Finite Group Databasebeta

← #13138677

Subgroups — #13138677

Order 1728.

644 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 91 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×#88646C211yesmaximal, characteristic, complemented
2C2×C3×D72C211yesmaximal, characteristic, complemented
3#11721007C211yesmaximal, characteristic, complemented
4C2×#88421C211yesmaximal, characteristic, complemented
5#11720542C211yesmaximal, characteristic, complemented
6C9×#821C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10590945—31nomaximal
9#10588453—31nomaximal
10C22×D3×D9—21no—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×C3×D36C2211yescharacteristic, complemented
15C2×#6194—21no—
16C2×#6192—21no—
17#88646—22no—
18#88646—22no—
19C2×C3×C8×C9C2211yescharacteristic, complemented
20C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
21C9×C4⋊Dic3C2211yescharacteristic, complemented
22C3×D72—22no—
23(C2×D9)⋊Dic3—21no—
24#88421—22no—
25C2×#3802—31no—
26C2×C3×D24—31no—
27#63916—31no—
28C2×#3713—31no—
29C2×D4×D9—31no—
30#63647—31no—
31C3×#821D311yescharacteristic
32#63562—31no—
33C2×#3673—31no—
34C2×D72D311yescharacteristic, complemented
35#63457—31no—
36C9×#102—31no—
37(C2×D9)⋊C8—31no—
38(C2×D9)⋊C8—61no—
39#63369—31no—
40C22×C3⋊D9D411yescharacteristic
41C22×D3×C9—21no—
42C22×C3×D9—21no—
43C22×C3×D9—21no—
44C2×D3×D9—24no—
45C2×D3×D9—42no—
46C2×D3×D9—42no—
47C2×C3×C4×C9C2311yescharacteristic
48C2×C3⋊Dic9D411yescharacteristic
49C4×C3⋊D9—21no—
50C4×C3⋊D9—21no—
51C2×C9×Dic3—21no—
52C9×D12—21no—
53C9×D12D412yescomplemented
54C3×D36—21no—
55C3×D36—21no—
56C3×D36—21no—
57C3×D36D412yescomplemented
58#6194—24no—
59#6192—42no—
60C3×C8×C9—21no—
61C3⋊(C9⋊C8)—21no—
62#5156—91no—
63#4773—91no—
64C22×D32—61no—
65C2×C4×C3⋊D3—31no—
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12C2×D311yescharacteristic
69C2×#626—61no—
70C2×#625—61no—
71#3802—62no—
72#3802—62no—
73C23×D9—61no—
74C2×C32×C8C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C3×C4⋊Dic3C2×D311yescharacteristic
77C3×D24—62no—
78(C2×D3)⋊Dic3—61no—
79#3713—62no—
80C2×D4×C9—31no—
81C2×#611—61no—
82D4×D9—62no—
83D4×D9—62no—
84C2×D36—61no—
85C2×D36C2×D311yescharacteristic
86C2×D36C2×D311yescharacteristic, complemented
87C2×C4×D9—31no—
88C2×C4×D9—61no—
89C2×C8×C9—21no—
90C2×C8×C9C2×D311yescharacteristic, complemented
91C9×C4⋊C4—31no—
92#3673—62no—
93(C2×D9)⋊C4—61no—
94C2×C9⋊C8—31no—
95C2×C9⋊C8—61no—
96D72—22no—
97C22×C3×C9C2×D411yescharacteristic
98C2×C3⋊D9—21no—
99C2×C3⋊D9—21no—
100C2×C3⋊D9D812yescomplemented
101C2×D3×C9—22no—
102C2×D3×C9—41no—
103C2×D3×C9—41no—
104C2×C3×D9—22no—
105C2×C3×D9—41no—
106C2×C3×D9—41no—
107C2×C3×D9—41no—
108C2×C3×D9—41no—
109C2×C3×D9—41no—
110D3×D9—48no—
111C3×C4×C9C2×D411yescharacteristic
112C3×C4×C9C2×D411yescharacteristic
113C3⋊Dic9—21no—
114C9×Dic3—41no—
115C2×D3×D4—91no—
116C2×D3×D4—91no—
117C2×C3×D8—91no—
118#938—91no—
119C2×#213—91no—
120C2×#213—91no—
121C2×D24—31no—
122#895—91no—
123C3×#102—31no—
124#832—91no—
125#821D911yescharacteristic, complemented
126(C2×D3)⋊C8—91no—
127(C2×D3)⋊C8—91no—
128(C2×D3)⋊C8—181no—
129#809—91no—
130C22×C3⋊D3—31no—
131C22×C3×D3—21no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C2×D32—64no—
135C2×D32—122no—
136C2×D32—122no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C3⋊Dic3—31no—
139C4×C3⋊D3—61no—
140C4×C3⋊D3—61no—
141C2×C3×Dic3—21no—
142C3×D12—21no—
143C3×D12—32no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12D1212yes—
148#626—122no—
149#625—64no—
