Finite Group Databasebeta

← #13159418

Subgroups — #13159418

Order 1728.

1328 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1328 conjugacy classes of subgroups — too many to draw. Only its 162 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×C3⋊D3×D12C211yesmaximal, characteristic
2C3×#64106C211yesmaximal, characteristic, complemented
3C2×#88797C211yesmaximal, characteristic, complemented
4C2×#88792C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592475—31nomaximal
9#10590596—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic, complemented
14C2×C32×D12C2211yescharacteristic
15C2×#6291—21no—
16C3⋊D3×D12—22no—
17C3⋊D3×D12—22no—
18C33×C4⋊C4C2211yescharacteristic, complemented
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
20C3×#3752—21no—
21C3×C4⋊(C3⋊Dic3)—21no—
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23#88797—22no—
24#88792—22no—
25C2×D4×C3⋊D3—31no—
26C2×D3×D12—34no—
27C2×#3754—31no—
28C2×#3744—31no—
29#64106D311yescharacteristic, complemented
30C3×#897—34no—
31C2×#3717—34no—
32C2×#3715—34no—
33C32×#102—31no—
34#63647—31no—
35#63636—31no—
36#63636—34no—
37#63636—64no—
38C3×#809D314yescomplemented
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C2×D3×C3⋊D3—24no—
44C2×D3×C3⋊D3—42no—
45C2×D3×C3⋊D3—42no—
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C4⋊(C3⋊(C3⋊D3))—21no—
49C4⋊(C3⋊(C3⋊D3))D412yes—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C3×#634—21no—
55C3×#634—21no—
56C32×D12—21no—
57C32×D12D412yes—
58#6291—24no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60C32×C3⋊C8—21no—
61#5046—91no—
62#4776—94no—
63C23×C3⋊D3—61no—
64C22×D32—64no—
65C2×C32×D4—31no—
66C2×#638—61no—
67D4×C3⋊D3—62no—
68D4×C3⋊D3—62no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—34no—
71C2×C4⋊(C3⋊D3)—64no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×#634C2×D311yescharacteristic, complemented
74C2×C3×D12C2×D314yes—
75C2×C3×C4×D3—34no—
76C2×C3×Dic6—34no—
77C2×#626—64no—
78D3×D12—68no—
79D3×D12—68no—
80C2×C32×C8—31no—
81C32×C4⋊C4—24no—
82C32×C4⋊C4C2×D314yescomplemented
83C32×C4⋊C4C2×D311yescharacteristic, complemented
84#3754—62no—
85#3752—21no—
86C4⋊(C3⋊Dic3)—21no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90#3744—62no—
91C3×(C2×D3)⋊C4—64no—
92C3×C4⋊Dic3—64no—
93C2×C3×C3⋊C8C2×D314yescomplemented
94#3717—68no—
95#3715—68no—
96C22×C33C2×D411yescharacteristic
97C2×C3⋊(C3⋊D3)—22no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3SD1612yescomplemented
103C2×C32×D3—22no—
104C2×C32×D3—41no—
105C2×C32×D3—41no—
106D3×C3⋊D3—48no—
107C33×C4—41no—
108C33×C4C2×D411yescharacteristic
109C33×C4C2×D411yescharacteristic
110C3×C3⋊Dic3—21no—
111C3×C3⋊Dic3—41no—
112C2×D3×D4—94no—
113C22×D12—91no—
114C3×C22⋊Q8—91no—
115C2×#215—91no—
116C2×#215—91no—
117C2×#214—94no—
118C2×#204—94no—
119#897—34no—
120C3×#102—34no—
121#830—91no—
122(C2×D3)⋊C8—94no—
123#809—94no—
124#809—91no—
125#809—184no—
126#809C3⋊D311yescharacteristic, complemented
127C23×C32—61no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—61no—
130C22×C3⋊D3—64no—
131C22×C3⋊D3—124no—
132C22×C3⋊D3—122no—
133C22×C3⋊D3—122no—
134C22×C3⋊D3#6711yescharacteristic
135C22×C3×D3—24no—
136C22×C3×D3—34no—
137C2×D32—616no—
138C2×D32—128no—
139C2×D32—128no—
140C32×D4—32no—
141C32×D4—61no—
142C2×C32×C4—21no—
143C2×C32×C4—24no—
144C2×C32×C4—24no—
145C2×C32×C4—44no—
146C2×C32×C4C22×D314yes—
147C2×C32×C4C22×D311yescharacteristic
