Finite Group Databasebeta

← #13138650

Subgroups — #13138650

Order 1728.

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

normal subgroups only

This group has 632 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
2C2×#88524C211yesmaximal, characteristic, complemented
3#11720961C211yesmaximal, characteristic, complemented
4C2×#88415C211yesmaximal, characteristic, complemented
5C9×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6C3×#63380C211yesmaximal, characteristic, complemented
7#11720284C211yesmaximal, characteristic, complemented
8#10590946—31nomaximal
9#10588413—31nomaximal
10C22×D3×D9—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×#6226C2211yescharacteristic, complemented
13C2×C4×D3×C9C2211yescharacteristic
14C2×C3×D36C2211yescharacteristic
15C2×#6192—21no—
16D3×D36—22no—
17D3×D36—22no—
18C2×C3×C8×C9C2211yescharacteristic, complemented
19#88524—22no—
20C3×C4⋊Dic9C2211yescharacteristic, complemented
21C2×C9×C3⋊C8C2211yescharacteristic, complemented
22(C2×D3)⋊Dic9—21no—
23Dic9⋊Dic3—21no—
24#88415—22no—
25C2×D3×D12—31no—
26C2×#3744—31no—
27#63857—31no—
28C2×#3716—31no—
29C22×D36—31no—
30C3×#821—31no—
31C3×(C2×D3)⋊C8D311yescharacteristic
32#63564—31no—
33#63486—31no—
34C2×#3664—31no—
35C2×#3664—31no—
36C2×#3664—61no—
37C9×C22⋊C8—31no—
38#63380—31no—
39#63380D311yescharacteristic, complemented
40C22×C3⋊D9—21no—
41C22×D3×C9D411yescharacteristic
42C22×C3×D9—21no—
43C2×D3×D9—24no—
44C2×D3×D9—42no—
45C2×D3×D9—42no—
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3⋊Dic9—21no—
48C4⋊(C3⋊D9)—21no—
49C4⋊(C3⋊D9)D412yes—
50#6226—21no—
51#6226—21no—
52C2×C9×Dic3D411yescharacteristic
53C2×C3×Dic9—21no—
54C4×D3×C9—21no—
55C4×D3×C9—21no—
56C3×D36—21no—
57C3×D36D412yes—
58#6192—24no—
59C3×C8×C9—21no—
60C9×C3⋊C8—21no—
61#4776—91no—
62#4731—91no—
63C22×D32—61no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×#634—31no—
66C2×C3×D12—31no—
67C2×C3×C4×D3C2×D311yescharacteristic
68C2×#626—61no—
69D3×D12—62no—
70D3×D12—62no—
71C23×D9—61no—
72C2×C32×C8C2×D311yescharacteristic
73#3744—62no—
74C3×C4⋊Dic3—31no—
75C2×C3×C3⋊C8C2×D311yescharacteristic
76Dic3⋊Dic3—61no—
77(C2×D3)⋊Dic3—61no—
78#3716—62no—
79C22×C4×C9—31no—
80C2×D36—31no—
81C2×D36—62no—
82C2×D36—61no—
83C2×D36—62no—
84C2×D36—61no—
85C2×D36C2×D311yescharacteristic
86C2×Dic18—31no—
87C2×Dic18—61no—
88C2×C8×C9—21no—
89C2×C8×C9—31no—
90C2×C8×C9C2×D311yescharacteristic, complemented
91#3676—61no—
92C4⋊Dic9C2×D311yescharacteristic, complemented
93Dic9⋊C4—61no—
94#3664—62no—
95#3664—64no—
96#3664—62no—
97C22×C3×C9C2×D411yescharacteristic
98C2×C3⋊D9—22no—
99C2×C3⋊D9—41no—
100C2×C3⋊D9—41no—
101C2×D3×C9—21no—
102C2×D3×C9—21no—
103C2×D3×C9SD1612yescomplemented
104C2×C3×D9—22no—
105C2×C3×D9—41no—
106C2×C3×D9—41no—
107D3×D9—48no—
108C3×C4×C9C2×D411yescharacteristic
109C3×C4×C9C2×D411yescharacteristic
110C3⋊Dic9—41no—
111C9×Dic3—21no—
112C3×Dic9—41no—
113C2×D3×D4—91no—
114C22×D12—91no—
115C2×#214—91no—
116#924—91no—
117C2×#204—91no—
118C2×#204—91no—
119C2×#204—91no—
120C2×#204—181no—
121#897—91no—
122C3×#102—91no—
123C3×C22⋊C8—31no—
124#821—31no—
125#821—91no—
126(C2×D3)⋊C8D911yescharacteristic, complemented
127#809—91no—
128C22×C3⋊D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3D1211yescharacteristic
131C2×D32—64no—
132C2×D32—122no—
133C2×D32—122no—
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—61no—
136C4⋊(C3⋊D3)—32no—
137C4⋊(C3⋊D3)—61no—
138#634—61no—
