Finite Group Databasebeta

← #13161198

Subgroups — #13161198

Order 1728.

1628 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1628 conjugacy classes of subgroups — too many to draw. Only its 168 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×#89080C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3#11723532C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592509—31nomaximal
9#10591593—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11#89080—22no—
12#89080—22no—
13C2×C4⋊(C3⋊(C3⋊D3))—21no—
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)—21no—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×#6292—21no—
19C3×#3755—22no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88803—21no—
24#88791—22no—
25C2×#3832—31no—
26C2×#3832—34no—
27C2×#3832—64no—
28C2×#3755D311yescharacteristic, complemented
29C2×#3745—31no—
30#64104—31no—
31C2×C3×#215—34no—
32#63916—34no—
33C2×#3714—34no—
34C32×#103—31no—
35#63647—31no—
36#63636—31no—
37C3×#835D314yescomplemented
38#63564—34no—
39#63557—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3⋊(C3⋊D3)D411yescharacteristic
42C22×C3×C3⋊D3—21no—
43C33×Q8—21no—
44C33×Q8D412yescomplemented
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)D411yescharacteristic
48C4⋊(C3⋊(C3⋊D3))—24no—
49C4⋊(C3⋊(C3⋊D3))—21no—
50C4⋊(C3⋊(C3⋊D3))D412yescomplemented
51C4×C3⋊(C3⋊D3)—24no—
52C4×C3⋊(C3⋊D3)—21no—
53C4×C3⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C2×C32×Dic3—21no—
57#6292—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#5172—91no—
61#4810—94no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D311yescharacteristic
65#3832—62no—
66#3832—68no—
67#3832—616no—
68#3832—616no—
69#3832—68no—
70#3832—62no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—34no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—64no—
75C2×C4⋊(C3⋊D3)—64no—
76C2×C4⋊(C3⋊D3)—68no—
77C2×C4⋊(C3⋊D3)—61no—
78C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
79C2×C4×C3⋊D3—31no—
80C2×C4×C3⋊D3—34no—
81C2×C4×C3⋊D3—61no—
82C2×C4×C3⋊D3—64no—
83C2×C4×C3⋊D3—64no—
84C2×C4×C3⋊D3—68no—
85C2×C3×D12—34no—
86C2×#626—64no—
87C2×C32×C8—31no—
88C32×C4⋊C4—31no—
89#3755—22no—
90#3752—61no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92#3745—62no—
93C3×#215—68no—
94C3×C4⋊Dic3C2×D314yes—
95C2×C3×C3⋊C8—34no—
96C2×C3×C3⋊C8C2×D314yes—
97(C2×D3)⋊Dic3—64no—
98#3714—68no—
99C22×C33C2×D411yescharacteristic
100C2×C3⋊(C3⋊D3)—21no—
101C2×C3⋊(C3⋊D3)—22no—
102C2×C3⋊(C3⋊D3)—21no—
103C2×C3⋊(C3⋊D3)—21no—
104C2×C3⋊(C3⋊D3)—41no—
105C2×C3⋊(C3⋊D3)—41no—
