Finite Group Databasebeta

← #13137835

Subgroups — #13137835

Order 1728.

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

normal subgroups only

This group has 688 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
2#11721751C211yesmaximal, characteristic, complemented
3C2×#88422C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic, complemented
5C9×#809C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic, complemented
7#11720282C211yesmaximal, characteristic, complemented
8#10590598—31nomaximal
9#10588453—31nomaximal
10C22×D3×D9—21no—
11C2×C4⋊(C3⋊D9)—21no—
12C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
13C2×C4×C3⋊D9C2211yescharacteristic
14C2×C9×D12C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×#6194—21no—
17#88646—22no—
18#88646—22no—
19C3×C9×C4⋊C4C2211yescharacteristic, complemented
20#88536—21no—
21C2×C9×C3⋊C8C2211yescharacteristic, complemented
22C2×C3×C9⋊C8C2211yescharacteristic, complemented
23#88422—22no—
24#88417—22no—
25C2×#3802—31no—
26#64104—31no—
27C2×#3714—31no—
28C2×#3714—31no—
29C2×D4×D9—31no—
30C3×#809—31no—
31C3×#809D311yescharacteristic
32#63557—31no—
33C2×#3673—31no—
34C2×D72—31no—
35#63457—31no—
36#63457—61no—
37C9×#102—31no—
38(C2×D9)⋊C8—31no—
39#63369D311yescharacteristic, complemented
40C22×C3⋊D9—21no—
41C22×C3⋊D9D411yescharacteristic
42C22×D3×C9—21no—
43C22×C3×D9—21no—
44C2×D3×D9—24no—
45C2×D3×D9—42no—
46C2×D3×D9—42no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9C2311yescharacteristic
49C2×C3⋊Dic9D411yescharacteristic
50C4⋊(C3⋊D9)—21no—
51C4⋊(C3⋊D9)—21no—
52C4⋊(C3⋊D9)—42no—
53C4×C3⋊D9—21no—
54C4×C3⋊D9—21no—
55C9×D12—21no—
56C9×D12D412yescomplemented
57C3×D36—21no—
58C3×D36D412yescomplemented
59#6194—24no—
60C9×C3⋊C8—21no—
61C3×C9⋊C8—21no—
62#4773—91no—
63#4773—91no—
64C22×D32—61no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—61no—
67C2×C4×C3⋊D3—31no—
68C2×C3×D12—31no—
69C2×C3×D12C2×D311yescharacteristic
70C2×#625—61no—
71#3802—62no—
72#3802—62no—
73C23×D9—61no—
74C32×C4⋊C4C2×D311yescharacteristic
75#3752—61no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8C2×D311yescharacteristic
78#3714—62no—
79#3714—62no—
80C2×D4×C9—31no—
81C2×#611—61no—
82D4×D9—62no—
83D4×D9—62no—
84C2×D36—31no—
85C2×D36—61no—
86C2×D36—61no—
87C2×D36—62no—
88C2×D36C2×D311yescharacteristic
89C2×C4×D9—31no—
90C2×C4×D9—61no—
91C2×C8×C9—31no—
92C9×C4⋊C4—21no—
93C9×C4⋊C4C2×D311yescharacteristic, complemented
94#3673—62no—
95(C2×D9)⋊C4—61no—
96(C2×D9)⋊C4—62no—
97C2×C9⋊C8C2×D311yescharacteristic, complemented
98D72—62no—
99C22×C3×C9C2×D411yescharacteristic
100C2×C3⋊D9—21no—
101C2×C3⋊D9—21no—
102C2×C3⋊D9—41no—
103C2×C3⋊D9—41no—
104C2×C3⋊D9—41no—
105C2×C3⋊D9D812yescomplemented
106C2×D3×C9—22no—
