Finite Group Databasebeta

← #13140452

Subgroups — #13140452

Order 1728.

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

normal subgroups only

This group has 650 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×#88742C211yesmaximal, characteristic, complemented
2C2×C9×#213C211yesmaximal, characteristic, complemented
3#11721045C211yesmaximal, characteristic, complemented
4C2×#88424C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic, complemented
7#11720277C211yesmaximal, characteristic, complemented
8#10591457—31nomaximal
9#10588838—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C22×C3⋊Dic9—21no—
13#88742—22no—
14#88742—22no—
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×#6226C2211yescharacteristic
17C2×C9×D12C2211yescharacteristic
18C2×#6195—21no—
19C9×#213—22no—
20C3×C4⋊Dic9C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23(C2×D3)⋊Dic9—21no—
24#88424—22no—
25C2×#3830—31no—
26C2×C3×#213D311yescharacteristic
27#63916—31no—
28C2×#3716—31no—
29C2×#3699—31no—
30C2×#3699—61no—
31C3×#832—31no—
32#63567—31no—
33#63557—31no—
34C2×C9×D8—31no—
35#63500—31no—
36C2×#3672—31no—
37#63392D311yescharacteristic, complemented
38(C2×D9)⋊C8—31no—
39#63376—31no—
40C23×C3×C9—21no—
41C22×C3⋊D9D411yescharacteristic
42C22×D3×C9—21no—
43C3×D4×C9—21no—
44C3×D4×C9D412yescomplemented
45C2×C3×C4×C9C2311yescharacteristic
46#6233—24no—
47C2×C3⋊Dic9—24no—
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9D411yescharacteristic
50C4×C3⋊D9—21no—
51C4×C3⋊D9—21no—
52#6226—21no—
53#6226D412yescomplemented
54C2×C3×Dic9—21no—
55C9×D12—21no—
56C9×D12—21no—
57#6195—42no—
58C9×C3⋊C8—21no—
59C3×C9⋊C8—21no—
60#5158—91no—
61#4781—91no—
62C2×C32×D4C2×D311yescharacteristic
63C2×#638—61no—
64C22×C3⋊Dic3—61no—
65#3830—62no—
66#3830—62no—
67C2×C4×C3⋊D3—31no—
68C2×#634—31no—
69C2×C3×D12C2×D311yescharacteristic
70C2×#626—61no—
71C3×#213—22no—
72C3×C4⋊Dic3—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75(C2×D3)⋊Dic3—61no—
76#3716—62no—
77C2×D4×C9—21no—
78C2×D4×C9—31no—
79C2×D4×C9C2×D311yescharacteristic
80C2×#611—61no—
81C2×#611—61no—
82C2×#611—62no—
83C22×Dic9—61no—
84C22×Dic9—62no—
85#3699—62no—
86#3699—64no—
87#3699—64no—
88#3699—62no—
89C2×C4×D9—31no—
90C2×C4×D9—61no—
91C2×Dic18—31no—
92C2×Dic18—61no—
93C9×D8—62no—
94C2×C8×C9—31no—
95#3676—61no—
96#3672—62no—
97C4⋊Dic9C2×D311yescharacteristic
98C2×C9⋊C8C2×D311yescharacteristic
99C22×C3×C9—22no—
100C22×C3×C9—41no—
101C22×C3×C9—41no—
102C22×C3×C9C2×D411yescharacteristic
103C2×C3⋊D9—21no—
104C2×C3⋊D9—21no—
105C2×C3⋊D9—21no—
106C2×D3×C9—41no—
107C2×D3×C9—41no—
108C2×D3×C9—41no—
109C3×C4×C9C2×D411yescharacteristic
110C3×C4×C9C2×D411yescharacteristic
111C3⋊Dic9—22no—
112C3⋊Dic9—21no—
113C3×Dic9—41no—
114C2×#237—91no—
115C2×#237—91no—
116C2×#237—181no—
117C2×C3×D8—31no—
118#938—91no—
119C2×#214—91no—
120C2×#213D911yescharacteristic, complemented
121C2×#204—91no—
122#895—91no—
123C3×#102—91no—
124#832—31no—
125(C2×D3)⋊C8—91no—
126(C2×D3)⋊C8—91no—
127#816—91no—
128#810—91no—
129C23×C32—21no—
130C22×C3⋊D3—31no—
131C22×C3×D3—21no—
132C32×D4—21no—
133C32×D4#6712yes—
134C2×C32×C4C22×D311yescharacteristic
135#638—64no—
136C2×C3⋊Dic3—31no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—61no—
139C4×C3⋊D3—61no—
140C4×C3⋊D3—61no—
141#634—32no—
142#634—61no—
143C2×C3×Dic3—61no—
144C3×D12—21no—
