Finite Group Databasebeta

← #13137836

Subgroups — #13137836

Order 1728.

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

normal subgroups only

This group has 638 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
2#11721753C211yesmaximal, characteristic, complemented
3C2×#88424C211yesmaximal, characteristic, complemented
4C2×#88415C211yesmaximal, characteristic, complemented
5C9×#809C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic, complemented
7#11720282C211yesmaximal, characteristic, complemented
8#10590597—31nomaximal
9#10588456—31nomaximal
10C22×D3×D9—21no—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×#6226C2211yescharacteristic, complemented
13C2×C9×D12C2211yescharacteristic
14C2×C3×D36C2211yescharacteristic
15C2×#6194—21no—
16#88646—22no—
17#88646—22no—
18C3×C9×C4⋊C4C2211yescharacteristic, complemented
19#88536—21no—
20C4⋊(C3⋊Dic9)—21no—
21C2×C9×C3⋊C8C2211yescharacteristic, complemented
22C2×C3×C9⋊C8C2211yescharacteristic, complemented
23#88424—22no—
24#88415—22no—
25C2×#3802—31no—
26#64106—31no—
27C2×#3716—31no—
28C2×#3716—31no—
29C2×D4×D9—31no—
30C3×#809—31no—
31C3×#809D311yescharacteristic
32#63557—31no—
33C2×#3672—31no—
34C2×#3664—31no—
35#63459—31no—
36#63459—61no—
37C9×#102—31no—
38(C2×D9)⋊C8—31no—
39#63369D311yescharacteristic, complemented
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×C9—21no—
47C2×C3×C4×C9C2311yescharacteristic
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9D411yescharacteristic
50C4×C3⋊D9—21no—
51C4×C3⋊D9—21no—
52#6226—21no—
53#6226—21no—
54C9×D12—21no—
55C9×D12D412yes—
56C3×D36—21no—
57C3×D36D412yes—
58#6194—24no—
59C9×C3⋊C8—21no—
60C3×C9⋊C8—21no—
61#4776—91no—
62#4776—91no—
63C22×D32—61no—
64C2×C4×C3⋊D3—31no—
65C2×#634—31no—
66C2×C3×D12—31no—
67C2×C3×D12C2×D311yescharacteristic
68C2×#625—61no—
69#3802—62no—
70#3802—62no—
71C23×D9—61no—
72C32×C4⋊C4C2×D311yescharacteristic
73#3752—61no—
74C4⋊(C3⋊Dic3)—61no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8C2×D311yescharacteristic
77#3716—62no—
78#3716—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×Dic18—31no—
87C2×Dic18—61no—
88C2×C8×C9—31no—
89C9×C4⋊C4—21no—
90C9×C4⋊C4C2×D311yescharacteristic, complemented
91#3672—62no—
92(C2×D9)⋊C4—61no—
93(C2×D9)⋊C4—62no—
94C4⋊Dic9—61no—
95C4⋊Dic9—62no—
96C2×C9⋊C8C2×D311yescharacteristic, complemented
97#3664—62no—
98C22×C3×C9C2×D411yescharacteristic
99C2×C3⋊D9—21no—
100C2×C3⋊D9—21no—
101C2×C3⋊D9SD1612yescomplemented
102C2×D3×C9—22no—
103C2×D3×C9—41no—
104C2×D3×C9—41no—
105C2×C3×D9—22no—
106C2×C3×D9—41no—
107C2×C3×D9—41no—
108D3×D9—48no—
109C3×C4×C9—41no—
110C3×C4×C9C2×D411yescharacteristic
111C3×C4×C9C2×D411yescharacteristic
112C3⋊Dic9—21no—
113C3⋊Dic9—41no—
114C2×D3×D4—91no—
115C2×D3×D4—91no—
116C2×#214—91no—
117C2×#214—91no—
118C2×#204—91no—
119C2×#204—91no—
120#897—91no—
121#897—91no—
122#897—181no—
123C3×#102—31no—
124C3×#102—91no—
125(C2×D3)⋊C8—91no—
126(C2×D3)⋊C8—91no—
127#809—31no—
128#809D911yescharacteristic, complemented
129C22×C3⋊D3—31no—
130C22×C3×D3—21no—
131C22×C3×D3—61no—
132C2×D32—64no—
133C2×D32—122no—
134C2×D32—122no—
135C2×C32×C4—21no—
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—31no—
138C2×C3⋊Dic3—61no—
139C4×C3⋊D3—61no—
140C4×C3⋊D3—61no—
141#634—61no—
142#634—61no—
143C3×D12—21no—
144C3×D12—32no—
145C3×D12—61no—
146C3×D12D1212yes—
147#625—64no—
148C23×C9—61no—
149C22×D9—21no—
