Finite Group Databasebeta

← #72220078

Subgroups — #72220078

Order 2592.

517 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 83 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×#67×3−1+2C211yesmaximal, characteristic
2#13047994C211yesmaximal, characteristic, complemented
3C3×#3676⋊C3C211yesmaximal, characteristic, complemented
4C3×#63580—31nomaximal
5#63393⋊C3—31nomaximal
6#11720332C311yesmaximal, characteristic, complemented
7C23×C3×3−1+2C2211yescharacteristic
8C22×D3×3−1+2C411yescharacteristic, complemented
9C2×C3⋊(3−1+2⋊C4)—21no—
10#67×3−1+2—21no—
11#67×3−1+2—21no—
12C2×Dic3×3−1+2C411yescharacteristic, complemented
13C2×C3×3−1+2⋊C4—21no—
14C2×C32×#67D311yescharacteristic
15C3×#3757—31no—
16C32×#217—31no—
17C2×D4×3−1+2—31no—
18C2×C9×#67—21no—
19C2×C9×#67C2×C311yescharacteristic
20#88551C2×C311yescharacteristic, complemented
21#3676⋊C3—21no—
22#3676⋊C3—31no—
23#3676⋊C3—61no—
24#3676⋊C3D311yescharacteristic, complemented
25C3×#3676C2×C311yescharacteristic, complemented
26C22×C3×3−1+2—41no—
27C22×C3×3−1+2C2×C411yescharacteristic
28C22×C3×3−1+2D411yescharacteristic
29C22×C3×3−1+2D411yescharacteristic
30C2×D3×3−1+2—21no—
31C2×D3×3−1+2—41no—
32C3⋊(3−1+2⋊C4)—41no—
33Dic3×3−1+2—21no—
34C3×3−1+2⋊C4—41no—
35C3×#834—91no—
36C3×#806—91no—
37#63580—31no—
38#63393—31no—
39C23×C33C2×D311yescharacteristic
40C22×C32×D3Dic311yescharacteristic
41C2×C3×C3⋊Dic3—61no—
42C32×#67—21no—
43C32×#67—21no—
44C2×C32×Dic3—61no—
45C2×C32×Dic3Dic311yescharacteristic
46C23×3−1+2—21no—
47C23×3−1+2—21no—
48C23×3−1+2—31no—
49C23×3−1+2—41no—
50C23×3−1+2C2×D311yescharacteristic
51C23×C3×C9—21no—
52C23×C3×C9C22×C311yescharacteristic
53C22×D3×C9—21no—
54C22×D3×C9C3×C411yescharacteristic, complemented
55D4×3−1+2—61no—
56D4×3−1+2—61no—
57C2×C4×3−1+2—31no—
58C2×C3⋊Dic9—21no—
59C2×3−1+2⋊C4—22no—
60C2×3−1+2⋊C4—21no—
61C2×3−1+2⋊C4—62no—
62C2×3−1+2⋊C4—61no—
63C9×#67—22no—
64C9×#67—22no—
65C9×#67—21no—
66C9×#67—21no—
67C2×C9×Dic3—21no—
68C2×C9×Dic3C3×C411yescharacteristic, complemented
69C2×C3×Dic9—21no—
70C2×C3×3−1+2—21no—
71C2×C3×3−1+2—21no—
72C2×C3×3−1+2—21no—
73C2×C3×3−1+2C22⋊C411yescharacteristic
74D3×3−1+2—41no—
75C2×C32×D4—31no—
76C2×C3×#67—31no—
77C2×C3×#67C3×D311yescharacteristic
78C32×C22⋊C4—91no—
79#3757—31no—
80C3×#217—31no—
81C3×#217—31no—
82C3×#217—61no—
83C3×#217—91no—
84C3×#217—91no—
85C3×#217—181no—
86C2×D4×C9—31no—
87C2×D4×C9—61no—
88#3676—31no—
89#3676—61no—
90#3676C3×D311yescharacteristic, complemented
91C22×C33—41no—
92C22×C33C2×Dic311yescharacteristic
93C22×C33#6711yescharacteristic
94C22×C33#6711yescharacteristic
95C2×C32×D3—21no—
96C2×C32×D3—41no—
97C3×C3⋊Dic3—121no—
98C32×Dic3—21no—