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3—41no—
108C2×C3×C3⋊D3—41no—
109C33×C4—22no—
110C33×C4C2×D411yescharacteristic
111C33×C4C2×D411yescharacteristic
112C3⋊(C3⋊Dic3)—21no—
113C32×Dic3—41no—
114C2×#239—94no—
115C2×#239—91no—
116C2×#239—184no—
117C2×C3×SD16—91no—
118C2×#215—34no—
119#938—91no—
120C2×#213—91no—
121C2×D24—94no—
122#895—94no—
123C3×#103—34no—
124#835C3⋊D311yescharacteristic, complemented
125#821—91no—
126(C2×D3)⋊C8—94no—
127(C2×D3)⋊C8—91no—
128#809—94no—
129C22×C3⋊D3—21no—
130C22×C3⋊D3—31no—
131C22×C3⋊D3—34no—
132C22×C3⋊D3—64no—
133C22×C3⋊D3—68no—
134C22×C3⋊D3—64no—
135C22×C3⋊D3—61no—
136C22×C3×D3—64no—
137C32×Q8—28no—
138C32×Q8—24no—
139C32×Q8—21no—
140C32×Q8—28no—
141C32×Q8D1218yescomplemented
142C32×Q8#6712yescomplemented
143C2×C32×C4—28no—
144C2×C32×C4—24no—
145C2×C32×C4—21no—
146C2×C32×C4—24no—
147C2×C32×C4—61no—
148C2×C32×C4C22×D314yes—
149C2×C32×C4C22×D311yescharacteristic
150C2×C3⋊Dic3—31no—
151C2×C3⋊Dic3—34no—
152C2×C3⋊Dic3—64no—
153C4⋊(C3⋊D3)—21no—
154C4⋊(C3⋊D3)—21no—
155C4⋊(C3⋊D3)—32no—
156C4⋊(C3⋊D3)—38no—
157C4⋊(C3⋊D3)—68no—
158C4⋊(C3⋊D3)—632no—
159C4⋊(C3⋊D3)—616no—
160C4⋊(C3⋊D3)—64no—
161C4⋊(C3⋊D3)—68no—
162C4⋊(C3⋊D3)—64no—
163C4⋊(C3⋊D3)—61no—
164C4⋊(C3⋊D3)—122no—
165C4×C3⋊D3—632no—
166C4×C3⋊D3—616no—
167C4×C3⋊D3—64no—
168C4×C3⋊D3—68no—
169C4×C3⋊D3—68no—
170C4×C3⋊D3—64no—
171C4×C3⋊D3—64no—
172C4×C3⋊D3—61no—
173C4×C3⋊D3—61no—
174C2×C3×Dic3—24no—
175C3×D12—64no—
176C3×D12—64no—
177#626—128no—
178C32×C8—61no—
179C3⋊(C3⋊C8)—21no—
180C3×C3⋊C8—24no—
181C3×C3⋊C8—64no—
182#449—271noSylow
183C2×C33#12011yescharacteristic
184C2×C33#13511yescharacteristic
185C2×C33AGL(1,ℤ/8)11yescharacteristic
186C3⋊(C3⋊D3)—41no—
187C3⋊(C3⋊D3)—42no—
188C3×C3⋊D3—81no—
189C2×C3×Q8—24no—
190C2×C3×Q8D3214yes—
191C2×C3×Q8C2×C3⋊D311yescharacteristic
192C2×C3×D4—91no—
193C2×#67—181no—
194#239—188no—
195#239—182no—
196#239—1816no—
197#239—1816no—
198#239—188no—
199#239—182no—
200C2×D12—34no—
201C2×D12—94no—
202C2×D12—91no—
203C2×D12—184no—
204C2×D12—184no—
205C2×D12—181no—
206C2×D12—188no—
207C2×D12—184no—
208C2×C4×D3—94no—
209C2×C4×D3—91no—
210C2×C4×D3—184no—
211C2×C4×D3—184no—
212C2×C4×D3—181no—
213C2×C4×D3—188no—
214C3×SD16—182no—
215C2×C3×C8—34no—
216C2×C3×C8—91no—
217C3×C4⋊C4—34no—
218#217—181no—
219#215—68no—
220#213—182no—
221(C2×D3)⋊C4—184no—
222C4⋊Dic3C2×C3⋊D311yescharacteristic
223C2×C3⋊C8—34no—
224C2×C3⋊C8C2×C3⋊D311yescharacteristic
225D24—188no—
226C22×C32—24no—
227C22×C32D3×D414yes—
228C22×C32D3×D411yescharacteristic
229C2×C3⋊D3—41no—
230C2×C3⋊D3—41no—
231C2×C3⋊D3—41no—
232C2×C3⋊D3—616no—
233C2×C3⋊D3—68no—
