Dic3—184no—
303Dic3—184no—
304Dic3—364no—
305Dic3—364no—
306Dic3—368no—
307Dic3—361no—
308Dic3—364no—
309C32—24no—
310C32#476414yescomplemented
311C32#476411yescharacteristic, complemented
312C23—61no—
313C23—181no—
314D4—61no—
315D4—61no—
316D4—181no—
317D4—181no—
318D4—1082no—
319D4—1082no—
320C2×C4—21no—
321C2×C4—271no—
322C2×C4—271no—
323C2×C4—541no—
324C2×C4C2×D3×C3⋊D311yescharacteristic
325C8—61no—
326C8—181no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—24no—
330C2×C3—244no—
331C2×C3—721no—
332C2×C3#6391314yes—
333C2×C3#6392714yes—
334C2×C3#6394114yes—
335C2×C3#6409411yescharacteristic
336C2×C3#6412011yescharacteristic
337C2×C3#6412311yescharacteristic
338D3—81no—
339D3—244no—
340C22—121no—
341C22—121no—
342C22—121no—
343C22—361no—
344C22—361no—
345C22—361no—
346C22#8903911yescharacteristic
347C4—41no—
348C4—541no—
349C4—541no—
350C4—1081no—
351C4#8901411yescharacteristic
352C4#8902111yescharacteristic
353C3—24no—
354C3#1059055414yescomplemented
355C3#1059246311yescharacteristic, complemented
356C2—241no—
357C2—721no—
358C2#1172352811yescharacteristic
359C2#1172354811yescharacteristic
360C2#1172357211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723529C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#490
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. #11723931C2semidirect
3. C3×#63636C2semidirect
4. C32×#809C2semidirect
5. #11722034C2semidirect
6. #11722032C2semidirect
7. #11722021C2semidirect
30. #63636D3semidirect
32. C3×#809D3semidirect
107. #809C3⋊D3semidirect
216. C33#490semidirect
310. C32#4764semidirect
311. C32#4764semidirect
354. C3#10590554semidirect
355. C3#10592463semidirect

Maximal subgroups

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

subgroupindex
#117239312
#117235292
C3×#636362
C32×#8092
#117220342
#117220322
#117220212
#105924633
#105905543