Finite Group Databasebeta

← #13138648

Subgroups — #13138648

Order 1728.

614 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 95 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×D36C211yesmaximal, characteristic
2C2×C3×#3664C211yesmaximal, characteristic, complemented
3#11720960C211yesmaximal, characteristic, complemented
4C2×#88416C211yesmaximal, characteristic, complemented
5#11720543C211yesmaximal, characteristic, complemented
6C9×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11720283C211yesmaximal, characteristic, complemented
8#10590942—31nomaximal
9#10588413—31nomaximal
10C22×D3×D9—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×D3×C9C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×C3×Dic18C2211yescharacteristic, complemented
15C2×#6192—21no—
16D3×D36—22no—
17D3×D36—22no—
18C2×C3×C8×C9C2211yescharacteristic, complemented
19C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
20C2×C9×C3⋊C8C2211yescharacteristic, complemented
21C3×#3664—22no—
22(C2×D3)⋊Dic9—21no—
23#88442—21no—
24#88416—22no—
25C2×D3×D12—31no—
26C2×C3×#204—31no—
27#63856—31no—
28C2×#3717—31no—
29C22×D36—31no—
30#63648—31no—
31C3×(C2×D3)⋊C8D311yescharacteristic
32#63563—31no—
33#63486—31no—
34C2×#3664—31no—
35C2×#3664D311yescharacteristic, complemented
36C9×C22⋊C8—31no—
37#63380—31no—
38#63380—31no—
39#63380—61no—
40C22×C3⋊D9—21no—
41C22×D3×C9D411yescharacteristic
42C22×C3×D9—21no—
43C2×D3×D9—24no—
44C2×D3×D9—42no—
45C2×D3×D9—42no—
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3⋊Dic9—21no—
48C4⋊(C3⋊D9)—21no—
49C4⋊(C3⋊D9)D412yes—
50C2×C9×Dic3D411yescharacteristic
51C2×C3×Dic9—21no—
52C4×D3×C9—21no—
53C4×D3×C9—21no—
54C3×D36—21no—
55C3×D36D412yes—
56C3×Dic18—21no—
57C3×Dic18—21no—
58#6192—24no—
59C3×C8×C9—21no—
60C9×C3⋊C8—21no—
61#5169—91no—
62#4731—91no—
63C22×D32—61no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C3×D12—31no—
66C2×C3×C4×D3C2×D311yescharacteristic
67C2×C3×Dic6—31no—
68C2×#626—61no—
69D3×D12—62no—
70D3×D12—62no—
71C23×D9—61no—
72C2×C32×C8C2×D311yescharacteristic
73C4⋊(C3⋊Dic3)—31no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75C3×#204—62no—
76#3724—61no—
77(C2×D3)⋊Dic3—61no—
78#3717—62no—
79C22×C4×C9—31no—
80C2×D36—31no—
81C2×D36—62no—
82C2×D36—61no—
83C2×D36—62no—
84C2×D36—61no—
85C2×D36C2×D311yescharacteristic
86C2×Dic18C2×D311yescharacteristic, complemented
87C2×C8×C9—21no—
88C2×C8×C9—31no—
89C2×C8×C9C2×D311yescharacteristic, complemented
90#3676—61no—
91C4⋊Dic9—31no—
92C4⋊Dic9—61no—
93Dic9⋊C4—61no—
94#3664—22no—
95#3664—62no—
96C22×C3×C9C2×D411yescharacteristic
97C2×C3⋊D9—22no—
98C2×C3⋊D9—41no—
99C2×C3⋊D9—41no—
100C2×D3×C9—21no—
101C2×D3×C9—21no—
102C2×D3×C9SD1612yescomplemented
103C2×C3×D9—22no—
104C2×C3×D9—41no—
105C2×C3×D9—41no—
106D3×D9—48no—
107C3×C4×C9C2×D411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C3⋊Dic9—41no—
110C9×Dic3—21no—
111C3×Dic9—41no—
112C2×D3×D4—91no—
113C22×D12—91no—
114C2×C3×SD16—91no—
115#946—91no—
116C2×#215—91no—
117#924—91no—
118C2×#204—31no—
119C2×#204—91no—
120C3×C22⋊C8—31no—
121#832—91no—
122#821—91no—
123#821—91no—
124#821—91no—
125#821—181no—
126(C2×D3)⋊C8D911yescharacteristic, complemented
127C22×C3⋊D3—61no—
128C22×C3×D3—61no—
129C22×C3×D3D1211yescharacteristic
130C2×D32—64no—
131C2×D32—122no—
132C2×D32—122no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—61no—
135C4⋊(C3⋊D3)—32no—
136C4⋊(C3⋊D3)—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3D1211yescharacteristic