234C2×C3⋊D3—68no—
235C2×C3⋊D3—64no—
236C2×C3⋊D3—61no—
237C2×C3⋊D3—61no—
238C2×C3⋊D3—62no—
239C2×C3⋊D3—61no—
240C2×C3⋊D3—64no—
241C2×C3⋊D3—64no—
242C2×C3⋊D3—124no—
243C2×C3⋊D3—124no—
244C2×C3⋊D3—128no—
245C2×C3⋊D3—128no—
246C2×C3⋊D3—124no—
247C2×C3⋊D3—121no—
248C2×C3⋊D3—121no—
249C2×C3⋊D3—124no—
250C2×C3×D3—124no—
251C2×C3×D3—124no—
252C2×C3×D3—124no—
253C32×C4—216no—
254C32×C4—28no—
255C32×C4—22no—
256C32×C4—24no—
257C32×C4—24no—
258C32×C4—121no—
259C32×C4C2×D1214yes—
260C32×C4D3×D414yes—
261C32×C4D3×D411yescharacteristic
262C32×C4C2×#6711yescharacteristic
263C3⋊Dic3—68no—
264C3⋊Dic3—64no—
265C3⋊Dic3—61no—
266C3×Dic3—44no—
267C2×#41—271no—
268C2×SD16—91no—
269C2×D8—271no—
270#121—271no—
271#103—31no—
272#102—271no—
273C22⋊C8—271no—
274C33#44911yescharacteristic, complemented, Sylow
275C23×C3—181no—
276C22×D3—64no—
277C22×D3—94no—
278C22×D3—91no—
279C22×D3—184no—
280C22×D3—188no—
281C22×D3—184no—
282C22×D3—181no—
283C3×Q8—28no—
284C3×Q8—24no—
285C3×Q8—21no—
286C3×Q8—28no—
287C3×Q8#62618yescomplemented
288C3×Q8C4⋊(C3⋊D3)12yescomplemented
289C3×D4—181no—
290C3×D4—181no—
291C2×C3×C4—28no—
292C2×C3×C4—24no—
293C2×C3×C4—21no—
294C2×C3×C4—24no—
295C2×C3×C4—64no—
296C2×C3×C4C2×D3214yes—
297C2×C3×C4C22×C3⋊D311yescharacteristic
298#67—362no—
299C2×Dic3—21no—
300C2×Dic3—94no—
301C2×Dic3—91no—
302C2×Dic3—184no—
303D12—64no—
304D12—64no—
305D12—98no—
306D12—92no—
307D12—188no—
308D12—1832no—
309D12—184no—
310D12—1816no—
311D12—188no—
312D12—184no—
313D12—181no—
314D12—368no—
315C4×D3—1832no—
316C4×D3—184no—
317C4×D3—1816no—
318C4×D3—188no—
319C4×D3—188no—
320C4×D3—184no—
321C4×D3—184no—
322C4×D3—181no—
323C4×D3—181no—
324C3×C8—64no—
325C3×C8—181no—
326C3⋊C8—21no—
327C3⋊C8—64no—
328C2×C32—24no—
329C2×C32—24no—
330C2×C32—24no—
331C2×C32#88214yes—
332C2×C32#91414yes—
333C2×C32#91411yescharacteristic
334C2×C32#91611yescharacteristic
335C2×C32#91914yes—
336C2×C32#93711yescharacteristic
337C3⋊D3—81no—
338C3⋊D3—128no—
339C3⋊D3—1216no—
340C3⋊D3—121no—
341C3⋊D3—122no—
342C3⋊D3—124no—
343C3⋊D3—128no—
344C3×D3—244no—
345#41—542no—
346#41—542no—
347C2×Q8D3×C3⋊D311yescharacteristic
348C2×D4—91no—
349C2×D4—271no—
350C2×D4—541no—
351C2×D4—541no—
352C22×C4—271no—
353C22×C4—541no—
354SD16—182no—
355D8—542no—
356C2×C8—31no—
357C2×C8—91no—
358C4⋊C4—31no—
359C22⋊C4—541no—
360C22×C3—24no—
361C22×C3—361no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3#381114yes—
365C22×C3D4×C3⋊D311yescharacteristic
366C2×D3—124no—
367C2×D3—124no—
368C2×D3—124no—
369C2×D3—1816no—
370C2×D3—188no—
371C2×D3—182no—
372C2×D3—181no—
373C2×D3—181no—
