Finite Group Databasebeta

← #13160981

Subgroups — #13160981

Order 1728.

2072 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 2072 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×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic, complemented
3#11723456C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592868—31nomaximal
9#10591467—34nomaximal
10C23×C3⋊(C3⋊D3)—21no—
11C2×C33×D4C2211yescharacteristic
12C2×#6312—21no—
13D4×C3⋊(C3⋊D3)—22no—
14D4×C3⋊(C3⋊D3)—22no—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic, complemented
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C32×#214—22no—
21C2×C32×C3⋊C8C2211yescharacteristic, complemented
22#88807—21no—
23#88804—21no—
24#88795—22no—
25C2×D4×C3⋊D3—31no—
26C2×D4×C3⋊D3—34no—
27C2×D4×C3⋊D3—64no—
28C2×C32×SD16—31no—
29#64155—31no—
30C2×#3755—31no—
31C2×C3×#214D314yescomplemented
32#63871—34no—
33C2×#3717—34no—
34#63659D311yescharacteristic, complemented
35#63648—31no—
36#63647—31no—
37C3×#832—34no—
38#63564—34no—
39#63557—34no—
40C23×C33—21no—
41C22×C3⋊(C3⋊D3)—24no—
42C22×C3⋊(C3⋊D3)—21no—
43C22×C3⋊(C3⋊D3)—42no—
44C22×C3⋊(C3⋊D3)—42no—
45C22×C3⋊(C3⋊D3)D411yescharacteristic
46C33×D4—21no—
47C33×D4D412yes—
48C2×C33×C4C2311yescharacteristic
49#6312—24no—
50C2×C3⋊(C3⋊Dic3)D411yescharacteristic
51C4⋊(C3⋊(C3⋊D3))—21no—
52C4⋊(C3⋊(C3⋊D3))D412yes—
53C4×C3⋊(C3⋊D3)—21no—
54C4×C3⋊(C3⋊D3)—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C32×Dic3—21no—
57C32×Dic6—21no—
58C32×Dic6—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61#5169—94no—
62#4776—91no—
63C23×C3⋊D3—61no—
64C23×C3⋊D3—64no—
65C23×C3⋊D3—68no—
66C2×C32×Q8—31no—
67C2×C32×D4—24no—
68C2×C32×D4C2×D311yescharacteristic
69C2×C32×D4C2×D314yes—
70C2×#638—61no—
71C2×#638—64no—
72C2×#638—68no—
73D4×C3⋊D3—62no—
74D4×C3⋊D3—68no—
75D4×C3⋊D3—616no—
76D4×C3⋊D3—616no—
77D4×C3⋊D3—62no—
78D4×C3⋊D3—68no—
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4⋊(C3⋊D3)—34no—
81C2×C4⋊(C3⋊D3)—64no—
82C2×C4×C3⋊D3—31no—
83C2×C4×C3⋊D3—34no—
84C2×C4×C3⋊D3—64no—
85C2×C3×Dic6C2×D314yescomplemented
86C32×SD16—62no—
87C2×C32×C8—31no—
88#3755—62no—
89#3752—61no—
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
91#3750—61no—
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
93C3×#214—28no—
94C3×C4⋊Dic3—34no—
95C2×C3×C3⋊C8—34no—
96C2×C3×C3⋊C8C2×D314yescomplemented
97Dic3⋊Dic3—64no—
98#3722—64no—
99#3717—68no—
100C22×C33—22no—
101C22×C33—41no—
102C22×C33—41no—
103C22×C33C2×D411yescharacteristic
104C2×C3⋊(C3⋊D3)—21no—
105C2×C3⋊(C3⋊D3)—21no—
106C2×C3⋊(C3⋊D3)—22no—
107C2×C3⋊(C3⋊D3)—48no—
108C2×C3⋊(C3⋊D3)—41no—
109C2×C3⋊(C3⋊D3)—41no—
110C2×C3⋊(C3⋊D3)SD1612yescomplemented
111C33×C4C2×D411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3⋊(C3⋊Dic3)—21no—