99C32×Dic3—121no—
100C22×3−1+2—21no—
101C22×3−1+2—21no—
102C22×3−1+2—21no—
103C22×3−1+2—21no—
104C22×3−1+2—21no—
105C22×3−1+2—21no—
106C22×3−1+2—41no—
107C22×3−1+2—42no—
108C22×3−1+2—42no—
109C22×3−1+2—41no—
110C22×3−1+2—41no—
111C22×3−1+2—41no—
112C22×3−1+2—44no—
113C22×3−1+2—61no—
114C22×3−1+2—121no—
115C22×3−1+2C4×D311yescharacteristic
116C22×3−1+2D1211yescharacteristic
117C22×3−1+2#6711yescharacteristic
118C22×C3×C9—21no—
119C22×C3×C9—21no—
120C22×C3×C9—21no—
121C22×C3×C9—41no—
122C22×C3×C9—42no—
123C22×C3×C9C2×C3×C411yescharacteristic
124C22×C3×C9C3×D411yescharacteristic
125C22×C3×C9C3×D411yescharacteristic
126C2×D3×C9—22no—
127C2×D3×C9—21no—
128C2×D3×C9—42no—
129C2×D3×C9—41no—
130C4×3−1+2—61no—
131C3⋊Dic9—41no—
1323−1+2⋊C4—42no—
1333−1+2⋊C4—41no—
1343−1+2⋊C4—122no—
1353−1+2⋊C4—121no—
136C9×Dic3—22no—
137C9×Dic3—21no—
138C3×Dic9—41no—
139C3×#99—271no—
140#834—91no—
141#806—91no—
142C3×3−1+2#9911yescharacteristic, complemented, Sylow
143C23×C32—21no—
144C23×C32—31no—
145C23×C32—31no—
146C23×C32—61no—
147C23×C32D3211yescharacteristic
148C23×C32C2×C3×D311yescharacteristic
149C22×C3×D3—31no—
150C22×C3×D3C3×Dic311yescharacteristic
151C32×D4—61no—
152C32×D4—61no—
153C2×C32×C4—31no—
154C2×C32×C4—181no—
155C2×C3⋊Dic3—61no—
156C3×#67—21no—
157C3×#67—21no—
158C3×#67—61no—
159C3×#67—61no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—181no—
165C2×C3×Dic3—182no—
166C2×C3×Dic3—181no—
167C2×C3×Dic3C3×Dic311yescharacteristic
168C23×C9—21no—
169C23×C9—21no—
170C23×C9—31no—
171C23×C9—41no—
172C23×C9—61no—
173C23×C9C2×C3×D311yescharacteristic
174D4×C9—62no—
175D4×C9—62no—
176D4×C9—61no—
177D4×C9—61no—
178C2×C4×C9—31no—
179C2×C4×C9—61no—
180C2×Dic9—21no—
181C2×Dic9—62no—
182C2×Dic9—61no—
183C2×C33—21no—
184C2×C33—21no—
185C2×C33—21no—
186C2×C33#21711yescharacteristic
187C32×D3—41no—
188C2×3−1+2—21no—
189C2×3−1+2—22no—
190C2×3−1+2—22no—
191C2×3−1+2—21no—
192C2×3−1+2—21no—
193C2×3−1+2—21no—
194C2×3−1+2—21no—
195C2×3−1+2—41no—
196C2×3−1+2—41no—
197C2×3−1+2—41no—
198C2×3−1+2—42no—
199C2×3−1+2—42no—
200C2×3−1+2—41no—
201C2×3−1+2—41no—
202C2×3−1+2—42no—
203C2×3−1+2—121no—
204C2×3−1+2(C2×D3)⋊C411yescharacteristic
205C2×C3×C9—21no—
206C2×C3×C9—22no—
207C2×C3×C9—21no—
208C2×C3×C9—21no—
209C2×C3×C9—21no—
210C2×C3×C9—41no—
211C2×C3×C9—41no—
212C2×C3×C9C3×C22⋊C411yescharacteristic
213D3×C9—42no—
214D3×C9—41no—
215C2×C3×D4—31no—
216C2×C3×D4—91no—
217C2×#67AGL(1,ℤ/9)11yescharacteristic
218C3×C22⋊C4—91no—
219C3×C22⋊C4—91no—
220C3×C22⋊C4—181no—
221C3×C22⋊C4—271no—
222#217—31no—
223#217—91no—
224#217—91no—
225#217—181no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—31no—
230C22×C32—31no—