150C23×C9—61no—
151C22×D9—21no—
152C22×D9—21no—
153C22×D9—31no—
154C22×D9—61no—
155C22×D9—64no—
156C22×D9—122no—
157C22×D9—122no—
158C32×C8—21no—
159C3⋊(C3⋊C8)—61no—
160D4×C9—32no—
161D4×C9—61no—
162C2×C4×C9—21no—
163C2×C4×C9—61no—
164C2×C4×C9C22×D311yescharacteristic
165#611—64no—
166C2×Dic9—31no—
167C2×Dic9—61no—
168D36—21no—
169D36—21no—
170D36—21no—
171D36—122no—
172D36#6712yescomplemented
173C4×D9—61no—
174C4×D9—61no—
175C4×D9—121no—
176C4×D9—121no—
177C8×C9—21no—
178C8×C9—41no—
179C9⋊C8—61no—
180C9⋊C8—121no—
181#447—271noSylow
182C2×C3×C9#12011yescharacteristic
183C2×C3×C9C2×D811yescharacteristic
184C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
185C3⋊D9—22no—
186D3×C9—42no—
187C3×D9—42no—
188C3×D9—81no—
189C23×D3—181no—
190C23×D3—181no—
191C2×C3×D4—31no—
192C2×C3×D4—91no—
193C2×C3×D4—91no—
194C2×#67—181no—
195C2×#67—181no—
196C2×#67—181no—
197D3×D4—182no—
198D3×D4—182no—
199D3×D4—182no—
200D3×D4—182no—
201C2×D12—31no—
202C2×D12—31no—
203C2×D12—181no—
204C2×D12C2×D911yescharacteristic
205C2×C4×D3—91no—
206C2×C4×D3—91no—
207C2×C4×D3—181no—
208C3×D8—182no—
209C2×C3×C8—21no—
210C2×C3×C8C2×D911yescharacteristic, complemented
211C2×C3×C8D3211yescharacteristic
212C3×C4⋊C4—31no—
213#217—181no—
214#213—182no—
215#213—182no—
216(C2×D3)⋊C4—181no—
217C4⋊Dic3C2×D911yescharacteristic, complemented
218C2×C3⋊C8—91no—
219C2×C3⋊C8—91no—
220C2×C3⋊C8—181no—
221D24—62no—
222C22×C32D3×D411yescharacteristic
223C2×C3⋊D3—32no—
224C2×C3⋊D3—61no—
225C2×C3⋊D3—61no—
226C2×C3×D3—22no—
227C2×C3×D3—41no—
228C2×C3×D3—41no—
229C2×C3×D3—62no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C2×C3×D3—121no—
235D32—128no—
236C32×C4C2×D1211yescharacteristic
237C32×C4D3×D411yescharacteristic
238C3⋊Dic3—61no—
239C3×Dic3—41no—
240C22×C9—21no—
241C22×C9—62no—
242C22×C9—121no—
243C22×C9—121no—
244C22×C9D3×D411yescharacteristic
245C2×D9—22no—
246C2×D9—32no—
247C2×D9—41no—
248C2×D9—41no—
249C2×D9—41no—
250C2×D9—41no—
251C2×D9—41no—
252C2×D9—62no—
253C2×D9—61no—
254C2×D9—61no—
255C2×D9—121no—
256C2×D9—121no—
257C2×D9—128no—
258C4×C9—21no—
259C4×C9—21no—
260C4×C9—121no—
261C4×C9D3×D411yescharacteristic
262C4×C9C2×#6711yescharacteristic
263Dic9—61no—
264Dic9—121no—
265C22×D4—271no—
266C2×D8—91no—
267C2×D8—271no—
268#121—271no—
269#102—31no—
270#102—271no—
271C22⋊C8—271no—
272C3×C9#44711yescharacteristic, complemented, Sylow
273C23×C3—61no—
274C23×C3—181no—
275C23×C3—181no—
276C22×D3—21no—
277C22×D3—61no—
278C22×D3—61no—
279C22×D3—91no—
280C22×D3—91no—
281C22×D3—181no—
282C22×D3—184no—
283C22×D3—184no—
284C22×D3—362no—
285C22×D3—362no—
286C22×D3—362no—
287C22×D3—362no—
288C3×D4—32no—
289C3×D4—61no—
290C3×D4—92no—
291C3×D4—181no—
292C3×D4—181no—
293C3×D4—181no—
294C2×C3×C4—21no—
295C2×C3×C4—61no—
296C2×C3×C4C22×D911yescharacteristic
297C2×C3×C4C2×D3211yescharacteristic
298#67—184no—
299#67—184no—
300#67—362no—
301C2×Dic3—21no—
302C2×Dic3—91no—
303C2×Dic3—91no—
304C2×Dic3—181no—
305D12—21no—
306D12—32no—
307D12—61no—
308D12—61no—
309D12—61no—
310D12—362no—
311D12D3612yescomplemented
312C4×D3—181no—
313C4×D3—181no—
314C4×D3—181no—
315C4×D3—181no—
316C4×D3—361no—
317C4×D3—361no—
318C3×C8—21no—
319C3×C8—21no—
320C3×C8—41no—
321C3⋊C8—181no—
322C3⋊C8—181no—
323C3⋊C8—361no—
324C2×C32#88211yescharacteristic
325C2×C32D3×D811yescharacteristic
326C2×C32#91411yescharacteristic
327C3⋊D3—62no—
328C3×D3—42no—
329C3×D3—122no—
330C3×D3—241no—
331C2×C9—21no—
332C2×C9—21no—
333C2×C9—21no—
334C2×C9—122no—
335C2×C9D3×D811yescharacteristic