139C3×D12—32no—
140C3×D12—61no—
141C3×C4×D3—21no—
142C3×C4×D3—21no—
143C3×Dic6—61no—
144C3×Dic6—61no—
145#626—64no—
146C23×C9—31no—
147C22×D9—21no—
148C22×D9—64no—
149C22×D9—61no—
150C22×D9—121no—
151C22×D9—122no—
152C22×D9—122no—
153C32×C8—21no—
154C3×C3⋊C8—21no—
155C2×C4×C9—21no—
156C2×C4×C9—31no—
157C2×C4×C9—61no—
158C2×C4×C9—61no—
159C2×C4×C9C22×D311yescharacteristic
160C2×Dic9—21no—
161C2×Dic9—61no—
162C2×Dic9—121no—
163D36—21no—
164D36—32no—
165D36—62no—
166D36—64no—
167D36—61no—
168D36—121no—
169D36#6712yes—
170Dic18—21no—
171Dic18—21no—
172C8×C9—21no—
173C8×C9—41no—
174C8×C9—61no—
175#450—271noSylow
176C2×C3×C9#12011yescharacteristic
177C2×C3×C9C2×SD1611yescharacteristic
178C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
179C3⋊D9—42no—
180D3×C9—22no—
181C3×D9—42no—
182C23×D3—181no—
183C23×D3—181no—
184C2×C3×Q8—91no—
185C2×C3×D4—91no—
186C22×C3×C4—31no—
187C2×#67—181no—
188D3×D4—182no—
189D3×D4—182no—
190C2×D12—31no—
191C2×D12—91no—
192C2×D12—91no—
193C2×D12—181no—
194C2×D12—182no—
195C2×D12—182no—
196C2×D12—181no—
197C2×C4×D3C2×D911yescharacteristic
198C2×Dic6—31no—
199C3×SD16—182no—
200C2×C3×C8—21no—
201C2×C3×C8—31no—
202C2×C3×C8C2×D911yescharacteristic, complemented
203C2×C3×C8D3211yescharacteristic
204#217—181no—
205#215—182no—
206(C2×D3)⋊C4—181no—
207C4⋊Dic3—91no—
208C4⋊Dic3—91no—
209C4⋊Dic3—181no—
210Dic3⋊C4—181no—
211Dic3⋊C4—181no—
212C2×C3⋊C8C2×D911yescharacteristic, complemented
213#204—62no—
214#204—182no—
215C22×C32C2×D1211yescharacteristic
216C2×C3⋊D3—62no—
217C2×C3⋊D3—121no—
218C2×C3⋊D3—121no—
219C2×C3×D3—21no—
220C2×C3×D3—21no—
221C2×C3×D3—62no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C2×C3×D3#20412yes—
225D32—128no—
226C32×C4D3×D411yescharacteristic
227C32×C4D3×D411yescharacteristic
228C3⋊Dic3—121no—
229C3×Dic3—21no—
230C3×Dic3—121no—
231C22×C9—21no—
232C22×C9—32no—
233C22×C9—61no—
234C22×C9—61no—
235C22×C9D3×D411yescharacteristic
236C2×D9—22no—
237C2×D9—41no—
238C2×D9—41no—
239C2×D9—62no—
240C2×D9—122no—
241C2×D9—128no—
242C2×D9—121no—
243C2×D9—121no—
244C2×D9—241no—
245C2×D9—241no—
246C4×C9—21no—
247C4×C9—21no—
248C4×C9—61no—
249C4×C9D3×D411yescharacteristic
250C4×C9C2×#6711yescharacteristic
251Dic9—41no—
252Dic9—121no—
253Dic9—241no—
254C22×D4—271no—
255C2×SD16—91no—
256C2×SD16—271no—
257C22⋊Q8—271no—
258#102—271no—
259#102—271no—
260C22⋊C8—31no—
261C3×C9#45011yescharacteristic, complemented, Sylow
262C23×C3—31no—
263C23×C3—181no—
264C22×D3—61no—
265C22×D3—184no—
266C22×D3—184no—
267C22×D3—181no—
268C22×D3—181no—
269C22×D3—361no—
270C22×D3—362no—
271C22×D3—362no—
272C22×D3—362no—
273C22×D3—362no—
274C22×D3D3611yescharacteristic
275C3×Q8—181no—
276C3×Q8—181no—
277C3×D4—92no—
278C3×D4—181no—
279C2×C3×C4—21no—
280C2×C3×C4—31no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—181no—
284C2×C3×C4C22×D911yescharacteristic
285C2×C3×C4C2×D3211yescharacteristic
286#67—184no—
287C2×Dic3—61no—
288C2×Dic3—181no—
289C2×Dic3—181no—
290C2×Dic3—361no—
291C2×Dic3D3611yescharacteristic
292D12—32no—
293D12—61no—
294D12—92no—
295D12—92no—
296D12—182no—
297D12—184no—
298D12—181no—
299D12—181no—
300D12—361no—
301C4×D3—21no—
302C4×D3—21no—
303Dic6—61no—
304Dic6—61no—
305C3×C8—21no—
306C3×C8—21no—
307C3×C8—41no—
308C3×C8—61no—
309C3⋊C8—21no—
310C2×C32#88211yescharacteristic
311C2×C32C2×#20411yescharacteristic
312C2×C32#90811yescharacteristic
313C3⋊D3—122no—
314C3×D3—22no—
315C3×D3—122no—