107C2×D3×C9—41no—
108C2×D3×C9—41no—
109C2×C3×D9—22no—
110C2×C3×D9—41no—
111C2×C3×D9—41no—
112D3×D9—48no—
113C3×C4×C9—41no—
114C3×C4×C9C2×D411yescharacteristic
115C3×C4×C9C2×D411yescharacteristic
116C3⋊Dic9—21no—
117C2×D3×D4—91no—
118C2×D3×D4—91no—
119C2×#213—91no—
120C2×#213—91no—
121C2×D24—91no—
122C2×D24—91no—
123#895—91no—
124#895—91no—
125#895—181no—
126C3×#102—31no—
127C3×#102—91no—
128(C2×D3)⋊C8—91no—
129(C2×D3)⋊C8—91no—
130#809—31no—
131#809D911yescharacteristic, complemented
132C22×C3⋊D3—31no—
133C22×C3⋊D3—61no—
134C22×C3×D3—21no—
135C22×C3×D3—61no—
136C2×D32—64no—
137C2×D32—122no—
138C2×D32—122no—
139C2×C32×C4—21no—
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—31no—
142C4⋊(C3⋊D3)—61no—
143C4⋊(C3⋊D3)—61no—
144C4⋊(C3⋊D3)—122no—
145C4×C3⋊D3—61no—
146C4×C3⋊D3—61no—
147C3×D12—21no—
148C3×D12—32no—
149C3×D12—61no—
150C3×D12D1212yes—
151#625—64no—
152C23×C9—61no—
153C22×D9—21no—
154C22×D9—31no—
155C22×D9—61no—
156C22×D9—64no—
157C22×D9—62no—
158C22×D9—61no—
159C22×D9—122no—
160C22×D9—122no—
161C3×C3⋊C8—21no—
162C3×C3⋊C8—61no—
163D4×C9—32no—
164D4×C9—61no—
165C2×C4×C9—22no—
166C2×C4×C9—21no—
167C2×C4×C9—21no—
168C2×C4×C9C22×D311yescharacteristic
169#611—64no—
170C2×Dic9—31no—
171C2×Dic9—61no—
172D36—21no—
173D36—61no—
174D36—61no—
175D36—121no—
176D36—121no—
177D36—122no—
178D36—124no—
179D36D1212yescomplemented
180C4×D9—61no—
181C4×D9—61no—
182C4×D9—121no—
183C4×D9—121no—
184C8×C9—61no—
185C9⋊C8—21no—
186#447—271noSylow
187C2×C3×C9#12011yescharacteristic
188C2×C3×C9C2×D811yescharacteristic
189C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
190C3⋊D9—22no—
191C3⋊D9—81no—
192D3×C9—42no—
193C3×D9—42no—
194C23×D3—181no—
195C23×D3—181no—
196C2×C3×D4—31no—
197C2×C3×D4—91no—
198C2×#67—181no—
199C2×#67—181no—
200D3×D4—182no—
201D3×D4—182no—
202D3×D4—182no—
203D3×D4—182no—
204C2×D12—31no—
205C2×D12—91no—
206C2×D12—91no—
207C2×D12—181no—
208C2×D12—181no—
209C2×D12—181no—
210C2×D12—182no—
211C2×D12C2×D911yescharacteristic
212C2×C4×D3—91no—
213C2×C4×D3—91no—
214C2×C4×D3—181no—
215C2×C3×C8—31no—
216C2×C3×C8—91no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4C2×D911yescharacteristic, complemented
219C3×C4⋊C4D3211yescharacteristic
220#213—182no—
221#213—182no—
222(C2×D3)⋊C4—181no—
223(C2×D3)⋊C4—181no—
224(C2×D3)⋊C4—182no—
225C2×C3⋊C8—31no—
226C2×C3⋊C8C2×D911yescharacteristic, complemented
227D24—182no—
228D24—182no—
229C22×C32D3×D411yescharacteristic
230C2×C3⋊D3—32no—
231C2×C3⋊D3—61no—
232C2×C3⋊D3—61no—
233C2×C3⋊D3—121no—
234C2×C3⋊D3—121no—
235C2×C3⋊D3—121no—
236C2×C3×D3—22no—
237C2×C3×D3—41no—
238C2×C3×D3—41no—
239C2×C3×D3—62no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242D32—128no—
243C32×C4—41no—
