Finite Group Databasebeta

← #13160046

Subgroups — #13160046

Order 1728.

1492 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 170 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×D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#88852C211yesmaximal, characteristic, complemented
3#11723516C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic, complemented
8#10592431—31nomaximal
9#10590947—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C32×C4×D3C2211yescharacteristic
15C2×#6292—21no—
16C2×#6291—21no—
17D3×C4⋊(C3⋊D3)—22no—
18D3×C4⋊(C3⋊D3)—22no—
19C2×C33×C8C2211yescharacteristic, complemented
20#88852—22no—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23#88804—21no—
24#88791—22no—
25C22×C4⋊(C3⋊D3)—31no—
26C2×D3×D12—34no—
27#64139—31no—
28C2×#3745—31no—
29C2×#3745—31no—
30C2×#3745—34no—
31C2×#3745—64no—
32#63909—34no—
33C2×#3714—34no—
34C32×C22⋊C8—31no—
35#63648—31no—
36#63648D311yescharacteristic, complemented
37C3×#821—34no—
38C3×(C2×D3)⋊C8D314yescomplemented
39#63564—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3×C3⋊D3—21no—
43C22×C32×D3D411yescharacteristic
44C2×D3×C3⋊D3—24no—
45C2×D3×C3⋊D3—42no—
46C2×D3×C3⋊D3—42no—
47C2×C33×C4C2311yescharacteristic
48C4⋊(C3⋊(C3⋊D3))—21no—
49C4⋊(C3⋊(C3⋊D3))—21no—
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))D412yescomplemented
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)D412yescomplemented
55C2×C32×Dic3D411yescharacteristic
56C32×C4×D3—21no—
57C32×C4×D3—21no—
58#6292—24no—
59#6291—42no—
60C33×C8—21no—
61C32×C3⋊C8—21no—
62#4773—91no—
63#4732—94no—
64C23×C3⋊D3—61no—
65C22×D32—64no—
66C22×C32×C4—31no—
67C2×#638—61no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—34no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—34no—
72C2×C4⋊(C3⋊D3)—62no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—62no—
75C2×C4⋊(C3⋊D3)—64no—
76C2×C4⋊(C3⋊D3)—64no—
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C2×C3×D12—34no—
79C2×C3×C4×D3C2×D314yes—
80C2×#626—64no—
81C2×#626—64no—
82D3×D12—68no—
83D3×D12—68no—
84C2×C32×C8—24no—
85C2×C32×C8—31no—
86C2×C32×C8C2×D314yescomplemented
87C2×C32×C8C2×D311yescharacteristic, complemented
88#3752—61no—
89C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
90#3745—62no—
91#3745—68no—
92#3745—616no—
93#3745—62no—
94C3×C4⋊Dic3—34no—
95C2×C3×C3⋊C8C2×D314yescomplemented
96#3722—64no—
97#3714—68no—
98C22×C33C2×D411yescharacteristic
99C2×C3⋊(C3⋊D3)—22no—
100C2×C3⋊(C3⋊D3)—41no—
101C2×C3⋊(C3⋊D3)—41no—
102C2×C3⋊(C3⋊D3)—41no—
103C2×C3⋊(C3⋊D3)—41no—
104C2×C3⋊(C3⋊D3)—41no—
105C2×C3×C3⋊D3—22no—
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3—41no—
108C2×C32×D3—21no—
109C2×C32×D3—21no—
110C2×C32×D3D812yescomplemented
111D3×C3⋊D3—48no—
112C33×C4C2×D411yescharacteristic