231C22×C32—31no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—42no—
235C22×C32—61no—
236C22×C32—61no—
237C22×C32—61no—
238C22×C32—61no—
239C22×C32—121no—
240C22×C32—122no—
241C22×C32—121no—
242C22×C32D3×Dic311yescharacteristic
243C22×C32#62511yescharacteristic
244C22×C32#62611yescharacteristic
245C22×C32C2×C3×Dic311yescharacteristic
246C22×C32C3×#6711yescharacteristic
247C22×C32C3×#6711yescharacteristic
248C2×C3×D3—21no—
249C2×C3×D3—41no—
250C2×C3×D3—61no—
251C2×C3×D3—121no—
252C32×C4—61no—
253C32×C4—361no—
254C3⋊Dic3—121no—
255C3×Dic3—21no—
256C3×Dic3—61no—
257C3×Dic3—121no—
258C3×Dic3—122no—
259C3×Dic3—121no—
260C3×Dic3—361no—
261C3×Dic3—361no—
262C3×Dic3—362no—
263C22×C9—21no—
264C22×C9—21no—
265C22×C9—21no—
266C22×C9—21no—
267C22×C9—21no—
268C22×C9—21no—
269C22×C9—41no—
270C22×C9—42no—
271C22×C9—42no—
272C22×C9—44no—
273C22×C9—41no—
274C22×C9—41no—
275C22×C9—41no—
276C22×C9—62no—
277C22×C9—61no—
278C22×C9—121no—
279C22×C9—122no—
280C22×C9C3×C4×D311yescharacteristic
281C22×C9C3×D1211yescharacteristic
282C22×C9C3×#6711yescharacteristic
283C4×C9—62no—
284C4×C9—61no—
285Dic9—41no—
286Dic9—122no—
287Dic9—121no—
288#99—271noSylow
289C33#83411yescharacteristic
2903−1+2—21no—
2913−1+2—21no—
2923−1+2—41no—
2933−1+2#80611yescharacteristic, complemented
294C3×C9—21no—
295C3×C9C3×#9911yescharacteristic, complemented
296C23×C3—21no—
297C23×C3—31no—
298C23×C3—31no—
299C23×C3—61no—
300C23×C3—91no—
301C23×C3C2×AGL(1,ℤ/9)11yescharacteristic
302C23×C3C3×D3211yescharacteristic
303C22×D33−1+2⋊C411yescharacteristic, complemented
304C3×D4—61no—
305C3×D4—61no—
306C3×D4—181no—
307C3×D4—181no—
308C2×C3×C4—31no—
309C2×C3×C4—91no—
310C2×C3×C4—181no—
311C2×C3×C4—182no—
312C2×C3×C4—181no—
313C2×C3×C4—541no—
314#67—21no—
315#67—21no—
316C2×Dic3—61no—
317C2×Dic3—182no—
318C2×Dic3—181no—
319C2×Dic3—181no—
320C2×Dic33−1+2⋊C411yescharacteristic, complemented
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—22no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—31no—
330C2×C32—41no—
331C2×C32—41no—
332C2×C32—61no—
333C2×C32—61no—
334C2×C32—61no—
335C2×C32—61no—
336C2×C32—62no—
337C2×C32—121no—
338C2×C32—121no—
339C2×C32—121no—
340C2×C32(C2×D3)⋊Dic311yescharacteristic
341C2×C32C3×#21711yescharacteristic
342C3×D3—41no—
343C3×D3—121no—
344C2×C9—22no—
345C2×C9—22no—
346C2×C9—21no—
347C2×C9—21no—
348C2×C9—21no—
349C2×C9—21no—
350C2×C9—21no—
351C2×C9—41no—
352C2×C9—41no—
353C2×C9—41no—
354C2×C9—41no—
355C2×C9—42no—
356C2×C9—42no—
357C2×C9—42no—
358C2×C9—41no—
359C2×C9—122no—
360C2×C9—121no—
361C2×C9C3×(C2×D3)⋊C411yescharacteristic
362C2×D4—31no—
363C22⋊C4—91no—
364C22⋊C4—271no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—31no—
