Finite Group Databasebeta

← #13140729

Subgroups — #13140729

Order 1728.

542 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 101 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×Q8×D9C211yesmaximal, characteristic, complemented
2C2×#88548C211yesmaximal, characteristic, complemented
3#11721004C211yesmaximal, characteristic
4C2×#88416C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10591584—31nomaximal
9#10588486—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C3×Q8×D9—22no—
13C3×Q8×D9—22no—
14C2×C3×C4×D9—21no—
15C2×C3×C4×D9C2211yescharacteristic
16C2×C3×Dic18—21no—
17C2×C3×Dic18C2211yescharacteristic
18C2×#6192—21no—
19#88548—22no—
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21C9×C4⋊Dic3C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23#88445—21no—
24#88416—22no—
25C2×C3×D3×Q8—31no—
26C2×#3755—31no—
27#63909—31no—
28C2×#3717—31no—
29C2×Q8×D9D311yescharacteristic, complemented
30C3×#835D311yescharacteristic
31#63565—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3675—31no—
34C2×#3675—61no—
35C2×#3664—31no—
36#63457—31no—
37C9×#103—31no—
38(C2×D9)⋊C8—31no—
39#63368—31no—
40C22×C3⋊D9—21no—
41C22×C3×D9D411yescharacteristic
42C3×Q8×C9—21no—
43C3×Q8×C9D412yescomplemented
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46C4⋊(C3⋊D9)—21no—
47C4⋊(C3⋊D9)—21no—
48C2×C9×Dic3—21no—
49C2×C3×Dic9—21no—
50C2×C3×Dic9D411yescharacteristic
51C3×C4×D9—24no—
52C3×C4×D9—21no—
53C3×C4×D9—21no—
54C3×Dic18—24no—
55C3×Dic18—21no—
56C3×Dic18D412yescomplemented
57#6192—42no—
58C3⋊(C9⋊C8)—21no—
59C9×C3⋊C8—21no—
60#5227—91no—
61#4806—91no—
62C2×C32×Q8C2×D311yescharacteristic
63C2×C4⋊(C3⋊D3)—31no—
64C3×D3×Q8—62no—
65C3×D3×Q8—62no—
66C2×C3×C4×D3—31no—
67C2×C3×C4×D3—61no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—61no—
70C2×#626—61no—
71#3755—62no—
72C2×C3⋊(C3⋊C8)—31no—
73C3×C4⋊Dic3C2×D311yescharacteristic
74C2×C3×C3⋊C8C2×D311yescharacteristic
75#3722—61no—
76#3717—62no—
77C2×Q8×C9—21no—
78C2×Q8×C9C2×D311yescharacteristic
79Q8×D9—22no—
80Q8×D9—22no—
81C2×D36—31no—
82C2×D36—61no—
83C2×D36—61no—
84C2×C4×D9—21no—
85C2×C4×D9C2×D311yescharacteristic
86C2×Dic18—21no—
87C2×Dic18C2×D311yescharacteristic
88C2×C8×C9—31no—
89C9×C4⋊C4—31no—
90#3675—64no—
91#3675—62no—
92(C2×D9)⋊C4—61no—
93C2×C9⋊C8—31no—
94C2×C9⋊C8—61no—
95#3664—62no—
96C22×C3×C9C2×D411yescharacteristic
97C2×C3⋊D9—41no—
98C2×C3⋊D9—41no—
99C2×C3⋊D9—41no—
100C2×C3×D9—21no—
101C2×C3×D9—21no—
102C2×C3×D9SD1612yescomplemented
103C3×C4×C9—22no—
104C3×C4×C9C2×D411yescharacteristic
105C3×C4×C9C2×D411yescharacteristic
106C9×Dic3—41no—
107C3×Dic9—21no—
108C3×Dic9—22no—
109C22×C3×Q8—91no—
110C2×D3×Q8—31no—
111C2×#215—91no—
112C2×#215—91no—
113C2×#215—91no—
114C2×#215—181no—
115#930—91no—
116C2×#204—91no—
117#895—91no—
118C3×#103—31no—
119#835—91no—
120#835D911yescharacteristic, complemented
121#830—91no—
122(C2×D3)⋊C8—91no—
123#810—91no—
124C22×C3⋊D3—61no—
125C22×C3×D3—31no—
126C32×Q8—21no—
127C32×Q8D1212yes—
128C2×C32×C4—21no—
129C2×C32×C4C22×D311yescharacteristic
130C4⋊(C3⋊D3)—61no—
131C4⋊(C3⋊D3)—61no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—61no—
135C3×C4×D3—64no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×Dic6—32no—