150C22×D9—31no—
151C22×D9—61no—
152C22×D9—64no—
153C22×D9—122no—
154C22×D9—122no—
155C3×C3⋊C8—21no—
156C3×C3⋊C8—61no—
157D4×C9—32no—
158D4×C9—61no—
159C2×C4×C9—22no—
160C2×C4×C9—21no—
161C2×C4×C9—21no—
162C2×C4×C9C22×D311yescharacteristic
163#611—64no—
164C2×Dic9—31no—
165C2×Dic9—61no—
166C2×Dic9—62no—
167C2×Dic9—61no—
168D36—21no—
169D36D1212yes—
170C4×D9—61no—
171C4×D9—61no—
172C4×D9—121no—
173C4×D9—121no—
174Dic18—61no—
175Dic18—61no—
176Dic18—121no—
177Dic18—121no—
178C8×C9—61no—
179C9⋊C8—21no—
180#450—271noSylow
181C2×C3×C9#12011yescharacteristic
182C2×C3×C9C2×SD1611yescharacteristic
183C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
184C3⋊D9—22no—
185D3×C9—42no—
186C3×D9—42no—
187C23×D3—181no—
188C23×D3—181no—
189C2×C3×D4—31no—
190C2×C3×D4—91no—
191C2×#67—181no—
192C2×#67—181no—
193D3×D4—182no—
194D3×D4—182no—
195D3×D4—182no—
196D3×D4—182no—
197C2×D12—31no—
198C2×D12C2×D911yescharacteristic
199C2×C4×D3—91no—
200C2×C4×D3—91no—
201C2×C4×D3—181no—
202C2×Dic6—91no—
203C2×Dic6—91no—
204C2×Dic6—181no—
205C2×C3×C8—31no—
206C2×C3×C8—91no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4C2×D911yescharacteristic, complemented
209C3×C4⋊C4D3211yescharacteristic
210#214—182no—
211#214—182no—
212(C2×D3)⋊C4—181no—
213(C2×D3)⋊C4—181no—
214(C2×D3)⋊C4—182no—
215C4⋊Dic3—181no—
216C4⋊Dic3—181no—
217C4⋊Dic3—182no—
218C2×C3⋊C8—31no—
219C2×C3⋊C8C2×D911yescharacteristic, complemented
220#204—182no—
221#204—182no—
222C22×C32D3×D411yescharacteristic
223C2×C3⋊D3—32no—
224C2×C3⋊D3—61no—
225C2×C3⋊D3—61no—
226C2×C3×D3—22no—
227C2×C3×D3—41no—
228C2×C3×D3—41no—
229C2×C3×D3—62no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232D32—128no—
233C32×C4—41no—
234C32×C4C2×D1211yescharacteristic
235C32×C4D3×D411yescharacteristic
236C3⋊Dic3—61no—
237C3⋊Dic3—121no—
238C22×C9—21no—
239C22×C9—62no—
240C22×C9—121no—
241C22×C9—121no—
242C22×C9D3×D411yescharacteristic
243C2×D9—22no—
244C2×D9—32no—
245C2×D9—41no—
246C2×D9—41no—
247C2×D9—62no—
248C2×D9—61no—
249C2×D9—61no—
250C2×D9—121no—
251C2×D9—121no—
252C2×D9—128no—
253C4×C9—21no—
254C4×C9—21no—
255C4×C9—42no—
256C4×C9—41no—
257C4×C9C2×D1211yescharacteristic
258C4×C9D3×D411yescharacteristic
259Dic9—61no—
260Dic9—121no—
261Dic9—122no—
262Dic9—121no—
263C22×D4—271no—
264C2×SD16—271no—
265C2×SD16—271no—
266C22⋊Q8—271no—
267#102—31no—
268#102—91no—
269C22⋊C8—271no—
270C3×C9#45011yescharacteristic, complemented, Sylow
271C23×C3—61no—
272C23×C3—181no—
273C22×D3—21no—
274C22×D3—61no—
275C22×D3—91no—
276C22×D3—91no—
277C22×D3—181no—
278C22×D3—184no—
279C22×D3—184no—
280C22×D3—362no—
281C22×D3—362no—
282C22×D3—362no—
283C22×D3—362no—
284C3×D4—32no—
285C3×D4—61no—
286C3×D4—92no—
287C3×D4—181no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4C22×D911yescharacteristic
293C2×C3×C4C2×D3211yescharacteristic
294#67—184no—
295#67—184no—
296C2×Dic3—91no—
297C2×Dic3—91no—
298C2×Dic3—181no—
299C2×Dic3—182no—
300C2×Dic3—181no—
301C2×Dic3—181no—
302D12—21no—
303D12—32no—
304D12—61no—
305D12D3612yes—
306C4×D3—181no—
307C4×D3—181no—
308C4×D3—181no—
309C4×D3—181no—
310C4×D3—361no—
311C4×D3—361no—
312Dic6—181no—
313Dic6—181no—
314Dic6—181no—
315Dic6—181no—
316Dic6—361no—
317Dic6—361no—
318C3×C8—61no—
319C3×C8—181no—
320C3⋊C8—21no—
321C3⋊C8—61no—
322C2×C32#88211yescharacteristic
323C2×C32#91111yescharacteristic
324C2×C32D3×SD1611yescharacteristic
325C3⋊D3—62no—
326C3×D3—42no—
327C3×D3—122no—
