Finite Group Databasebeta

← #13159690

Subgroups — #13159690

Order 1728.

992 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 184 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
1C2×#88851C211yesmaximal, characteristic
2#11723484C211yesmaximal, characteristic
3#11723399C211yesmaximal, characteristic, complemented
4C3×#63645C211yesmaximal, characteristic, complemented
5C32×#815C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic
8#10592409—31nomaximal
9#10590773—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×#6289—21no—
13C2×C33×C8C2211yescharacteristic
14#88851C414yescomplemented
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20#88804—21no—
21Dic3×C3⋊Dic3—21no—
22C2×#3744—31no—
23C2×#3744—34no—
24C2×#3744—64no—
25C4×C4⋊(C3⋊D3)—31no—
26C4×#634—31no—
27#63894—34no—
28#63844—34no—
29C32×#97—31no—
30#63648—31no—
31#63645D311yescharacteristic, complemented
32#63643—31no—
33C3×#818—34no—
34C3×#815D314yescomplemented
35#63567—34no—
36#63564—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C4⋊(C3⋊(C3⋊D3))—21no—
41C4⋊(C3⋊(C3⋊D3))C2×C412yes—
42#6308—21no—
43#6308C2×C412yescomplemented
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3D411yescharacteristic
47C2×C32×Dic3D411yescharacteristic
48#6289—42no—
49C33×C8C2×C412yescomplemented
50C32×C3⋊C8—21no—
51#4889—91no—
52#4710—94no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—34no—
55C2×C4⋊(C3⋊D3)—64no—
56C2×C4×C3⋊D3—61no—
57C2×#634—31no—
58C2×#634—34no—
59C2×#634—64no—
60C2×#624—64no—
61C2×C32×C8—24no—
62C2×C32×C8—31no—
63C2×C32×C8C2×D314yes—
64C2×C32×C8C2×D311yescharacteristic
65C32×C42—31no—
66#3752—61no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69#3750—61no—
70C4×C3⋊Dic3—61no—
71#3744—34no—
72#3744—316no—
73#3744—616no—
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3—34no—
76C3×C4×Dic3C2×D314yes—
77C2×C3×C3⋊C8C2×D314yes—
78#3724—64no—
79#3722—64no—
80Dic32—64no—
81C22×C33C2×D411yescharacteristic
82C2×C3⋊(C3⋊D3)—22no—
83C2×C3⋊(C3⋊D3)—41no—
84C2×C3⋊(C3⋊D3)—41no—
85C33×C4C22×C411yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—22no—
88C3×C3⋊Dic3—41no—
89C3×C3⋊Dic3—41no—
90C32×Dic3—21no—
91C32×Dic3—21no—
92C2×#204—94no—
93C2×#204—91no—
94C2×#204—184no—
95#893—91no—
96#887—91no—
97C4×D12—94no—
98C4×Dic6—94no—
99C3×#107—91no—
100C3×#97—34no—
101#821—94no—
102#818—34no—
103#816—94no—
104#815C3⋊D311yescharacteristic, complemented
105#810—91no—
106#809—91no—
107C22×C3⋊D3—61no—
108C22×C3⋊D3—64no—
109C22×C3⋊D3—124no—
110C2×C32×C4—24no—
111C2×C32×C4—31no—
112C2×C32×C4—31no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—124no—
120C4⋊(C3⋊D3)—32no—
121C4⋊(C3⋊D3)—38no—
122C4⋊(C3⋊D3)—68no—
123C4⋊(C3⋊D3)—61no—
124C4⋊(C3⋊D3)—64no—
125C4⋊(C3⋊D3)—124no—
126C4×C3⋊D3—122no—
127#634—32no—
128#634—38no—
129#634—68no—
130#634—61no—
131#634—64no—
132#634—124no—
133C2×C3×Dic3—64no—
134C2×C3×Dic3—64no—
135C2×C3×Dic3D1214yes—
136C2×C3×Dic3D1214yes—
137#624—128no—
138C32×C8—28no—
139C32×C8—61no—
140C32×C8C4×D318yescomplemented
141C32×C8C4×D312yescomplemented
142C3×C3⋊C8—24no—
143#440—271noSylow
144C2×C33C4×D411yescharacteristic
145C2×C33AGL(1,ℤ/8)11yescharacteristic
146C2×C33#13711yescharacteristic
147C3⋊(C3⋊D3)—42no—
148C2×D12—94no—
149C2×D12—91no—
150C2×D12—184no—
151C2×C4×D3—184no—
152C2×C4×D3—181no—
153C2×Dic6—94no—
154C2×Dic6—91no—
155C2×Dic6—184no—
156C2×C3×C8—24no—
157C2×C3×C8—34no—
158C2×C3×C8D3214yes—
159C2×C3×C8C2×C3⋊D311yescharacteristic
160C3×C4⋊C4—91no—
161C3×C4⋊C4—91no—
162C3×C42—34no—
163(C2×D3)⋊C4—184no—
164(C2×D3)⋊C4—181no—
165C4⋊Dic3—34no—
166C4⋊Dic3—34no—
167Dic3⋊C4—184no—
168Dic3⋊C4—181no—
169C4×Dic3—184no—
170C4×Dic3—181no—
171C4×Dic3C2×C3⋊D311yescharacteristic
172C2×C3⋊C8C2×C3⋊D311yescharacteristic
173#204—916no—
174#204—94no—
175#204—1816no—
176C22×C32—24no—
177C22×C32C2×D1214yes—
178C22×C32D3×D411yescharacteristic
179C2×C3⋊D3—62no—
180C2×C3⋊D3—68no—
181C2×C3⋊D3—128no—
182C2×C3⋊D3—121no—
183C2×C3⋊D3—121no—
184C2×C3⋊D3—124no—
185C2×C3⋊D3—124no—
186C2×C3⋊D3—244no—
187C2×C3⋊D3—244no—