244C32×C4C2×D1211yescharacteristic
245C32×C4D3×D411yescharacteristic
246C3⋊Dic3—61no—
247C22×C9—21no—
248C22×C9—62no—
249C22×C9—121no—
250C22×C9—121no—
251C22×C9D3×D411yescharacteristic
252C2×D9—22no—
253C2×D9—32no—
254C2×D9—41no—
255C2×D9—41no—
256C2×D9—62no—
257C2×D9—61no—
258C2×D9—61no—
259C2×D9—121no—
260C2×D9—121no—
261C2×D9—128no—
262C2×D9—122no—
263C2×D9—122no—
264C2×D9—122no—
265C2×D9—121no—
266C2×D9—121no—
267C2×D9—121no—
268C4×C9—21no—
269C4×C9—21no—
270C4×C9—42no—
271C4×C9—41no—
272C4×C9C2×D1211yescharacteristic
273C4×C9D3×D411yescharacteristic
274Dic9—61no—
275Dic9—121no—
276C22×D4—271no—
277C2×D8—271no—
278C2×D8—271no—
279#121—271no—
280#102—31no—
281#102—91no—
282C22⋊C8—271no—
283C3×C9#44711yescharacteristic, complemented, Sylow
284C23×C3—61no—
285C23×C3—181no—
286C22×D3—21no—
287C22×D3—61no—
288C22×D3—91no—
289C22×D3—91no—
290C22×D3—181no—
291C22×D3—184no—
292C22×D3—184no—
293C22×D3—182no—
294C22×D3—181no—
295C22×D3—181no—
296C22×D3—362no—
297C22×D3—362no—
298C22×D3—362no—
299C22×D3—362no—
300C3×D4—32no—
301C3×D4—61no—
302C3×D4—92no—
303C3×D4—181no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4C22×D911yescharacteristic
309C2×C3×C4C2×D3211yescharacteristic
310#67—184no—
311#67—184no—
312C2×Dic3—91no—
313C2×Dic3—91no—
314C2×Dic3—181no—
315D12—21no—
316D12—32no—
317D12—61no—
318D12—181no—
319D12—181no—
320D12—181no—
321D12—181no—
322D12—361no—
323D12—361no—
324D12—362no—
325D12—362no—
326D12—364no—
327D12D3612yescomplemented
328C4×D3—181no—
329C4×D3—181no—
330C4×D3—181no—
331C4×D3—181no—
332C4×D3—361no—
333C4×D3—361no—
334C3×C8—61no—
335C3×C8—181no—
336C3⋊C8—21no—
337C3⋊C8—61no—
338C2×C32#88211yescharacteristic
339C2×C32D3×D811yescharacteristic
340C2×C32#91411yescharacteristic
341C3⋊D3—62no—
342C3⋊D3—241no—
343C3×D3—42no—
344C3×D3—122no—
345C2×C9—21no—
346C2×C9—21no—
347C2×C9—21no—
348C2×C9—122no—
349C2×C9#88211yescharacteristic
350C2×C9D3×D811yescharacteristic
351C2×C9#91411yescharacteristic
352D9—42no—
353D9—62no—
354D9—122no—
355D9—242no—
356D9—241no—
357C24—541no—
358C2×D4—31no—
359C2×D4—91no—
360C2×D4—271no—
361C2×D4—541no—
362C2×D4—541no—
363C2×D4—542no—
364C2×D4—542no—
365C22×C4—271no—
366D8—542no—
367D8—542no—
368C2×C8—31no—
369C2×C8—91no—
370C4⋊C4D3×D911yescharacteristic, complemented
371C22⋊C4—541no—
372C22×C3—21no—
373C22×C3—62no—
374C22×C3—121no—
375C22×C3—121no—
376C22×C3—182no—
377C22×C3—361no—
378C22×C3—361no—
379C22×C3D4×D911yescharacteristic
380C22×C3#381111yescharacteristic
381C2×D3—22no—
382C2×D3—41no—
383C2×D3—41no—
384C2×D3—62no—
385C2×D3—92no—
386C2×D3—92no—
387C2×D3—121no—
388C2×D3—121no—
389C2×D3—182no—
390C2×D3—181no—
391C2×D3—181no—
392C2×D3—181no—
393C2×D3—181no—
394C2×D3—361no—
395C2×D3—361no—
