Finite Group Databasebeta

← #13139511

Subgroups — #13139511

Order 1728.

580 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 97 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×C3×#3697C211yesmaximal, characteristic, complemented
2#11721819C211yesmaximal, characteristic, complemented
3C2×#88417C211yesmaximal, characteristic, complemented
4C2×#88416C211yesmaximal, characteristic, complemented
5C9×#830C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720276C211yesmaximal, characteristic, complemented
8#10591163—31nomaximal
9#10588416—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×C3×#611—21no—
14C3×#3697—22no—
15C3×#3697—22no—
16C2×C3×D36C2211yescharacteristic
17C2×C3×C4×D9—21no—
18C2×C3×Dic18C2211yescharacteristic
19#88536—21no—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23#88417—22no—
24#88416—22no—
25C2×C3×#235—31no—
26#64139—31no—
27C2×#3717—31no—
28C2×#3714—31no—
29C2×#3697D311yescharacteristic, complemented
30C3×#830D311yescharacteristic
31#63566—31no—
32#63563—31no—
33#63492—31no—
34#63492—61no—
35C2×D72—31no—
36C2×#3664—31no—
37C9×C22⋊C8—31no—
38#63380—31no—
39#63376—31no—
40C23×C3×C9D411yescharacteristic
41C22×C3⋊D9—21no—
42C22×C3×D9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9D411yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47#6233—42no—
48C2×C3⋊Dic9—21no—
49C4⋊(C3⋊D9)—21no—
50C4⋊(C3⋊D9)—21no—
51C3×#611—24no—
52C2×C3×Dic9—21no—
53C3×D36—21no—
54C3×D36D412yescomplemented
55C3×C4×D9—24no—
56C3×Dic18—21no—
57C3×Dic18D412yescomplemented
58C9×C3⋊C8—21no—
59C9×C3⋊C8—21no—
60#5238—91no—
61#4734—91no—
62C22×C32×C4C2×D311yescharacteristic
63C2×#638—61no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C3×#67—61no—
66C3×#235—62no—
67C3×#235—62no—
68C2×C3×D12—31no—
69C2×C3×C4×D3—61no—
70C2×C3×Dic6—31no—
71#3752—61no—
72C4⋊(C3⋊Dic3)—31no—
73C2×C3×C3⋊C8C2×D311yescharacteristic
74C2×C3×C3⋊C8C2×D311yescharacteristic
75#3717—62no—
76#3714—62no—
77C22×C4×C9—21no—
78C22×C4×C9C2×D311yescharacteristic
79C2×#611—21no—
80C2×#611—62no—
81C2×#611—61no—
82#3697—22no—
83#3697—22no—
84C2×D36—31no—
85C2×D36—61no—
86C2×D36C2×D311yescharacteristic
87C2×C4×D9—21no—
88C2×Dic18C2×D311yescharacteristic
89C2×C8×C9—31no—
90C2×C8×C9—31no—
91(C2×D9)⋊C4—62no—
92(C2×D9)⋊C4—61no—
93C4⋊Dic9—31no—
94C4⋊Dic9—61no—
95D72—62no—
96#3664—62no—
97C22×C3×C9—21no—
98C22×C3×C9—21no—
99C22×C3×C9—21no—
100C22×C3×C9C2×D411yescharacteristic
101C2×C3⋊D9—41no—
102C2×C3⋊D9—41no—
103C2×C3⋊D9—41no—
104C2×C3×D9—22no—
105C2×C3×D9—41no—
106C2×C3×D9—41no—
107C3×C4×C9—21no—
108C3×C4×C9C2×D411yescharacteristic
109C3×C4×C9C2×D411yescharacteristic
110C3⋊Dic9—41no—
111C3×Dic9—22no—
112C2×C3×#41—91no—
113C2×#235—31no—
114C2×#215—91no—
115C2×#213—91no—
116#930—91no—
117#930—91no—
118#930—181no—
119C2×D24—91no—
120C2×#204—91no—
121C3×C22⋊C8—31no—
122#835—91no—
123#832—91no—
124#830D911yescharacteristic, complemented
125#821—91no—
126#816—91no—
127C23×C32D1211yescharacteristic
128C22×C3⋊D3—61no—
129C22×C3×D3—61no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4D1211yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134#638—122no—
135C2×C3⋊Dic3—61no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—61no—
138C3×#67—64no—
139C2×C3×Dic3—61no—
140C3×D12—32no—
141C3×D12—61no—
142C3×C4×D3—64no—