113C33×C4C2×D411yescharacteristic
114C3×C3⋊Dic3—41no—
115C32×Dic3—21no—
116C2×D3×D4—91no—
117C22×D12—94no—
118C2×#213—91no—
119#930—94no—
120C2×D24—94no—
121C2×D24—94no—
122C2×D24—91no—
123C2×D24—184no—
124#895—91no—
125C3×#102—91no—
126C3×C22⋊C8—34no—
127#821—34no—
128#821—94no—
129(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
130#809—91no—
131C23×C32—31no—
132C22×C3⋊D3—21no—
133C22×C3⋊D3—64no—
134C22×C3⋊D3—61no—
135C22×C3⋊D3—61no—
136C22×C3⋊D3—64no—
137C22×C3⋊D3—64no—
138C22×C3⋊D3—124no—
139C22×C3⋊D3—124no—
140C22×C3⋊D3—122no—
141C22×C3⋊D3—122no—
142C22×C3×D3—64no—
143C22×C3×D3D1214yes—
144C2×D32—616no—
145C2×D32—128no—
146C2×D32—128no—
147C2×C32×C4—24no—
148C2×C32×C4—31no—
149C2×C32×C4—61no—
150C2×C32×C4—61no—
151C2×C32×C4C22×D314yes—
152C2×C32×C4C22×D311yescharacteristic
153#638—122no—
154C2×C3⋊Dic3—21no—
155C4⋊(C3⋊D3)—21no—
156C4⋊(C3⋊D3)—32no—
157C4⋊(C3⋊D3)—38no—
158C4⋊(C3⋊D3)—68no—
159C4⋊(C3⋊D3)—68no—
160C4⋊(C3⋊D3)—64no—
161C4⋊(C3⋊D3)—61no—
162C4⋊(C3⋊D3)—61no—
163C4⋊(C3⋊D3)—64no—
164C4⋊(C3⋊D3)—61no—
165C4⋊(C3⋊D3)—64no—
166C4⋊(C3⋊D3)—64no—
167C4⋊(C3⋊D3)—124no—
168C4⋊(C3⋊D3)—124no—
169C4⋊(C3⋊D3)D1212yescomplemented
170C2×C3×Dic3—64no—
171C2×C3×Dic3D1214yes—
172C3×D12—38no—
173C3×D12—64no—
174C3×C4×D3—24no—
175C3×C4×D3—24no—
176#626—616no—
177#626—128no—
178C32×C8—28no—
179C32×C8—24no—
180C32×C8—21no—
181C32×C8—61no—
182C3×C3⋊C8—24no—
183#447—271noSylow
184C2×C33#12011yescharacteristic
185C2×C33C2×D811yescharacteristic
186C2×C33AGL(1,ℤ/8)11yescharacteristic
187C3⋊(C3⋊D3)—42no—
188C3⋊(C3⋊D3)—81no—
189C3×C3⋊D3—42no—
190C32×D3—22no—
191C23×D3—184no—
192C23×D3—181no—
193C2×C3×D4—91no—
194C22×C3×C4—34no—
195C2×#67—184no—
196C2×#67—181no—
197D3×D4—182no—
198D3×D4—182no—
199C2×D12—34no—
200C2×D12—94no—
201C2×D12—91no—
202C2×D12—94no—
203C2×D12—91no—
204C2×D12—184no—
205C2×D12—184no—
206C2×D12—188no—
207C2×D12—188no—
208C2×D12—184no—
209C2×D12—181no—
210C2×C4×D3C2×C3⋊D311yescharacteristic
211C2×C3×C8—24no—
212C2×C3×C8—34no—
213C2×C3×C8D3214yescomplemented
214C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
215C3×C4⋊C4—91no—
216#213—182no—
217(C2×D3)⋊C4—184no—
218(C2×D3)⋊C4—181no—
219C4⋊Dic3—34no—
220C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
221D24—188no—
222D24—182no—
223D24—1816no—
224D24—188no—
225C22×C32—24no—
226C22×C32—32no—
227C22×C32—61no—
228C22×C32—61no—
229C22×C32C2×D1214yes—
230C22×C32D3×D411yescharacteristic
231C2×C3⋊D3—22no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3—41no—
234C2×C3⋊D3—62no—
235C2×C3⋊D3—68no—
236C2×C3⋊D3—128no—
237C2×C3⋊D3—128no—
238C2×C3⋊D3—121no—
239C2×C3⋊D3—121no—
240C2×C3⋊D3—128no—
241C2×C3⋊D3—124no—
242C2×C3⋊D3—121no—
243C2×C3⋊D3—121no—
244C2×C3⋊D3—121no—
245C2×C3⋊D3—124no—
246C2×C3⋊D3—124no—
247C2×C3⋊D3—124no—
248C2×C3⋊D3—124no—