396C2×D3—368no—
397C2×D3—368no—
398C2×D3—362no—
399C2×D3—362no—
400C2×D3—362no—
401C2×D3—361no—
402C2×D3—361no—
403C2×D3—361no—
404C2×D3—361no—
405C2×D3—361no—
406C2×D3—361no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—42no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4C2×D3611yescharacteristic
413C3×C4D4×D911yescharacteristic
414C3×C4D3×D1211yescharacteristic
415C3×C4D3×D1211yescharacteristic
416Dic3—181no—
417Dic3—181no—
418Dic3—361no—
419C32#477311yescharacteristic
420C9—21no—
421C9#477311yescharacteristic, complemented
422C23—61no—
423C23—181no—
424C23—271no—
425C23—544no—
426C23—541no—
427C23—1082no—
428C23—1082no—
429D4—32no—
430D4—61no—
431D4—92no—
432D4—181no—
433D4—544no—
434D4—541no—
435D4—541no—
436D4—1082no—
437C2×C4—21no—
438C2×C4—271no—
439C2×C4—541no—
440C2×C4—541no—
441C2×C4C2×D3×D911yescharacteristic
442C8—61no—
443C8—181no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—122no—
448C2×C3—362no—
449C2×C3#6344411yescharacteristic
450C2×C3D8×D911yescharacteristic
451C2×C3#6347611yescharacteristic
452C2×C3#6392311yescharacteristic
453C2×C3#6392611yescharacteristic
454C2×C3#6394111yescharacteristic
455D3—42no—
456D3—122no—
457D3—182no—
458D3—182no—
459D3—362no—
460D3—722no—
461D3—721no—
462D3—721no—
463C22—62no—
464C22—121no—
465C22—121no—
466C22—182no—
467C22—272no—
468C22—361no—
469C22—361no—
470C22—541no—
471C22—541no—
472C22—1088no—
473C22—1081no—
474C22—1081no—
475C22—1081no—
476C22#8866811yescharacteristic
477C4—41no—
478C4—541no—
479C4D3×D3611yescharacteristic
480C4D9×D1211yescharacteristic
481C3—21no—
482C3#1058845311yescharacteristic, complemented
483C3#1059059811yescharacteristic
484C2—122no—
485C2—362no—
486C2—542no—
487C2—2161no—
488C2#1172105111yescharacteristic
489C2#1172117411yescharacteristic
490C2#1172121311yescharacteristic
subgroupG/H
centerC22#88668
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×#88646C2semidirect
2. #11721751C2semidirect
3. C2×#88422C2semidirect
4. C2×#88417C2semidirect
5. C9×#809C2semidirect
6. C3×#63369C2semidirect
7. #11720282C2semidirect
12. C2×C4⋊(C3⋊D9)C22semidirect
19. C3×C9×C4⋊C4C22semidirect
21. C2×C9×C3⋊C8C22semidirect
22. C2×C3×C9⋊C8C22semidirect
39. #63369D3semidirect
56. C9×D12D4semidirect
58. C3×D36D4semidirect
93. C9×C4⋊C4C2×D3semidirect
97. C2×C9⋊C8C2×D3semidirect
105. C2×C3⋊D9D8semidirect
131. #809D9semidirect
179. D36D12semidirect
218. C3×C4⋊C4C2×D9semidirect
226. C2×C3⋊C8C2×D9semidirect
283. C3×C9#447semidirect
327. D12D36semidirect
370. C4⋊C4D3×D9semidirect
421. C9#4773semidirect
482. C3#10588453semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
#117217512
C2×#884222
C2×#884172
C9×#8092
C3×#633692
#117202822
#105905983
#105884533