143C3×Dic6—32no—
144C3×Dic6—61no—
145C23×C9—21no—
146C23×C9#6711yescharacteristic
147C22×D9—21no—
148C22×D9—62no—
149C22×D9—61no—
150C3×C3⋊C8—21no—
151C3×C3⋊C8—21no—
152C2×C4×C9—21no—
153C2×C4×C9—21no—
154C2×C4×C9—21no—
155C2×C4×C9—21no—
156C2×C4×C9—41no—
157C2×C4×C9—41no—
158C2×C4×C9#6711yescharacteristic
159C2×C4×C9C22×D311yescharacteristic
160#611—24no—
161#611—124no—
162#611—122no—
163C2×Dic9—21no—
164C2×Dic9—62no—
165C2×Dic9—61no—
166D36—21no—
167D36—61no—
168D36—61no—
169D36—121no—
170D36—121no—
171D36#6712yescomplemented
172C4×D9—24no—
173Dic18—21no—
174Dic18#6712yescomplemented
175C8×C9—61no—
176C8×C9—61no—
177#449—271noSylow
178C2×C3×C9—41no—
179C2×C3×C9#12011yescharacteristic
180C2×C3×C9#13511yescharacteristic
181C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
182C3⋊D9—81no—
183C3×D9—42no—
184C3×#41—182no—
185C3×#41—182no—
186C2×C3×Q8—91no—
187C2×C3×D4—91no—
188C2×C3×D4—181no—
189C22×C3×C4—21no—
190C22×C3×C4—181no—
191C22×C3×C4C2×D911yescharacteristic
192C22×C3×C4D3211yescharacteristic
193C2×#67—61no—
194C2×#67—182no—
195C2×#67—181no—
196C2×#67—181no—
197#235—62no—
198#235—62no—
199C2×D12—31no—
200C2×D12—91no—
201C2×D12—91no—
202C2×D12—181no—
203C2×C4×D3—61no—
204C2×Dic6—31no—
205C2×C3×C8—31no—
206C2×C3×C8—31no—
207#215—182no—
208#213—182no—
209(C2×D3)⋊C4—182no—
210(C2×D3)⋊C4—181no—
211(C2×D3)⋊C4—181no—
212C4⋊Dic3—91no—
213C4⋊Dic3—91no—
214C4⋊Dic3—181no—
215C2×C3⋊C8C2×D911yescharacteristic
216C2×C3⋊C8C2×D911yescharacteristic
217D24—182no—
218#204—182no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32C2×D1211yescharacteristic
223C2×C3⋊D3—121no—
224C2×C3⋊D3—121no—
225C2×C3⋊D3—121no—
226C2×C3×D3—62no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C32×C4—21no—
230C32×C4D3×D411yescharacteristic
231C32×C4D3×D411yescharacteristic
232C3⋊Dic3—121no—
233C3×Dic3—62no—
234C22×C9—22no—
235C22×C9—21no—
236C22×C9—21no—
237C22×C9—21no—
238C22×C9—21no—
239C22×C9—41no—
240C22×C9—41no—
241C22×C9C2×#6711yescharacteristic
242C2×D9—22no—
243C2×D9—41no—
244C2×D9—41no—
245C2×D9—122no—
246C2×D9—122no—
247C2×D9—122no—
248C2×D9—121no—
249C2×D9—121no—
250C2×D9—121no—
251C4×C9—21no—
252C4×C9—21no—
253C4×C9—21no—
254C4×C9—41no—
255C4×C9C2×#6711yescharacteristic
256C4×C9C2×#6711yescharacteristic
257Dic9—22no—
258Dic9—122no—
259Dic9—121no—
260C2×#41—91no—
261C2×SD16—271no—
262C2×D8—271no—
263#121—271no—
264#103—271no—
265#102—271no—
266C22⋊C8—31no—
267C3×C9#44911yescharacteristic, complemented, Sylow
268C23×C3—21no—
269C23×C3—181no—
270C23×C3D3611yescharacteristic
271C23×C3#62611yescharacteristic
272C22×D3—61no—
273C22×D3—182no—
274C22×D3—181no—
275C22×D3—181no—
276C3×Q8—92no—
277C3×Q8—181no—
278C3×D4—92no—
279C3×D4—184no—
280C3×D4—181no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—184no—
290C2×C3×C4—181no—
291C2×C3×C4D3611yescharacteristic
292C2×C3×C4C22×D911yescharacteristic
293C2×C3×C4#62611yescharacteristic
294C2×C3×C4C2×D3211yescharacteristic
295#67—64no—
296#67—362no—
297#67—364no—
298#67—362no—
299C2×Dic3—61no—
300C2×Dic3—182no—
301C2×Dic3—181no—
302C2×Dic3—181no—
303D12—32no—
304D12—61no—
305D12—181no—
306D12—181no—
307D12—181no—
308D12—181no—
309D12—361no—
310D12—361no—
311C4×D3—64no—
312Dic6—32no—
313Dic6—61no—
314C3×C8—61no—
315C3×C8—61no—
316C3⋊C8—21no—
