Finite Group Databasebeta

← #13159667

Subgroups — #13159667

Order 1728.

1048 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1048 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
1C2×#88852C211yesmaximal, characteristic
2#11723494C211yesmaximal, characteristic, complemented
3#11723440C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic, complemented
5C3×#63641C211yesmaximal, characteristic, complemented
6C32×#816C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592518—31nomaximal
9#10590770—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6289—21no—
14C2×C33×C8C2211yescharacteristic
15#88852—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88804—21no—
20#88804—21no—
21#88804—21no—
22#88795—22no—
23#64156—31no—
24C2×#3755—31no—
25C2×#3745—31no—
26C2×#3745—34no—
27C2×#3745—64no—
28#64102—31no—
29#63894—34no—
30#63864—34no—
31C2×#3717—34no—
32C32×#103—31no—
33#63641D311yescharacteristic, complemented
34#63636—31no—
35C3×#816D314yescomplemented
36C3×Dic3⋊C8—34no—
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⋊Dic3D411yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C2×C32×Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51#6289—42no—
52C33×C8—21no—
53C3×C3⋊(C3⋊C8)—21no—
54#4844—91no—
55#4819—94no—
56C2×C32×Q8—31no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—34no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C4⋊(C3⋊D3)—34no—
61C2×C4⋊(C3⋊D3)—64no—
62C2×C4⋊(C3⋊D3)—64no—
63C2×C4×C3⋊D3—61no—
64C2×C3×Dic6C2×D314yes—
65C2×#624—64no—
66C2×C32×C8—24no—
67C2×C32×C8C2×D314yes—
68C2×C32×C8C2×D311yescharacteristic
69C32×C4⋊C4—31no—
70#3755—62no—
71#3752—61no—
72#3752—61no—
73#3752—61no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76#3745—62no—
77#3745—68no—
78#3745—616no—
79C3×C4⋊Dic3C2×D314yes—
80C3×C4×Dic3—34no—
81C2×C3×C3⋊C8—34no—
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—21no—
96C3×C3⋊Dic3—21no—
97C32×Dic3—41no—
98C32×Dic3—41no—
99#947—94no—
100C2×#215—94no—
101C2×D24—94no—
102C2×D24—91no—
103C2×D24—184no—
104C2×#204—91no—
105#893—94no—
106#875—91no—
107C4×D12—91no—
108C3×C4⋊C8—91no—
109C3×#103—34no—
110#821—91no—
111#816C3⋊D311yescharacteristic, complemented
112Dic3⋊C8—34no—
113#809—94no—
114C22×C3⋊D3—61no—
115C22×C3⋊D3—61no—
116C22×C3⋊D3—64no—
117C22×C3⋊D3—64no—
118C22×C3⋊D3—124no—
119C22×C3⋊D3—124no—
120C32×Q8—61no—
121C32×Q8—61no—
122C2×C32×C4—24no—
123C2×C32×C4—61no—
124C2×C32×C4—61no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3D1211yescharacteristic
128C2×C3⋊Dic3D1211yescharacteristic
129C4⋊(C3⋊D3)—32no—
130C4⋊(C3⋊D3)—38no—
131C4⋊(C3⋊D3)—68no—
132C4⋊(C3⋊D3)—68no—
133C4⋊(C3⋊D3)—64no—
134C4⋊(C3⋊D3)—61no—
135C4⋊(C3⋊D3)—61no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—64no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—124no—
140C4⋊(C3⋊D3)—124no—
141C4×C3⋊D3—122no—
142C2×C3×Dic3—24no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—34no—
146C3×Dic6—24no—
147C3×Dic6—24no—
148#624—128no—
149C32×C8—28no—
150C32×C8—24no—
151C32×C8—21no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—64no—
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—34no—
161C2×D12—94no—
162C2×D12—91no—
163C2×D12—94no—
164C2×D12—91no—
165C2×D12—184no—
166C2×D12—184no—
167C2×C4×D3—181no—
168C2×C4×D3—184no—
169C2×Dic6C2×C3⋊D311yescharacteristic
170C2×C3×C8—24no—
171C2×C3×C8—91no—
172C2×C3×C8D3214yes—
173C2×C3×C8C2×C3⋊D311yescharacteristic
174C3×C4⋊C4—34no—
175C3×C42—91no—
176#215—188no—
177(C2×D3)⋊C4—181no—
178(C2×D3)⋊C4—181no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—184no—
181(C2×D3)⋊C4—184no—
182(C2×D3)⋊C4—184no—
183C4⋊Dic3C2×C3⋊D311yescharacteristic
184C4×Dic3—34no—
185C2×C3⋊C8—34no—