369C22×C3—31no—
370C22×C3—31no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—42no—
374C22×C3—61no—
375C22×C3—61no—
376C22×C3—61no—
377C22×C3—61no—
378C22×C3—121no—
379C22×C3—122no—
380C22×C3—121no—
381C22×C3—181no—
382C22×C3—361no—
383C22×C3C2×3−1+2⋊C411yescharacteristic
384C22×C3#611⋊C311yescharacteristic
385C22×C3#611⋊C311yescharacteristic
386C22×C3C3×D3×Dic311yescharacteristic
387C22×C3C3×#62511yescharacteristic
388C22×C3C3×#62611yescharacteristic
389C2×D3—21no—
390C2×D3—41no—
391C3×C4—61no—
392C3×C4—181no—
393C3×C4—361no—
394C3×C4—361no—
395C3×C4—362no—
396C3×C4—1081no—
397Dic3—21no—
398Dic3—121no—
399Dic3—361no—
400Dic3—362no—
401Dic3—361no—
402C32—21no—
403C32—31no—
404C32—61no—
405C32#6358011yescharacteristic
406C32C3×#83411yescharacteristic
407C9—21no—
408C9—21no—
409C9—41no—
410C9C3×#80611yescharacteristic, complemented
411C23—31no—
412C23D3×AGL(1,ℤ/9)11yescharacteristic
413D4—61no—
414D4—61no—
415C2×C4—31no—
416C2×C4—181no—
417C2×C4—541no—
418C2×C3—22no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—31no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—61no—
430C2×C3—61no—
431C2×C3—61no—
432C2×C3—62no—
433C2×C3—61no—
434C2×C3—121no—
435C2×C3—121no—
436C2×C3—121no—
437C2×C3—361no—
438C2×C3#3676⋊C311yescharacteristic
439C2×C3C3×(C2×D3)⋊Dic311yescharacteristic
440D3—41no—
441C22—41no—
442C22—61no—
443C22—121no—
444C22D3×3−1+2⋊C411yescharacteristic
445C22#6194⋊C311yescharacteristic
446C22#6195⋊C311yescharacteristic
447C4—61no—
448C4—361no—
449C4—1081no—
450C3—21no—
451C3—31no—
452C3—61no—
453C3#63393⋊C311yescharacteristic, complemented
454C3C3×#6358011yescharacteristic
455C2—21no—
456C2—21no—
457C2—21no—
458C2—121no—
459C2(C2×D3)⋊(3−1+2⋊C4)11yescharacteristic
subgroupG/H
centerC2(C2×D3)⋊(3−1+2⋊C4)
commutatorC22×C3×C9C2×C3×C4
FrattiniC23×C3C3×D32
FittingC23×C3×3−1+2C22
socleC2×C32C3×#217
prime radicalC2×C32×#67D3
cosocleC23×C3×C9C22×C3
hypercenterC23D3×AGL(1,ℤ/9)
hypocenterC3×C9C3×#99
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #13047994C2semidirect
3. C3×#3676⋊C3C2semidirect
6. #11720332C3semidirect
8. C22×D3×3−1+2C4semidirect
12. C2×Dic3×3−1+2C4semidirect
20. #88551C2×C3semidirect
24. #3676⋊C3D3semidirect
25. C3×#3676C2×C3semidirect
54. C22×D3×C9C3×C4semidirect
68. C2×C9×Dic3C3×C4semidirect
90. #3676C3×D3semidirect
142. C3×3−1+2#99semidirect
293. 3−1+2#806semidirect
295. C3×C9C3×#99semidirect
303. C22×D33−1+2⋊C4semidirect
320. C2×Dic33−1+2⋊C4semidirect
410. C9C3×#806semidirect
453. C3#63393⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#67×3−1+22
#130479942
C3×#3676⋊C32
C3×#635803
#63393⋊C33
#117203323