Finite Group Databasebeta

← #13138651

Subgroups — #13138651

Order 1728.

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

normal subgroups only

This group has 670 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, complemented
2C2×#88525C211yesmaximal, characteristic, complemented
3#11721038C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic, complemented
5C9×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6C3×#63380C211yesmaximal, characteristic, complemented
7#11720284C211yesmaximal, characteristic, complemented
8#10590947—31nomaximal
9#10588414—31nomaximal
10C22×D3×D9—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
13C2×C4×D3×C9C2211yescharacteristic
14C2×C3×D36C2211yescharacteristic
15C2×#6195—21no—
16C2×#6192—21no—
17D3×D36—22no—
18D3×D36—22no—
19C2×C3×C8×C9C2211yescharacteristic, complemented
20#88525—22no—
21C3×C4⋊Dic9C2211yescharacteristic, complemented
22C2×C9×C3⋊C8C2211yescharacteristic, complemented
23#88445—21no—
24#88417—22no—
25C2×D3×D12—31no—
26C2×#3745—31no—
27#63909—31no—
28C2×#3714—31no—
29C22×D36—31no—
30C3×#821—31no—
31C3×(C2×D3)⋊C8D311yescharacteristic
32#63564—31no—
33#63492—31no—
34C2×D72—31no—
35C2×D72—31no—
36C2×D72—61no—
37C9×C22⋊C8—31no—
38#63380—31no—
39#63380D311yescharacteristic, complemented
40C22×C3⋊D9—21no—
41C22×C3⋊D9—21no—
42C22×D3×C9D411yescharacteristic
43C22×C3×D9—21no—
44C2×D3×D9—24no—
45C2×D3×D9—42no—
46C2×D3×D9—42no—
47C2×C3×C4×C9C2311yescharacteristic
48C4⋊(C3⋊D9)—21no—
49C4⋊(C3⋊D9)—21no—
50C4⋊(C3⋊D9)—21no—
51C4⋊(C3⋊D9)D412yescomplemented
52C2×C9×Dic3D411yescharacteristic
53C2×C3×Dic9—21no—
54C4×D3×C9—21no—
55C4×D3×C9—21no—
56C3×D36—21no—
57C3×D36D412yescomplemented
58#6195—42no—
59#6192—24no—
60C3×C8×C9—21no—
61C9×C3⋊C8—21no—
62#4773—91no—
63#4732—91no—
64C22×D32—61no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—31no—
67C2×C3×D12—31no—
68C2×C3×C4×D3C2×D311yescharacteristic
69C2×#626—61no—
70C2×#626—61no—
71D3×D12—62no—
72D3×D12—62no—
73C23×D9—61no—
74C2×C32×C8C2×D311yescharacteristic
75#3745—62no—
76C3×C4⋊Dic3—31no—
77C2×C3×C3⋊C8C2×D311yescharacteristic
78#3722—61no—
79#3714—62no—
80C22×C4×C9—31no—
81C2×#611—61no—
82C2×D36—31no—
83C2×D36—31no—
84C2×D36—62no—
85C2×D36—61no—
86C2×D36—62no—
87C2×D36—61no—
88C2×D36—61no—
89C2×D36C2×D311yescharacteristic
90C2×C8×C9—21no—
91C2×C8×C9—31no—
92C2×C8×C9C2×D311yescharacteristic, complemented
93(C2×D9)⋊C4—61no—
94C4⋊Dic9C2×D311yescharacteristic, complemented
95D72—62no—
96D72—64no—
97D72—62no—
98C22×C3×C9C2×D411yescharacteristic
99C2×C3⋊D9—22no—
100C2×C3⋊D9—41no—
101C2×C3⋊D9—41no—
102C2×C3⋊D9—41no—
103C2×C3⋊D9—41no—
104C2×C3⋊D9—41no—
105C2×D3×C9—21no—
106C2×D3×C9—21no—
107C2×D3×C9D812yescomplemented
