Finite Group Databasebeta

← #13159420

Subgroups — #13159420

Order 1728.

1328 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1328 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
2C32×#897C211yesmaximal, characteristic, complemented
3C2×#88795C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592745—31nomaximal
9#10590596—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic, complemented
15C2×#6292—21no—
16D3×C4⋊(C3⋊D3)—22no—
17D3×C4⋊(C3⋊D3)—22no—
18C33×C4⋊C4C2211yescharacteristic, complemented
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
21C32×(C2×D3)⋊C4—21no—
22C32×C4⋊Dic3—21no—
23#88795—22no—
24#88793—22no—
25C22×C4⋊(C3⋊D3)—31no—
26C2×D3×D12—34no—
27C32×C22⋊Q8—31no—
28C2×#3755—31no—
29C2×#3755—31no—
30C3×#897D314yescomplemented
31C2×#3717—34no—
32C2×#3715—34no—
33#63657—31no—
34#63636—31no—
35#63636—34no—
36#63636—64no—
37#63636D311yescharacteristic, complemented
38C3×#809—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3—21no—
42C22×C32×D3D411yescharacteristic
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—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)D412yes—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C32×C4×D3—21no—
55C32×C4×D3—21no—
56C32×Dic6—21no—
57C32×Dic6—21no—
58#6292—24no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60C3×C3⋊(C3⋊C8)—21no—
61#5046—94no—
62#4776—91no—
63C23×C3⋊D3—61no—
64C22×D32—64no—
65C2×C32×Q8—31no—
66C22×C32×C4—31no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—34no—
69C2×C4⋊(C3⋊D3)—62no—
70C2×C4⋊(C3⋊D3)—61no—
71C2×C4⋊(C3⋊D3)—62no—
72C2×C4⋊(C3⋊D3)—64no—
73C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
74C2×C3×D12—34no—
75C2×C3×C4×D3C2×D314yes—
76C2×C3×Dic6C2×D314yescomplemented
77C2×#626—64no—
78D3×D12—68no—
79D3×D12—68no—
80C32×C4⋊C4—24no—
81C32×C4⋊C4—61no—
82C32×C4⋊C4C2×D314yescomplemented
83C32×C4⋊C4C2×D311yescharacteristic, complemented
84C32×C22⋊C4—61no—
85#3755—62no—
86#3755—62no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
91C3×(C2×D3)⋊C4—24no—
92C3×C4⋊Dic3—24no—
93C2×C3×C3⋊C8—34no—
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—22no—
101C2×C3×C3⋊D3—41no—
102C2×C3×C3⋊D3—41no—
103C2×C32×D3—21no—
104C2×C32×D3—21no—
105C2×C32×D3SD1612yescomplemented
106D3×C3⋊D3—48no—
107C33×C4—41no—
108C33×C4C2×D411yescharacteristic
109C33×C4C2×D411yescharacteristic
110C32×Dic3—21no—
111C32×Dic3—41no—
112C2×D3×D4—91no—
113C22×D12—94no—
114C3×C22⋊Q8—34no—
115C2×#215—94no—
116C2×#215—94no—
117C2×#214—91no—
118C2×#204—91no—
119#897C3⋊D311yescharacteristic, complemented
120C3×#102—91no—
121#830—94no—
122(C2×D3)⋊C8—91no—
123#809—34no—
124#809—94no—
125#809—91no—
126#809—184no—
127C23×C32—31no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—61no—
131C22×C3⋊D3—64no—
132C22×C3⋊D3—124no—
133C22×C3⋊D3—122no—
134C22×C3⋊D3—122no—
135C22×C3×D3—64no—
136C22×C3×D3#6714yes—
137C2×D32—616no—
138C2×D32—128no—
139C2×D32—128no—
140C32×Q8—61no—
141C32×Q8—61no—
142C2×C32×C4—21no—
143C2×C32×C4—24no—
144C2×C32×C4—24no—
145C2×C32×C4—31no—
146C2×C32×C4—44no—
147C2×C32×C4—61no—
148C2×C32×C4—61no—
149C2×C32×C4—61no—
150C2×C32×C4C22×D314yes—
151C2×C32×C4C22×D311yescharacteristic
152C4⋊(C3⋊D3)—21no—
153C4⋊(C3⋊D3)—32no—
154C4⋊(C3⋊D3)—38no—
155C4⋊(C3⋊D3)—68no—
156C4⋊(C3⋊D3)—64no—
157C4⋊(C3⋊D3)—61no—
158C4⋊(C3⋊D3)—64no—
159C4⋊(C3⋊D3)—124no—
