Finite Group Databasebeta

← #13137833

Subgroups — #13137833

Order 1728.

644 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 93 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×D9×D12C211yesmaximal, characteristic, complemented
2C3×#63457C211yesmaximal, characteristic, complemented
3C2×#88421C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic, complemented
5#11720528C211yesmaximal, characteristic, complemented
6C9×#809C211yesmaximal, characteristic, complemented
7(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10590595—31nomaximal
9#10588453—31nomaximal
10C22×D3×D9—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C3×D36—21no—
14C2×C3×D36C2211yescharacteristic, complemented
15C2×C3×C4×D9C2211yescharacteristic
16C2×#6195—21no—
17D9×D12—22no—
18D9×D12—22no—
19C3×C9×C4⋊C4C2211yescharacteristic, complemented
20C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
21C3×(C2×D9)⋊C4—21no—
22C2×C9×C3⋊C8C2211yescharacteristic, complemented
23#88421—22no—
24#88417—22no—
25C2×D3×D12—31no—
26C3×#895—31no—
27C2×#3714—31no—
28C2×#3713—31no—
29C2×D4×D9—31no—
30#63636—31no—
31C3×#809D311yescharacteristic
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3673—31no—
34C2×D72—31no—
35#63457D311yescharacteristic, complemented
36C9×#102—31no—
37(C2×D9)⋊C8—31no—
38#63369—31no—
39#63369—61no—
40C22×C3⋊D9—21no—
41C22×D3×C9—21no—
42C22×C3×D9—21no—
43C22×C3×D9D411yescharacteristic
44C2×D3×D9—24no—
45C2×D3×D9—42no—
46C2×D3×D9—42no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9C2311yescharacteristic
49C4⋊(C3⋊D9)—21no—
50C4⋊(C3⋊D9)D412yescomplemented
51C2×C3×Dic9D411yescharacteristic
52C9×D12—21no—
53C9×D12D412yescomplemented
54C3×D36—21no—
55C3×D36—21no—
56C3×D36—42no—
57C3×C4×D9—21no—
58C3×C4×D9—21no—
59#6195—24no—
60C3⋊(C9⋊C8)—21no—
61C9×C3⋊C8—21no—
62#5036—91no—
63#4773—91no—
64C22×D32—61no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C3×D12—31no—
67C2×C3×D12—61no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×C4×D3—31no—
70C2×#626—61no—
71D3×D12—62no—
72D3×D12—62no—
73C23×D9—61no—
74C32×C4⋊C4C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C3×(C2×D3)⋊C4—61no—
77C2×C3×C3⋊C8C2×D311yescharacteristic
78#3714—62no—
79#3713—62no—
80C2×D4×C9—31no—
81C2×#611—61no—
82D4×D9—62no—
83D4×D9—62no—
84C2×D36—21no—
85C2×D36—31no—
86C2×D36—61no—
87C2×D36C2×D311yescharacteristic, complemented
88C2×C4×D9C2×D311yescharacteristic
89C2×C8×C9—31no—
90C9×C4⋊C4—21no—
91C9×C4⋊C4C2×D311yescharacteristic, complemented
92#3673—62no—
93(C2×D9)⋊C4—21no—
94C2×C9⋊C8—31no—
95C2×C9⋊C8—61no—
96D72—62no—
97C22×C3×C9C2×D411yescharacteristic
98C2×C3⋊D9—22no—
99C2×C3⋊D9—41no—
100C2×C3⋊D9—41no—
101C2×D3×C9—22no—
102C2×D3×C9—41no—
103C2×D3×C9—41no—
104C2×C3×D9—21no—
105C2×C3×D9—21no—
106C2×C3×D9—41no—
107C2×C3×D9—41no—
108C2×C3×D9—41no—
109C2×C3×D9D812yescomplemented
110D3×D9—48no—
111C3×C4×C9—41no—
112C3×C4×C9C2×D411yescharacteristic
113C3×C4×C9C2×D411yescharacteristic
114C3×Dic9—21no—
115C2×D3×D4—91no—
116C22×D12—91no—
117C3×#121—91no—
118C2×#213—91no—
119C2×#213—91no—
120C2×#213—91no—
121C2×D24—91no—
122#895—31no—
123C3×#102—31no—
124#830—91no—
125(C2×D3)⋊C8—91no—
126#809—91no—
127#809—91no—
128#809—181no—
129#809D911yescharacteristic, complemented
130C22×C3⋊D3—61no—
131C22×C3×D3—21no—
132C22×C3×D3—31no—
133C22×C3×D3—61no—
134C2×D32—64no—
135C2×D32—122no—
136C2×D32—122no—
137C2×C32×C4—21no—
138C2×C32×C4C22×D311yescharacteristic
139C4⋊(C3⋊D3)—32no—
140C4⋊(C3⋊D3)—61no—
141C2×C3×Dic3—31no—
142C3×D12—21no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—122no—
146C3×D12D1212yes—