108C2×C3×D9—22no—
109C2×C3×D9—41no—
110C2×C3×D9—41no—
111D3×D9—48no—
112C3×C4×C9C2×D411yescharacteristic
113C3×C4×C9C2×D411yescharacteristic
114C9×Dic3—21no—
115C3×Dic9—41no—
116C2×D3×D4—91no—
117C22×D12—91no—
118C2×#213—91no—
119#930—91no—
120C2×D24—91no—
121C2×D24—91no—
122C2×D24—91no—
123C2×D24—181no—
124#895—91no—
125C3×#102—91no—
126C3×C22⋊C8—31no—
127#821—31no—
128#821—91no—
129(C2×D3)⋊C8D911yescharacteristic, complemented
130#809—91no—
131C22×C3⋊D3—61no—
132C22×C3⋊D3—61no—
133C22×C3×D3—61no—
134C22×C3×D3D1211yescharacteristic
135C2×D32—64no—
136C2×D32—122no—
137C2×D32—122no—
138C2×C32×C4C22×D311yescharacteristic
139C4⋊(C3⋊D3)—32no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—61no—
142C4⋊(C3⋊D3)—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3D1211yescharacteristic
145C3×D12—32no—
146C3×D12—61no—
147C3×C4×D3—21no—
148C3×C4×D3—21no—
149#626—64no—
150#626—122no—
151C23×C9—31no—
152C22×D9—21no—
153C22×D9—64no—
154C22×D9—61no—
155C22×D9—61no—
156C22×D9—121no—
157C22×D9—121no—
158C22×D9—122no—
159C22×D9—122no—
160C32×C8—21no—
161C3×C3⋊C8—21no—
162C2×C4×C9—21no—
163C2×C4×C9—31no—
164C2×C4×C9—61no—
165C2×C4×C9—61no—
166C2×C4×C9C22×D311yescharacteristic
167#611—122no—
168C2×Dic9—21no—
169D36—21no—
170D36—32no—
171D36—62no—
172D36—62no—
173D36—64no—
174D36—61no—
175D36—61no—
176D36—61no—
177D36—121no—
178D36—121no—
179D36D1212yescomplemented
180C8×C9—22no—
181C8×C9—21no—
182C8×C9—61no—
183#447—271noSylow
184C2×C3×C9#12011yescharacteristic
185C2×C3×C9C2×D811yescharacteristic
186C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
187C3⋊D9—42no—
188C3⋊D9—81no—
189D3×C9—22no—
190C3×D9—42no—
191C23×D3—181no—
192C23×D3—181no—
193C2×C3×D4—91no—
194C22×C3×C4—31no—
195C2×#67—181no—
196C2×#67—181no—
197D3×D4—182no—
198D3×D4—182no—
199C2×D12—31no—
200C2×D12—91no—
201C2×D12—91no—
202C2×D12—91no—
203C2×D12—91no—
204C2×D12—181no—
205C2×D12—181no—
206C2×D12—182no—
207C2×D12—182no—
208C2×D12—181no—
209C2×D12—181no—
210C2×C4×D3C2×D911yescharacteristic
211C2×C3×C8—21no—
212C2×C3×C8—31no—
213C2×C3×C8C2×D911yescharacteristic, complemented
214C2×C3×C8D3211yescharacteristic
215C3×C4⋊C4—91no—
216#213—182no—
217(C2×D3)⋊C4—181no—
218(C2×D3)⋊C4—181no—
219C4⋊Dic3—31no—
220C2×C3⋊C8C2×D911yescharacteristic, complemented
221D24—182no—
222D24—182no—
223D24—184no—
224D24—182no—
225C22×C32C2×D1211yescharacteristic
226C2×C3⋊D3—62no—
227C2×C3⋊D3—121no—
228C2×C3⋊D3—121no—
229C2×C3⋊D3—121no—
230C2×C3⋊D3—121no—
231C2×C3⋊D3—121no—
232C2×C3×D3—21no—
233C2×C3×D3—21no—
234C2×C3×D3—62no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C2×C3×D3D2412yes—
238D32—128no—
239C32×C4D3×D411yescharacteristic
240C32×C4D3×D411yescharacteristic
241C3×Dic3—21no—
