Finite Group Databasebeta

← #13138674

Subgroups — #13138674

Order 1728.

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

normal subgroups only

This group has 614 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
2C2×C3×#3664C211yesmaximal, characteristic, complemented
3#11720936C211yesmaximal, characteristic, complemented
4C2×#88423C211yesmaximal, characteristic, complemented
5#11720542C211yesmaximal, characteristic, complemented
6C9×#821C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10590943—31nomaximal
9#10588456—31nomaximal
10C22×D3×D9—21no—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×C3×Dic18C2211yescharacteristic, complemented
15C2×#6194—21no—
16#88646—22no—
17#88646—22no—
18C2×C3×C8×C9C2211yescharacteristic, complemented
19C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
20C9×C4⋊Dic3C2211yescharacteristic, complemented
21C3×#3664—22no—
22#88445—21no—
23Dic3⋊Dic9—21no—
24#88423—22no—
25C2×#3802—31no—
26C2×C3×#204—31no—
27#63871—31no—
28C2×#3715—31no—
29C2×D4×D9—31no—
30#63647—31no—
31C3×#821D311yescharacteristic
32#63562—31no—
33C2×#3672—31no—
34C2×#3664D311yescharacteristic, complemented
35#63459—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—
43C2×D3×D9—24no—
44C2×D3×D9—42no—
45C2×D3×D9—42no—
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3⋊Dic9D411yescharacteristic
48C4×C3⋊D9—21no—
49C4×C3⋊D9—21no—
50C2×C9×Dic3—21no—
51C2×C3×Dic9—21no—
52C9×D12—21no—
53C9×D12D412yes—
54C3×D36—21no—
55C3×D36D412yes—
56C3×Dic18—21no—
57C3×Dic18—21no—
58#6194—24no—
59C3×C8×C9—21no—
60C3⋊(C9⋊C8)—21no—
61#5169—91no—
62#4776—91no—
63C22×D32—61no—
64C2×C4×C3⋊D3—31no—
65C2×C3×D12—31no—
66C2×C3×D12C2×D311yescharacteristic
67C2×C3×Dic6—31no—
68C2×#625—61no—
69#3802—62no—
70#3802—62no—
71C23×D9—61no—
72C2×C32×C8C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)—31no—
74C3×C4⋊Dic3C2×D311yescharacteristic
75C3×#204—62no—
76Dic3⋊Dic3—61no—
77#3722—61no—
78#3715—62no—
79C2×D4×C9—31no—
80C2×#611—61no—
81D4×D9—62no—
82D4×D9—62no—
83C2×D36C2×D311yescharacteristic
84C2×C4×D9—31no—
85C2×C4×D9—61no—
86C2×Dic18C2×D311yescharacteristic, complemented
87C2×C8×C9—21no—
88C2×C8×C9C2×D311yescharacteristic, complemented
89C9×C4⋊C4—31no—
90#3672—62no—
91(C2×D9)⋊C4—61no—
92C4⋊Dic9—61no—
93C2×C9⋊C8—31no—
94C2×C9⋊C8—61no—
95#3664—22no—
96C22×C3×C9C2×D411yescharacteristic
97C2×C3⋊D9—21no—
98C2×C3⋊D9—21no—
99C2×C3⋊D9SD1612yescomplemented
100C2×D3×C9—22no—
101C2×D3×C9—41no—
102C2×D3×C9—41no—
103C2×C3×D9—22no—
104C2×C3×D9—41no—
105C2×C3×D9—41no—
106D3×D9—48no—
107C3×C4×C9C2×D411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C3⋊Dic9—21no—
110C9×Dic3—41no—
111C3×Dic9—41no—
112C2×D3×D4—91no—
113C2×D3×D4—91no—
114C2×C3×SD16—91no—
115#946—91no—
116C2×#215—91no—
117C2×#214—91no—
118C2×#204—31no—
119#897—91no—
120C3×#102—31no—
121#832—91no—
122#821D911yescharacteristic, complemented
123(C2×D3)⋊C8—91no—
124(C2×D3)⋊C8—91no—
125(C2×D3)⋊C8—181no—
126#809—91no—
127C22×C3⋊D3—31no—
128C22×C3×D3—21no—
129C22×C3×D3—61no—
130C2×D32—64no—
131C2×D32—122no—
132C2×D32—122no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—31no—
135C4×C3⋊D3—61no—
136C4×C3⋊D3—61no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—61no—
139C3×D12—21no—
140C3×D12—32no—
141C3×D12—61no—
142C3×D12D1212yes—
143C3×Dic6—61no—
144C3×Dic6—61no—
145#625—64no—