148#638—64no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3#6711yescharacteristic
151C4⋊(C3⋊D3)—32no—
152C4⋊(C3⋊D3)—38no—
153C4⋊(C3⋊D3)—68no—
154C4⋊(C3⋊D3)—61no—
155C4⋊(C3⋊D3)—64no—
156C4⋊(C3⋊D3)—124no—
157C4×C3⋊D3—21no—
158C4×C3⋊D3—21no—
159#634—21no—
160#634—21no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—64no—
163C3×D12—24no—
164C3×D12D1218yes—
165C3×C4×D3—64no—
166C3×C4×D3—64no—
167C3×Dic6—64no—
168C3×Dic6—64no—
169#626—616no—
170C32×C8—61no—
171C3⋊(C3⋊C8)—61no—
172C3⋊(C3⋊C8)—64no—
173C3⋊(C3⋊C8)—124no—
174C3×C3⋊C8—24no—
175#450—271noSylow
176C2×C33#12011yescharacteristic
177C2×C33C2×SD1611yescharacteristic
178C2×C33AGL(1,ℤ/8)11yescharacteristic
179C3⋊(C3⋊D3)—42no—
180C3×C3⋊D3—22no—
181C32×D3—42no—
182C23×D3—181no—
183C23×D3—184no—
184C2×C3×Q8—91no—
185C2×C3×D4—34no—
186C22×C3×C4—91no—
187C2×#67—184no—
188D3×D4—188no—
189D3×D4—188no—
190C2×D12—94no—
191C2×D12—91no—
192C2×D12—184no—
193C2×D12—182no—
194C2×D12—181no—
195C2×D12—182no—
196C2×D12C2×C3⋊D311yescharacteristic
197C2×C4×D3—34no—
198C2×Dic6—34no—
199C2×C3×C8—34no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—181no—
202C3×C4⋊C4D3214yescomplemented
203C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
204C3×C22⋊C4—181no—
205#215—182no—
206#215—182no—
207#214—188no—
208(C2×D3)⋊C4—64no—
209C4⋊Dic3—64no—
210C2×C3⋊C8—94no—
211C2×C3⋊C8—91no—
212C2×C3⋊C8—184no—
213C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
214#204—188no—
215C22×C32—24no—
216C22×C32—62no—
217C22×C32—121no—
218C22×C32—121no—
219C22×C32D3×D414yes—
220C22×C32C2×#6711yescharacteristic
221C2×C3⋊D3—21no—
222C2×C3⋊D3—21no—
223C2×C3⋊D3—62no—
224C2×C3⋊D3—68no—
225C2×C3⋊D3—128no—
226C2×C3⋊D3—128no—
227C2×C3⋊D3—121no—
228C2×C3⋊D3—121no—
229C2×C3⋊D3—124no—
230C2×C3⋊D3—124no—
231C2×C3⋊D3—244no—
232C2×C3⋊D3—244no—
233C2×C3⋊D3#21512yescomplemented
234C2×C3×D3—28no—
235C2×C3×D3—38no—
236C2×C3×D3—44no—
237C2×C3×D3—44no—
238C2×C3×D3—64no—
239C2×C3×D3—64no—
240D32—1232no—
241C32×C4—24no—
242C32×C4—24no—
243C32×C4—41no—
244C32×C4—44no—
245C32×C4—84no—
246C32×C4C2×D1214yes—
247C32×C4D3×D414yes—
248C32×C4D3×D411yescharacteristic
249C32×C4D3×D411yescharacteristic
250C3⋊Dic3—21no—
251C3⋊Dic3—41no—
252C3×Dic3—64no—
253C3×Dic3—124no—
254C22×D4—271no—
255C2×SD16—271no—
256C2×SD16—271no—
257C22⋊Q8—91no—
258#102—31no—
259#102—271no—
260C22⋊C8—271no—
261C33#45011yescharacteristic, complemented, Sylow
262C23×C3—64no—
263C23×C3—91no—
264C22×D3—21no—
265C22×D3—34no—
266C22×D3—184no—
267C22×D3—1816no—
268C22×D3—184no—
269C22×D3—181no—
270C22×D3—364no—
271C22×D3—368no—
272C22×D3—362no—
273C22×D3—362no—
274C22×D3—368no—
275C3×Q8—181no—
276C3×Q8—181no—
277C3×D4—38no—
278C3×D4—64no—
279C2×C3×C4—24no—
280C2×C3×C4—21no—
281C2×C3×C4—24no—
282C2×C3×C4—44no—
283C2×C3×C4—91no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4—181no—
287C2×C3×C4C2×D3214yes—
288C2×C3×C4C22×C3⋊D311yescharacteristic
289#67—1816no—
290C2×Dic3—34no—
291C2×Dic3—64no—
292D12—21no—
293D12—98no—
294D12—92no—
295D12—188no—
296D12—184no—
297D12—184no—
298D12—181no—
299D12—364no—
300D12C4⋊(C3⋊D3)12yes—
301C4×D3—64no—
302C4×D3—64no—
303Dic6—64no—
304Dic6—64no—
305C3×C8—64no—
306C3⋊C8—21no—
307C3⋊C8—184no—
308C3⋊C8—181no—
309C3⋊C8—364no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—122no—
314C2×C32#88214yes—
315C2×C32#91114yes—
316C2×C32D3×SD1614yes—
317C2×C32#93711yescharacteristic
318C2×C32C2×#21511yescharacteristic
319C2×C32#94911yescharacteristic