145C3×D12—21no—
146#626—122no—
147C23×C9—22no—
148C23×C9—21no—
149C23×C9—61no—
150C22×D9—31no—
151C22×D9—61no—
152C3×C3⋊C8—21no—
153C3×C3⋊C8—61no—
154D4×C9—22no—
155D4×C9—21no—
156D4×C9—22no—
157D4×C9—61no—
158D4×C9—61no—
159D4×C9D1212yescomplemented
160C2×C4×C9—21no—
161C2×C4×C9C22×D311yescharacteristic
162#611—68no—
163#611—64no—
164#611—122no—
165C2×Dic9—21no—
166C2×Dic9—31no—
167C2×Dic9—61no—
168C2×Dic9—62no—
169C2×Dic9—68no—
170C2×Dic9—64no—
171C2×Dic9—61no—
172C4×D9—62no—
173C4×D9—62no—
174C4×D9—61no—
175C4×D9—61no—
176Dic18—32no—
177Dic18—62no—
178Dic18—62no—
179Dic18—61no—
180C8×C9—61no—
181C9⋊C8—21no—
182#449—271noSylow
183C2×C3×C9—42no—
184C2×C3×C9#12011yescharacteristic
185C2×C3×C9#13511yescharacteristic
186C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
187C3⋊D9—41no—
188D3×C9—81no—
189C2×C3×D4—21no—
190C2×C3×D4—31no—
191C2×C3×D4C2×D911yescharacteristic
192C2×C3×D4D3211yescharacteristic
193C2×#67—181no—
194C2×#67—181no—
195C2×#67—181no—
196C2×#67—182no—
197C22×Dic3—181no—
198C22×Dic3—181no—
199C22×Dic3—182no—
200#237—182no—
201#237—182no—
202#237—184no—
203#237—184no—
204#237—182no—
205#237—182no—
206C2×D12—181no—
207C2×D12C2×D911yescharacteristic
208C2×C4×D3—91no—
209C2×C4×D3—91no—
210C2×C4×D3—181no—
211C2×Dic6—91no—
212C2×Dic6—91no—
213C2×Dic6—181no—
214C3×D8—62no—
215C2×C3×C8—31no—
216C2×C3×C8—91no—
217C3×C4⋊C4—91no—
218#217—181no—
219#214—182no—
220#213—22no—
221(C2×D3)⋊C4—181no—
222C4⋊Dic3—31no—
223C2×C3⋊C8—31no—
224C2×C3⋊C8C2×D911yescharacteristic
225#204—182no—
226C22×C32—22no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32D3×D411yescharacteristic
230C2×C3⋊D3—61no—
231C2×C3⋊D3—61no—
232C2×C3⋊D3—61no—
233C2×C3×D3—41no—
234C2×C3×D3—41no—
235C2×C3×D3—41no—
236C32×C4D3×D411yescharacteristic
237C32×C4C2×#6711yescharacteristic
238C3⋊Dic3—62no—
239C3⋊Dic3—61no—
240C3×Dic3—121no—
241C22×C9—24no—
242C22×C9—22no—
243C22×C9—21no—
244C22×C9—42no—
245C22×C9—42no—
246C22×C9—41no—
247C22×C9—41no—
248C22×C9—121no—
249C22×C9—121no—
250C22×C9—121no—
251C22×C9D3×D411yescharacteristic
252C2×D9—62no—
253C2×D9—61no—
254C2×D9—61no—
255C2×D9—61no—
256C2×D9—121no—
257C2×D9—121no—
258C4×C9—21no—
259C4×C9—21no—
260C4×C9C2×D1211yescharacteristic
261C4×C9D3×D411yescharacteristic
262Dic9—41no—
263Dic9—64no—
264Dic9—62no—
265Dic9—62no—
266Dic9—61no—
267C2×#41—271no—
268C2×SD16—271no—
269C2×D8—31no—
270#121—271no—
271#103—271no—
272#102—91no—
273C22⋊C8—271no—
274C3×C9#44911yescharacteristic, complemented, Sylow
275C23×C3—22no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—61no—
279C22×D3—21no—
280C22×D3—91no—
281C22×D3—91no—
282C22×D3—181no—
283C3×D4—22no—
284C3×D4—21no—
285C3×D4—21no—
286C3×D4—22no—
287C3×D4—61no—
288C3×D4—61no—
289C3×D4#61112yescomplemented
290C3×D4#62612yes—
291C2×C3×C4—21no—
292C2×C3×C4—181no—
293C2×C3×C4C22×D911yescharacteristic
294C2×C3×C4C2×D3211yescharacteristic
295#67—184no—
296#67—188no—
297#67—184no—
298#67—362no—
299C2×Dic3—61no—
300C2×Dic3—91no—
301C2×Dic3—91no—
302C2×Dic3—181no—
303C2×Dic3—184no—
304C2×Dic3—182no—
305C2×Dic3—188no—
306C2×Dic3—184no—
307C2×Dic3—181no—
308C2×Dic3—181no—
309D12—21no—
310D12—21no—
311D12—362no—
312C4×D3—182no—
313C4×D3—182no—
314C4×D3—181no—
315C4×D3—181no—
316C4×D3—181no—
317C4×D3—181no—
318Dic6—92no—
319Dic6—92no—
320Dic6—182no—
321Dic6—182no—
322Dic6—181no—