374C2×D3—181no—
375C2×D3—184no—
376C2×D3—184no—
377C2×D3—188no—
378C2×D3—184no—
379C2×D3—364no—
380C2×D3—364no—
381C2×D3—368no—
382C2×D3—368no—
383C2×D3—361no—
384C2×D3—361no—
385C2×D3—364no—
386C2×D3—364no—
387C3×C4—216no—
388C3×C4—22no—
389C3×C4—28no—
390C3×C4—24no—
391C3×C4—24no—
392C3×C4—124no—
393C3×C4C2×#62614yes—
394C3×C4#381114yes—
395C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
396C3×C4D4×C3⋊D311yescharacteristic
397Dic3—41no—
398Dic3—188no—
399Dic3—181no—
400Dic3—184no—
401C32—24no—
402C32#481014yescomplemented
403C32#517211yescharacteristic, complemented
404C23—181no—
405C23—271no—
406C23—541no—
407Q8—21no—
408Q8#629212yescomplemented
409D4—181no—
410D4—181no—
411D4—272no—
412D4—544no—
413D4—541no—
414D4—1082no—
415C2×C4—21no—
416C2×C4—61no—
417C2×C4—271no—
418C2×C4—544no—
419C2×C4—541no—
420C2×C4—541no—
421C2×C4C2×D3×C3⋊D311yescharacteristic
422C8—61no—
423C8—181no—
424C2×C3—24no—
425C2×C3—24no—
426C2×C3—24no—
427C2×C3—721no—
428C2×C3#6394114yes—
429C2×C3#6395014yes—
430C2×C3#6398114yes—
431C2×C3#6409111yescharacteristic
432C2×C3#6412311yescharacteristic
433C2×C3#6412811yescharacteristic
434D3—244no—
435D3—368no—
436D3—3616no—
437D3—364no—
438D3—368no—
439D3—361no—
440D3—362no—
441C22—361no—
442C22—361no—
443C22—361no—
444C22—541no—
445C22—541no—
446C22—542no—
447C22—541no—
448C22—1081no—
449C22—1081no—
450C22#8903911yescharacteristic
451C4—22no—
452C4—121no—
453C4—541no—
454C4C2×#629211yescharacteristic
455C4#8903911yescharacteristic
456C3—24no—
457C3#1059159314yescomplemented
458C3#1059250911yescharacteristic, complemented
459C2—721no—
460C2—1081no—
461C2—1082no—
462C2#1172357211yescharacteristic
463C2#1172358911yescharacteristic
464C2#1172363911yescharacteristic
subgroupG/H
centerC22#89039
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×#89080C2semidirect
2. C2×C3×#3755C2semidirect
3. #11723532C2semidirect
4. C2×#88791C2semidirect
5. C32×#835C2semidirect
6. #11722035C2semidirect
7. #11722024C2semidirect
28. C2×#3755D3semidirect
37. C3×#835D3semidirect
44. C33×Q8D4semidirect
50. C4⋊(C3⋊(C3⋊D3))D4semidirect
124. #835C3⋊D3semidirect
141. C32×Q8D12semidirect
142. C32×Q8#67semidirect
274. C33#449semidirect
287. C3×Q8#626semidirect
288. C3×Q8C4⋊(C3⋊D3)semidirect
402. C32#4810semidirect
403. C32#5172semidirect
408. Q8#6292semidirect
457. C3#10591593semidirect
458. C3#10592509semidirect

Maximal subgroups

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

subgroupindex
C2×#890802
C2×C3×#37552
#117235322
C2×#887912
C32×#8352
#117220352
#117220242
#105925093
#105915933