Finite Group Databasebeta

← #13159683

Subgroups — #13159683

Order 1728.

1048 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 154 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×#88852C211yesmaximal, characteristic
2#11723484C211yesmaximal, characteristic, complemented
3#11723441C211yesmaximal, characteristic, complemented
4C2×#88797C211yesmaximal, characteristic, complemented
5C3×#63643C211yesmaximal, characteristic, complemented
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic, complemented
8#10592543—31nomaximal
9#10590770—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×#6289—21no—
14C2×C33×C8C2211yescharacteristic
15#88852—22no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88804—21no—
21#88804—21no—
22#88797—22no—
23C2×#3745—31no—
24C2×#3745—34no—
25C2×#3745—64no—
26C2×#3744—31no—
27#64084—31no—
28C4×C4⋊(C3⋊D3)—31no—
29#63894—34no—
30#63864—34no—
31C2×#3717—34no—
32C32×C4⋊C8—31no—
33#63648—31no—
34#63643D311yescharacteristic, complemented
35C3×#816—34no—
36C3×Dic3⋊C8D314yescomplemented
37#63564—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C2×C33×C4C2311yescharacteristic
41C4⋊(C3⋊(C3⋊D3))—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))D412yes—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634—21no—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51#6289—42no—
52C33×C8—21no—
53C32×C3⋊C8—21no—
54#4844—94no—
55#4819—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—34no—
60C2×C4⋊(C3⋊D3)—64no—
61C2×C4⋊(C3⋊D3)—64no—
62C2×C4×C3⋊D3—61no—
63C2×#634C2×D311yescharacteristic
64C2×C3×Dic6—34no—
65C2×#624—64no—
66C2×C32×C8—24no—
67C2×C32×C8—31no—
68C2×C32×C8C2×D314yes—
69C2×C32×C8C2×D311yescharacteristic
70C32×C42—31no—
71#3752—61no—
72#3752—61no—
73#3752—61no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75#3745—62no—
76#3745—68no—
77#3745—616no—
78#3744—62no—
79C3×C4⋊Dic3—34no—
80C3×C4×Dic3C2×D314yes—
81C2×C3×C3⋊C8C2×D314yes—
82#3722—64no—
83#3722—64no—
84#3722—64no—
85#3717—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—22no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3⋊(C3⋊D3)—41no—
91C2×C3⋊(C3⋊D3)—41no—
92C2×C3⋊(C3⋊D3)—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3×C3⋊Dic3—41no—
96C3×C3⋊Dic3—41no—
97C32×Dic3—21no—
98C32×Dic3—21no—
99#947—91no—
100C2×#215—91no—
101C2×D24—94no—
102C2×D24—91no—
103C2×D24—184no—
104C2×#204—94no—
105#893—91no—
106#875—94no—
107C4×D12—94no—
108C3×C4⋊C8—34no—
109C3×#103—91no—
110#821—94no—
111#816—34no—
112Dic3⋊C8C3⋊D311yescharacteristic, complemented
113#809—91no—
114C22×C3⋊D3—61no—
115C22×C3⋊D3—61no—
116C22×C3⋊D3—64no—
117C22×C3⋊D3—64no—
118C22×C3⋊D3—124no—
119C22×C3⋊D3—124no—
120C2×C32×C4—24no—
121C2×C32×C4—31no—
122C2×C32×C4—31no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C4⋊(C3⋊D3)—32no—
128C4⋊(C3⋊D3)—38no—
129C4⋊(C3⋊D3)—68no—
130C4⋊(C3⋊D3)—68no—
131C4⋊(C3⋊D3)—61no—
132C4⋊(C3⋊D3)—61no—
133C4⋊(C3⋊D3)—64no—