249C2×C3⋊D3—244no—
250C2×C3⋊D3—244no—
251C2×C3⋊D3—244no—
252C2×C3⋊D3—244no—
253C2×C3×D3—24no—
254C2×C3×D3—24no—
255C2×C3×D3—68no—
256C2×C3×D3—124no—
257C2×C3×D3—124no—
258C2×C3×D3D2418yescomplemented
259D32—1232no—
260C32×C4—24no—
261C32×C4—24no—
262C32×C4—61no—
263C32×C4C2×D1211yescharacteristic
264C32×C4D3×D414yes—
265C32×C4D3×D414yes—
266C32×C4D3×D411yescharacteristic
267C3⋊Dic3—41no—
268C3×Dic3—24no—
269C3×Dic3—124no—
270C22×D4—271no—
271C2×D8—271no—
272C2×D8—271no—
273#121—271no—
274#102—91no—
275#102—271no—
276C22⋊C8—31no—
277C33#44711yescharacteristic, complemented, Sylow
278C23×C3—34no—
279C23×C3—181no—
280C22×D3—64no—
281C22×D3—184no—
282C22×D3—1816no—
283C22×D3—184no—
284C22×D3—181no—
285C22×D3—184no—
286C22×D3—181no—
287C22×D3—364no—
288C22×D3—364no—
289C22×D3—362no—
290C22×D3—368no—
291C22×D3—368no—
292C22×D3—362no—
293C22×D3C4⋊(C3⋊D3)11yescharacteristic
294C3×D4—92no—
295C3×D4—181no—
296C2×C3×C4—24no—
297C2×C3×C4—34no—
298C2×C3×C4—64no—
299C2×C3×C4—64no—
300C2×C3×C4—181no—
301C2×C3×C4C2×D3214yes—
302C2×C3×C4C22×C3⋊D311yescharacteristic
303#67—184no—
304#67—368no—
305C2×Dic3—64no—
306C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
307D12—38no—
308D12—64no—
309D12—98no—
310D12—92no—
311D12—188no—
312D12—188no—
313D12—1816no—
314D12—184no—
315D12—184no—
316D12—184no—
317D12—181no—
318D12—181no—
319D12—181no—
320D12—364no—
321D12—364no—
322D12—362no—
323C4×D3—21no—
324C4×D3—21no—
325C3×C8—28no—
326C3×C8—24no—
327C3×C8—21no—
328C3×C8—64no—
329C3⋊C8—21no—
330C2×C32—24no—
331C2×C32—24no—
332C2×C32—24no—
333C2×C32—62no—
334C2×C32#88214yes—
335C2×C32#88211yescharacteristic
336C2×C32C2×D2414yes—
337C2×C32#90814yes—
338C2×C32D3×D811yescharacteristic
339C2×C32#91411yescharacteristic
340C3⋊D3—42no—
341C3⋊D3—122no—
342C3⋊D3—128no—
343C3⋊D3—248no—
344C3⋊D3—248no—
345C3⋊D3—244no—
346C3⋊D3—241no—
347C3×D3—28no—
348C3×D3—128no—
349C24—541no—
350C2×D4—91no—
351C2×D4—271no—
352C2×D4—271no—
353C2×D4—541no—
354C2×D4—542no—
355C2×D4—542no—
356C2×D4—541no—
357C22×C4—31no—
358D8—542no—
359D8—542no—
360C2×C8—31no—
361C2×C8D3×C3⋊D311yescharacteristic, complemented
362C4⋊C4—91no—
363C22⋊C4—541no—
364C22×C3—24no—
365C22×C3—38no—
366C22×C3—64no—
367C22×C3—64no—
368C22×C3—182no—
369C22×C3—361no—
370C22×C3—361no—
371C22×C3D3×D1214yes—
372C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
373C2×D3—21no—
374C2×D3—21no—
375C2×D3—68no—
376C2×D3—124no—
377C2×D3—124no—
378C2×D3—182no—
379C2×D3—188no—
380C2×D3—368no—
381C2×D3—368no—
382C2×D3—3632no—
383C2×D3—364no—
384C2×D3—364no—
385C2×D3—361no—
386C2×D3—361no—
387C2×D3—368no—
388C2×D3—364no—
389C2×D3—364no—
390C2×D3—361no—
391C2×D3—361no—
392C2×D3—361no—
393C2×D3—364no—
394C2×D3—724no—
395C2×D3—724no—
396C2×D3—724no—
397C2×D3—724no—
398C2×D3#374512yescomplemented
