Finite Group Databasebeta

← #13161115

Subgroups — #13161115

Order 1728.

1276 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 158 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×#89018C211yesmaximal, characteristic, complemented
2C2×#88860C211yesmaximal, characteristic, complemented
3C3×#64147C211yesmaximal, characteristic, complemented
4C2×C32×#214C211yesmaximal, characteristic, complemented
5#11722037C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592871—31nomaximal
9#10591521—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6292—21no—
16C2×#6289—21no—
17#89018—22no—
18#89018—22no—
19#88860—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#3757—21no—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C32×#214—22no—
24C2×C32×C3⋊C8C2211yescharacteristic
25C2×#3832—31no—
26C2×#3799—34no—
27C2×C32×SD16—31no—
28#64147D311yescharacteristic, complemented
29C2×#3753—31no—
30C2×#3753—34no—
31C2×#3753—64no—
32C3×#938—34no—
33C2×C3×#214D314yescomplemented
34#63662—31no—
35#63648—31no—
36#63647—31no—
37#63565—34no—
38#63564—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C33×D4—21no—
44C33×D4—21no—
45C2×C33×C4C2311yescharacteristic
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))D412yescomplemented
48C3×#638—42no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C2×C32×Dic3—21no—
54C32×Dic6—21no—
55C32×Dic6D412yescomplemented
56#6292—24no—
57#6289—24no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5172—94no—
61#5037—91no—
62C2×C32×Q8—31no—
63C2×C32×D4—24no—
64C2×C32×D4C2×D314yes—
65C2×C32×D4C2×D311yescharacteristic
66C2×#638—21no—
67#3832—62no—
68#3832—62no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—34no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)—64no—
73C2×C4×C3⋊D3—61no—
74C2×C4×C3⋊D3C2×D311yescharacteristic
75C2×C3×#67—64no—
76C2×C3×C4×D3—34no—
77C2×C3×Dic6C2×D314yes—
78C2×#626—64no—
79C2×#624—64no—
80#3799—68no—
81#3799—68no—
82C32×SD16—62no—
83C2×C32×C8—31no—
84#3757—21no—
85#3753—62no—
86#3753—68no—
87#3753—616no—
88C4⋊(C3⋊Dic3)C2×D311yescharacteristic
89C2×C3⋊(C3⋊C8)—31no—
90C2×C3⋊(C3⋊C8)—34no—
91C2×C3⋊(C3⋊C8)—64no—
92C3×#217—64no—
93C3×#214—28no—
94C3×C4⋊Dic3—34no—
95C2×C3×C3⋊C8C2×D314yes—
96C22×C33—41no—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33C2×D411yescharacteristic
100C2×C3⋊(C3⋊D3)—22no—
101C2×C3⋊(C3⋊D3)—41no—
102C2×C3⋊(C3⋊D3)—41no—
103C2×C3×C3⋊D3—21no—
104C2×C3×C3⋊D3—21no—
105C2×C3×C3⋊D3—21no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3—21no—
109C3×C3⋊Dic3—41no—
110C32×Dic3—22no—
111C2×#239—94no—
112C2×#235—91no—
113C2×C3×SD16—34no—
114C3×#121—91no—
115#938—34no—
116C2×#214C3⋊D311yescharacteristic, complemented
117C2×#213—94no—
118C2×#213—91no—
119C2×#213—184no—
120#835—94no—
121#830—91no—
122#821—94no—
123(C2×D3)⋊C8—94no—
124#810—91no—
125#809—91no—
126C23×C32—21no—
127C23×C32—24no—
128C23×C32—44no—
129C22×C3⋊D3—61no—
130C22×C3⋊D3—64no—
131C22×C3⋊D3—124no—
132C22×C3⋊D3#6711yescharacteristic
133C22×C3×D3—34no—
134C32×Q8—32no—
135C32×Q8—61no—
136C32×D4—28no—
137C32×D4—24no—
138C32×D4—21no—
139C32×D4—21no—
140C32×D4—24no—
141C32×D4—44no—
142C2×C32×C4—24no—
143C2×C32×C4—61no—
144C2×C32×C4C22×D314yes—
145C2×C32×C4C22×D311yescharacteristic