316C2×C9—21no—
317C2×C9—21no—
318C2×C9—21no—
319C2×C9—62no—
320C2×C9D3×SD1611yescharacteristic
321C2×C9#91411yescharacteristic
322C2×C9#93711yescharacteristic
323D9—42no—
324D9—122no—
325D9—242no—
326C24—541no—
327C2×Q8—91no—
328C2×D4—91no—
329C2×D4—271no—
330C2×D4—541no—
331C2×D4—542no—
332C2×D4—542no—
333C22×C4—31no—
334SD16—182no—
335SD16—542no—
336C2×C8—31no—
337C2×C8D3×D911yescharacteristic, complemented
338C4⋊C4—271no—
339C4⋊C4—541no—
340C22⋊C4—541no—
341C22×C3—21no—
342C22×C3—32no—
343C22×C3—61no—
344C22×C3—61no—
345C22×C3—182no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3C2×D3611yescharacteristic
349C22×C3D3×D1211yescharacteristic
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—62no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—182no—
356C2×D3—182no—
357C2×D3—362no—
358C2×D3—368no—
359C2×D3—368no—
360C2×D3—361no—
361C2×D3—361no—
362C2×D3—361no—
363C2×D3—361no—
364C2×D3—721no—
365C2×D3—721no—
366C2×D3#366412yescomplemented
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—61no—
370C3×C4—361no—
371C3×C4D4×D911yescharacteristic
372C3×C4D4×D911yescharacteristic
373C3×C4#380211yescharacteristic
374C3×C4D3×#6711yescharacteristic
375Dic3—21no—
376Dic3—121no—
377Dic3—361no—
378Dic3—361no—
379Dic3—721no—
380C32#473111yescharacteristic
381C9—21no—
382C9#516911yescharacteristic, complemented
383C23—31no—
384C23—181no—
385C23—544no—
386C23—541no—
387C23—1082no—
388C23—1082no—
389Q8—181no—
390Q8—181no—
391D4—92no—
392D4—181no—
393D4—272no—
394D4—544no—
395D4—541no—
396C2×C4—31no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—181no—
400C2×C4—541no—
401C2×C4C2×D3×D911yescharacteristic
402C8—21no—
403C8—61no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—62no—
408C2×C3—362no—
409C2×C3#6344411yescharacteristic
410C2×C3C2×#366411yescharacteristic
411C2×C3#6347011yescharacteristic
412C2×C3D3×#20411yescharacteristic
413C2×C3#6379411yescharacteristic
414C2×C3#6392211yescharacteristic
415D3—22no—
416D3—122no—
417D3—362no—
418D3—362no—
419D3—722no—
420C22—32no—
421C22—61no—
422C22—61no—
423C22—182no—
424C22—361no—
425C22—361no—
426C22—542no—
427C22—1088no—
428C22—1081no—
429C22—1081no—
430C22D3×D3611yescharacteristic
431C4—61no—
432C4—361no—
433C4—1081no—
434C4#8864611yescharacteristic
435C4D9×#6711yescharacteristic
436C3—21no—
437C3#1058841311yescharacteristic, complemented
438C3#1059094211yescharacteristic
439C2—62no—
440C2—362no—
441C2—1082no—
442C2D3×#366411yescharacteristic
443C2#1172080411yescharacteristic
444C2#1172104911yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radicalC2×D3×D36C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#450
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3664C2semidirect
3. #11720960C2semidirect
4. C2×#88416C2semidirect
5. #11720543C2semidirect
6. C9×(C2×D3)⋊C8C2semidirect
7. #11720283C2semidirect
14. C2×C3×Dic18C22semidirect
18. C2×C3×C8×C9C22semidirect
19. C4⋊(C3⋊Dic9)C22semidirect
20. C2×C9×C3⋊C8C22semidirect
35. C2×#3664D3semidirect
86. C2×Dic18C2×D3semidirect
89. C2×C8×C9C2×D3semidirect
102. C2×D3×C9SD16semidirect
126. (C2×D3)⋊C8D9semidirect
202. C2×C3×C8C2×D9semidirect
212. C2×C3⋊C8C2×D9semidirect
261. C3×C9#450semidirect
337. C2×C8D3×D9semidirect
366. C2×D3#3664semidirect
382. C9#5169semidirect
437. C3#10588413semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D362
C2×C3×#36642
#117209602
C2×#884162
#117205432
C9×(C2×D3)⋊C82
#117202832
#105909423
#105884133