186D24—188no—
187D24—182no—
188D24—1816no—
189#204—182no—
190C22×C32—24no—
191C22×C32C2×D1211yescharacteristic
192C22×C32D3×D414yes—
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—124no—
200C2×C3⋊D3—121no—
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—121no—
214C32×C4—121no—
215C32×C4#23514yes—
216C32×C4D3×D414yes—
217C32×C4D3×D411yescharacteristic
218C32×C4#23911yescharacteristic
219C3⋊Dic3—21no—
220C3⋊Dic3—21no—
221C3×Dic3—44no—
222C3×Dic3—44no—
223C3×Dic3—64no—
224C3×Dic3—64no—
225C2×SD16—271no—
226C2×D8—271no—
227#124—271no—
228C4×D4—271no—
229C4⋊C8—91no—
230#103—31no—
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—64no—
240C3×Q8—64no—
241C2×C3×C4—24no—
242C2×C3×C4—64no—
243C2×C3×C4—64no—
244C2×C3×C4—91no—
245C2×C3×C4—91no—
246C2×C3×C4C2×D3214yes—
247C2×C3×C4C22×C3⋊D311yescharacteristic
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—34no—
251C2×Dic3—34no—
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—21no—
267Dic6—21no—
268C3×C8—28no—
269C3×C8—24no—
270C3×C8—21no—
271C3×C8—181no—
272C3⋊C8—64no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32#88414yes—
277C2×C32#89511yescharacteristic
278C2×C32#90411yescharacteristic
279C2×C32#90811yescharacteristic
280C2×C32#91414yes—
281C2×C32#91914yes—
282C3⋊D3—122no—
283C3⋊D3—128no—
284C3⋊D3—248no—
285C3⋊D3—248no—
286C3⋊D3—244no—
287C3⋊D3—241no—
288C2×Q8—31no—
289C2×D4—271no—
290C2×D4—271no—
291C22×C4—541no—
292SD16—542no—
293D8—542no—
294C2×C8—91no—
295C2×C8D3×C3⋊D311yescharacteristic
296C4⋊C4—31no—
297C22⋊C4—541no—
298C22⋊C4—541no—
299C22⋊C4—541no—
300C42—91no—
301C22×C3—24no—
302C22×C3D3×D1214yes—
303C22×C3D4×C3⋊D311yescharacteristic
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—124no—
325C3×C4—124no—
326C3×C4—181no—
327C3×C4—181no—
328C3×C4#379914yes—
329C3×C4#381114yes—
330C3×C4#382811yescharacteristic
331C3×C4D4×C3⋊D311yescharacteristic
332Dic3—41no—
333Dic3—41no—
334Dic3—64no—
335Dic3—64no—
336C32—24no—
337C32#481914yescomplemented
338C32#484411yescharacteristic, complemented
339C23—541no—
340C23—541no—
341Q8—61no—
342Q8—61no—
343D4—272no—
344D4—541no—
345D4—541no—
346D4—541no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—1082no—
352C2×C4C2×D3×C3⋊D311yescharacteristic
353C8—21no—
354C8—181no—
355C2×C3—24no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3#6379414yes—
359C2×C3#6380814yes—
360C2×C3#6391014yes—
361C2×C3#6409311yescharacteristic
362C2×C3#6412311yescharacteristic
363C2×C3#6412811yescharacteristic
364D3—362no—
365D3—368no—
366D3—728no—
367D3—728no—
368D3—721no—
369D3—724no—
370C22—542no—
371C22—1081no—
372C22—1081no—
373C22—1081no—
374C22—1081no—
375C22—1081no—
376C22C3⋊D3×D1211yescharacteristic
377C4—121no—
378C4—121no—
379C4—181no—
380C4—181no—
381C4#8902211yescharacteristic
382C4#8903911yescharacteristic
383C3—24no—
384C3#1059077014yescomplemented
385C3#1059251811yescharacteristic, complemented
386C2—1082no—
387C2—2161no—
388C2#1172331911yescharacteristic
389C2#1172334111yescharacteristic
390C2#1172352511yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
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. #11723494C2semidirect
3. #11723440C2semidirect
4. C2×#88795C2semidirect
5. C3×#63641C2semidirect
6. C32×#816C2semidirect
7. #11722035C2semidirect
33. #63641D3semidirect
35. C3×#816D3semidirect
111. #816C3⋊D3semidirect
232. C33#463semidirect
337. C32#4819semidirect
338. C32#4844semidirect
384. C3#10590770semidirect
385. C3#10592518semidirect

Maximal subgroups

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

subgroupindex
C2×#888522
#117234942
#117234402
C2×#887952
C3×#636412
C32×#8162
#117220352
#105925183
#105907703