139C3×Dic6—64no—
140C3×Dic6—61no—
141#626—122no—
142C22×D9—61no—
143C22×D9—121no—
144C22×D9#6711yescharacteristic
145C3⋊(C3⋊C8)—61no—
146C3×C3⋊C8—21no—
147Q8×C9—21no—
148Q8×C9—22no—
149Q8×C9—41no—
150Q8×C9#6712yescomplemented
151C2×C4×C9—21no—
152C2×C4×C9—21no—
153C2×C4×C9—41no—
154C2×C4×C9—61no—
155C2×C4×C9C22×D311yescharacteristic
156C2×Dic9—21no—
157C2×Dic9#6711yescharacteristic
158D36—62no—
159D36—61no—
160D36—61no—
161D36—121no—
162D36—122no—
163C4×D9—24no—
164C4×D9—21no—
165C4×D9—21no—
166Dic18—24no—
167Dic18—21no—
168Dic18#6712yescomplemented
169C8×C9—61no—
170C9⋊C8—62no—
171C9⋊C8—61no—
172#448—271noSylow
173C2×C3×C9#12011yescharacteristic
174C2×C3×C9C2×SD1611yescharacteristic
175C2×C3×C9#13711yescharacteristic
176C3⋊D9—81no—
177C3×D9—22no—
178C2×C3×Q8—21no—
179C2×C3×Q8—91no—
180C2×C3×Q8—181no—
181C2×C3×Q8—182no—
182C2×C3×Q8—182no—
183C2×C3×Q8C2×D911yescharacteristic
184C2×C3×Q8D3211yescharacteristic
185C22×C3×C4—91no—
186C22×C3×C4—181no—
187C2×#67—181no—
188D3×Q8—62no—
189D3×Q8—62no—
190C2×D12—91no—
191C2×D12—91no—
192C2×D12—181no—
193C2×D12—181no—
194C2×C4×D3—31no—
195C2×C4×D3—61no—
196C2×Dic6—31no—
197C2×Dic6—61no—
198C2×C3×C8—31no—
199C3×C4⋊C4—31no—
200#215—182no—
201#215—184no—
202#215—182no—
203#215—182no—
204(C2×D3)⋊C4—181no—
205(C2×D3)⋊C4—181no—
206C4⋊Dic3C2×D911yescharacteristic
207C2×C3⋊C8—91no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—181no—
210C2×C3⋊C8C2×D911yescharacteristic
211#204—182no—
212C22×C32D3×D411yescharacteristic
213C2×C3⋊D3—121no—
214C2×C3⋊D3—121no—
215C2×C3⋊D3—121no—
216C2×C3×D3—32no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C32×C4—22no—
220C32×C4C2×D1211yescharacteristic
221C32×C4D3×D411yescharacteristic
222C3×Dic3—41no—
223C3×Dic3—62no—
224C3×Dic3—61no—
225C22×C9—21no—
226C22×C9C2×#6711yescharacteristic
227C2×D9—21no—
228C2×D9—21no—
229C2×D9—122no—
230C2×D9—121no—
231C2×D9—121no—
232C2×D9—121no—
233C2×D9—241no—
234C2×D9—241no—
235C2×D9#21512yescomplemented
236C4×C9—22no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—42no—
240C4×C9—121no—
241C4×C9C2×#6711yescharacteristic
242C4×C9C2×#6711yescharacteristic
243Dic9—21no—
244Dic9—22no—
245C22×Q8—91no—
246C2×SD16—271no—
247C2×SD16—271no—
248#121—271no—
249#103—31no—
250#103—271no—
251C22⋊C8—271no—
252C3×C9#44811yescharacteristic, complemented, Sylow
253C23×C3—91no—
254C22×D3—31no—
255C22×D3—181no—
256C22×D3—181no—
257C22×D3—361no—
258C3×Q8—21no—
259C3×Q8—21no—
260C3×Q8—22no—
261C3×Q8—41no—
262C3×Q8—92no—
263C3×Q8—184no—
264C3×Q8—181no—
265C3×Q8D3612yescomplemented
266C3×Q8#62612yes—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—41no—
271C2×C3×C4—61no—
272C2×C3×C4—91no—
273C2×C3×C4—184no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4C22×D911yescharacteristic
278C2×C3×C4C2×D3211yescharacteristic
279#67—362no—
280C2×Dic3—21no—
281C2×Dic3—31no—
282C2×Dic3—61no—
283D12—182no—
284D12—181no—
285D12—181no—
286D12—181no—
287D12—181no—
288D12—361no—
289D12—362no—
290C4×D3—64no—
291C4×D3—61no—
292C4×D3—61no—
293Dic6—32no—
294Dic6—64no—
295Dic6—61no—
296C3×C8—61no—
297C3⋊C8—21no—
298C3⋊C8—182no—
299C3⋊C8—181no—
300C3⋊C8—181no—