328C2×C9—21no—
329C2×C9—21no—
330C2×C9—21no—
331C2×C9—122no—
332C2×C9#88211yescharacteristic
333C2×C9#91111yescharacteristic
334C2×C9D3×SD1611yescharacteristic
335D9—42no—
336D9—62no—
337D9—122no—
338C24—541no—
339C2×Q8—271no—
340C2×D4—31no—
341C2×D4—91no—
342C2×D4—541no—
343C2×D4—542no—
344C2×D4—542no—
345C22×C4—271no—
346SD16—542no—
347SD16—542no—
348C2×C8—31no—
349C2×C8—91no—
350C4⋊C4—541no—
351C4⋊C4D3×D911yescharacteristic, complemented
352C22⋊C4—541no—
353C22×C3—21no—
354C22×C3—62no—
355C22×C3—121no—
356C22×C3—121no—
357C22×C3—182no—
358C22×C3—361no—
359C22×C3—361no—
360C22×C3D4×D911yescharacteristic
361C22×C3#381111yescharacteristic
362C2×D3—22no—
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—62no—
366C2×D3—92no—
367C2×D3—92no—
368C2×D3—121no—
369C2×D3—121no—
370C2×D3—182no—
371C2×D3—181no—
372C2×D3—181no—
373C2×D3—181no—
374C2×D3—181no—
375C2×D3—361no—
376C2×D3—361no—
377C2×D3—368no—
378C2×D3—368no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—42no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4C2×D3611yescharacteristic
385C3×C4D4×D911yescharacteristic
386C3×C4D3×D1211yescharacteristic
387C3×C4D3×D1211yescharacteristic
388Dic3—181no—
389Dic3—181no—
390Dic3—361no—
391Dic3—362no—
392Dic3—361no—
393Dic3—361no—
394C32#477611yescharacteristic
395C9—21no—
396C9#477611yescharacteristic, complemented
397C23—61no—
398C23—181no—
399C23—271no—
400C23—544no—
401C23—1082no—
402C23—1082no—
403Q8—541no—
404Q8—541no—
405D4—32no—
406D4—61no—
407D4—92no—
408D4—181no—
409D4—544no—
410C2×C4—21no—
411C2×C4—271no—
412C2×C4—541no—
413C2×C4—541no—
414C2×C4—541no—
415C2×C4C2×D3×D911yescharacteristic
416C8—61no—
417C8—181no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—122no—
422C2×C3—362no—
423C2×C3#6344411yescharacteristic
424C2×C3#6347311yescharacteristic
425C2×C3SD16×D911yescharacteristic
426C2×C3#6392311yescharacteristic
427C2×C3#6392711yescharacteristic
428C2×C3#6394011yescharacteristic
429D3—42no—
430D3—122no—
431D3—182no—
432D3—182no—
433D3—362no—
434C22—62no—
435C22—121no—
436C22—121no—
437C22—182no—
438C22—272no—
439C22—361no—
440C22—361no—
441C22—541no—
442C22—541no—
443C22—1088no—
444C22#8866811yescharacteristic
445C4—41no—
446C4—541no—
447C4—1081no—
448C4D3×D3611yescharacteristic
449C4D9×D1211yescharacteristic
450C3—21no—
451C3#1058845611yescharacteristic, complemented
452C3#1059059711yescharacteristic
453C2—122no—
454C2—362no—
455C2—542no—
456C2#1172105111yescharacteristic
457C2#1172117511yescharacteristic
458C2#1172121211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88646C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#450
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
2. #11721753C2semidirect
3. C2×#88424C2semidirect
4. C2×#88415C2semidirect
5. C9×#809C2semidirect
6. C3×#63369C2semidirect
7. #11720282C2semidirect
12. C2×#6226C22semidirect
18. C3×C9×C4⋊C4C22semidirect
21. C2×C9×C3⋊C8C22semidirect
22. C2×C3×C9⋊C8C22semidirect
39. #63369D3semidirect
90. C9×C4⋊C4C2×D3semidirect
96. C2×C9⋊C8C2×D3semidirect
101. C2×C3⋊D9SD16semidirect
128. #809D9semidirect
208. C3×C4⋊C4C2×D9semidirect
219. C2×C3⋊C8C2×D9semidirect
270. C3×C9#450semidirect
351. C4⋊C4D3×D9semidirect
396. C9#4776semidirect
451. C3#10588456semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
#117217532
C2×#884242
C2×#884152
C9×#8092
C3×#633692
#117202822
#105905973
#105884563