Finite Group Databasebeta

← #13161193

Subgroups — #13161193

Order 1728.

1140 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 188 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×C32×#239C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3#11723515C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722205C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592937—31nomaximal
9#10591586—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C32×#239—22no—
13C32×#239—22no—
14C2×C32×D12—21no—
15C2×C32×D12C2211yescharacteristic
16C2×C32×C4×D3—21no—
17C2×C32×C4×D3C2211yescharacteristic
18C2×#6290—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#3755—22no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23#88803—21no—
24#88789—22no—
25C2×C32×#41—31no—
26C2×C3×#239D314yescomplemented
27C2×#3755D311yescharacteristic, complemented
28C2×#3753—31no—
29#64139—31no—
30C2×C3×#215—34no—
31#63908—34no—
32C2×#3713—34no—
33#63662—31no—
34#63662—34no—
35#63662—64no—
36#63659—31no—
37#63657—31no—
38#63563—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3×C3⋊D3—21no—
41C22×C32×D3—21no—
42C22×C32×D3D411yescharacteristic
43C33×Q8—21no—
44C33×Q8D412yescomplemented
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C2×C32×Dic3D411yescharacteristic
51C32×D12—24no—
52C32×D12—21no—
53C32×D12D412yescomplemented
54C32×C4×D3—24no—
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57#6290—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5238—94no—
61#5172—91no—
62C32×#41—62no—
63C32×#41—62no—
64C2×C32×Q8—24no—
65C2×C32×Q8C2×D314yes—
66C2×C32×Q8C2×D311yescharacteristic
67C2×C32×D4—31no—
68C2×C32×D4—61no—
69C22×C32×C4—31no—
70C22×C32×C4—61no—
71C2×#638—61no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C3×#239—28no—
74C3×#239—28no—
75C2×C3×D12—24no—
76C2×C3×D12—34no—
77C2×C3×D12C2×D314yes—
78C2×C3×C4×D3—24no—
79C2×C3×C4×D3C2×D314yes—
80C2×#625—64no—
81#3755—22no—
82#3753—62no—
83#3752—61no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—34no—
86C4⋊(C3⋊Dic3)—64no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C3×#215—68no—
92C2×C3×C3⋊C8—34no—
93(C2×D3)⋊Dic3—64no—
94#3713—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3—41no—
98C2×C3×C3⋊D3—41no—
99C2×C32×D3—21no—
100C2×C32×D3—22no—
101C2×C32×D3—21no—
102C2×C32×D3—21no—
103C2×C32×D3—41no—
104C2×C32×D3—41no—
105C33×C4—22no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3⋊(C3⋊Dic3)—41no—
109C32×Dic3—21no—
110C2×C3×#41—34no—
111C2×#239C3⋊D311yescharacteristic, complemented
112C2×C3×SD16—91no—
113C2×#215—34no—
114#938—91no—
115C2×#213—94no—
116C2×#213—91no—
117#930—94no—
118#835—94no—
119#835—91no—
120#835—184no—
121#832—94no—
122#830—94no—
123#821—91no—
124(C2×D3)⋊C8—91no—
125C23×C32—31no—
126C23×C32—61no—
127C22×C3⋊D3—21no—
128C22×C3×D3—24no—
129C22×C3×D3—64no—
130C22×C3×D3#6714yes—
131C32×Q8—21no—
132C32×Q8—28no—
133C32×Q8—24no—
134C32×Q8—44no—
135C32×Q8#6718yescomplemented
136C32×Q8#6712yescomplemented
137C32×D4—32no—
138C32×D4—64no—
139C32×D4—61no—
140C2×C32×C4—21no—
141C2×C32×C4—24no—
142C2×C32×C4—24no—
143C2×C32×C4—31no—
144C2×C32×C4—44no—
145C2×C32×C4—64no—
146C2×C32×C4—61no—
147C2×C32×C4—61no—