146#638—42no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3#6711yescharacteristic
149C4⋊(C3⋊D3)—32no—
150C4⋊(C3⋊D3)—38no—
151C4⋊(C3⋊D3)—68no—
152C4⋊(C3⋊D3)—64no—
153C4⋊(C3⋊D3)—61no—
154C4⋊(C3⋊D3)—64no—
155C4⋊(C3⋊D3)—124no—
156C4×C3⋊D3—21no—
157C4×C3⋊D3—21no—
158C4×C3⋊D3—64no—
159C3×#67—128no—
160C2×C3×Dic3—24no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—64no—
163C3×C4×D3—64no—
164C3×C4×D3—64no—
165C3×Dic6—24no—
166C3×Dic6#6718yescomplemented
167#626—616no—
168#624—616no—
169C32×C8—61no—
170C3⋊(C3⋊C8)—68no—
171C3⋊(C3⋊C8)—64no—
172C3⋊(C3⋊C8)—61no—
173C3×C3⋊C8—24no—
174#449—271noSylow
175C2×C33—81no—
176C2×C33#12011yescharacteristic
177C2×C33#13511yescharacteristic
178C2×C33AGL(1,ℤ/8)11yescharacteristic
179C3⋊(C3⋊D3)—42no—
180C3×C3⋊D3—41no—
181C2×C3×Q8—34no—
182C2×C3×D4—24no—
183C2×C3×D4—181no—
184C2×C3×D4D3214yes—
185C2×C3×D4C2×C3⋊D311yescharacteristic
186C22×C3×C4—91no—
187C2×#67—64no—
188C2×#67—181no—
189#239—188no—
190#239—188no—
191#235—182no—
192#235—182no—
193C2×D12—94no—
194C2×D12—91no—
195C2×D12—184no—
196C2×D12—184no—
197C2×C4×D3—34no—
198C2×C4×D3—181no—
199C2×C4×D3—184no—
200C2×Dic6C2×C3⋊D311yescharacteristic
201C3×SD16—68no—
202C2×C3×C8—34no—
203C3×C4⋊C4—91no—
204C3×C22⋊C4—181no—
205#217—64no—
206#214—22no—
207#213—188no—
208#213—182no—
209#213—1816no—
210C4⋊Dic3—34no—
211C2×C3⋊C8—94no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—184no—
214C2×C3⋊C8C2×C3⋊D311yescharacteristic
215C22×C32—24no—
216C22×C32—48no—
217C22×C32—41no—
218C22×C32—44no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—44no—
222C22×C32—44no—
223C22×C32—84no—
224C22×C32—84no—
225C22×C32D3×D414yes—
226C22×C32C2×#6711yescharacteristic
227C2×C3⋊D3—21no—
228C2×C3⋊D3—21no—
229C2×C3⋊D3—21no—
230C2×C3⋊D3—62no—
231C2×C3⋊D3—68no—
232C2×C3⋊D3—128no—
233C2×C3⋊D3—121no—
234C2×C3⋊D3—121no—
235C2×C3⋊D3—124no—
236C2×C3⋊D3—124no—
237C2×C3⋊D3—244no—
238C2×C3⋊D3—244no—
239C2×C3×D3—64no—
240C2×C3×D3—64no—
241C2×C3×D3—64no—
242C32×C4—24no—
243C32×C4—24no—
244C32×C4—62no—
245C32×C4D3×D414yes—
246C32×C4D3×D411yescharacteristic
247C32×C4D3×D411yescharacteristic
248C32×C4C2×#6714yes—
249C3⋊Dic3—21no—
250C3⋊Dic3—41no—
251C3×Dic3—28no—
252C3×Dic3—64no—
253C3×Dic3—124no—
254C2×#41—271no—
255C2×SD16—31no—
256C2×D8—271no—
257#121—91no—
258#103—271no—
259#102—271no—
260C22⋊C8—271no—
261C33#44911yescharacteristic, complemented, Sylow
262C23×C3—24no—
263C23×C3—21no—
264C23×C3—44no—
265C23×C3—91no—
266C22×D3—34no—
267C22×D3—184no—
268C22×D3—181no—
269C22×D3—364no—
270C3×Q8—38no—
271C3×Q8—64no—
272C3×D4—28no—
273C3×D4—24no—
274C3×D4—24no—
275C3×D4—21no—
276C3×D4—21no—
277C3×D4—44no—
278C3×D4—362no—
279C2×C3×C4—24no—
280C2×C3×C4—64no—
281C2×C3×C4—91no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4C2×D3214yes—
286C2×C3×C4C22×C3⋊D311yescharacteristic
287#67—128no—
288#67—184no—
289C2×Dic3—21no—
290C2×Dic3—34no—
291C2×Dic3—64no—
292D12—98no—
293D12—92no—
294D12—188no—
295D12—1816no—
296D12—184no—
297D12—181no—
298D12—364no—
299C4×D3—64no—
300C4×D3—64no—
301C4×D3—184no—
302C4×D3—1816no—
303Dic6—21no—
304Dic6#63812yescomplemented
305C3×C8—64no—
306C3⋊C8—21no—
307C3⋊C8—188no—
308C3⋊C8—184no—
309C3⋊C8—181no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—88no—
