Finite Group Databasebeta

← #13138699

Subgroups — #13138699

Order 1728.

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

normal subgroups only

This group has 524 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×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#3664C211yesmaximal, characteristic, complemented
3#11721007C211yesmaximal, characteristic
4C2×#88425C211yesmaximal, characteristic, complemented
5#11720542C211yesmaximal, characteristic, complemented
6C9×#816C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10590824—31nomaximal
9#10588486—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×#6192—21no—
15C2×(C3⋊D9)⋊C4—21no—
16C2×#6189—21no—
17(C3⋊D9)⋊Q8—22no—
18(C3⋊D9)⋊Q8—22no—
19C2×C3×C8×C9C2211yescharacteristic
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21C9×C4⋊Dic3C2211yescharacteristic
22C3×#3664—22no—
23(C2×D9)⋊Dic3—21no—
24#88425—22no—
25C2×#3797—31no—
26C2×C3×#204—31no—
27#63916—31no—
28C2×#3715—31no—
29C2×Q8×D9—31no—
30#63647—31no—
31C3×#816D311yescharacteristic
32#63565—31no—
33C2×#3675—31no—
34C2×#3664D311yescharacteristic, complemented
35#63457—31no—
36C9×#103—31no—
37(C2×D9)⋊C8—31no—
38(C2×D9)⋊C8—61no—
39#63368—31no—
40C22×C3⋊D9D411yescharacteristic
41C22×C3×D9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9D411yescharacteristic
44C4×C3⋊D9—21no—
45C4×C3⋊D9—21no—
46C2×C9×Dic3—21no—
47C2×C9×Dic3—21no—
48C2×C3×Dic9—21no—
49C3×D36—21no—
50C3×D36—21no—
51C9×Dic6—21no—
52C9×Dic6D412yescomplemented
53C3×Dic18—21no—
54C3×Dic18D412yescomplemented
55#6192—42no—
56(C3⋊D9)⋊C4—24no—
57#6189—24no—
58C3×C8×C9—21no—
59C3⋊(C9⋊C8)—21no—
60#5170—91no—
61#4806—91no—
62C2×C4×C3⋊D3—31no—
63C2×C3×D12—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6C2×D311yescharacteristic
66C2×#627—61no—
67C2×#626—61no—
68C2×#624—61no—
69#3797—62no—
70#3797—62no—
71C2×C32×C8C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—31no—
73C3×C4⋊Dic3C2×D311yescharacteristic
74C3×#204—62no—
75(C2×D3)⋊Dic3—61no—
76#3715—62no—
77C2×Q8×C9—31no—
78Q8×D9—62no—
79Q8×D9—62no—
80C2×D36—61no—
81C2×D36C2×D311yescharacteristic
82C2×C4×D9—31no—
83C2×C4×D9—61no—
84C2×C4×D9—61no—
85C2×Dic18—61no—
86C2×Dic18C2×D311yescharacteristic
87C2×C8×C9—21no—
88C2×C8×C9C2×D311yescharacteristic
89C9×C4⋊C4—31no—
90#3675—62no—
91(C2×D9)⋊C4—61no—
92C2×C9⋊C8—31no—
93C2×C9⋊C8—61no—
94#3664—22no—
95C22×C3×C9C2×D411yescharacteristic
96C2×C3⋊D9—21no—
97C2×C3⋊D9—21no—
98C2×C3⋊D9SD1612yescomplemented
99C2×C3×D9—41no—
100C2×C3×D9—41no—
101C2×C3×D9—41no—
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—21no—
105C9×Dic3—22no—
106C9×Dic3—41no—
107C3×Dic9—22no—
108C2×D3×Q8—91no—
109C2×D3×Q8—91no—
110C2×C3×SD16—91no—
111C2×#215—91no—
112#938—91no—
113C2×#214—91no—
114C2×#204—31no—
115#895—91no—
116C3×#103—31no—
117#835—91no—
118(C2×D3)⋊C8—91no—
119(C2×D3)⋊C8—91no—
120(C2×D3)⋊C8—181no—
121#816D911yescharacteristic, complemented
122#810—91no—
123C22×C3⋊D3—31no—
124C22×C3×D3—61no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—31no—
127C4×C3⋊D3—61no—
128C4×C3⋊D3—61no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—61no—
132C3×D12—61no—