242C3×Dic3—121no—
243C22×C9—21no—
244C22×C9—32no—
245C22×C9—61no—
246C22×C9—61no—
247C22×C9D3×D411yescharacteristic
248C2×D9—22no—
249C2×D9—41no—
250C2×D9—41no—
251C2×D9—62no—
252C2×D9—122no—
253C2×D9—128no—
254C2×D9—121no—
255C2×D9—121no—
256C2×D9—122no—
257C2×D9—121no—
258C2×D9—121no—
259C2×D9—121no—
260C2×D9—241no—
261C2×D9—241no—
262C2×D9—241no—
263C2×D9—241no—
264C4×C9—21no—
265C4×C9—21no—
266C4×C9—61no—
267C4×C9C2×D1211yescharacteristic
268C4×C9D3×D411yescharacteristic
269Dic9—41no—
270C22×D4—271no—
271C2×D8—271no—
272C2×D8—271no—
273#121—271no—
274#102—91no—
275#102—271no—
276C22⋊C8—31no—
277C3×C9#44711yescharacteristic, complemented, Sylow
278C23×C3—31no—
279C23×C3—181no—
280C22×D3—61no—
281C22×D3—184no—
282C22×D3—184no—
283C22×D3—181no—
284C22×D3—181no—
285C22×D3—181no—
286C22×D3—181no—
287C22×D3—361no—
288C22×D3—361no—
289C22×D3—362no—
290C22×D3—362no—
291C22×D3—362no—
292C22×D3—362no—
293C22×D3D3611yescharacteristic
294C3×D4—92no—
295C3×D4—181no—
296C2×C3×C4—21no—
297C2×C3×C4—31no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—181no—
301C2×C3×C4C22×D911yescharacteristic
302C2×C3×C4C2×D3211yescharacteristic
303#67—184no—
304#67—362no—
305C2×Dic3—61no—
306C2×Dic3D3611yescharacteristic
307D12—32no—
308D12—61no—
309D12—92no—
310D12—92no—
311D12—182no—
312D12—182no—
313D12—184no—
314D12—181no—
315D12—181no—
316D12—181no—
317D12—181no—
318D12—181no—
319D12—181no—
320D12—361no—
321D12—361no—
322D12—362no—
323C4×D3—21no—
324C4×D3—21no—
325C3×C8—22no—
326C3×C8—21no—
327C3×C8—21no—
328C3×C8—61no—
329C3⋊C8—21no—
330C2×C32#88211yescharacteristic
331C2×C32C2×D2411yescharacteristic
332C2×C32#90811yescharacteristic
333C3⋊D3—122no—
334C3⋊D3—241no—
335C3×D3—22no—
336C3×D3—122no—
337C2×C9—21no—
338C2×C9—21no—
339C2×C9—21no—
340C2×C9—62no—
341C2×C9#88211yescharacteristic
342C2×C9D3×D811yescharacteristic
343C2×C9#91411yescharacteristic
344D9—42no—
345D9—122no—
346D9—242no—
347D9—242no—
348D9—241no—
349C24—541no—
350C2×D4—91no—
351C2×D4—271no—
352C2×D4—271no—
353C2×D4—541no—
354C2×D4—542no—
355C2×D4—542no—
356C2×D4—541no—
357C22×C4—31no—
358D8—542no—
359D8—542no—
360C2×C8—31no—
361C2×C8D3×D911yescharacteristic, complemented
362C4⋊C4—91no—
363C22⋊C4—541no—
364C22×C3—21no—
365C22×C3—32no—
366C22×C3—61no—
367C22×C3—61no—
368C22×C3—182no—
369C22×C3—361no—
370C22×C3—361no—
371C22×C3C2×D3611yescharacteristic
372C22×C3D3×D1211yescharacteristic
373C2×D3—21no—
374C2×D3—21no—
375C2×D3—62no—
376C2×D3—121no—
377C2×D3—121no—
378C2×D3—182no—
379C2×D3—182no—
380C2×D3—362no—
381C2×D3—368no—
382C2×D3—368no—
383C2×D3—361no—
384C2×D3—361no—
385C2×D3—361no—
386C2×D3—361no—
387C2×D3—362no—
388C2×D3—361no—
389C2×D3—361no—
390C2×D3—361no—
391C2×D3—361no—