139#634—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3D1211yescharacteristic
142C3×D12—32no—
143C3×D12—61no—
144C3×C4×D3—21no—
145C3×C4×D3—21no—
146#626—64no—
147C23×C9—31no—
148C22×D9—21no—
149C22×D9—64no—
150C22×D9—61no—
151C22×D9—121no—
152C22×D9—122no—
153C22×D9—122no—
154C32×C8—21no—
155C3×C3⋊C8—21no—
156C2×C4×C9—21no—
157C2×C4×C9—31no—
158C2×C4×C9—61no—
159C2×C4×C9—61no—
160C2×C4×C9C22×D311yescharacteristic
161C2×Dic9—21no—
162C2×Dic9—61no—
163C2×Dic9—121no—
164D36—21no—
165D36—32no—
166D36—62no—
167D36—64no—
168D36—61no—
169D36—121no—
170D36D1212yes—
171Dic18—62no—
172Dic18—61no—
173Dic18—61no—
174Dic18—121no—
175C8×C9—22no—
176C8×C9—21no—
177C8×C9—61no—
178#450—271noSylow
179C2×C3×C9#12011yescharacteristic
180C2×C3×C9C2×SD1611yescharacteristic
181C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
182C3⋊D9—42no—
183D3×C9—22no—
184C3×D9—42no—
185C23×D3—181no—
186C23×D3—181no—
187C2×C3×D4—91no—
188C22×C3×C4—31no—
189C2×#67—181no—
190D3×D4—182no—
191D3×D4—182no—
192C2×D12—31no—
193C2×D12—91no—
194C2×D12—91no—
195C2×D12—181no—
196C2×D12—182no—
197C2×D12—182no—
198C2×D12—181no—
199C2×C4×D3C2×D911yescharacteristic
200C2×Dic6—91no—
201C2×Dic6—91no—
202C2×Dic6—181no—
203C2×C3×C8—21no—
204C2×C3×C8—31no—
205C2×C3×C8C2×D911yescharacteristic, complemented
206C2×C3×C8D3211yescharacteristic
207C3×C4⋊C4—91no—
208#217—181no—
209#214—182no—
210(C2×D3)⋊C4—181no—
211C4⋊Dic3—31no—
212C4⋊Dic3—181no—
213Dic3⋊C4—181no—
214C2×C3⋊C8C2×D911yescharacteristic, complemented
215#204—182no—
216#204—182no—
217#204—184no—
218#204—182no—
219C22×C32C2×D1211yescharacteristic
220C2×C3⋊D3—62no—
221C2×C3⋊D3—121no—
222C2×C3⋊D3—121no—
223C2×C3×D3—21no—
224C2×C3×D3—21no—
225C2×C3×D3—62no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3#20412yes—
229D32—128no—
230C32×C4D3×D411yescharacteristic
231C32×C4D3×D411yescharacteristic
232C3⋊Dic3—121no—
233C3×Dic3—21no—
234C3×Dic3—121no—
235C22×C9—21no—
236C22×C9—32no—
237C22×C9—61no—
238C22×C9—61no—
239C22×C9D3×D411yescharacteristic
240C2×D9—22no—
241C2×D9—41no—
242C2×D9—41no—
243C2×D9—62no—
244C2×D9—122no—
245C2×D9—128no—
246C2×D9—121no—
247C2×D9—121no—
248C2×D9—241no—
249C2×D9—241no—
250C4×C9—21no—
251C4×C9—21no—
252C4×C9—61no—
253C4×C9C2×D1211yescharacteristic
254C4×C9D3×D411yescharacteristic
255Dic9—41no—
256Dic9—122no—
257Dic9—121no—
258C22×D4—271no—
259C2×SD16—271no—
260C2×SD16—271no—
261C22⋊Q8—271no—
262#102—91no—
263#102—271no—
264C22⋊C8—31no—
265C3×C9#45011yescharacteristic, complemented, Sylow
266C23×C3—31no—
267C23×C3—181no—
268C22×D3—61no—
269C22×D3—184no—
270C22×D3—184no—
271C22×D3—181no—
272C22×D3—181no—
273C22×D3—361no—
274C22×D3—362no—
275C22×D3—362no—
276C22×D3—362no—
277C22×D3—362no—
278C22×D3D3611yescharacteristic
279C3×D4—92no—
280C3×D4—181no—
281C2×C3×C4—21no—
282C2×C3×C4—31no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—181no—
286C2×C3×C4C22×D911yescharacteristic
287C2×C3×C4C2×D3211yescharacteristic
288#67—184no—
289C2×Dic3—61no—
290C2×Dic3—181no—
291C2×Dic3—181no—
292C2×Dic3—361no—
293C2×Dic3D3611yescharacteristic
294D12—32no—
295D12—61no—
296D12—92no—
297D12—92no—
298D12—182no—
299D12—184no—
300D12—181no—
301D12—181no—
302D12—361no—
303C4×D3—21no—
304C4×D3—21no—
305Dic6—182no—
306Dic6—181no—
307Dic6—181no—
308Dic6—181no—
309Dic6—181no—
310Dic6—361no—
311C3×C8—22no—
312C3×C8—21no—
313C3×C8—21no—
314C3×C8—61no—
315C3⋊C8—21no—
316C2×C32#88211yescharacteristic
317C2×C32C2×#20411yescharacteristic
318C2×C32#90811yescharacteristic
319C3⋊D3—122no—