399C3×C4—24no—
400C3×C4—24no—
401C3×C4—64no—
402C3×C4—361no—
403C3×C4D3×D1214yes—
404C3×C4#381114yes—
405C3×C4D4×C3⋊D311yescharacteristic
406C3×C4D4×C3⋊D311yescharacteristic
407Dic3—21no—
408Dic3—124no—
409C32—24no—
410C32#473214yescomplemented
411C32#477311yescharacteristic, complemented
412C23—31no—
413C23—181no—
414C23—544no—
415C23—541no—
416C23—541no—
417C23—1082no—
418C23—1082no—
419D4—92no—
420D4—181no—
421D4—272no—
422D4—544no—
423D4—541no—
424D4—541no—
425D4—541no—
426D4—1082no—
427C2×C4—31no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—181no—
431C2×C4C2×D3×C3⋊D311yescharacteristic
432C8—21no—
433C8—61no—
434C2×C3—24no—
435C2×C3—24no—
436C2×C3—24no—
437C2×C3—68no—
438C2×C3—362no—
439C2×C3D3×D2414yes—
440C2×C3#6379414yes—
441C2×C3#6392314yes—
442C2×C3#6409111yescharacteristic
443C2×C3C2×#374511yescharacteristic
444C2×C3#6411711yescharacteristic
445D3—22no—
446D3—128no—
447D3—362no—
448D3—368no—
449D3—728no—
450D3—728no—
451D3—721no—
452D3—724no—
453C22—32no—
454C22—61no—
455C22—61no—
456C22—182no—
457C22—361no—
458C22—361no—
459C22—542no—
460C22—1088no—
461C22—1081no—
462C22—1081no—
463C22—1081no—
464C22—1081no—
465C22—1081no—
466C22D3×C4⋊(C3⋊D3)11yescharacteristic
467C4—61no—
468C4—361no—
469C4C3⋊D3×D1211yescharacteristic
470C4#8903911yescharacteristic
471C3—24no—
472C3#1059094714yescomplemented
473C3#1059243111yescharacteristic, complemented
474C2—62no—
475C2—362no—
476C2—1082no—
477C2—2161no—
478C2D3×#374511yescharacteristic
479C2#1172331511yescharacteristic
480C2#1172354211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#447
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
1. C2×D3×C4⋊(C3⋊D3)C2semidirect
2. C2×#88852C2semidirect
3. #11723516C2semidirect
4. C2×#88791C2semidirect
5. C3×#63648C2semidirect
6. C32×(C2×D3)⋊C8C2semidirect
7. #11722036C2semidirect
12. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C2×C33×C8C22semidirect
21. C3×C4⋊(C3⋊Dic3)C22semidirect
22. C2×C32×C3⋊C8C22semidirect
36. #63648D3semidirect
38. C3×(C2×D3)⋊C8D3semidirect
51. C4⋊(C3⋊(C3⋊D3))D4semidirect
54. C3×C4⋊(C3⋊D3)D4semidirect
86. C2×C32×C8C2×D3semidirect
87. C2×C32×C8C2×D3semidirect
89. C4⋊(C3⋊Dic3)C2×D3semidirect
95. C2×C3×C3⋊C8C2×D3semidirect
110. C2×C32×D3D8semidirect
129. (C2×D3)⋊C8C3⋊D3semidirect
169. C4⋊(C3⋊D3)D12semidirect
213. C2×C3×C8D32semidirect
214. C2×C3×C8C2×C3⋊D3semidirect
220. C2×C3⋊C8C2×C3⋊D3semidirect
258. C2×C3×D3D24semidirect
277. C33#447semidirect
361. C2×C8D3×C3⋊D3semidirect
398. C2×D3#3745semidirect
410. C32#4732semidirect
411. C32#4773semidirect
472. C3#10590947semidirect
473. C3#10592431semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊(C3⋊D3)2
C2×#888522
#117235162
C2×#887912
C3×#636482
C32×(C2×D3)⋊C82
#117220362
#105924313
#105909473