133C3×D12—61no—
134C3×Dic6—21no—
135C3×Dic6—32no—
136C3×Dic6—61no—
137C3×Dic6D1212yes—
138#627—64no—
139#626—122no—
140#624—64no—
141C22×D9—21no—
142C22×D9—31no—
143C22×D9—61no—
144C32×C8—21no—
145C3⋊(C3⋊C8)—61no—
146Q8×C9—32no—
147Q8×C9—61no—
148C2×C4×C9—21no—
149C2×C4×C9—61no—
150C2×C4×C9—61no—
151C2×C4×C9C22×D311yescharacteristic
152C2×Dic9—21no—
153C2×Dic9—31no—
154C2×Dic9—61no—
155D36—21no—
156D36—21no—
157D36—122no—
158C4×D9—64no—
159C4×D9—61no—
160C4×D9—61no—
161C4×D9—121no—
162C4×D9—121no—
163Dic18—21no—
164Dic18—64no—
165Dic18#6712yescomplemented
166C8×C9—21no—
167C8×C9—41no—
168C9⋊C8—61no—
169C9⋊C8—121no—
170#448—271noSylow
171C2×C3×C9#12011yescharacteristic
172C2×C3×C9C2×SD1611yescharacteristic
173C2×C3×C9#13711yescharacteristic
174C3⋊D9—22no—
175C3×D9—81no—
176C2×C3×Q8—31no—
177C2×C3×Q8—91no—
178C2×C3×D4—91no—
179C2×#67—181no—
180D3×Q8—182no—
181D3×Q8—182no—
182D3×Q8—182no—
183D3×Q8—182no—
184C2×D12—31no—
185C2×D12—181no—
186C2×C4×D3—91no—
187C2×C4×D3—91no—
188C2×C4×D3—181no—
189C2×C4×D3—181no—
190C2×C4×D3—181no—
191C2×Dic6—31no—
192C2×Dic6—181no—
193C2×Dic6—181no—
194C2×Dic6C2×D911yescharacteristic
195C3×SD16—182no—
196C2×C3×C8—21no—
197C2×C3×C8C2×D911yescharacteristic
198C2×C3×C8D3211yescharacteristic
199C3×C4⋊C4—31no—
200#217—181no—
201#215—182no—
202#214—182no—
203(C2×D3)⋊C4—181no—
204C4⋊Dic3C2×D911yescharacteristic
205C2×C3⋊C8—91no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—181no—
208#204—62no—
209C22×C32D3×D411yescharacteristic
210C2×C3⋊D3—32no—
211C2×C3⋊D3—61no—
212C2×C3⋊D3—61no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C32×C4C2×D1211yescharacteristic
217C32×C4D3×D411yescharacteristic
218C3⋊Dic3—61no—
219C3×Dic3—22no—
220C3×Dic3—41no—
221C3×Dic3—62no—
222C22×C9—21no—
223C22×C9D3×D411yescharacteristic
224C2×D9—32no—
225C2×D9—41no—
226C2×D9—41no—
227C2×D9—41no—
228C2×D9—62no—
229C2×D9—61no—
230C2×D9—61no—
231C2×D9—121no—
232C2×D9—121no—
233C4×C9—21no—
234C4×C9—21no—
235C4×C9—62no—
236C4×C9—121no—
237C4×C9D3×D411yescharacteristic
238C4×C9C2×#6711yescharacteristic
239Dic9—22no—
240Dic9—61no—
241Dic9—121no—
242C22×Q8—271no—
243C2×SD16—91no—
244C2×SD16—271no—
245#121—271no—
246#103—31no—
247#103—271no—
248C22⋊C8—271no—
249C3×C9#44811yescharacteristic, complemented, Sylow
250C23×C3—181no—
251C22×D3—61no—
252C22×D3—91no—
253C22×D3—91no—
254C22×D3—181no—
255C3×Q8—32no—
256C3×Q8—61no—
257C3×Q8—92no—
258C3×Q8—181no—
259C3×D4—181no—
260C3×D4—181no—
261C2×C3×C4—21no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—181no—
265C2×C3×C4C22×D911yescharacteristic
266C2×C3×C4C2×D3211yescharacteristic
267#67—362no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—61no—
271C2×Dic3—91no—
272C2×Dic3—91no—
273C2×Dic3—181no—
274D12—61no—
275D12—61no—
276D12—362no—
277C4×D3—184no—
278C4×D3—184no—
279C4×D3—181no—
280C4×D3—181no—
281C4×D3—181no—
282C4×D3—181no—
283C4×D3—361no—
284C4×D3—361no—
285Dic6—21no—
286Dic6—32no—
287Dic6—61no—
288Dic6—184no—
289Dic6—184no—
290Dic6D3612yescomplemented
291C3×C8—21no—
292C3×C8—21no—
293C3×C8—41no—
294C3⋊C8—181no—
295C3⋊C8—181no—
296C3⋊C8—361no—
297C2×C32#88211yescharacteristic