320C3×D3—22no—
321C3×D3—122no—
322C2×C9—21no—
323C2×C9—21no—
324C2×C9—21no—
325C2×C9—62no—
326C2×C9#88211yescharacteristic
327C2×C9#91111yescharacteristic
328C2×C9D3×SD1611yescharacteristic
329D9—42no—
330D9—122no—
331D9—242no—
332C24—541no—
333C2×Q8—271no—
334C2×D4—91no—
335C2×D4—271no—
336C2×D4—541no—
337C2×D4—542no—
338C2×D4—542no—
339C22×C4—31no—
340SD16—542no—
341SD16—542no—
342C2×C8—31no—
343C2×C8D3×D911yescharacteristic, complemented
344C4⋊C4—91no—
345C4⋊C4—541no—
346C22⋊C4—541no—
347C22×C3—21no—
348C22×C3—32no—
349C22×C3—61no—
350C22×C3—61no—
351C22×C3—182no—
352C22×C3—361no—
353C22×C3—361no—
354C22×C3C2×D3611yescharacteristic
355C22×C3D3×D1211yescharacteristic
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—62no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—182no—
362C2×D3—182no—
363C2×D3—362no—
364C2×D3—368no—
365C2×D3—368no—
366C2×D3—361no—
367C2×D3—361no—
368C2×D3—361no—
369C2×D3—361no—
370C2×D3—721no—
371C2×D3—721no—
372C2×D3#366412yescomplemented
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—61no—
376C3×C4—361no—
377C3×C4D4×D911yescharacteristic
378C3×C4D4×D911yescharacteristic
379C3×C4D3×D1211yescharacteristic
380C3×C4#381111yescharacteristic
381Dic3—21no—
382Dic3—121no—
383Dic3—362no—
384Dic3—361no—
385Dic3—361no—
386C32#473111yescharacteristic
387C9—21no—
388C9#477611yescharacteristic, complemented
389C23—31no—
390C23—181no—
391C23—544no—
392C23—541no—
393C23—1082no—
394C23—1082no—
395Q8—541no—
396Q8—541no—
397D4—92no—
398D4—181no—
399D4—272no—
400D4—544no—
401D4—541no—
402C2×C4—31no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—181no—
406C2×C4—541no—
407C2×C4C2×D3×D911yescharacteristic
408C8—21no—
409C8—61no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—62no—
414C2×C3—362no—
415C2×C3#6344411yescharacteristic
416C2×C3C2×#366411yescharacteristic
417C2×C3#6347011yescharacteristic
418C2×C3D3×#20411yescharacteristic
419C2×C3#6379511yescharacteristic
420C2×C3#6392311yescharacteristic
421D3—22no—
422D3—122no—
423D3—362no—
424D3—362no—
425D3—722no—
426C22—32no—
427C22—61no—
428C22—61no—
429C22—182no—
430C22—361no—
431C22—361no—
432C22—542no—
433C22—1088no—
434C22—1081no—
435C22—1081no—
436C22D3×D3611yescharacteristic
437C4—61no—
438C4—361no—
439C4—1081no—
440C4D9×D1211yescharacteristic
441C4#8866811yescharacteristic
442C3—21no—
443C3#1058841311yescharacteristic, complemented
444C3#1059094611yescharacteristic
445C2—62no—
446C2—362no—
447C2—1082no—
448C2D3×#366411yescharacteristic
449C2#1172080511yescharacteristic
450C2#1172105111yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radicalC2×D3×D36C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#450
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
2. C2×#88524C2semidirect
3. #11720961C2semidirect
4. C2×#88415C2semidirect
5. C9×(C2×D3)⋊C8C2semidirect
6. C3×#63380C2semidirect
7. #11720284C2semidirect
12. C2×#6226C22semidirect
18. C2×C3×C8×C9C22semidirect
20. C3×C4⋊Dic9C22semidirect
21. C2×C9×C3⋊C8C22semidirect
39. #63380D3semidirect
90. C2×C8×C9C2×D3semidirect
92. C4⋊Dic9C2×D3semidirect
103. C2×D3×C9SD16semidirect
126. (C2×D3)⋊C8D9semidirect
205. C2×C3×C8C2×D9semidirect
214. C2×C3⋊C8C2×D9semidirect
265. C3×C9#450semidirect
343. C2×C8D3×D9semidirect
372. C2×D3#3664semidirect
388. C9#4776semidirect
443. C3#10588413semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D362
C2×#885242
#117209612
C2×#884152
C9×(C2×D3)⋊C82
C3×#633802
#117202842
#105909463
#105884133