160C4⋊(C3⋊D3)D1212yes—
161C2×C3×Dic3—24no—
162C2×C3×Dic3#6714yes—
163C3×D12—38no—
164C3×D12—64no—
165C3×C4×D3—24no—
166C3×C4×D3—24no—
167C3×Dic6—24no—
168C3×Dic6—24no—
169#626—616no—
170C3⋊(C3⋊C8)—21no—
171C3⋊(C3⋊C8)—61no—
172C3⋊(C3⋊C8)—64no—
173C3⋊(C3⋊C8)—124no—
174C3×C3⋊C8—64no—
175#450—271noSylow
176C2×C33#12011yescharacteristic
177C2×C33C2×SD1611yescharacteristic
178C2×C33AGL(1,ℤ/8)11yescharacteristic
179C3⋊(C3⋊D3)—42no—
180C3×C3⋊D3—42no—
181C32×D3—22no—
182C23×D3—184no—
183C23×D3—181no—
184C2×C3×Q8—34no—
185C2×C3×D4—91no—
186C22×C3×C4—34no—
187C2×#67—181no—
188D3×D4—182no—
189D3×D4—182no—
190C2×D12—34no—
191C2×D12—94no—
192C2×D12—91no—
193C2×D12—184no—
194C2×D12—188no—
195C2×D12—188no—
196C2×D12—184no—
197C2×C4×D3C2×C3⋊D311yescharacteristic
198C2×Dic6C2×C3⋊D311yescharacteristic, complemented
199C2×C3×C8—91no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—64no—
202C3×C4⋊C4D3214yescomplemented
203C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
204C3×C22⋊C4—64no—
205#215—188no—
206#215—188no—
207#214—182no—
208(C2×D3)⋊C4—21no—
209C4⋊Dic3—21no—
210C2×C3⋊C8—34no—
211C2×C3⋊C8—94no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—184no—
214#204—182no—
215C22×C32—24no—
216C22×C32—32no—
217C22×C32—61no—
218C22×C32—61no—
219C22×C32D3×D411yescharacteristic
220C22×C32C2×#6714yes—
221C2×C3⋊D3—22no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—62no—
225C2×C3⋊D3—68no—
226C2×C3⋊D3—128no—
227C2×C3⋊D3—128no—
228C2×C3⋊D3—121no—
229C2×C3⋊D3—121no—
230C2×C3⋊D3—124no—
231C2×C3⋊D3—124no—
232C2×C3⋊D3—244no—
233C2×C3⋊D3—244no—
234C2×C3×D3—24no—
235C2×C3×D3—24no—
236C2×C3×D3—68no—
237C2×C3×D3—124no—
238C2×C3×D3—124no—
239C2×C3×D3#21518yescomplemented
240D32—1232no—
241C32×C4—24no—
242C32×C4—24no—
243C32×C4—41no—
244C32×C4—44no—
245C32×C4—61no—
246C32×C4—84no—
247C32×C4—121no—
248C32×C4C2×D1211yescharacteristic
249C32×C4D3×D414yes—
250C32×C4D3×D414yes—
251C32×C4D3×D411yescharacteristic
252C3×Dic3—24no—
253C3×Dic3—44no—
254C22×D4—271no—
255C2×SD16—271no—
256C2×SD16—271no—
257C22⋊Q8—31no—
258#102—91no—
259#102—271no—
260C22⋊C8—271no—
261C33#45011yescharacteristic, complemented, Sylow
262C23×C3—34no—
263C23×C3—181no—
264C22×D3—64no—
265C22×D3—184no—
266C22×D3—1816no—
267C22×D3—184no—
268C22×D3—181no—
269C22×D3—364no—
270C22×D3—362no—
271C22×D3—368no—
272C22×D3—368no—
273C22×D3—362no—
274C22×D3#63811yescharacteristic
275C3×Q8—64no—
276C3×Q8—64no—
277C3×D4—92no—
278C3×D4—181no—
279C2×C3×C4—24no—
280C2×C3×C4—21no—
281C2×C3×C4—24no—
282C2×C3×C4—34no—
283C2×C3×C4—44no—
284C2×C3×C4—64no—
285C2×C3×C4—64no—
286C2×C3×C4—64no—
287C2×C3×C4C2×D3214yes—
288C2×C3×C4C22×C3⋊D311yescharacteristic
289#67—184no—
290C2×Dic3—21no—
291C2×Dic3#63811yescharacteristic
292D12—38no—
293D12—64no—
294D12—98no—
295D12—92no—
296D12—188no—
297D12—1816no—
298D12—184no—
299D12—181no—
300D12—364no—
301C4×D3—21no—
302C4×D3—21no—
303Dic6—21no—
304Dic6—21no—
305C3×C8—181no—
306C3⋊C8—64no—
307C3⋊C8—184no—
308C3⋊C8—181no—
309C3⋊C8—364no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—62no—
314C2×C32#88211yescharacteristic
315C2×C32#91111yescharacteristic
316C2×C32D3×SD1611yescharacteristic
317C2×C32#93714yes—
318C2×C32C2×#21514yes—
319C2×C32#94914yes—
320C3⋊D3—42no—
321C3⋊D3—122no—
322C3⋊D3—128no—
323C3⋊D3—248no—
324C3×D3—28no—
325C3×D3—128no—
326C24—541no—
327C2×Q8—31no—