301C2×C32#88211yescharacteristic
302C2×C32D3×SD1611yescharacteristic
303C2×C32#91811yescharacteristic
304C3⋊D3—241no—
305C3×D3—62no—
306C2×C9—21no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9C2×#21511yescharacteristic
310C2×C9#94211yescharacteristic
311C2×C9#95311yescharacteristic
312D9—22no—
313D9—242no—
314D9—241no—
315C2×Q8—91no—
316C2×Q8—181no—
317C2×Q8—182no—
318C2×Q8—182no—
319C2×Q8D3×D911yescharacteristic
320C2×D4—271no—
321C2×D4—541no—
322C22×C4—91no—
323C22×C4—181no—
324SD16—542no—
325SD16—542no—
326C2×C8—31no—
327C2×C8—271no—
328C4⋊C4—31no—
329C22⋊C4—541no—
330C22×C3—21no—
331C22×C3—92no—
332C22×C3—181no—
333C22×C3—181no—
334C22×C3D4×D911yescharacteristic
335C22×C3D3×#6711yescharacteristic
336C2×D3—32no—
337C2×D3—61no—
338C2×D3—61no—
339C2×D3—362no—
340C2×D3—361no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—361no—
344C2×D3—361no—
345C2×D3—361no—
346C2×D3—721no—
347C2×D3—721no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—42no—
353C3×C4—121no—
354C3×C4—181no—
355C3×C4—182no—
356C3×C4C2×D3611yescharacteristic
357C3×C4D4×D911yescharacteristic
358C3×C4C2×#62611yescharacteristic
359C3×C4D3×#6711yescharacteristic
360Dic3—41no—
361Dic3—62no—
362Dic3—61no—
363C32#480611yescharacteristic
364C9—21no—
365C9#522711yescharacteristic, complemented
366C23—91no—
367C23—541no—
368Q8—21no—
369Q8—92no—
370Q8—184no—
371Q8—181no—
372Q8#619212yescomplemented
373D4—541no—
374D4—541no—
375D4—1082no—
376C2×C4—21no—
377C2×C4—61no—
378C2×C4—91no—
379C2×C4—184no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4C2×D3×D911yescharacteristic
384C8—61no—
385C8—541no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—182no—
390C2×C3#6344411yescharacteristic
391C2×C3SD16×D911yescharacteristic
392C2×C3#6348011yescharacteristic
393C2×C3D3×#21511yescharacteristic
394C2×C3#6394811yescharacteristic
395C2×C3#6397711yescharacteristic
396D3—62no—
397D3—722no—
398D3—721no—
399D3—721no—
400C22—92no—
401C22—181no—
402C22—181no—
403C22—1081no—
404C22—1081no—
405C22—1081no—
406C22D9×#6711yescharacteristic
407C4—22no—
408C4—121no—
409C4—181no—
410C4—182no—
411C4C2×#619211yescharacteristic
412C4D9×#6711yescharacteristic
413C3—21no—
414C3#1058848611yescharacteristic, complemented
415C3#1059158411yescharacteristic
416C2—182no—
417C2—2161no—
418C2D9×#21511yescharacteristic
419C2#1172122211yescharacteristic
420C2#1172127611yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radical#11721004C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#448
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×D9C2semidirect
2. C2×#88548C2semidirect
4. C2×#88416C2semidirect
5. C9×#835C2semidirect
6. (C2×D9)⋊(C3⋊C8)C2semidirect
7. #11720268C2semidirect
29. C2×Q8×D9D3semidirect
43. C3×Q8×C9D4semidirect
56. C3×Dic18D4semidirect
102. C2×C3×D9SD16semidirect
120. #835D9semidirect
150. Q8×C9#67semidirect
168. Dic18#67semidirect
235. C2×D9#215semidirect
252. C3×C9#448semidirect
265. C3×Q8D36semidirect
365. C9#5227semidirect
372. Q8#6192semidirect
414. C3#10588486semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D92
C2×#885482
#117210042
C2×#884162
C9×#8352
(C2×D9)⋊(C3⋊C8)2
#117202682
#105915843
#105884863