148C2×C32×C4C22×D314yes—
149C2×C32×C4C22×D311yescharacteristic
150#638—122no—
151C2×C3⋊Dic3—61no—
152C2×C3⋊Dic3—64no—
153C2×C3⋊Dic3—124no—
154C4⋊(C3⋊D3)—21no—
155C4⋊(C3⋊D3)—21no—
156C2×C3×Dic3#6714yes—
157C3×D12—216no—
158C3×D12—24no—
159C3×D12—64no—
160C3×D12—64no—
161C3×D12#6718yescomplemented
162C3×C4×D3—216no—
163C3×C4×D3—24no—
164C3×C4×D3—24no—
165#625—128no—
166C3⋊(C3⋊C8)—21no—
167C3⋊(C3⋊C8)—61no—
168C3⋊(C3⋊C8)—64no—
169C3⋊(C3⋊C8)—124no—
170C3×C3⋊C8—64no—
171#449—271noSylow
172C2×C33#12011yescharacteristic
173C2×C33#13511yescharacteristic
174C2×C33AGL(1,ℤ/8)11yescharacteristic
175C3×C3⋊D3—81no—
176C32×D3—41no—
177C32×D3—42no—
178C3×#41—68no—
179C3×#41—68no—
180C2×C3×Q8—24no—
181C2×C3×Q8D3214yes—
182C2×C3×Q8C2×C3⋊D311yescharacteristic
183C2×C3×D4—34no—
184C2×C3×D4—64no—
185C2×C3×D4—91no—
186C22×C3×C4—34no—
187C22×C3×C4—64no—
188C2×#67—184no—
189C2×#67—181no—
190#239—22no—
191#239—22no—
192C2×D12—21no—
193C2×D12—34no—
194C2×D12C2×C3⋊D311yescharacteristic
195C2×C4×D3—21no—
196C2×C4×D3C2×C3⋊D311yescharacteristic
197C3×SD16—182no—
198C2×C3×C8—91no—
199#217—181no—
200#215—68no—
201#213—188no—
202#213—182no—
203(C2×D3)⋊C4—184no—
204C4⋊Dic3—94no—
205C4⋊Dic3—91no—
206C4⋊Dic3—184no—
207C2×C3⋊C8—34no—
208C2×C3⋊C8—94no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—184no—
211C22×C32—24no—
212C22×C32—61no—
213C22×C32—61no—
214C22×C32—62no—
215C22×C32—61no—
216C22×C32—121no—
217C22×C32—121no—
218C22×C32D3×D411yescharacteristic
219C22×C32C2×#6714yes—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3×D3—24no—
224C2×C3×D3—24no—
225C2×C3×D3—24no—
226C2×C3×D3—28no—
227C2×C3×D3—44no—
228C2×C3×D3—44no—
229C2×C3×D3—124no—
230C2×C3×D3—124no—
231C2×C3×D3—124no—
232C32×C4—22no—
233C32×C4—24no—
234C32×C4—24no—
235C32×C4—28no—
236C32×C4—48no—
237C32×C4—61no—
238C32×C4D3×D411yescharacteristic
239C32×C4C2×#6714yes—
240C32×C4C2×#6714yes—
241C32×C4C2×#6711yescharacteristic
242C3⋊Dic3—121no—
243C3⋊Dic3—124no—
244C3⋊Dic3—244no—
245C3×Dic3—24no—
246C2×#41—31no—
247C2×SD16—91no—
248C2×D8—271no—
249#121—271no—
250#103—271no—
251#102—271no—
252C22⋊C8—271no—
253C33#44911yescharacteristic, complemented, Sylow
254C23×C3—34no—
255C23×C3—64no—
256C23×C3—181no—
257C22×D3—21no—
258C22×D3—64no—
259C22×D3#63811yescharacteristic
260C3×Q8—24no—
261C3×Q8—21no—
262C3×Q8—28no—
263C3×Q8—44no—
264C3×Q8#62518yescomplemented
265C3×Q8#63812yescomplemented
266C3×D4—38no—
267C3×D4—616no—
268C3×D4—64no—
269C3×D4—181no—
270C3×D4—181no—
271C2×C3×C4—24no—
272C2×C3×C4—21no—
273C2×C3×C4—24no—
274C2×C3×C4—34no—
275C2×C3×C4—44no—
276C2×C3×C4—616no—
277C2×C3×C4—64no—
278C2×C3×C4—64no—
279C2×C3×C4C2×D3214yes—
280C2×C3×C4C22×C3⋊D311yescharacteristic
281#67—362no—
282#67—368no—
283C2×Dic3—184no—
284C2×Dic3—181no—
285C2×Dic3—364no—
286C2×Dic3#63811yescharacteristic
287D12—24no—
288D12—21no—
289D12—64no—
290D12—64no—
291D12#63812yescomplemented
292C4×D3—24no—
293C4×D3—21no—
294C4×D3—21no—
295C3×C8—181no—
296C3⋊C8—64no—
297C3⋊C8—184no—
298C3⋊C8—181no—
299C3⋊C8—364no—
300C2×C32—24no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—121no—
304C2×C32—122no—
305C2×C32#91411yescharacteristic
306C2×C32#91611yescharacteristic