147C3×C4×D3—61no—
148C3×C4×D3—61no—
149#626—64no—
150C23×C9—61no—
151C22×D9—21no—
152C22×D9—64no—
153C22×D9—61no—
154C22×D9—121no—
155C22×D9—122no—
156C22×D9—122no—
157C22×D9#6711yescharacteristic
158C3⋊(C3⋊C8)—61no—
159C3×C3⋊C8—21no—
160D4×C9—32no—
161D4×C9—61no—
162C2×C4×C9—21no—
163C2×C4×C9—21no—
164C2×C4×C9—41no—
165C2×C4×C9C22×D311yescharacteristic
166#611—64no—
167C2×Dic9#6711yescharacteristic
168D36—21no—
169D36—21no—
170D36—32no—
171D36—42no—
172D36—62no—
173D36—61no—
174D36—121no—
175C4×D9—21no—
176C4×D9—21no—
177C8×C9—61no—
178C9⋊C8—61no—
179C9⋊C8—121no—
180#447—271noSylow
181C2×C3×C9#12011yescharacteristic
182C2×C3×C9C2×D811yescharacteristic
183C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
184C3⋊D9—42no—
185D3×C9—42no—
186C3×D9—22no—
187C3×D9—81no—
188C23×D3—181no—
189C23×D3—181no—
190C2×C3×D4—31no—
191C2×C3×D4—91no—
192C2×C3×D4—181no—
193C22×C3×C4—91no—
194C2×#67—181no—
195D3×D4—182no—
196D3×D4—182no—
197C2×D12—31no—
198C2×D12—61no—
199C2×D12—91no—
200C2×D12—91no—
201C2×D12—181no—
202C2×D12—182no—
203C2×D12—181no—
204C2×D12—182no—
205C2×D12C2×D911yescharacteristic
206C2×C4×D3—31no—
207C2×C3×C8—31no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4C2×D911yescharacteristic, complemented
210C3×C4⋊C4D3211yescharacteristic
211C3×C22⋊C4—181no—
212#213—182no—
213#213—182no—
214#213—182no—
215(C2×D3)⋊C4—61no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—91no—
218C2×C3⋊C8—181no—
219C2×C3⋊C8C2×D911yescharacteristic, complemented
220D24—182no—
221C22×C32D3×D411yescharacteristic
222C2×C3⋊D3—62no—
223C2×C3⋊D3—121no—
224C2×C3⋊D3—121no—
225C2×C3×D3—22no—
226C2×C3×D3—32no—
227C2×C3×D3—41no—
228C2×C3×D3—41no—
229C2×C3×D3—61no—
230C2×C3×D3—61no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234D32—128no—
235C32×C4—41no—
236C32×C4C2×D1211yescharacteristic
237C32×C4D3×D411yescharacteristic
238C3×Dic3—61no—
239C22×C9—21no—
240C22×C9—62no—
241C22×C9—121no—
242C22×C9—121no—
243C22×C9C2×#6711yescharacteristic
244C2×D9—21no—
245C2×D9—21no—
246C2×D9—41no—
247C2×D9—41no—
248C2×D9—41no—
249C2×D9—62no—
250C2×D9—122no—
251C2×D9—128no—
252C2×D9—121no—
253C2×D9—121no—
254C2×D9—241no—
255C2×D9—241no—
256C2×D9#21312yescomplemented
257C4×C9—21no—
258C4×C9—21no—
259C4×C9—41no—
260C4×C9—81no—
261C4×C9D3×D411yescharacteristic
262C4×C9D3×D411yescharacteristic
263Dic9—21no—
264C22×D4—271no—
265C2×D8—271no—
266C2×D8—271no—
267#121—91no—
268#102—31no—
269#102—271no—
270C22⋊C8—271no—
271C3×C9#44711yescharacteristic, complemented, Sylow
272C23×C3—61no—
273C23×C3—91no—
274C23×C3—181no—
275C22×D3—21no—
276C22×D3—31no—
277C22×D3—61no—
278C22×D3—184no—
279C22×D3—184no—
280C22×D3—181no—
281C22×D3—181no—
282C22×D3—361no—
283C22×D3—362no—
284C22×D3—362no—
285C22×D3—362no—
286C22×D3—362no—
287C3×D4—32no—
288C3×D4—61no—
289C3×D4—181no—
290C3×D4—181no—
291C3×D4—362no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—41no—
296C2×C3×C4—91no—
297C2×C3×C4—181no—
298C2×C3×C4—181no—
299C2×C3×C4C22×D911yescharacteristic
300C2×C3×C4C2×D3211yescharacteristic
301#67—184no—
302C2×Dic3—31no—
303D12—21no—
304D12—61no—
305D12—61no—
306D12—92no—
307D12—92no—
308D12—122no—
309D12—182no—
310D12—184no—
311D12—181no—
312D12—181no—
313D12—361no—
314D12D3612yescomplemented
315C4×D3—61no—
316C4×D3—61no—
317C3×C8—61no—
318C3⋊C8—21no—
319C3⋊C8—181no—
320C3⋊C8—181no—
321C3⋊C8—361no—
322C2×C32#88211yescharacteristic
323C2×C32D3×D811yescharacteristic
324C2×C32#91411yescharacteristic
325C3⋊D3—122no—
326C3×D3—42no—
327C3×D3—62no—
328C3×D3—241no—
329C2×C9—21no—
330C2×C9—21no—
331C2×C9—21no—