146C23×C9—61no—
147C22×D9—21no—
148C22×D9—31no—
149C22×D9—61no—
150C22×D9—64no—
151C22×D9—122no—
152C22×D9—122no—
153C32×C8—21no—
154C3⋊(C3⋊C8)—61no—
155D4×C9—32no—
156D4×C9—61no—
157C2×C4×C9—21no—
158C2×C4×C9—61no—
159C2×C4×C9C22×D311yescharacteristic
160#611—64no—
161C2×Dic9—21no—
162C2×Dic9—31no—
163C2×Dic9—61no—
164D36—21no—
165D36#6712yes—
166C4×D9—61no—
167C4×D9—61no—
168C4×D9—121no—
169C4×D9—121no—
170Dic18—21no—
171Dic18—21no—
172C8×C9—21no—
173C8×C9—41no—
174C9⋊C8—61no—
175C9⋊C8—121no—
176#450—271noSylow
177C2×C3×C9#12011yescharacteristic
178C2×C3×C9C2×SD1611yescharacteristic
179C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
180C3⋊D9—22no—
181D3×C9—42no—
182C3×D9—42no—
183C23×D3—181no—
184C23×D3—181no—
185C2×C3×Q8—91no—
186C2×C3×D4—31no—
187C2×C3×D4—91no—
188C2×#67—181no—
189C2×#67—181no—
190D3×D4—182no—
191D3×D4—182no—
192D3×D4—182no—
193D3×D4—182no—
194C2×D12—31no—
195C2×D12C2×D911yescharacteristic
196C2×C4×D3—91no—
197C2×C4×D3—91no—
198C2×C4×D3—181no—
199C2×Dic6—31no—
200C3×SD16—182no—
201C2×C3×C8—21no—
202C2×C3×C8C2×D911yescharacteristic, complemented
203C2×C3×C8D3211yescharacteristic
204C3×C4⋊C4—31no—
205#215—182no—
206#214—182no—
207(C2×D3)⋊C4—181no—
208(C2×D3)⋊C4—181no—
209C4⋊Dic3—181no—
210C4⋊Dic3C2×D911yescharacteristic, complemented
211Dic3⋊C4—181no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—91no—
214C2×C3⋊C8—181no—
215#204—62no—
216C22×C32D3×D411yescharacteristic
217C2×C3⋊D3—32no—
218C2×C3⋊D3—61no—
219C2×C3⋊D3—61no—
220C2×C3×D3—22no—
221C2×C3×D3—41no—
222C2×C3×D3—41no—
223C2×C3×D3—62no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226D32—128no—
227C32×C4C2×D1211yescharacteristic
228C32×C4D3×D411yescharacteristic
229C3⋊Dic3—61no—
230C3×Dic3—41no—
231C3×Dic3—121no—
232C22×C9—21no—
233C22×C9—62no—
234C22×C9—121no—
235C22×C9—121no—
236C22×C9D3×D411yescharacteristic
237C2×D9—22no—
238C2×D9—32no—
239C2×D9—41no—
240C2×D9—41no—
241C2×D9—62no—
242C2×D9—61no—
243C2×D9—61no—
244C2×D9—121no—
245C2×D9—121no—
246C2×D9—128no—
247C4×C9—21no—
248C4×C9—21no—
249C4×C9—121no—
250C4×C9D3×D411yescharacteristic
251C4×C9C2×#6711yescharacteristic
252Dic9—41no—
253Dic9—61no—
254Dic9—121no—
255C22×D4—271no—
256C2×SD16—91no—
257C2×SD16—271no—
258C22⋊Q8—271no—
259#102—31no—
260#102—271no—
261C22⋊C8—271no—
262C3×C9#45011yescharacteristic, complemented, Sylow
263C23×C3—61no—
264C23×C3—181no—
265C22×D3—21no—
266C22×D3—61no—
267C22×D3—91no—
268C22×D3—91no—
269C22×D3—181no—
270C22×D3—184no—
271C22×D3—184no—
272C22×D3—362no—
273C22×D3—362no—
274C22×D3—362no—
275C22×D3—362no—
276C3×Q8—181no—
277C3×Q8—181no—
278C3×D4—32no—
279C3×D4—61no—
280C3×D4—92no—
281C3×D4—181no—
282C2×C3×C4—21no—
283C2×C3×C4—61no—
284C2×C3×C4—181no—
285C2×C3×C4C22×D911yescharacteristic
286C2×C3×C4C2×D3211yescharacteristic
287#67—184no—
288#67—184no—
289C2×Dic3—21no—
290C2×Dic3—61no—
291C2×Dic3—91no—
292C2×Dic3—91no—
293C2×Dic3—181no—
294D12—21no—
295D12—32no—
296D12—61no—
297D12D3612yes—
298C4×D3—181no—
299C4×D3—181no—
300C4×D3—181no—
301C4×D3—181no—
302C4×D3—361no—
303C4×D3—361no—
304Dic6—61no—
305Dic6—61no—
306C3×C8—21no—
307C3×C8—21no—
308C3×C8—41no—
309C3⋊C8—181no—
310C3⋊C8—181no—
311C3⋊C8—361no—
312C2×C32#88211yescharacteristic
313C2×C32#91111yescharacteristic
314C2×C32D3×SD1611yescharacteristic
315C3⋊D3—62no—
316C3×D3—42no—
317C3×D3—122no—
318C2×C9—21no—