134C4⋊(C3⋊D3)—61no—
135C4⋊(C3⋊D3)—64no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)—124no—
138C4⋊(C3⋊D3)—124no—
139C4×C3⋊D3—122no—
140#634—21no—
141#634—21no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3D1214yes—
145C2×C3×Dic3D1214yes—
146C3×Dic6—64no—
147C3×Dic6—64no—
148#624—128no—
149C32×C8—28no—
150C32×C8—24no—
151C32×C8—21no—
152C32×C8—61no—
153C3×C3⋊C8—24no—
154#463—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33#13511yescharacteristic
157C2×C33AGL(1,ℤ/8)11yescharacteristic
158C3⋊(C3⋊D3)—42no—
159C3⋊(C3⋊D3)—81no—
160C2×C3×Q8—91no—
161C2×D12—94no—
162C2×D12—91no—
163C2×D12—94no—
164C2×D12—91no—
165C2×D12—184no—
166C2×D12—184no—
167C2×C4×D3—184no—
168C2×C4×D3—181no—
169C2×Dic6—34no—
170C2×C3×C8—24no—
171C2×C3×C8—34no—
172C2×C3×C8D3214yes—
173C2×C3×C8C2×C3⋊D311yescharacteristic
174C3×C4⋊C4—91no—
175C3×C42—34no—
176#215—182no—
177(C2×D3)⋊C4—184no—
178(C2×D3)⋊C4—184no—
179(C2×D3)⋊C4—184no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—181no—
182(C2×D3)⋊C4—181no—
183C4⋊Dic3—34no—
184C4×Dic3C2×C3⋊D311yescharacteristic
185C2×C3⋊C8C2×C3⋊D311yescharacteristic
186D24—188no—
187D24—182no—
188D24—1816no—
189#204—188no—
190C22×C32—24no—
191C22×C32C2×D1214yes—
192C22×C32D3×D411yescharacteristic
193C2×C3⋊D3—62no—
194C2×C3⋊D3—68no—
195C2×C3⋊D3—128no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—121no—
198C2×C3⋊D3—128no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—124no—
201C2×C3⋊D3—121no—
202C2×C3⋊D3—121no—
203C2×C3⋊D3—124no—
204C2×C3⋊D3—124no—
205C2×C3⋊D3—124no—
206C2×C3⋊D3—124no—
207C2×C3⋊D3—244no—
208C2×C3⋊D3—244no—
209C2×C3⋊D3—244no—
210C2×C3⋊D3—244no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—61no—
214C32×C4—61no—
215C32×C4#23511yescharacteristic
216C32×C4D3×D414yes—
217C32×C4D3×D411yescharacteristic
218C32×C4#23914yes—
219C3⋊Dic3—41no—
220C3⋊Dic3—41no—
221C3×Dic3—24no—
222C3×Dic3—24no—
223C3×Dic3—124no—
224C3×Dic3—124no—
225C2×SD16—271no—
226C2×D8—271no—
227#124—271no—
228C4×D4—271no—
229C4⋊C8—31no—
230#103—91no—
231#102—271no—
232C33#46311yescharacteristic, complemented, Sylow
233C22×D3—184no—
234C22×D3—181no—
235C22×D3—184no—
236C22×D3—181no—
237C22×D3—364no—
238C22×D3—364no—
239C3×Q8—181no—
240C3×Q8—181no—
241C2×C3×C4—24no—
242C2×C3×C4—34no—
243C2×C3×C4—34no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4C2×D3214yes—
247C2×C3×C4C22×C3⋊D311yescharacteristic
248C2×Dic3—64no—
249C2×Dic3—64no—
250C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
251C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
252D12—98no—
253D12—92no—
254D12—188no—
255D12—188no—
256D12—184no—
257D12—184no—
258D12—184no—
259D12—181no—
260D12—181no—
261D12—181no—
262D12—364no—
263D12—364no—
264C4×D3—362no—
265C4×D3—368no—
266Dic6—64no—
267Dic6—64no—
268C3×C8—28no—
269C3×C8—24no—
270C3×C8—21no—
271C3×C8—64no—
272C3⋊C8—21no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32#88411yescharacteristic
277C2×C32#89514yes—