317C3⋊C8—21no—
318C2×C32—41no—
319C2×C32#88211yescharacteristic
320C2×C32#90411yescharacteristic
321C2×C32#90811yescharacteristic
322C3⋊D3—241no—
323C3×D3—122no—
324C2×C9—21no—
325C2×C9—21no—
326C2×C9—21no—
327C2×C9—42no—
328C2×C9—41no—
329C2×C9#94911yescharacteristic
330C2×C9#95011yescharacteristic
331C2×C9#95311yescharacteristic
332D9—42no—
333D9—242no—
334D9—241no—
335#41—182no—
336#41—182no—
337C2×Q8—91no—
338C2×D4—91no—
339C2×D4—181no—
340C2×D4—271no—
341C2×D4—541no—
342C22×C4—181no—
343C22×C4D3×D911yescharacteristic
344SD16—542no—
345D8—542no—
346C2×C8—31no—
347C2×C8—31no—
348C4⋊C4—271no—
349C22⋊C4—541no—
350C22×C3—22no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—182no—
361C22×C3—361no—
362C22×C3—361no—
363C22×C3C2×D3611yescharacteristic
364C22×C3C2×#62611yescharacteristic
365C2×D3—62no—
366C2×D3—121no—
367C2×D3—121no—
368C2×D3—362no—
369C2×D3—362no—
370C2×D3—362no—
371C2×D3—361no—
372C2×D3—361no—
373C2×D3—361no—
374C2×D3—361no—
375C2×D3—361no—
376C2×D3—361no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—41no—
382C3×C4—182no—
383C3×C4D4×D911yescharacteristic
384C3×C4D4×D911yescharacteristic
385C3×C4D3×#6711yescharacteristic
386C3×C4D3×#6711yescharacteristic
387Dic3—62no—
388Dic3—362no—
389Dic3—361no—
390Dic3—361no—
391C32#473411yescharacteristic
392C9—21no—
393C9#523811yescharacteristic, complemented
394C23—181no—
395C23—541no—
396C23#619211yescharacteristic
397Q8—92no—
398Q8—181no—
399D4—92no—
400D4—184no—
401D4—181no—
402D4—541no—
403D4—541no—
404D4—1082no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—184no—
408C2×C4—181no—
409C2×C4—541no—
410C2×C4#619211yescharacteristic
411C2×C4C2×D3×D911yescharacteristic
412C8—61no—
413C8—61no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—42no—
418C2×C3—41no—
419C2×C3—41no—
420C2×C3—362no—
421C2×C3#6344411yescharacteristic
422C2×C3#6346711yescharacteristic
423C2×C3#6347011yescharacteristic
424C2×C3#6382511yescharacteristic
425C2×C3#6382911yescharacteristic
426C2×C3#6397711yescharacteristic
427D3—122no—
428D3—722no—
429D3—721no—
430D3—721no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—182no—
435C22—361no—
436C22—361no—
437C22—1081no—
438C22—1081no—
439C22—1081no—
440C22C2×#619211yescharacteristic
441C4—21no—
442C4—182no—
443C4—1081no—
444C4D9×#6711yescharacteristic
445C4D9×#6711yescharacteristic
446C3—21no—
447C3#1058841611yescharacteristic, complemented
448C3#1059116311yescharacteristic
449C2—41no—
450C2—362no—
451C2—2161no—
452C2#1172086711yescharacteristic
453C2#1172087311yescharacteristic
454C2#1172127611yescharacteristic
subgroupG/H
centerC22C2×#6192
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×D12
hypercenterC22×C4D3×D9
hypocenterC3×C9#449
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3697C2semidirect
2. #11721819C2semidirect
3. C2×#88417C2semidirect
4. C2×#88416C2semidirect
5. C9×#830C2semidirect
6. #11720283C2semidirect
7. #11720276C2semidirect
29. C2×#3697D3semidirect
54. C3×D36D4semidirect
57. C3×Dic18D4semidirect
124. #830D9semidirect
171. D36#67semidirect
174. Dic18#67semidirect
267. C3×C9#449semidirect
393. C9#5238semidirect
447. C3#10588416semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
#117218192
C2×#884172
C2×#884162
C9×#8302
#117202832
#117202762
#105911633
#105884163