323Dic6—181no—
324C3×C8—61no—
325C3×C8—181no—
326C3⋊C8—21no—
327C3⋊C8—61no—
328C2×C32—42no—
329C2×C32#91111yescharacteristic
330C2×C32#91211yescharacteristic
331C2×C32#93711yescharacteristic
332C3⋊D3—121no—
333C3×D3—81no—
334C2×C9—21no—
335C2×C9—21no—
336C2×C9—21no—
337C2×C9—44no—
338C2×C9—42no—
339C2×C9—241no—
340C2×C9#88211yescharacteristic
341C2×C9#91111yescharacteristic
342C2×C9#91611yescharacteristic
343D9—122no—
344D9—121no—
345#41—542no—
346#41—542no—
347C2×Q8—271no—
348C2×D4—31no—
349C2×D4—541no—
350C2×D4—541no—
351C2×D4D3×D911yescharacteristic
352C22×C4—271no—
353C22×C4—541no—
354SD16—542no—
355D8—62no—
356C2×C8—31no—
357C2×C8—91no—
358C4⋊C4—91no—
359C22⋊C4—541no—
360C22×C3—24no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—121no—
371C22×C3—121no—
372C22×C3—121no—
373C22×C3D4×D911yescharacteristic
374C22×C3#381111yescharacteristic
375C2×D3—41no—
376C2×D3—41no—
377C2×D3—41no—
378C2×D3—182no—
379C2×D3—181no—
380C2×D3—181no—
381C2×D3—181no—
382C2×D3—181no—
383C2×D3—181no—
384C2×D3—181no—
385C2×D3—361no—
386C2×D3—361no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—361no—
390C3×C4D4×D911yescharacteristic
391C3×C4C2×#61111yescharacteristic
392C3×C4C2×#62611yescharacteristic
393C3×C4#381111yescharacteristic
394Dic3—121no—
395Dic3—184no—
396Dic3—182no—
397Dic3—182no—
398Dic3—181no—
399Dic3—182no—
400Dic3—181no—
401C32#515811yescharacteristic
402C9—21no—
403C9#478111yescharacteristic, complemented
404C23—21no—
405C23—61no—
406C23—271no—
407Q8—272no—
408Q8—541no—
409D4—21no—
410D4—61no—
411D4—61no—
412D4—544no—
413D4—1082no—
414D4#619512yescomplemented
415C2×C4—181no—
416C2×C4—271no—
417C2×C4—544no—
418C2×C4—541no—
419C2×C4—541no—
420C2×C4—541no—
421C2×C4C2×D3×D911yescharacteristic
422C8—61no—
423C8—181no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—44no—
428C2×C3—42no—
429C2×C3—42no—
430C2×C3—241no—
431C2×C3#6347311yescharacteristic
432C2×C3#6347411yescharacteristic
433C2×C3#6349911yescharacteristic
434C2×C3#6392711yescharacteristic
435C2×C3#6393611yescharacteristic
436C2×C3#6398111yescharacteristic
437D3—81no—
438D3—362no—
439D3—361no—
440D3—361no—
441C22—22no—
442C22—41no—
443C22—41no—
444C22—121no—
445C22—121no—
446C22—121no—
447C22—541no—
448C22—541no—
449C22—541no—
450C22#8866811yescharacteristic
451C4—361no—
452C4—541no—
453C4—542no—
454C4C2×#619511yescharacteristic
455C4#8866811yescharacteristic
456C3—21no—
457C3#1058883811yescharacteristic, complemented
458C3#1059145711yescharacteristic
459C2—42no—
460C2—241no—
461C2—1081no—
462C2#1172117511yescharacteristic
463C2#1172117811yescharacteristic
464C2#1172129111yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
hypercenterC2×D4D3×D9
hypocenterC3×C9#449
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88742C2semidirect
2. C2×C9×#213C2semidirect
3. #11721045C2semidirect
4. C2×#88424C2semidirect
5. C3×#63392C2semidirect
6. #11720282C2semidirect
7. #11720277C2semidirect
37. #63392D3semidirect
44. C3×D4×C9D4semidirect
53. #6226D4semidirect
120. C2×#213D9semidirect
159. D4×C9D12semidirect
274. C3×C9#449semidirect
289. C3×D4#611semidirect
403. C9#4781semidirect
414. D4#6195semidirect
457. C3#10588838semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
C2×C9×#2132
#117210452
C2×#884242
C3×#633922
#117202822
#117202772
#105914573
#105888383