114C3×C3⋊Dic3—41no—
115C32×Dic3—41no—
116C2×D3×D4—94no—
117C2×D3×D4—91no—
118C2×D3×D4—184no—
119C2×C3×SD16—34no—
120#946—94no—
121C2×#215—94no—
122C2×#214C3⋊D311yescharacteristic, complemented
123C2×#204—91no—
124#897—91no—
125C3×#102—91no—
126#832—34no—
127#821—94no—
128(C2×D3)⋊C8—94no—
129(C2×D3)⋊C8—91no—
130#809—91no—
131C23×C32—28no—
132C23×C32—24no—
133C23×C32—21no—
134C22×C3⋊D3—31no—
135C22×C3⋊D3—34no—
136C22×C3⋊D3—64no—
137C22×C3⋊D3—68no—
138C22×C3⋊D3—632no—
139C22×C3⋊D3—64no—
140C22×C3⋊D3—616no—
141C22×C3⋊D3—64no—
142C22×C3⋊D3—61no—
143C22×C3⋊D3—122no—
144C22×C3⋊D3—128no—
145C22×C3⋊D3—1216no—
146C22×C3⋊D3—1216no—
147C22×C3⋊D3—122no—
148C22×C3⋊D3—128no—
149C32×Q8—61no—
150C32×Q8—61no—
151C32×D4—28no—
152C32×D4—24no—
153C32×D4—21no—
154C32×D4—28no—
155C32×D4D1212yes—
156C32×D4#6718yes—
157C2×C32×C4—24no—
158C2×C32×C4—61no—
159C2×C32×C4C22×D314yes—
160C2×C32×C4C22×D311yescharacteristic
161#638—632no—
162#638—64no—
163#638—616no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—31no—
166C2×C3⋊Dic3—34no—
167C2×C3⋊Dic3—64no—
168C4⋊(C3⋊D3)—32no—
169C4⋊(C3⋊D3)—38no—
170C4⋊(C3⋊D3)—68no—
171C4⋊(C3⋊D3)—68no—
172C4⋊(C3⋊D3)—61no—
173C4⋊(C3⋊D3)—64no—
174C4×C3⋊D3—68no—
175C4×C3⋊D3—68no—
176C4×C3⋊D3—61no—
177C4×C3⋊D3—64no—
178C4×C3⋊D3—61no—
179C4×C3⋊D3—64no—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—64no—
182C3×Dic6—24no—
183C3×Dic6—24no—
184C32×C8—61no—
185C3⋊(C3⋊C8)—21no—
186C3×C3⋊C8—24no—
187C3×C3⋊C8—64no—
188#450—271noSylow
189C2×C33—42no—
190C2×C33#12011yescharacteristic
191C2×C33C2×SD1611yescharacteristic
192C2×C33AGL(1,ℤ/8)11yescharacteristic
193C3⋊(C3⋊D3)—22no—
194C3⋊(C3⋊D3)—42no—
195C23×D3—184no—
196C23×D3—181no—
197C23×D3—188no—
198C2×C3×Q8—34no—
199C2×C3×D4—24no—
200C2×C3×D4D3214yes—
201C2×C3×D4C2×C3⋊D311yescharacteristic
202C2×#67—184no—
203C2×#67—181no—
204C2×#67—188no—
205D3×D4—188no—
206D3×D4—182no—
207D3×D4—1816no—
208D3×D4—1816no—
209D3×D4—188no—
210D3×D4—182no—
211C2×D12—94no—
212C2×D12—91no—
213C2×D12—184no—
214C2×C4×D3—94no—
215C2×C4×D3—91no—
216C2×C4×D3—184no—
217C2×Dic6C2×C3⋊D311yescharacteristic, complemented
218C3×SD16—68no—
219C2×C3×C8—34no—
220C2×C3×C8—91no—
221C3×C4⋊C4—91no—
222#215—188no—
223#214—22no—
224(C2×D3)⋊C4—181no—
225(C2×D3)⋊C4—184no—
226C4⋊Dic3—34no—
227C4⋊Dic3—181no—
228Dic3⋊C4—184no—
229C2×C3⋊C8—34no—
230C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
231#204—182no—
232C22×C32—216no—
233C22×C32—22no—
234C22×C32—28no—
235C22×C32—24no—
236C22×C32—48no—
237C22×C32—48no—
238C22×C32—41no—
239C22×C32—44no—
240C22×C32—41no—
241C22×C32—44no—
242C22×C32D3×D414yes—
243C22×C32D3×D411yescharacteristic
244C2×C3⋊D3—32no—
245C2×C3⋊D3—38no—
246C2×C3⋊D3—68no—
247C2×C3⋊D3—68no—