314C2×C32—84no—
315C2×C32—81no—
316C2×C32#91414yes—
317C2×C32#91614yes—
318C2×C32#93211yescharacteristic
319C2×C32#93714yes—
320C2×C32#93711yescharacteristic
321C2×C32#95011yescharacteristic
322C3⋊D3—41no—
323C3⋊D3—122no—
324C3⋊D3—128no—
325C3⋊D3—248no—
326C3×D3—124no—
327#41—542no—
328#41—542no—
329C2×Q8—31no—
330C2×D4—181no—
331C2×D4—271no—
332C2×D4—541no—
333C2×D4D3×C3⋊D311yescharacteristic
334C22×C4—91no—
335C22×C4—541no—
336SD16—62no—
337D8—542no—
338C2×C8—31no—
339C2×C8—271no—
340C4⋊C4—91no—
341C22⋊C4—181no—
342C22×C3—24no—
343C22×C3—48no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—44no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—84no—
351C22×C3—84no—
352C22×C3—181no—
353C22×C3—181no—
354C22×C3—181no—
355C22×C3D3×#6714yes—
356C22×C3D4×C3⋊D311yescharacteristic
357C2×D3—64no—
358C2×D3—64no—
359C2×D3—64no—
360C2×D3—182no—
361C2×D3—188no—
362C2×D3—368no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—364no—
366C2×D3—364no—
367C2×D3—724no—
368C2×D3—724no—
369C3×C4—24no—
370C3×C4—24no—
371C3×C4—68no—
372C3×C4—181no—
373C3×C4—361no—
374C3×C4D3×#6714yes—
375C3×C4#381114yes—
376C3×C4D4×C3⋊D311yescharacteristic
377C3×C4C2×#63811yescharacteristic
378Dic3—22no—
379Dic3—64no—
380Dic3—124no—
381C32—24no—
382C32#503711yescharacteristic, complemented
383C32#517214yescomplemented
384C23—21no—
385C23—91no—
386C23—541no—
387Q8—32no—
388Q8—61no—
389D4—21no—
390D4—21no—
391D4—272no—
392D4—362no—
393D4—544no—
394D4—541no—
395C2×C4—61no—
396C2×C4—91no—
397C2×C4—181no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—544no—
401C2×C4C2×D3×C3⋊D311yescharacteristic
402C8—61no—
403C8—541no—
404C2×C3—24no—
405C2×C3—24no—
406C2×C3—24no—
407C2×C3—88no—
408C2×C3—84no—
409C2×C3—81no—
410C2×C3—361no—
411C2×C3#6393414yes—
412C2×C3#6393514yes—
413C2×C3#6398314yes—
414C2×C3#6412311yescharacteristic
415C2×C3#6412511yescharacteristic
416C2×C3#6414611yescharacteristic
417D3—124no—
418D3—368no—
419D3—362no—
420D3—728no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—181no—
425C22—181no—
426C22—181no—
427C22—542no—
428C22—1081no—
429C22—1081no—
430C22C3⋊D3×#6711yescharacteristic
431C4—62no—
432C4—181no—
433C4—361no—
434C4D3×#63811yescharacteristic
435C4#8903911yescharacteristic
436C3—24no—
437C3#1059152114yescomplemented
438C3#1059287111yescharacteristic, complemented
439C2—81no—
440C2—361no—
441C2—1082no—
442C2#1172356111yescharacteristic
443C2#1172356211yescharacteristic
444C2#1172364311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#449
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
1. C2×#89018C2semidirect
2. C2×#88860C2semidirect
3. C3×#64147C2semidirect
4. C2×C32×#214C2semidirect
5. #11722037C2semidirect
6. #11722036C2semidirect
7. #11722023C2semidirect
28. #64147D3semidirect
33. C2×C3×#214D3semidirect
47. C4⋊(C3⋊(C3⋊D3))D4semidirect
55. C32×Dic6D4semidirect
116. C2×#214C3⋊D3semidirect
166. C3×Dic6#67semidirect
261. C33#449semidirect
304. Dic6#638semidirect
382. C32#5037semidirect
383. C32#5172semidirect
437. C3#10591521semidirect
438. C3#10592871semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C2×#888602
C3×#641472
C2×C32×#2142
#117220372
#117220362
#117220232
#105928713
#105915213