278C2×C32#90414yes—
279C2×C32#90814yes—
280C2×C32#91411yescharacteristic
281C2×C32#91911yescharacteristic
282C3⋊D3—122no—
283C3⋊D3—128no—
284C3⋊D3—248no—
285C3⋊D3—248no—
286C3⋊D3—241no—
287C3⋊D3—244no—
288C2×Q8—91no—
289C2×D4—271no—
290C2×D4—271no—
291C22×C4—541no—
292SD16—542no—
293D8—542no—
294C2×C8—31no—
295C2×C8D3×C3⋊D311yescharacteristic
296C4⋊C4—91no—
297C22⋊C4—541no—
298C22⋊C4—541no—
299C22⋊C4—541no—
300C42—31no—
301C22×C3—24no—
302C22×C3D3×D1214yes—
303C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
304C2×D3—182no—
305C2×D3—188no—
306C2×D3—368no—
307C2×D3—364no—
308C2×D3—364no—
309C2×D3—361no—
310C2×D3—361no—
311C2×D3—368no—
312C2×D3—364no—
313C2×D3—364no—
314C2×D3—361no—
315C2×D3—361no—
316C2×D3—361no—
317C2×D3—364no—
318C2×D3—724no—
319C2×D3—724no—
320C2×D3—724no—
321C2×D3—724no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—64no—
325C3×C4—64no—
326C3×C4—361no—
327C3×C4—361no—
328C3×C4#379914yes—
329C3×C4#381114yes—
330C3×C4D4×C3⋊D311yescharacteristic
331C3×C4#383211yescharacteristic
332Dic3—21no—
333Dic3—21no—
334Dic3—124no—
335Dic3—124no—
336C32—24no—
337C32#481911yescharacteristic, complemented
338C32#484414yescomplemented
339C23—541no—
340C23—541no—
341Q8—181no—
342Q8—181no—
343D4—272no—
344D4—541no—
345D4—541no—
346D4—541no—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4—1082no—
352C2×C4C2×D3×C3⋊D311yescharacteristic
353C8—21no—
354C8—61no—
355C2×C3—24no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3#6379414yes—
359C2×C3#6380814yes—
360C2×C3#6391014yes—
361C2×C3#6410411yescharacteristic
362C2×C3#6411311yescharacteristic
363C2×C3#6411711yescharacteristic
364D3—362no—
365D3—368no—
366D3—728no—
367D3—728no—
368D3—721no—
369D3—724no—
370C22—542no—
371C22—1081no—
372C22—1081no—
373C22—1081no—
374C22—1081no—
375C22—1081no—
376C22D3×C4⋊(C3⋊D3)11yescharacteristic
377C4—61no—
378C4—61no—
379C4—361no—
380C4—361no—
381C4#8901811yescharacteristic
382C4#8903911yescharacteristic
383C3—24no—
384C3#1059077014yescomplemented
385C3#1059254311yescharacteristic, complemented
386C2—1082no—
387C2—2161no—
388C2#1172331511yescharacteristic
389C2#1172333711yescharacteristic
390C2#1172348611yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#88852C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#463
LCSG ▷ C2×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
2. #11723484C2semidirect
3. #11723441C2semidirect
4. C2×#88797C2semidirect
5. C3×#63643C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. #11722036C2semidirect
34. #63643D3semidirect
36. C3×Dic3⋊C8D3semidirect
112. Dic3⋊C8C3⋊D3semidirect
232. C33#463semidirect
337. C32#4819semidirect
338. C32#4844semidirect
384. C3#10590770semidirect
385. C3#10592543semidirect

Maximal subgroups

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

subgroupindex
C2×#888522
#117234842
#117234412
C2×#887972
C3×#636432
C32×Dic3⋊C82
#117220362
#105925433
#105907703