307C2×C32#93711yescharacteristic
308C2×C32#94914yes—
309C2×C32#95014yes—
310C2×C32#95314yes—
311C3⋊D3—81no—
312C3×D3—44no—
313C3×D3—48no—
314C3×D3—244no—
315#41—62no—
316#41—62no—
317C2×Q8D3×C3⋊D311yescharacteristic
318C2×D4—31no—
319C2×D4—61no—
320C2×D4—91no—
321C2×D4—541no—
322C22×C4—31no—
323C22×C4—61no—
324SD16—182no—
325D8—542no—
326C2×C8—91no—
327C2×C8—271no—
328C4⋊C4—271no—
329C22⋊C4—541no—
330C22×C3—24no—
331C22×C3—64no—
332C22×C3—64no—
333C22×C3—68no—
334C22×C3—64no—
335C22×C3—124no—
336C22×C3—124no—
337C22×C3—361no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3D3×#6714yes—
341C22×C3C2×#63811yescharacteristic
342C2×D3—21no—
343C2×D3—21no—
344C2×D3—21no—
345C2×D3—22no—
346C2×D3—41no—
347C2×D3—41no—
348C2×D3—124no—
349C2×D3—124no—
350C2×D3—124no—
351C3×C4—22no—
352C3×C4—28no—
353C3×C4—24no—
354C3×C4—24no—
355C3×C4—48no—
356C3×C4—64no—
357C3×C4C2×#62514yes—
358C3×C4D3×#6714yes—
359C3×C4C2×#63811yescharacteristic
360C3×C4C2×#63811yescharacteristic
361Dic3—21no—
362Dic3—361no—
363Dic3—364no—
364Dic3—724no—
365C32—24no—
366C32#517211yescharacteristic, complemented
367C32#523814yescomplemented
368C23—31no—
369C23—61no—
370C23—181no—
371Q8—21no—
372Q8#629012yescomplemented
373D4—32no—
374D4—64no—
375D4—61no—
376D4—181no—
377D4—181no—
378D4—1082no—
379C2×C4—21no—
380C2×C4—31no—
381C2×C4—64no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—541no—
385C2×C4C2×D3×C3⋊D311yescharacteristic
386C8—181no—
387C8—541no—
388C2×C3—24no—
389C2×C3—24no—
390C2×C3—24no—
391C2×C3—124no—
392C2×C3—128no—
393C2×C3—721no—
394C2×C3#6393914yes—
395C2×C3#6394714yes—
396C2×C3#6397614yes—
397C2×C3#6415811yescharacteristic
398C2×C3#6415911yescharacteristic
399C2×C3#6416211yescharacteristic
400D3—41no—
401D3—42no—
402D3—244no—
403C22—61no—
404C22—61no—
405C22—62no—
406C22—61no—
407C22—121no—
408C22—121no—
409C22—361no—
410C22—361no—
411C22—361no—
412C22D3×#63811yescharacteristic
413C4—22no—
414C4—61no—
415C4—1081no—
416C4C2×#629011yescharacteristic
417C4D3×#63811yescharacteristic
418C3—24no—
419C3#1059158614yescomplemented
420C3#1059293711yescharacteristic, complemented
421C2—121no—
422C2—122no—
423C2—721no—
424C2#1172356811yescharacteristic
425C2#1172358011yescharacteristic
426C2#1172362911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#449
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
1. C2×C32×#239C2semidirect
2. C2×C3×#3755C2semidirect
3. #11723515C2semidirect
4. C2×#88789C2semidirect
5. #11722205C2semidirect
6. #11722032C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
26. C2×C3×#239D3semidirect
27. C2×#3755D3semidirect
44. C33×Q8D4semidirect
53. C32×D12D4semidirect
111. C2×#239C3⋊D3semidirect
135. C32×Q8#67semidirect
136. C32×Q8#67semidirect
161. C3×D12#67semidirect
253. C33#449semidirect
264. C3×Q8#625semidirect
265. C3×Q8#638semidirect
291. D12#638semidirect
366. C32#5172semidirect
367. C32#5238semidirect
372. Q8#6290semidirect
419. C3#10591586semidirect
420. C3#10592937semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
C2×C3×#37552
#117235152
C2×#887892
#117222052
#117220322
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105929373
#105915863