319C2×C9—21no—
320C2×C9—21no—
321C2×C9—122no—
322C2×C9D3×SD1611yescharacteristic
323C2×C9#91411yescharacteristic
324C2×C9#93711yescharacteristic
325D9—42no—
326D9—62no—
327D9—122no—
328C24—541no—
329C2×Q8—91no—
330C2×D4—31no—
331C2×D4—91no—
332C2×D4—541no—
333C2×D4—542no—
334C2×D4—542no—
335C22×C4—271no—
336SD16—182no—
337SD16—542no—
338C2×C8—271no—
339C2×C8D3×D911yescharacteristic, complemented
340C4⋊C4—31no—
341C4⋊C4—541no—
342C22⋊C4—541no—
343C22×C3—21no—
344C22×C3—62no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—182no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3D4×D911yescharacteristic
351C22×C3#380211yescharacteristic
352C2×D3—22no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—62no—
356C2×D3—92no—
357C2×D3—92no—
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—182no—
361C2×D3—181no—
362C2×D3—181no—
363C2×D3—181no—
364C2×D3—181no—
365C2×D3—361no—
366C2×D3—361no—
367C2×D3—368no—
368C2×D3—368no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—121no—
372C3×C4—361no—
373C3×C4C2×D3611yescharacteristic
374C3×C4D4×D911yescharacteristic
375C3×C4D3×D1211yescharacteristic
376C3×C4D3×#6711yescharacteristic
377Dic3—41no—
378Dic3—121no—
379Dic3—181no—
380Dic3—181no—
381Dic3—361no—
382C32#477611yescharacteristic
383C9—21no—
384C9#516911yescharacteristic, complemented
385C23—61no—
386C23—181no—
387C23—271no—
388C23—544no—
389C23—1082no—
390C23—1082no—
391Q8—181no—
392Q8—181no—
393D4—32no—
394D4—61no—
395D4—92no—
396D4—181no—
397D4—544no—
398C2×C4—61no—
399C2×C4—181no—
400C2×C4—271no—
401C2×C4—541no—
402C2×C4—541no—
403C2×C4C2×D3×D911yescharacteristic
404C8—21no—
405C8—541no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—122no—
410C2×C3—362no—
411C2×C3#6344411yescharacteristic
412C2×C3#6347311yescharacteristic
413C2×C3SD16×D911yescharacteristic
414C2×C3#6379611yescharacteristic
415C2×C3#6379811yescharacteristic
416C2×C3#6392211yescharacteristic
417D3—42no—
418D3—122no—
419D3—182no—
420D3—182no—
421D3—362no—
422C22—62no—
423C22—121no—
424C22—121no—
425C22—182no—
426C22—272no—
427C22—361no—
428C22—361no—
429C22—541no—
430C22—541no—
431C22—1088no—
432C22#8864611yescharacteristic
433C4—121no—
434C4—361no—
435C4—541no—
436C4D3×D3611yescharacteristic
437C4D9×#6711yescharacteristic
438C3—21no—
439C3#1058845611yescharacteristic, complemented
440C3#1059094311yescharacteristic
441C2—122no—
442C2—362no—
443C2—542no—
444C2#1172081711yescharacteristic
445C2#1172081911yescharacteristic
446C2#1172104911yescharacteristic
subgroupG/H
centerC22#88646
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88646C2
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×C3×#3664C2semidirect
3. #11720936C2semidirect
4. C2×#88423C2semidirect
5. #11720542C2semidirect
6. C9×#821C2semidirect
7. #11720266C2semidirect
14. C2×C3×Dic18C22semidirect
18. C2×C3×C8×C9C22semidirect
19. C2×C3⋊(C9⋊C8)C22semidirect
20. C9×C4⋊Dic3C22semidirect
34. C2×#3664D3semidirect
86. C2×Dic18C2×D3semidirect
88. C2×C8×C9C2×D3semidirect
99. C2×C3⋊D9SD16semidirect
122. #821D9semidirect
202. C2×C3×C8C2×D9semidirect
210. C4⋊Dic3C2×D9semidirect
262. C3×C9#450semidirect
339. C2×C8D3×D9semidirect
384. C9#5169semidirect
439. C3#10588456semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
C2×C3×#36642
#117209362
C2×#884232
#117205422
C9×#8212
#117202662
#105909433
#105884563