248C2×C3⋊D3—68no—
249C2×C3⋊D3—61no—
250C2×C3⋊D3—61no—
251C2×C3⋊D3—64no—
252C2×C3⋊D3—64no—
253C2×C3⋊D3—616no—
254C2×C3⋊D3—62no—
255C2×C3⋊D3—68no—
256C2×C3⋊D3—1264no—
257C2×C3⋊D3—1232no—
258C2×C3⋊D3—128no—
259C2×C3⋊D3—128no—
260C2×C3⋊D3—128no—
261C2×C3⋊D3—121no—
262C2×C3⋊D3—124no—
263C2×C3⋊D3—121no—
264C2×C3⋊D3—124no—
265C32×C4—24no—
266C32×C4—24no—
267C32×C4—121no—
268C32×C4C2×D1211yescharacteristic
269C32×C4D3×D414yes—
270C32×C4D3×D411yescharacteristic
271C32×C4C2×#6714yes—
272C3⋊Dic3—41no—
273C3⋊Dic3—68no—
274C3⋊Dic3—61no—
275C3⋊Dic3—64no—
276C3×Dic3—44no—
277C3×Dic3—124no—
278C22×D4—271no—
279C2×SD16—31no—
280C2×SD16—271no—
281C22⋊Q8—271no—
282#102—91no—
283#102—271no—
284C22⋊C8—271no—
285C33#45011yescharacteristic, complemented, Sylow
286C23×C3—28no—
287C23×C3—24no—
288C23×C3—21no—
289C22×D3—94no—
290C22×D3—91no—
291C22×D3—184no—
292C22×D3—188no—
293C22×D3—1832no—
294C22×D3—184no—
295C22×D3—1816no—
296C22×D3—184no—
297C22×D3—181no—
298C22×D3—362no—
299C22×D3—368no—
300C22×D3—3616no—
301C22×D3—3616no—
302C22×D3—362no—
303C22×D3—368no—
304C3×Q8—64no—
305C3×Q8—64no—
306C3×D4—28no—
307C3×D4—24no—
308C3×D4—21no—
309C3×D4—28no—
310C3×D4#62618yes—
311C3×D4#63812yes—
312C2×C3×C4—24no—
313C2×C3×C4—64no—
314C2×C3×C4—181no—
315C2×C3×C4C2×D3214yes—
316C2×C3×C4C22×C3⋊D311yescharacteristic
317#67—1832no—
318#67—184no—
319#67—1816no—
320C2×Dic3—21no—
321C2×Dic3—64no—
322C2×Dic3—94no—
323C2×Dic3—91no—
324C2×Dic3—184no—
325D12—98no—
326D12—92no—
327D12—188no—
328D12—188no—
329D12—184no—
330D12—181no—
331C4×D3—188no—
332C4×D3—188no—
333C4×D3—184no—
334C4×D3—184no—
335C4×D3—181no—
336C4×D3—181no—
337Dic6—21no—
338Dic6—21no—
339C3×C8—64no—
340C3×C8—181no—
341C3⋊C8—21no—
342C3⋊C8—64no—
343C2×C32—24no—
344C2×C32—24no—
345C2×C32—24no—
346C2×C32—416no—
347C2×C32—48no—
348C2×C32—42no—
349C2×C32#88211yescharacteristic
350C2×C32#91111yescharacteristic
351C2×C32D3×SD1614yes—
352C2×C32D3×SD1611yescharacteristic
353C2×C32#91414yes—
354C2×C32#93714yes—
355C3⋊D3—616no—
356C3⋊D3—68no—
357C3⋊D3—62no—
358C3⋊D3—1216no—
359C3⋊D3—128no—
360C3⋊D3—122no—
361C24—541no—
362C2×Q8—31no—
363C2×D4—271no—
364C2×D4—541no—
365C2×D4—542no—
366C2×D4—542no—
367C2×D4D3×C3⋊D311yescharacteristic
368C22×C4—271no—
369SD16—62no—
370SD16—542no—
371C2×C8—31no—
372C2×C8—91no—
373C4⋊C4—91no—
374C4⋊C4—541no—
375C22⋊C4—541no—
376C22×C3—216no—
377C22×C3—22no—
378C22×C3—28no—
379C22×C3—24no—
380C22×C3—48no—
381C22×C3—48no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—44no—
385C22×C3—44no—
386C22×C3#381114yes—
387C22×C3D4×C3⋊D311yescharacteristic
388C2×D3—92no—
389C2×D3—98no—
390C2×D3—188no—
391C2×D3—188no—
392C2×D3—188no—
393C2×D3—181no—
394C2×D3—181no—
395C2×D3—184no—
396C2×D3—184no—