298C2×C32D3×SD1611yescharacteristic
299C2×C32#91811yescharacteristic
300C3⋊D3—62no—
301C3×D3—241no—
302C2×C9—21no—
303C2×C9—21no—
304C2×C9—21no—
305C2×C9D3×SD1611yescharacteristic
306C2×C9#91511yescharacteristic
307C2×C9#93711yescharacteristic
308D9—62no—
309D9—81no—
310D9—122no—
311C2×Q8—31no—
312C2×Q8—91no—
313C2×Q8—541no—
314C2×Q8—542no—
315C2×Q8—542no—
316C2×D4—91no—
317C2×D4—541no—
318C22×C4—271no—
319C22×C4—541no—
320SD16—182no—
321SD16—542no—
322C2×C8—271no—
323C2×C8D3×D911yescharacteristic
324C4⋊C4—31no—
325C22⋊C4—541no—
326C22×C3—21no—
327C22×C3—361no—
328C22×C3—361no—
329C22×C3—361no—
330C22×C3D4×D911yescharacteristic
331C22×C3#380211yescharacteristic
332C2×D3—92no—
333C2×D3—92no—
334C2×D3—121no—
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—182no—
338C2×D3—181no—
339C2×D3—181no—
340C2×D3—181no—
341C2×D3—181no—
342C2×D3—361no—
343C2×D3—361no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—62no—
347C3×C4—121no—
348C3×C4—182no—
349C3×C4C2×D3611yescharacteristic
350C3×C4D4×D911yescharacteristic
351C3×C4D3×D1211yescharacteristic
352C3×C4D3×#6711yescharacteristic
353Dic3—22no—
354Dic3—41no—
355Dic3—62no—
356Dic3—181no—
357Dic3—181no—
358Dic3—361no—
359C32#480611yescharacteristic
360C9—21no—
361C9#517011yescharacteristic, complemented
362C23—181no—
363C23—271no—
364Q8—32no—
365Q8—61no—
366Q8—92no—
367Q8—181no—
368Q8—544no—
369D4—181no—
370D4—181no—
371D4—1082no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—181no—
375C2×C4—271no—
376C2×C4—544no—
377C2×C4—541no—
378C2×C4—541no—
379C2×C4C2×D3×D911yescharacteristic
380C8—21no—
381C8—541no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—721no—
386C2×C3#6344411yescharacteristic
387C2×C3SD16×D911yescharacteristic
388C2×C3#6348011yescharacteristic
389C2×C3#6379611yescharacteristic
390C2×C3#6381111yescharacteristic
391C2×C3#6392211yescharacteristic
392D3—182no—
393D3—182no—
394D3—241no—
395D3—362no—
396C22—272no—
397C22—361no—
398C22—361no—
399C22—361no—
400C22—541no—
401C22—541no—
402C22#8864611yescharacteristic
403C4—62no—
404C4—121no—
405C4—182no—
406C4—541no—
407C4D3×D3611yescharacteristic
408C4D9×#6711yescharacteristic
409C3—21no—
410C3#1058848611yescharacteristic, complemented
411C3#1059082411yescharacteristic
412C2—542no—
413C2—721no—
414C2#1172081711yescharacteristic
415C2#1172083211yescharacteristic
416C2#1172104911yescharacteristic
subgroupG/H
centerC22#88646
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radical#11721007C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#448
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
1. C2×(C3⋊D9)⋊Q8C2semidirect
2. C2×C3×#3664C2semidirect
4. C2×#88425C2semidirect
5. #11720542C2semidirect
6. C9×#816C2semidirect
7. #11720268C2semidirect
34. C2×#3664D3semidirect
52. C9×Dic6D4semidirect
54. C3×Dic18D4semidirect
98. C2×C3⋊D9SD16semidirect
121. #816D9semidirect
165. Dic18#67semidirect
249. C3×C9#448semidirect
290. Dic6D36semidirect
361. C9#5170semidirect
410. C3#10588486semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
C2×C3×#36642
#117210072
C2×#884252
#117205422
C9×#8162
#117202682
#105908243
#105884863