320C3⋊D3—22no—
321C3⋊D3—122no—
322C3⋊D3—128no—
323C3⋊D3—248no—
324C3×D3—48no—
325C3×D3—68no—
326C24—541no—
327C2×Q8—91no—
328C2×D4—31no—
329C2×D4—271no—
330C2×D4—541no—
331C2×D4—542no—
332C2×D4—542no—
333C22×C4—91no—
334SD16—542no—
335SD16—542no—
336C2×C8—31no—
337C2×C8—271no—
338C4⋊C4—181no—
339C4⋊C4D3×C3⋊D311yescharacteristic, complemented
340C22⋊C4—181no—
341C22×C3—24no—
342C22×C3—68no—
343C22×C3—92no—
344C22×C3—124no—
345C22×C3—124no—
346C22×C3—181no—
347C22×C3—181no—
348C22×C3D3×#6714yes—
349C22×C3D4×C3⋊D311yescharacteristic
350C2×D3—22no—
351C2×D3—38no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—64no—
355C2×D3—64no—
356C2×D3—182no—
357C2×D3—188no—
358C2×D3—368no—
359C2×D3—368no—
360C2×D3—3632no—
361C2×D3—361no—
362C2×D3—361no—
363C2×D3—364no—
364C2×D3—364no—
365C2×D3—724no—
366C2×D3—724no—
367C3×C4—24no—
368C3×C4—24no—
369C3×C4—41no—
370C3×C4—44no—
371C3×C4—84no—
372C3×C4—181no—
373C3×C4—361no—
374C3×C4D3×D1214yes—
375C3×C4#380214yes—
376C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
377C3×C4D4×C3⋊D311yescharacteristic
378Dic3—64no—
379Dic3—124no—
380C32—24no—
381C32#477614yescomplemented
382C32#504611yescharacteristic, complemented
383C23—61no—
384C23—91no—
385C23—544no—
386C23—541no—
387C23—1082no—
388C23—1082no—
389Q8—181no—
390Q8—181no—
391D4—32no—
392D4—61no—
393D4—272no—
394D4—544no—
395D4—541no—
396C2×C4—21no—
397C2×C4—91no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—181no—
401C2×C4C2×D3×C3⋊D311yescharacteristic
402C8—61no—
403C8—541no—
404C2×C3—24no—
405C2×C3—24no—
406C2×C3—24no—
407C2×C3—128no—
408C2×C3—182no—
409C2×C3#6392214yes—
410C2×C3#6392514yes—
411C2×C3D3×#21514yes—
412C2×C3#6409111yescharacteristic
413C2×C3#6412011yescharacteristic
414C2×C3SD16×C3⋊D311yescharacteristic
415D3—42no—
416D3—68no—
417D3—368no—
418D3—362no—
419D3—728no—
420C22—62no—
421C22—92no—
422C22—121no—
423C22—121no—
424C22—181no—
425C22—181no—
426C22—542no—
427C22—1088no—
428C22—1081no—
429C22—1081no—
430C22C3⋊D3×#6711yescharacteristic
431C4—41no—
432C4—181no—
433C4—361no—
434C4D3×C4⋊(C3⋊D3)11yescharacteristic
435C4#8902611yescharacteristic
436C3—24no—
437C3#1059059614yescomplemented
438C3#1059247511yescharacteristic, complemented
439C2—122no—
440C2—182no—
441C2—1082no—
442C2#1172354111yescharacteristic
443C2#1172354611yescharacteristic
444C2C3⋊D3×#21511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3⋊D3×D12C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#450
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64106C2semidirect
3. C2×#88797C2semidirect
4. C2×#88792C2semidirect
5. #11722179C2semidirect
6. C32×#809C2semidirect
7. #11722023C2semidirect
13. C2×C3×#634C22semidirect
18. C33×C4⋊C4C22semidirect
19. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
22. C2×C32×C3⋊C8C22semidirect
29. #64106D3semidirect
38. C3×#809D3semidirect
73. C2×#634C2×D3semidirect
82. C32×C4⋊C4C2×D3semidirect
83. C32×C4⋊C4C2×D3semidirect
93. C2×C3×C3⋊C8C2×D3semidirect
102. C2×C3×C3⋊D3SD16semidirect
126. #809C3⋊D3semidirect
202. C3×C4⋊C4D32semidirect
203. C3×C4⋊C4C2×C3⋊D3semidirect
213. C2×C3⋊C8C2×C3⋊D3semidirect
233. C2×C3⋊D3#215semidirect
261. C33#450semidirect
339. C4⋊C4D3×C3⋊D3semidirect
381. C32#4776semidirect
382. C32#5046semidirect
437. C3#10590596semidirect
438. C3#10592475semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
C3×#641062
C2×#887972
C2×#887922
#117221792
C32×#8092
#117220232
#105924753
#105905963