336C2×C9#91111yescharacteristic
337C2×C9#93711yescharacteristic
338D9—42no—
339D9—62no—
340D9—81no—
341D9—122no—
342C24—541no—
343C2×D4—31no—
344C2×D4—91no—
345C2×D4—91no—
346C2×D4—541no—
347C2×D4—541no—
348C2×D4—542no—
349C2×D4—542no—
350C22×C4—271no—
351D8—182no—
352D8—542no—
353C2×C8—271no—
354C2×C8D3×D911yescharacteristic, complemented
355C4⋊C4—31no—
356C22⋊C4—541no—
357C22×C3—21no—
358C22×C3—62no—
359C22×C3—121no—
360C22×C3—121no—
361C22×C3—182no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3—361no—
365C22×C3—361no—
366C22×C3—361no—
367C22×C3D4×D911yescharacteristic
368C22×C3#380211yescharacteristic
369C2×D3—22no—
370C2×D3—41no—
371C2×D3—41no—
372C2×D3—62no—
373C2×D3—92no—
374C2×D3—92no—
375C2×D3—121no—
376C2×D3—121no—
377C2×D3—121no—
378C2×D3—121no—
379C2×D3—121no—
380C2×D3—182no—
381C2×D3—181no—
382C2×D3—181no—
383C2×D3—181no—
384C2×D3—181no—
385C2×D3—361no—
386C2×D3—361no—
387C2×D3—368no—
388C2×D3—368no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—121no—
392C3×C4C2×D3611yescharacteristic
393C3×C4D4×D911yescharacteristic
394C3×C4D3×D1211yescharacteristic
395C3×C4D3×#6711yescharacteristic
396Dic3—41no—
397Dic3—181no—
398Dic3—181no—
399Dic3—361no—
400C32#477311yescharacteristic
401C9—21no—
402C9#515611yescharacteristic, complemented
403C23—61no—
404C23—181no—
405C23—181no—
406C23—271no—
407C23—544no—
408C23—1082no—
409C23—1082no—
410D4—32no—
411D4—61no—
412D4—92no—
413D4—181no—
414D4—181no—
415D4—181no—
416D4—544no—
417D4—1082no—
418C2×C4—61no—
419C2×C4—271no—
420C2×C4—541no—
421C2×C4—541no—
422C2×C4C2×D3×D911yescharacteristic
423C8—21no—
424C8—541no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—122no—
429C2×C3—362no—
430C2×C3—721no—
431C2×C3#6344411yescharacteristic
432C2×C3D8×D911yescharacteristic
433C2×C3#6347611yescharacteristic
434C2×C3#6379711yescharacteristic
435C2×C3#6379811yescharacteristic
436C2×C3#6392211yescharacteristic
437D3—42no—
438D3—122no—
439D3—182no—
440D3—182no—
441D3—241no—
442D3—362no—
443C22—62no—
444C22—121no—
445C22—121no—
446C22—182no—
447C22—272no—
448C22—361no—
449C22—361no—
450C22—361no—
451C22—361no—
452C22—361no—
453C22—541no—
454C22—541no—
455C22—1088no—
456C22#8864611yescharacteristic
457C4—121no—
458C4—541no—
459C4D3×D3611yescharacteristic
460C4D9×#6711yescharacteristic
461C3—21no—
462C3#1058845311yescharacteristic, complemented
463C3#1059094511yescharacteristic
464C2—122no—
465C2—362no—
466C2—542no—
467C2—721no—
468C2#1172081811yescharacteristic
469C2#1172082011yescharacteristic
470C2#1172104911yescharacteristic
subgroupG/H
centerC22#88646
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
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×#88646C2semidirect
2. C2×C3×D72C2semidirect
3. #11721007C2semidirect
4. C2×#88421C2semidirect
5. #11720542C2semidirect
6. C9×#821C2semidirect
7. #11720266C2semidirect
14. C2×C3×D36C22semidirect
19. C2×C3×C8×C9C22semidirect
20. C2×C3⋊(C9⋊C8)C22semidirect
21. C9×C4⋊Dic3C22semidirect
34. C2×D72D3semidirect
53. C9×D12D4semidirect
57. C3×D36D4semidirect
86. C2×D36C2×D3semidirect
90. C2×C8×C9C2×D3semidirect
100. C2×C3⋊D9D8semidirect
125. #821D9semidirect
172. D36#67semidirect
210. C2×C3×C8C2×D9semidirect
217. C4⋊Dic3C2×D9semidirect
272. C3×C9#447semidirect
311. D12D36semidirect
354. C2×C8D3×D9semidirect
402. C9#5156semidirect
462. C3#10588453semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
C2×C3×D722
#117210072
C2×#884212
#117205422
C9×#8212
#117202662
#105909453
#105884533