392C2×D3—361no—
393C2×D3—361no—
394C2×D3—721no—
395C2×D3—721no—
396C2×D3—721no—
397C2×D3—721no—
398C2×D3D7212yescomplemented
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—61no—
402C3×C4—361no—
403C3×C4D4×D911yescharacteristic
404C3×C4D4×D911yescharacteristic
405C3×C4D3×D1211yescharacteristic
406C3×C4#381111yescharacteristic
407Dic3—21no—
408Dic3—121no—
409C32#473211yescharacteristic
410C9—21no—
411C9#477311yescharacteristic, complemented
412C23—31no—
413C23—181no—
414C23—544no—
415C23—541no—
416C23—541no—
417C23—1082no—
418C23—1082no—
419D4—92no—
420D4—181no—
421D4—272no—
422D4—544no—
423D4—541no—
424D4—541no—
425D4—541no—
426D4—1082no—
427C2×C4—31no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—181no—
431C2×C4C2×D3×D911yescharacteristic
432C8—21no—
433C8—61no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—62no—
438C2×C3—362no—
439C2×C3#6344411yescharacteristic
440C2×C3C2×D7211yescharacteristic
441C2×C3#6347011yescharacteristic
442C2×C3D3×D2411yescharacteristic
443C2×C3#6379411yescharacteristic
444C2×C3#6392311yescharacteristic
445D3—22no—
446D3—122no—
447D3—362no—
448D3—362no—
449D3—722no—
450D3—722no—
451D3—721no—
452D3—721no—
453C22—32no—
454C22—61no—
455C22—61no—
456C22—182no—
457C22—361no—
458C22—361no—
459C22—542no—
460C22—1088no—
461C22—1081no—
462C22—1081no—
463C22—1081no—
464C22—1081no—
465C22—1081no—
466C22D3×D3611yescharacteristic
467C4—61no—
468C4—361no—
469C4D9×D1211yescharacteristic
470C4#8866811yescharacteristic
471C3—21no—
472C3#1058841411yescharacteristic, complemented
473C3#1059094711yescharacteristic
474C2—62no—
475C2—362no—
476C2—1082no—
477C2—2161no—
478C2D3×D7211yescharacteristic
479C2#1172080411yescharacteristic
480C2#1172105111yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
hypercenterC2×C8D3×D9
hypocenterC3×C9#447
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
1. C2×D3×D36C2semidirect
2. C2×#88525C2semidirect
3. #11721038C2semidirect
4. C2×#88417C2semidirect
5. C9×(C2×D3)⋊C8C2semidirect
6. C3×#63380C2semidirect
7. #11720284C2semidirect
12. C2×C4⋊(C3⋊D9)C22semidirect
19. C2×C3×C8×C9C22semidirect
21. C3×C4⋊Dic9C22semidirect
22. C2×C9×C3⋊C8C22semidirect
39. #63380D3semidirect
51. C4⋊(C3⋊D9)D4semidirect
57. C3×D36D4semidirect
92. C2×C8×C9C2×D3semidirect
94. C4⋊Dic9C2×D3semidirect
107. C2×D3×C9D8semidirect
129. (C2×D3)⋊C8D9semidirect
179. D36D12semidirect
213. C2×C3×C8C2×D9semidirect
220. C2×C3⋊C8C2×D9semidirect
277. C3×C9#447semidirect
361. C2×C8D3×D9semidirect
398. C2×D3D72semidirect
411. C9#4773semidirect
472. C3#10588414semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D362
C2×#885252
#117210382
C2×#884172
C9×(C2×D3)⋊C82
C3×#633802
#117202842
#105909473
#105884143