188C32×C4—24no—
189C32×C4—24no—
190C32×C4—61no—
191C32×C4—61no—
192C32×C4C2×C4×D314yes—
193C32×C4C2×C4×D311yescharacteristic
194C32×C4#23514yes—
195C32×C4#23911yescharacteristic
196C3⋊Dic3—41no—
197C3⋊Dic3—41no—
198C3⋊Dic3—62no—
199C3⋊Dic3—68no—
200C3⋊Dic3—128no—
201C3×Dic3—24no—
202C3×Dic3—24no—
203C3×Dic3—124no—
204C3×Dic3—124no—
205C2×SD16—271no—
206C4×Q8—271no—
207C4×D4—271no—
208#107—91no—
209#103—271no—
210#102—271no—
211#97—31no—
212C33#44011yescharacteristic, complemented, Sylow
213C22×D3—184no—
214C22×D3—181no—
215C22×D3—364no—
216C2×C3×C4—24no—
217C2×C3×C4—34no—
218C2×C3×C4—34no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4C2×D3214yes—
222C2×C3×C4C22×C3⋊D311yescharacteristic
223C2×Dic3—64no—
224C2×Dic3—64no—
225C2×Dic3—184no—
226C2×Dic3—181no—
227C2×Dic3—364no—
228C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
229C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
230D12—98no—
231D12—92no—
232D12—188no—
233D12—184no—
234D12—181no—
235D12—364no—
236C4×D3—362no—
237C4×D3—368no—
238Dic6—98no—
239Dic6—92no—
240Dic6—188no—
241Dic6—184no—
242Dic6—181no—
243Dic6—364no—
244C3×C8—28no—
245C3×C8—64no—
246C3×C8#62418yescomplemented
247C3×C8C4×C3⋊D312yescomplemented
248C3⋊C8—21no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32C4×D1214yes—
253C2×C32#89311yescharacteristic
254C2×C32#90814yes—
255C2×C32#90914yes—
256C2×C32#91111yescharacteristic
257C2×C32#91811yescharacteristic
258C3⋊D3—122no—
259C3⋊D3—128no—
260C3⋊D3—248no—
261C2×Q8—271no—
262C2×D4—271no—
263C22×C4—541no—
264SD16—274no—
265C2×C8—31no—
266C2×C8D3×C3⋊D311yescharacteristic
267C4⋊C4—91no—
268C4⋊C4—91no—
269C4⋊C4—541no—
270C22⋊C4—541no—
271C42—31no—
272C42—541no—
273C22×C3—24no—
274C22×C3D3×D1214yes—
275C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
276C2×D3—182no—
277C2×D3—188no—
278C2×D3—368no—
279C2×D3—364no—
280C2×D3—364no—
281C2×D3—361no—
282C2×D3—361no—
283C2×D3—724no—
284C2×D3—724no—
285C3×C4—24no—
286C3×C4—24no—
287C3×C4—64no—
288C3×C4—64no—
289C3×C4—361no—
290C3×C4—361no—
291C3×C4#379914yes—
292C3×C4C2×#62414yes—
293C3×C4C2×C4×C3⋊D311yescharacteristic
294C3×C4#382811yescharacteristic
295Dic3—21no—
296Dic3—21no—
297Dic3—124no—
298Dic3—124no—
299Dic3—182no—
300Dic3—188no—
301Dic3—368no—
302C32—24no—
303C32#471014yescomplemented
304C32#488911yescharacteristic, complemented
305C23—541no—
306Q8—272no—
307Q8—541no—
308D4—272no—
309D4—541no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—181no—
313C2×C4—181no—
314C2×C4—541no—
315C2×C4—1082no—
316C2×C4C2×D3×C3⋊D311yescharacteristic
317C8—61no—
318C8#628912yescomplemented
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3#6379514yes—
323C2×C3#6380114yes—
324C2×C3#6389414yes—
325C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
326C2×C3#6411711yescharacteristic
327C2×C3#6411811yescharacteristic
328D3—368no—
329D3—362no—
330D3—728no—
331C22—542no—
332C22—1081no—
333C22—1081no—
334C22D3×C4⋊(C3⋊D3)11yescharacteristic
335C4—61no—
336C4—61no—
337C4—361no—
338C4—361no—
339C4—542no—
340C4#8902211yescharacteristic
341C4C2×#628911yescharacteristic
342C3—24no—
343C3#1059077314yescomplemented
344C3#1059240911yescharacteristic, complemented
345C2—1082no—
346C2#1172331611yescharacteristic
347C2#1172332711yescharacteristic
348C2#1172348411yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #11723399C2semidirect
4. C3×#63645C2semidirect
5. C32×#815C2semidirect
6. #11722042C2semidirect
14. #88851C4semidirect
31. #63645D3semidirect
34. C3×#815D3semidirect
43. #6308C2×C4semidirect
49. C33×C8C2×C4semidirect
104. #815C3⋊D3semidirect
140. C32×C8C4×D3semidirect
141. C32×C8C4×D3semidirect
212. C33#440semidirect
246. C3×C8#624semidirect
247. C3×C8C4×C3⋊D3semidirect
303. C32#4710semidirect
304. C32#4889semidirect
318. C8#6289semidirect
343. C3#10590773semidirect
344. C3#10592409semidirect

Maximal subgroups

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

subgroupindex
C2×#888512
#117234842
#117233992
C3×#636452
C32×#8152
#117220422
#117220362
#105924093
#105907733