328C2×D4—91no—
329C2×D4—271no—
330C2×D4—541no—
331C2×D4—542no—
332C2×D4—542no—
333C22×C4—31no—
334SD16—542no—
335SD16—542no—
336C2×C8—91no—
337C2×C8—271no—
338C4⋊C4—61no—
339C4⋊C4D3×C3⋊D311yescharacteristic, complemented
340C22⋊C4—61no—
341C22×C3—24no—
342C22×C3—38no—
343C22×C3—64no—
344C22×C3—64no—
345C22×C3—182no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3D3×#6714yes—
349C22×C3C2×#63811yescharacteristic
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—68no—
353C2×D3—124no—
354C2×D3—124no—
355C2×D3—182no—
356C2×D3—188no—
357C2×D3—368no—
358C2×D3—368no—
359C2×D3—3632no—
360C2×D3—361no—
361C2×D3—361no—
362C2×D3—364no—
363C2×D3—364no—
364C2×D3—724no—
365C2×D3—724no—
366C2×D3#375512yescomplemented
367C3×C4—24no—
368C3×C4—24no—
369C3×C4—41no—
370C3×C4—44no—
371C3×C4—64no—
372C3×C4—84no—
373C3×C4—124no—
374C3×C4D3×D1214yes—
375C3×C4#380214yes—
376C3×C4D4×C3⋊D311yescharacteristic
377C3×C4D4×C3⋊D311yescharacteristic
378Dic3—21no—
379Dic3—41no—
380C32—24no—
381C32#477611yescharacteristic, complemented
382C32#504614yescomplemented
383C23—31no—
384C23—181no—
385C23—544no—
386C23—541no—
387C23—1082no—
388C23—1082no—
389Q8—61no—
390Q8—61no—
391D4—92no—
392D4—181no—
393D4—272no—
394D4—544no—
395D4—541no—
396C2×C4—21no—
397C2×C4—31no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4C2×D3×C3⋊D311yescharacteristic
402C8—181no—
403C8—541no—
404C2×C3—24no—
405C2×C3—24no—
406C2×C3—24no—
407C2×C3—68no—
408C2×C3—362no—
409C2×C3#6392214yes—
410C2×C3#6392514yes—
411C2×C3D3×#21514yes—
412C2×C3#6414611yescharacteristic
413C2×C3C2×#375511yescharacteristic
414C2×C3#6415811yescharacteristic
415D3—22no—
416D3—128no—
417D3—368no—
418D3—362no—
419D3—728no—
420C22—32no—
421C22—61no—
422C22—61no—
423C22—182no—
424C22—361no—
425C22—361no—
426C22—542no—
427C22—1088no—
428C22—1081no—
429C22—1081no—
430C22D3×#63811yescharacteristic
431C4—41no—
432C4—61no—
433C4—121no—
434C4C3⋊D3×D1211yescharacteristic
435C4#8902611yescharacteristic
436C3—24no—
437C3#1059059614yescomplemented
438C3#1059274511yescharacteristic, complemented
439C2—62no—
440C2—362no—
441C2—1082no—
442C2#1172354011yescharacteristic
443C2#1172354411yescharacteristic
444C2D3×#375511yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×D3×C4⋊(C3⋊D3)C2
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. C32×#897C2semidirect
3. C2×#88795C2semidirect
4. C2×#88793C2semidirect
5. #11722179C2semidirect
6. C3×#63636C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
14. C2×C32×Dic6C22semidirect
18. C33×C4⋊C4C22semidirect
19. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
20. C2×C3×C3⋊(C3⋊C8)C22semidirect
30. C3×#897D3semidirect
37. #63636D3semidirect
76. C2×C3×Dic6C2×D3semidirect
82. C32×C4⋊C4C2×D3semidirect
83. C32×C4⋊C4C2×D3semidirect
90. C2×C3⋊(C3⋊C8)C2×D3semidirect
105. C2×C32×D3SD16semidirect
119. #897C3⋊D3semidirect
198. C2×Dic6C2×C3⋊D3semidirect
202. C3×C4⋊C4D32semidirect
203. C3×C4⋊C4C2×C3⋊D3semidirect
239. C2×C3×D3#215semidirect
261. C33#450semidirect
339. C4⋊C4D3×C3⋊D3semidirect
366. C2×D3#3755semidirect
381. C32#4776semidirect
382. C32#5046semidirect
437. C3#10590596semidirect
438. C3#10592745semidirect

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
C32×#8972
C2×#887952
C2×#887932
#117221792
C3×#636362
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927453
#105905963