332C2×C9—122no—
333C2×C9C2×#21311yescharacteristic
334C2×C9#93711yescharacteristic
335C2×C9#94911yescharacteristic
336D9—22no—
337D9—81no—
338D9—122no—
339D9—242no—
340C24—541no—
341C2×D4—31no—
342C2×D4—91no—
343C2×D4—181no—
344C2×D4—271no—
345C2×D4—541no—
346C2×D4—542no—
347C2×D4—542no—
348C22×C4—91no—
349D8—542no—
350D8—542no—
351C2×C8—31no—
352C2×C8—271no—
353C4⋊C4D3×D911yescharacteristic, complemented
354C22⋊C4—181no—
355C22×C3—21no—
356C22×C3—62no—
357C22×C3—92no—
358C22×C3—121no—
359C22×C3—121no—
360C22×C3—181no—
361C22×C3—181no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3—361no—
365C22×C3D4×D911yescharacteristic
366C22×C3D3×#6711yescharacteristic
367C2×D3—22no—
368C2×D3—32no—
369C2×D3—41no—
370C2×D3—41no—
371C2×D3—61no—
372C2×D3—61no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—121no—
376C2×D3—182no—
377C2×D3—182no—
378C2×D3—362no—
379C2×D3—368no—
380C2×D3—368no—
381C2×D3—361no—
382C2×D3—361no—
383C2×D3—361no—
384C2×D3—361no—
385C2×D3—721no—
386C2×D3—721no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—41no—
390C3×C4—41no—
391C3×C4—81no—
392C3×C4—181no—
393C3×C4C2×D3611yescharacteristic
394C3×C4D4×D911yescharacteristic
395C3×C4D3×D1211yescharacteristic
396C3×C4#380211yescharacteristic
397Dic3—61no—
398C32#477311yescharacteristic
399C9—21no—
400C9#503611yescharacteristic, complemented
401C23—61no—
402C23—91no—
403C23—181no—
404C23—544no—
405C23—541no—
406C23—1082no—
407C23—1082no—
408D4—32no—
409D4—61no—
410D4—181no—
411D4—181no—
412D4—272no—
413D4—362no—
414D4—544no—
415D4—541no—
416C2×C4—21no—
417C2×C4—91no—
418C2×C4—181no—
419C2×C4—181no—
420C2×C4C2×D3×D911yescharacteristic
421C8—61no—
422C8—541no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—122no—
427C2×C3—182no—
428C2×C3—721no—
429C2×C3#6344411yescharacteristic
430C2×C3D8×D911yescharacteristic
431C2×C3#6347611yescharacteristic
432C2×C3#6392211yescharacteristic
433C2×C3D3×#21311yescharacteristic
434C2×C3#6393911yescharacteristic
435D3—42no—
436D3—62no—
437D3—241no—
438D3—362no—
439D3—362no—
440D3—722no—
441C22—62no—
442C22—92no—
443C22—121no—
444C22—121no—
445C22—181no—
446C22—181no—
447C22—361no—
448C22—361no—
449C22—361no—
450C22—542no—
451C22—1088no—
452C22—1081no—
453C22—1081no—
454C22D9×#6711yescharacteristic
455C4—41no—
456C4—181no—
457C4D3×D3611yescharacteristic
458C4#8864611yescharacteristic
459C3—21no—
460C3#1058845311yescharacteristic, complemented
461C3#1059059511yescharacteristic
462C2—122no—
463C2—182no—
464C2—721no—
465C2—1082no—
466C2#1172104911yescharacteristic
467C2D9×#21311yescharacteristic
468C2#1172121111yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#447
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D9×D12C2semidirect
2. C3×#63457C2semidirect
3. C2×#88421C2semidirect
4. C2×#88417C2semidirect
5. #11720528C2semidirect
6. C9×#809C2semidirect
7. (C2×D9)⋊(C3⋊C8)C2semidirect
14. C2×C3×D36C22semidirect
19. C3×C9×C4⋊C4C22semidirect
20. C2×C3⋊(C9⋊C8)C22semidirect
22. C2×C9×C3⋊C8C22semidirect
35. #63457D3semidirect
50. C4⋊(C3⋊D9)D4semidirect
53. C9×D12D4semidirect
87. C2×D36C2×D3semidirect
91. C9×C4⋊C4C2×D3semidirect
109. C2×C3×D9D8semidirect
129. #809D9semidirect
209. C3×C4⋊C4C2×D9semidirect
219. C2×C3⋊C8C2×D9semidirect
256. C2×D9#213semidirect
271. C3×C9#447semidirect
314. D12D36semidirect
353. C4⋊C4D3×D9semidirect
400. C9#5036semidirect
460. C3#10588453semidirect

Maximal subgroups

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

subgroupindex
C2×D9×D122
C3×#634572
C2×#884212
C2×#884172
#117205282
C9×#8092
(C2×D9)⋊(C3⋊C8)2
#105905953
#105884533