397C2×D3—1816no—
398C2×D3—182no—
399C2×D3—188no—
400C2×D3—3664no—
401C2×D3—368no—
402C2×D3—3632no—
403C2×D3—368no—
404C2×D3—368no—
405C2×D3—361no—
406C2×D3—361no—
407C2×D3—364no—
408C2×D3—364no—
409C3×C4—24no—
410C3×C4—24no—
411C3×C4—124no—
412C3×C4—361no—
413C3×C4C2×#62614yes—
414C3×C4#381114yes—
415C3×C4D4×C3⋊D311yescharacteristic
416C3×C4C2×#63811yescharacteristic
417Dic3—41no—
418Dic3—124no—
419Dic3—188no—
420Dic3—181no—
421Dic3—184no—
422C32—24no—
423C32#477611yescharacteristic, complemented
424C32#516914yescomplemented
425C23—21no—
426C23—271no—
427C23—544no—
428C23—541no—
429C23—1082no—
430C23—1082no—
431Q8—61no—
432Q8—61no—
433D4—21no—
434D4—272no—
435D4—544no—
436D4—541no—
437D4#629112yes—
438C2×C4—61no—
439C2×C4—181no—
440C2×C4—271no—
441C2×C4—541no—
442C2×C4—541no—
443C2×C4C2×D3×C3⋊D311yescharacteristic
444C8—61no—
445C8—181no—
446C2×C3—24no—
447C2×C3—24no—
448C2×C3—24no—
449C2×C3—416no—
450C2×C3—42no—
451C2×C3—48no—
452C2×C3#6393014yes—
453C2×C3#6393714yes—
454C2×C3#6398114yes—
455C2×C3SD16×C3⋊D311yescharacteristic
456C2×C3#6412311yescharacteristic
457C2×C3#6414611yescharacteristic
458D3—1816no—
459D3—182no—
460D3—188no—
461D3—3616no—
462D3—362no—
463D3—368no—
464C22—22no—
465C22—41no—
466C22—41no—
467C22—272no—
468C22—541no—
469C22—541no—
470C22—542no—
471C22—1088no—
472C22—1081no—
473C22—1081no—
474C22#8903911yescharacteristic
475C4—121no—
476C4—361no—
477C4—541no—
478C4C2×#629111yescharacteristic
479C4#8903911yescharacteristic
480C3—24no—
481C3#1059146714yescomplemented
482C3#1059286811yescharacteristic, complemented
483C2—42no—
484C2—542no—
485C2—1082no—
486C2#1172355311yescharacteristic
487C2#1172356611yescharacteristic
488C2#1172363811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×D4×C3⋊(C3⋊D3)C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#450
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C32×#214C2semidirect
3. #11723456C2semidirect
4. C2×#88795C2semidirect
5. C3×#63659C2semidirect
6. #11722036C2semidirect
7. #11722024C2semidirect
17. C2×C32×Dic6C22semidirect
18. C3×C4⋊(C3⋊Dic3)C22semidirect
19. C2×C3×C3⋊(C3⋊C8)C22semidirect
21. C2×C32×C3⋊C8C22semidirect
31. C2×C3×#214D3semidirect
34. #63659D3semidirect
85. C2×C3×Dic6C2×D3semidirect
90. C4⋊(C3⋊Dic3)C2×D3semidirect
92. C2×C3⋊(C3⋊C8)C2×D3semidirect
96. C2×C3×C3⋊C8C2×D3semidirect
110. C2×C3⋊(C3⋊D3)SD16semidirect
122. C2×#214C3⋊D3semidirect
217. C2×Dic6C2×C3⋊D3semidirect
230. C2×C3⋊C8C2×C3⋊D3semidirect
285. C33#450semidirect
423. C32#4776semidirect
424. C32#5169semidirect
481. C3#10591467semidirect
482. C3#10592868semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
C2×C32×#2142
#117234562
C2×#887952
C3×#636592
#117220362
#117220242
#105928683
#105914673